# Enzyme kinetics

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K

KINETICS, ENZYMES

ATHEL CORNISH-BOWDEN

National Center for Scientiﬁc Research Marseilles, France

MICHAELIS–MENTEN KINETICS The starting point for any discussion of enzyme kinetics is the Michaelis–Menten equation, which expresses the initial rate v of a reaction at a concentration a of the substrate transformed in a reaction catalyzed by an enzyme at total concentration e0: v k0e0a Km a Va Km a (1)

KEY WORDS Enzymes Inhibition Kinetics Manipulation of enzyme activity Multienzyme systems OUTLINE Introduction Michaelis–Menten Kinetics Graphical Analysis Two-Substrate Reactions Inhibition and Activation Inhibition Speciﬁcity Activation Irreversible Inhibition Inhibitory Effects in Metabolic Systems Non-Michaelis–Menten Behavior Kinetics of Multienzyme Systems Further Reading Bibliography INTRODUCTION The kinetic behavior of enzymes has been studied in detail for a century, beginning with the classic work of Henri (1) and Michaelis and Menten (2). The objectives have been threefold: to gain an understanding of the mechanisms of enzyme action; to illuminate the physiological roles of enzyme-catalyzed reactions; and, mainly in recent years, to manipulate enzyme properties for biotechnological ends. Experimental practice has been overwhelmingly dominated by the ﬁrst of these aims; most experiments have been designed as if shedding light on the mechanism was the principal, or even the only, objective. However, although much valuable information has been obtained in this way, there are some important aspects of enzyme function that are obscured when working mainly with isolated enzymes in conditions far removed from those that exist in the cell. For this reason the latter part of this chapter will be devoted to a discussion of enzymes as they behave in complex mixtures.

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The parameters are k0, the catalytic constant, and Km, the Michaelis constant. The form shown in the middle is more fundamental than that on the right, but the second form, in which k0e0 is written as the limiting rate V, is often used because the enzyme concentration in meaningful units is often not known. V is the limit that v approaches as the enzyme becomes saturated (that is, when a becomes very large) and Km is the value of a at which v 0.5V (that is, at which the rate is half-maximal). The ratio k0 /Km is called the speciﬁcity constant and given the symbol kA: It is a more fundamental constant than Km in the analysis of enzyme mechanisms (i.e., it has a simpler mechanistic meaning), but the equation is usually written in terms of V (or k0) and Km nonetheless. The principal assumption implied by equation 1 is that the rate of the reverse reaction is negligible: This may be because the reaction is irreversible for practical purposes, but even with reversible reactions the reverse reaction can be made negligible by measuring the rate in the absence of products and by extrapolating the rate back to zero time, that is, by estimating the initial rate at a time when no products have accumulated. Even if there is no signiﬁcant reverse reaction, products can still affect the rate of the forward reaction because product inhibition (discussed later) is a common phenomenon. Another assumption is that the reaction is allowed sufﬁcient time to reach a steady state because all reactions pass through an initial acceleration phase known as the transient state. This phase is normally very brief (a few milliseconds), and in practice enzymes are usually studied under steady-state conditions. Note, however, that this is made experimentally possible by working with extremely small enzyme concentrations compared with those that may exist in the cell. The use of very low enzyme concentrations has two important consequences: First, it normally means that the enzyme concentration can be neglected in comparison with the substrate concentration, and second it makes the steady-state rate sufﬁciently slow to be easily measured and the steady-state phase long enough to be meaningful. If high enzyme concentrations were used, the transient state would be as brief as before, but the steady state would also be very brief, so that there would be no period in which one could adequately treat the rate as constant. Equation 1 may be derived from the following model

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KINETICS, ENZYMES

A

E i EA i EP i E

k

1

k1

k2

k3

P

(2)

v

V

k

2

k

3

which assumes that the reaction passes through an enzyme–substrate complex EA, which undergoes catalytic transformation to an enzyme–product complex EP, which then breaks down to form products. Although real enzyme mechanisms may be more complicated than this, every reaction passes through steps of substrate binding, chemical transformation, and product release. In simple introductory treatments the second and third steps are often treated as a single step, but conceptually they are clearly distinct. In reactions of more than one substrate, the steps do not necessarily occur in the order one might guess; that is, some products may be released before all substrates have bound, but the general principle that any reaction involves the same three kinds of step remains valid. The Michaelis–Menten parameters can be deﬁned as follows in terms of the rate constants shown in equation 2: k0 k2k3 k2 ; kA k k1k2k3 ; k 1k3 k2k3 k 1k3 k2k3 (3) k2 k3)

0.5V

0 0 Km

5 Km

10 Km a

15 Km

20 Km

Figure 1. Michaelis–Menten dependence of rate on substrate concentration. The curve is a rectangular hyperbola through the origin, approaching a limit of v V at saturation. The rate is 0.5 V at a substrate concentration equal to the Michaelis constant, Km, but note that the approach to the limit is slow, so that, for example, even at a 10Km the rate is still nearly 10% less than V.

k

2

k3

k

1k 1k 2

2

Km

k1(k

2

Note that none of the three parameters has a simple transparent meaning. The interpretations commonly attributed to them depend on additional simplifying assumptions that are not always correct. For example, Km is often said to be equal to the equilibrium constant k 1 /k1 for dissociation of A from EA, but the expression in equation 3 does not take this form unless k2 is very small. As there is no good reason for k2 to be small, and indeed ideas of evolutionary optimization of enzyme function lead one to expect the opposite when the enzyme is acting on its natural physiological substrate, it follows that Km should not, in general, be regarded as a measure of the equilibrium dissociation constant. Despite these difﬁculties in providing a detailed mechanistic meaning to Km, it does provide a measure of the tightness of substrate binding in the steady state, as it is quite correct to take a/Km as equal to the ratio of the sum of concentrations of all enzyme complexes (i.e., both EA and EP) over the concentration of free enzyme. Similarly k0 provides a valid measure of the capacity of the enzyme– substrate complex to react to give products, even if it cannot be interpreted as the rate constant for a unique step in the mechanism. The reason for the term speciﬁcity constant for kA, that is, its relationship to enzyme speciﬁcity, will become clear once inhibition has been considered. Graphical Analysis Equation 1 deﬁnes a hyperbolic dependence of rate on substrate concentration, as illustrated in Figure 1. The initial steep rise in v as a increases from zero is rapidly transformed into the phenomenon of saturation, whereby further increases in a produce smaller and smaller increases in v, which approaches, but does not reach or exceed, the limiting rate V. The rectangular hyperbola makes this type

of plot inconvenient for estimating the values of the kinetic parameters (because the line does not approach the saturation limit closely enough at reasonable values of a to allow direct measurement of V). For this purpose, therefore, it is usual to transform equation 1 into one of the following three forms, which underlie the three straight-line plots illustrated in Figures 2 through 4: 1 v a v v 1 V Km V V Km 1 V a 1 a V Km v a (4) (5) (6)

The double-reciprocal plot, illustrated in Figure 2 and based on equation 4, is the mostly widely used, but it is also the least satisfactory because it distorts the effect of experimental error to such an extent that it is difﬁcult to form any visual judgement of where the best line should be drawn. The other two plots are better, and the plot of v

1/v

Km 1 1 1 v = V + V a

Slope = Km/V 1/V

–1/Km

0

1/a

Figure 2. The double-reciprocal plot. This is the most widely used method of plotting the Michaelis–Menten equation as a straight line. However, the severe distortion of any experimental errors in the original data causes it to give a misleading impression.

KINETICS, ENZYMES

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a/v

a Km 1 a v= V + V

v v = V – Km V

v a

Slope = 1/V Km /V –Km 0 a

Slope = –Km

0

Km /V

v/a

Figure 3. The plot of a/v against a. This alternative to the plot shown in Figure 2 produces much less distortion of the experimental error.

against v/a (Fig. 4, equation 6) has the particular advantage that the entire observable range of v values, from 0 to V, is mapped onto a ﬁnite range of graph; this makes it easy to judge by eye if an experiment has been well designed. On the other hand, it has the disadvantage that v, normally the less reliable measurement, contributes to both coordinates, and errors in v cause deviations along lines through the origin, rather than parallel with one or the other axis. In modern practice it is usually best to regard these plots as for illustration purposes only, and to use suitable computer programs for the actual parameter estimation. For this purpose it is not sufﬁcient just to apply unweighted linear regression to the straight-line plots, as this suffers from the same statistical distortions as the plots themselves. Full treatment would require more space than is available here, but see Ref. 3. The following two equations for calculating best-ﬁt values of Km and V give satisfactory results if the v values have uniform coefﬁcient of variation (uniform standard deviation expressed as a percentage), as is usually at least approximately correct: Km V Rv2R(v/a) R(v2 /a2)Rv R(v2 /a)Rv R(v2 /a)R(v/a) (7) (8)

Figure 4. The plot of v against v/a. The third way of plotting the Michaelis–Menten equation as a straight line also avoids the error-distorting property of the plot shown in Figure 2, and maps the entire range of observable rates (from 0 to V) onto a ﬁnite range of paper. This is a desirable property because it makes it impossible to disguise deﬁciencies in the experimental design.

equation 1, it can be arranged in the same form if one of the two substrate concentrations, for example b, is treated as a constant: k0b ea KmB b 0 KiAKmB KmB KmAb b a

v

(10)

The two fractions in this equation can be regarded as “apparent” values of the Michaelis–Menten parameters for A, that is, the equation can be written as v with kapp 0 k0b ; kapp KmB b A (k0 /KmA)b ; (KiAKmB /KmA) b KiAKmB KmAb Kapp mA KmB b kappe0a 0 Kapp a mA (11)

R(v2 /a2)Rv2 [R(v2 /a)]2 2 2 R(v /a )Rv R(v2 /a)R(v/a)

(12)

Each summation is made over all observations. Two-Substrate Reactions Enzymes that catalyze reactions of a single substrate are only a small minority of all the enzymes known, but the Michaelis–Menten equation remains useful for examining the kinetics of the more common case of a reaction with two substrates and (often but not necessarily) two products, because such a reaction normally obeys Michaelis– Menten kinetics when only one substrate concentration is varied at a time. This is illustrated by the following typical equation for such a reaction: v k0e0ab KmBa KmAb (9)

Notice that the expressions for the apparent values of k0 and kA are both individually of Michaelis–Menten form with respect to b, whereas that for the apparent value of Km is more complicated: This behavior is quite typical and is one of the reasons why k0 and kA should regarded as more fundamental parameters than Km. More generally, the concept of apparent parameters pervades the analysis of simple cases in steady-state enzyme kinetics, being important for the study of reactions with multiple substrates and inhibition. INHIBITION AND ACTIVATION Inhibition For most enzyme-catalyzed reactions, molecules exist that resemble the substrate closely enough to bind to the en-

KiAKmB

ab

Although at ﬁrst sight this appears quite different from

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KINETICS, ENZYMES

zyme, but not closely enough to undergo a chemical reaction. Such a molecule is known as a competitive inhibitor and causes competitive inhibition, characterized by a rate equation of the following form: v k0e0a i/Kic) (13)

Km(1

a

in which i is the concentration of the inhibitor and Kic is the competitive inhibition constant. (The qualiﬁcation “competitive” and the second subscript c are usually omitted if only this simplest kind of inhibition is being considered.) Inhibitors can interfere with catalysis as well as with substrate binding. In the simplest case, an inhibitory term affects the variable term in the denominator of the Michaelis–Menten equation, instead of the constant term: v k0e0a a(1 i/Kiu) (14)

Km

This is called uncompetitive inhibition, and the inhibition constant Kiu is the uncompetitive inhibition constant. This is important as a limiting case of inhibition, but in its pure form it is not at all common. Much more often one has mixed inhibition, when both competitive and uncompetitive effects occur simultaneously: v k0e0a i/Kic) a(1 (15)

the inhibition is noncompetitive. In general it is simplest to regard kA as the parameter affected by competitive inhibition, negligibly so when the competitive component is negligible, k0 as the parameter affected by uncompetitive inhibition, negligibly so when the uncompetitive component is negligible, and Km just as the ratio of the two, so Km k0 /kA. The effects of the different kinds of inhibition on the common plots as illustrated in Figures 2 through 4 follows naturally from equation 16. Any competitive effect affects the apparent value of kA, hence, it increases the slope of the plot of 1/v against 1/a (Fig. 2), it increases the ordinate intercept of the plot of a/v against a (Fig. 3), and it decreases the abscissa intercept of the plot of v against v/a (Fig. 4). Conversely, any uncompetitive effect increases the ordinate intercept of the plot of 1/v against 1/a, increases the slope of the plot of a/v against a, and decreases the ordinate intercept of the plot of v against v/a. When both components of the inhibition are present, both kinds of effects occur. As an illustration we may consider just one example, the effect of competitive inhibition on the plot of 1/v against 1/a: Plots made at various different inhibitor concentrations produce a family of straight lines intersecting on the ordinate axis, as shown in Figure 5, the lack of effect on the ordinate intercept being a direct consequence of the lack of effect on the apparent value of V. Speciﬁcity Speciﬁcity is the most fundamentally important property of enzymes. Although one is often impressed by the catalytic effectiveness of enzymes, accelerating a reaction is not, in reality, difﬁcult: Heating the reaction mixture in a sealed tube is an efﬁcient way of accelerating virtually any reaction, essentially without limit. What is difﬁcult is to accelerate one selected reaction without at the same time accelerating a great mass of unwanted reactions. What is important about an enzyme, therefore, is not that it is an excellent catalyst for a small set of reactions, but that it is an extremely bad catalyst—virtually without any activity—for all other reactions. In other words, the essential properties of enzymes are that they act under very mild conditions and are highly speciﬁc. In the past, speciﬁcity was often assessed by comparing the kinetic parameters for different reactions measured in

Km(1

i/Kiu)

There is no particular reason for the two inhibition constants Kic and Kiu to be equal, and most of the mechanisms one might propose to account for mixed inhibition lead one to expect them to be different, yet the case where Kic Kiu is often given an undeserved prominence in discussions of inhibition, largely because experiments done many years ago suggested that it was a more common phenomenon than it is. This is called noncompetitive inhibition and its rate equation is the same as equation 15, but with both Kic and Kiu written simply as Ki. All of these kinds of inhibition are conveniently discussed in terms of apparent Michaelis–Menten parameters. In the general case (equation 15), these are as follows: kapp 0 k0 ; kapp i/Kiu A kA ; i/Kic Kapp m Km(1 1 i/Kic) i/Kiu (16)

1

1

1/v Increasing i

Note that the ﬁrst two expressions have the same form, and both simplify to independence of i in the event that one or other inhibition term is negligible. The expression for the apparent value of Km is more complicated, especially when one considers how it varies with the different types of inhibition: It increases with the concentration of a competitive inhibitor, it decreases as the concentration of an uncompetitive inhibitor increases, it may change in either direction as the concentration of a mixed inhibitor increases, or it is independent of inhibitor concentration if

1/V

1 1 Km(1 + i/Kic) 1 v = V + a V 0 1/a

Figure 5. Effect of competitive inhibition on the double-reciprocal plot.

KINETICS, ENZYMES

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isolation from one another, which led to sterile arguments as to whether speciﬁcity was best measured in terms of k0, Km, or some combination of the two. This type of argument was resolved once it was realized that the only meaningful way of deﬁning speciﬁcity is as a property of an enzyme that allows it to discriminate between substrates that are mixed together. The simplest way to consider this is with a model similar to that for competitive inhibition, except that one assumes that both molecules are capable of reacting. The equation for reaction of one substrate A in the presence of a competing substrate A follows an equation similar to that for competitive inhibition (equation 13): v k0e0a a /Km) (17)

Km(1

a

with the inhibitor concentration replaced by a , the concentration of A , and the inhibition constant by Km, the Michaelis constant for the reaction of A considered in isolation. The rate of reaction of A is given by the same equation with an obvious transposition of symbols: v k0e0a a/Km) (18)

Km(1

a

NAD, but although consideration of the metabolic pathway in which the reaction occurs may make it convenient to regard ethanol as the substrate and NAD as the coenzyme, this is meaningless when the reaction is considered in isolation. So far as alcohol dehydrogenase is concerned, it catalyzes a reaction that requires two substrates, ethanol and oxidized NAD; the reaction will not proceed unless both are present, and neither has any more reason to be called the substrate than the other. When such improper uses of the term are excluded, there remain a number of enzymes for which the true inverse of inhibition occurs. In the simplest cases the equations are just the inverse of inhibition equations, with terms of the form i/Ki replaced by ones of the form Kx /x (for an activator X with activation constant Kx). However, the simplest cases constitute a smaller proportion of the whole than they do for inhibition. This is because whereas most inhibitors inhibit completely, in the sense that enzyme species with inhibitor bound retain no activity as long as the inhibitor remains bound, many enzymes subject to activation retain some activity in the absence of the activator. As a result, full analysis of activation is often more complicated than it is for inhibition, but this will not be discussed further here. Irreversible Inhibition The types of inhibition considered so far are examples of reversible inhibition; the inhibitor binds reversibly and catalytic activity returns when the inhibitor is released. Irreversible inhibition also occurs, in which the inhibitor either binds so tightly that for practical purposes it cannot be removed, or reacts with the enzyme and converts it irreversibly to a form that has no catalytic activity. These two cases are conceptually different, and the former is more correctly called tight-binding inhibition, rather than irreversible inhibition. However, they are not easy to distinguish in practice, and have similar practical effects and, hence, similar practical uses. Although irreversible inhibition has played a smaller part than reversible inhibition in the academic study of enzyme mechanisms, it has far greater industrial and pharmacological importance. This is because competitive inhibitors, the most common kind of reversible inhibitors, are almost completely ineffective in complete physiological systems, for reasons to be considered shortly. By contrast, whenever irreversible (or tight-binding) inhibition occurs in a physiological system, it can be expected to have profound effects. Many toxic and pharmacologically active substances owe their effects to irreversible inhibition. Both tight-binding and irreversible inhibition manifest themselves in ways that allow them to be confused with noncompetitive inhibition, as in equation 15 with the two inhibition constants equal. This is because the practical effect of irreversible inhibition is not on any of the kinetic parameters in the Michaelis–Menten equation, but on e0, the total enzyme concentration. However, a decrease in e0 can be confused with a decrease in the apparent value of k0, as they occur as a product in equation 1. Although uncompetitive inhibition affects k0, it does so without affecting kA, and thus also changes Km. Decreasing k0 without

It can then be seen that the ratio of rates is the ratio of substrate concentrations multiplied by the ratio of speciﬁcity constants: v v k0 /Km a k0 /Km a kAa kAa (19)

This result, which is still less well known than its importance merits, is the reason for the term speciﬁcity constant. Note that although inspection of equation 1 suggests that kA is no more than the parameter that deﬁnes the rate at very low substrate concentrations, no assumption about the magnitudes of the concentrations was made in arriving at equation 19. It is thus valid at all concentrations, and the speciﬁcity constant measures speciﬁcity at all concentrations, not just low ones. Activation Activation is the opposite from inhibition, in which a reaction proceeds more rapidly in the presence of a particular molecule than in its absence. It is less common than inhibition, and discussion is complicated by the fact that a variety of quite different phenomena have been termed “activation.” The most important of these is a confusion between true activation and the case where the activator is really a component of the substrate. Numerous ATPhandling enzymes are said to be activated by magnesium ions, when in reality the complex MgATP is the true substrate, that is, the species that reacts with the enzyme. Other metal ions, such as the zinc in a number of enzymes, may be true activators as they bind to the enzyme itself and confer catalytic properties on it. Another misuse of the term activation relates to coenzymes such as NAD in many dehydrogenases: Alcohol dehydrogenase, for example, may be said to be activated by

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affecting Km, similar to what one would observe if e0 decreases, is a deﬁnition of noncompetitive inhibition. Inhibitory Effects in Metabolic Systems Competitive and uncompetitive inhibition are sufﬁciently similar in their effects in artiﬁcial experiments on isolated enzymes that they are often not distinguished, and an uncompetitive component in mixed inhibition often passes unnoticed. Many inhibitors are described in the literature as competitive inhibitors in the absence of any real evidence of the type of inhibition. This sort of confusion can easily lead to the entirely false idea that they are similar in their effects in systems where the inhibited enzyme is mixed with other enzymes and catalyzes a step in the middle of a pathway. In a typical experiment in vitro, one decides the concentrations of the various components in advance and measures the rate that results; however, this is very different from what happens in the cell. To a ﬁrst approximation, an enzyme catalyzing a step in the middle of a pathway must transform its substrate at the rate at which it arrives, that is, within certain limits it has little or no effect on the rate of its reaction, but instead determines the concentrations of the metabolites around it. (This is an oversimpliﬁcation, but is useful for discussion.) It is useful therefore to transform equations 13 and 14 into expressions for a in terms of i: vKm(1 k0e0 k0e0 vKm v(1 i/Kic) v a/Km

6 4 2 0 0 1 i/Kic (a) 6 a/Km 4 2 0 0 1 i/Kiu (b)

Figure 6. Effects of (a) competitive and (b) uncompetitive inhibition on the concentration of substrate in a constant-rate system. Both curves are drawn for the case of a Km in the absence of inhibitor. Note that both kinds of inhibitor have quantitatively equal effects at very low concentrations, but the initial slope is maintained indeﬁnitely if the inhibition is competitive, whereas it rapidly becomes inﬁnite if the inhibition is uncompetitive.

2

2

a

(20) NON-MICHAELIS–MENTEN BEHAVIOR All of the cases considered so far can be regarded as generalizations of the Michaelis–Menten equation (equation 1). However, although many enzymes do behave in this way, at least as a ﬁrst approximation, there are some important exceptions. It is simple to calculate from equation 1 that if a Km /9 then v 0.1V and if a 9Km then v 0.9V; in other words, spanning the 10–90% range of available rates requires an 81-fold range of substrate concentrations, almost two orders of magnitude. Similar calculations may be done with any of the equations of the Michaelis–Menten type for additional substrates, inhibitors, or activators. Their implication is that as long as enzymes follow Michaelis–Menten kinetics, their rates cannot be adequately varied by manipulating concentrations of substrates, for example, because effective regulation will often require sensitivity to small changes in signals— certainly changes much smaller than two orders of magnitude. A second difﬁculty arises from the fact that inhibition of the types considered commonly derives from structural similarities between inhibitors and substrates or products, whereas there is no reason to expect the molecules needed for metabolic signals to resemble the substrates or products of the enzymes that need to respond to the signals. In reality, the concentration of the end product of a pathway often serves as such a signal: too low, and the pathway

a

i/Kiu)

(21)

However similar equations 13 and 14 may seem, their transformed versions are drastically different. Equation 20 shows a linear dependence of a on i, which means that increasing i can never result in uncontrolled increases in a. This is illustrated in Fig. 6a. Even at an inhibition concentration equal to the inhibition constant, the substrate concentration is only doubled. By contrast, the curve deﬁned by equation 21 is a rectangular hyperbola (Fig. 6b) that produces a steep and uncontrolled rise in substrate concentration at quite moderate inhibitor concentrations (4). The point is that in competitive inhibition, rises in substrate and inhibitor concentrations oppose one another— not only does the inhibitor compete with the substrate, but equally, the substrate competes with the inhibitor. In uncompetitive inhibition, however, these effects potentiate one another. It follows that although it is relatively easy to ﬁnd molecules that will act as competitive inhibitors, it is also largely useless as a strategy for designing pesticides or drugs because it is correspondingly easy for the organism to counteract the effect of the inhibition. To produce major metabolic effects one needs uncompetitive inhibitors, irreversible inhibitors, or tight-binding inhibitors: None of these are as easy to produce as weakly binding competitive inhibitors, but they are far more effective.

KINETICS, ENZYMES

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needs to be activated; too high, and it needs to be inhibited. It is often found, therefore, that the enzyme that catalyzes the ﬁrst committed step of a pathway, that is, the ﬁrst step after a branch point, in the branch that leads to the end product in question, is inhibited by that end product. For inhibition of this kind to be possible, the enzyme must have a speciﬁc binding site for the end product, independent of the binding sites for substrates and products. Such a site is called an allosteric site, and the phenomenon is called allosteric inhibition. Because the need for it often coincides with the need for higher sensitivity than is provided by Michaelis–Menten kinetics and the common kinds of inhibition, allosteric inhibition is often cooperative: This means that the equations that deﬁne it are more complicated than those considered above, allowing, for example, a change from 10% to 90% inhibited over a concentration range much smaller than 81-fold, and typically less than 10-fold (though rarely, if ever, less than 3-fold) (Fig. 7). The fact that allosteric and cooperative behavior are often found together has led many authors to treat the two terms as synonymous, a tendency encouraged by the fact that in one of the most widely accepted models of nonMichaelis–Menten behavior, that of Monod, Wyman, and Changeux (5), both result from the same structural properties attributed to the enzyme. Nonetheless, most careful authors consider the two properties to be conceptually distinct and not necessarily occurring together, so the two terms should be considered distinct as well. KINETICS OF MULTIENZYME SYSTEMS As noted in the introduction, nearly all kinetic studies of enzymes have been carried out using isolated enzymes, and although this has been very valuable for arriving at a good understanding of the nature of enzyme catalysis, it is quite inadequate as a guide to how systems of enzymes will behave. One cannot assume that the ﬂux through a met-

v V 0.9V

0.1V 0 81× 9×

abolic pathway is a property of a unique enzyme catalyzing the rate-limiting step, and that the properties of the pathway as a whole can be deduced from studies of the kinetics of this one enzyme in isolation. Space does not permit a full analysis of this subject, which has developed from a classic paper by Kacser and Burns (6), but it should sufﬁce to examine an example of a pathway in which biotechnologists have attempted to increase the ﬂux by identifying the rate-limiting enzyme and using genetic manipulation to increase its activity. Tryptophan biosynthesis in yeast provides such an example, tryptophan being a commercially valuable metabolite for which increased production would be very desirable. The tryptophan pathway consists of ﬁve enzymes, and in the classical model any one of these could be the “key” enzyme catalyzing the rate-limiting step. However, when the activity of each of these enzymes was increased in turn, either singly or at the same time as others in the pathway, the results were trivial: Increases of enzyme activity of 20fold or greater produced ﬂux increases of perhaps 30%. Only when all ﬁve enzyme activities were increased (or all but one, apparently unimportant, activity) was there a substantial increase in ﬂux, which even then was much smaller than the increase in enzyme activity (7). The object here is not to analyze in detail why manipulation of tryptophan biosynthesis did not produce the desired results, but to use it to illustrate the point that the whole approach is misconceived. One cannot treat the kinetic behavior of systems of enzymes as if it were determined by the properties of a single component. Moreover, abundant evidence exists to show that all organisms have evolved regulatory mechanisms to control metabolic ﬂuxes and concentrations to satisfy their own requirements. Artiﬁcially trying to force more activity in a pathway by increasing the activities of certain enzymes simply stimulates the regulatory mechanisms to resist. Tryptophan biosynthesis is just one example of a general result, and similar efforts in other pathways—for example, increased alcohol production in yeast and increased starch production in potatoes—have produced similar results. The essential point is that ﬂux control is not a property of a single enzyme in a pathway, but is shared among all the enzymes in the pathway. Strictly speaking, it is shared among all the enzymes in the cell, but it is usually safe to assume that enzymes catalyzing reactions remote from the pathway of interest have very little effect on the ﬂux through it, so to a ﬁrst approximation one can consider ﬂux control to be shared among the enzymes of the pathway. To express this idea in quantitative terms, one can deﬁne a ﬂux control coefﬁcient for any enzyme by the following equation: lnJ p lnvi p

a CJ i (22)

Figure 7. Non-Michaelis–Menten kinetics. For an enzyme obeying the Michaelis–Menten equation (cf. Fig. 1), an 81-fold increase in substrate concentration is needed to bring the rate from 10% to 90% of V. If the enzyme shows positive cooperativity the curve typically becomes sigmoid (S-shaped), and this range of substrate concentrations is decreased (to nine-fold in the example, but in strongly cooperative cases it can be as small as three-fold).

This deﬁnition compares the effect on the ﬂux J through a pathway of some perturbation of the activity of the ith enzyme, represented by a change in the parameter p, with the effect the same perturbation would have on the rate vi of the same reaction if it were considered in isolation. The

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KINETICS, ENZYMES

identity of the parameter p does not have to be speciﬁed, because as long as it affects only one enzyme, the control coefﬁcient deﬁned by equation 22 is independent of the manner in which the ﬂux and isolated rate are perturbed. However, to make the deﬁnition more concrete, consider the case where p is the logarithm of the enzyme concentration, ei. As most reactions are considered under conditions where the rate is proportional to the enzyme concentration, it will often be true to write dlnvi dlnei, so that the denominator in equation 22 has a value of unity and the whole equation simpliﬁes to the following: lnJ lnei

FURTHER READING The topic of this article is referred to in all textbooks of biochemistry, but unfortunately the treatment is nearly always very superﬁcial and potentially misleading. Since Mahler and Cordes (8), there has been no serious attempt to cover enzyme kinetics in a general biochemistry text. More specialized textbooks were abundant in the past, but have become much less so. There are, however, two recent ones (9,10) that may be consulted for more information about most of the points discussed here, as well as topics such as the pH- and temperature-dependence of enzymecatalyzed reactions, which require more detail than can be covered adequately in a short article. Another source is the book by Segel (11), which was originally published in 1975, but has been reissued recently. Fast reactions are not dealt with in this article, and as they require both techniques and methods of analysis different from those of steady-state kinetics, it is best to refer to a specialized source, such as Hiromi (12), who provides a thorough treatment. Two recent books deal with the kinetics of multienzyme systems, one (13) at a very readable level, the other (14) much more advanced. This is also covered much more brieﬂy in two of the books mentioned earlier (9,10).

CJ i

(23)

This equation is less abstract and simpler to understand than equation 22, and in the past was often used as a primary deﬁnition of a control coefﬁcient. However, this is not recommended, ﬁrst because it is not always true that the isolated rate is proportional to the enzyme concentration, and second because equation 23 can give the false impression that control coefﬁcients are concerned only with effects brought about by changes in enzyme concentration. Flux control coefﬁcients are measures of how much the ﬂux through a pathway is dependent on the activities of the individual enzymes. Mathematical analysis shows that they satisfy a property called the summation relationship: RC J i 1 (24)

BIBLIOGRAPHY

1. V. Henri, Lois Generales de l’Action des Diastases, Hermann, ´ ´ Paris, 1903. 2. L. Michaelis and M.L. Menten, Biochem. Z. 49, 333–369 (1913). 3. A. Cornish-Bowden, Analysis of Enzyme Kinetic Data, Oxford University Press, Oxford, 1995. 4. A. Cornish-Bowden, FEBS Lett. 203, 3–6 (1986). 5. J. Monod, J. Wyman and J.-P. Changeux, J. Molec. Biol. 12, 88–118 (1965). 6. H. Kacser and J.A. Burns, Symp. Soc. Exp. Biol. 27, 65–104 (1973). 7. P. Niederberger, R. Prasad, G. Miozzari, and H. Kacser, Biochem. J. 287, 473–479 (1992). 8. H.R. Mahler and E.H. Cordes, Biological Chemistry, 2nd ed., Harper & Row, New York, 1972. 9. A.R. Schulz, Enzyme Kinetics: From Diastase to MultiEnzyme Systems, Cambridge University Press, Cambridge, 1994. 10. A. Cornish-Bowden, Fundamentals of Enzyme Kinetics, 2nd ed., Portland Press, London, 1995. 11. I.H. Segel, Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems, Wiley, New York, 1993. 12. K. Hiromi, Kinetics of Fast Enzyme Reactions: Theory and Practice, Kodansha, Tokyo 1979. 13. D. Fell, Understanding the Control of Metabolism, Portland Press, London, 1997. 14. R. Heinrich and S. Schuster, The Regulation of Cellular Systems, Chapman & Hall, New York, 1996.

The interpretation of this relationship would be straightforward if all ﬂux control coefﬁcients were always positive, but in reality negative ﬂux control coefﬁcients also occur; for example, if a pathway contains a branchpoint, the enzymes in one branch normally have negative ﬂux control coefﬁcients for ﬂux through the other. Nonetheless, in practice, negative ﬂux control coefﬁcients are nearly always small in magnitude, so although the sum of positive ﬂux control coefﬁcients may be greater than 1, it is rarely much greater than 1, so that the idea of sharing ﬂux control among all the enzymes of a system is reasonably accurate. It follows that we should not expect any enzyme to have complete control over ﬂux, and the closer to complete control any enzyme approaches, the less control all of the others have, taken together. Moreover, a ﬂux control coefﬁcient is not a constant, but tends to decrease when the activity of an enzyme is increased. In other words, even if one can identify one enzyme that has a large proportion of the total ﬂux control, increasing its activity will tend to decrease that proportion, so that the amount the ﬂux can be increased by increasing the activity of one enzyme will always be small. Once this is understood, the failure to increase the ﬂux to tryptophan (in the example mentioned) or to achieve signiﬁcant ﬂux increases (in other similar examples), ceases to be a mysterious result, but is simply what was to be expected.

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KINETICS, MICROBIAL GROWTH

NICOLAI S. PANIKOV

Institute of Microbiology, Russian Academy of Sciences Moscow, Russian Federation

Colonies Bibliography INTRODUCTION Kinetics (Greek jimesijor, forcing to move) is a branch of natural science that deals with the rates and mechanisms of any processes—physical, chemical, or biological. Kinetic studies in microbiology cover all dynamic manifestations of microbial life: growth itself, survival and death, product formation, adaptations, mutations, cell cycles, environmental effects, and biological interactions. Kinetics provides a theoretical framework for optimal design in biotechnologies based on fermentation and enzyme catalysis, as well as on employment of outdoor activity of natural microbial populations (wastewater treatment, soil bioremediation, etc.) Contrary to simple rates measurements, kinetic studies require the perception of the underlying basic mechanisms of studied processes. We will deﬁne mechanistic studies as those that interpret some complex process as an interplay of several simpler reactions, for example, cell growth can be explained through activity of enzymes and microbial community dynamics can be interpreted through behavior of individual cells and populations. Ideally, mechanistic studies infer the coupling of experimental measurements with analysis of simulating mathematical models. The models formalize postulated mechanisms, so that the comparison of observations and the model’s predictions allows one to discard an incorrect hypotheses. The quantitative studies in microbiology often involve the assessment of growth stoichiometry. Stoichiometry [Greek rsoijgeiom, element] is the quantitative relationship between reactants and products in a chemical reaction. In microbiology, stoichiometry stands for a quantitative relationship between substrates and products of microbial processes, including biomass formation (the consequence of complying with mass and energy conservation laws). In practical terms, kinetic and stoichiometry are tightly linked to each other, but stoichiometry mainly addresses problems of a static nature (how much? in what proportion?), whereas kinetics considers the dynamics questions (at what rate? by which mechanism?). GROWTH STOICHIOMETRY Macrostoichiometry of Microbial Growth By analogy to simple chemical reactions, we can represent growth as a conversion of a number of substrates (medium components) into cell mass and products. Growth of aerobic heterotrophic microorganisms can be approximated by the following stoichiometric equation (substrates biomass products) (1,2): CHmO1

2 a1NH3 a2HPO4 YCHpOnNqPoKv . . .

KEY WORDS Cell size distribution Colonies Energy and conserved substrates Growth models Macrostoichiometry Maintenance Microstoichiometry Physiological state Steady-state and transient dynamics Yield OUTLINE Introduction Growth Stoichiometry Macrostoichiometry of Microbial Growth Growth Yield: Catabolic and Conserved Substrates Yield Variation as Dependent on the Chemical Nature of Organic Substrates Variations in Yield from Energy Source, Maintenance Requirements Experimental Determination of m Maintenance Requirements and Wasteful Catabolism Variation in Biomass Yield from Conserved Substrates Microscopic Approach in Studies of Growth Stoichiometry Basic Principles of Growth Kinetics Kinetics of Chemical and Enzyme Reactions Simple Models of Microbial and Cell Growth Structured Models Cell Cycle Population Dynamics (Mutations, Autoselection, Plasmid Transfer) Microbial Growth as Dependent on Cultivation Systems 1a —Homogeneous Continuous Culture (Continuous-Flow Fermenters with Complete Mixing) 1ab—Continuous Cultivation without Cell Washout 2a —Continuous Cultivation with a Discontinuous Supply of Limiting Substrate 2ab—Simple Batch Culture 1b —Plug-Flow (Tubular) Culture 1bb—Continuous-Flow Reactors with Microbes Attached

a3K a4CO2

... bO2 a5H2O (1)

Here, microbial biomass is empirically expressed by the gross formula CHpOnNqPoKv . . . , for example, if some av-

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KINETICS, MICROBIAL GROWTH D A B C

Table 1. Selected Macrostoichiometric Equations Describing Growth of Microorganisms with Different Types of Energy Generation

Growth Yield: Catabolic and Conserved Substrates The growth yield is one of the main stoichiometric parameters. It is deﬁned as follows Y dx ds Dx Ds (2)

Heterotrophic growth and by-product formation

Microbial process

Phototrophic growth (algae or plant cells)

where Dx is the increase in microbial biomass consequent on utilization of the amount Ds of substrate, and dx and ds are respective inﬁnitely small increments. Rigorous deﬁnition of Y as derivative dx/ds stems from the fact that Y can vary in time, the negative sign being introduced because x and s vary in opposite senses. Sometimes, it is used as the reciprocal of Y: 1/Y, which is called the economic coefﬁcient. It expresses explicitly the nutrient requirements for growth: how many mass units of a particular substrate should be consumed to produce one unit mass of cell material. The growth efﬁciency depends generally on the partitioning of consumed element between new cell biomass and extracellular products. The mass balance (total element consumed amount incorporated into cell plus amount incorporated into extracellular products) is as follows: dEs dEx dEp (3)

Methanogenesisa

There are two groups of substrates for microbial growth: (1) catabolic substrates, which are sources of energy; and (2) anabolic or conserved substrates, which are sources of

Nitriﬁcationb

a

b

Particular example of H2-utilizing methanogenic bacteria. The total growth balance for two phases of nitriﬁcation: oxidation of ammonium to nitrite and subsequent oxidation of nitrite to nitrate. Y is growth yield of bacterial mass per mass unit of oxidized N.

YPCHrOsNt

a3CHrOsNt

YCHpOnNq

YCHpOnNq

YCHpOnNq

a2NH3

a2NH3

Substrates

a1O2

a1H2O

a2NH3

CHmOl

a1H2

YCO2

CO2

CO2

a1O2

a2NH4

YCHpOnNq

Biomass

a3NO3

a3CH4

erage microbial cell contains per dry cell weight (%) C 46, H 7.5, O 31, N 11; and P, 1.3, then the biomass formula is CH1.9O0.5N0.2P0.01. The stoichiometric quotients a1–a5 . . . , b and Y (biomass yield) specify quantities of substrate and products of microbial growth. If we know biomass yield and gross formulas of all substrates and products, then quotients a1–a6 . . . are easily calculated from conservation conditions. There are at least two such conditions. First, the mass of each element (C, H, O, N, P, K, . . . ) on the left side of equation 1 should be equal to that on the right side (mass balance). Second, if ionized substances are involved, we should take into account the balance of charges to satisfy the condition of electroneutrality. Table 1 demonstrates some examples of stoichiometric growth reactions relevant to biotechnology. Described formalism is useful as a ﬁrst step in biotechnological studies aimed at planning and optimizing microbial growth. It estimates how much nutrient should be supplied to the fermenter to obtain the required amount of biomass or target product. However, it should be absolutely clear that stoichiometric equations like equation 1 are no more than an approximation to reality. The most severe deviation stems from the fact that unlike chemical reagents, microbial cells are characterized by changeable composition, and stoichiometric coefﬁcients are not true constants. One task of contemporary microbial stoichiometry is to ﬁnd out the functional relationships between stoichiometric parameters and internal (physiological) and external (environmental) factors.

a4H2O

Products

a3CO2

a4O2

a4H2O

a4H

a5H2O

a1 a2 a4 a1 a2 a4 a1 a2 a4 a1 a2 a4

0.5(Yn Ys l a4) a3 Yq Y t, a3 1 Y Y 0.5[m Y(3q p) Y (3t r)] 0.5[Y( p 3n) (r 3t)(1 Y)] Y(n t) t, a3 1 Y 0.5(2 a1 Yn a3s) 4 0.5Y(4 2n 3q p) Yq, a3 1 Y 2 Yn 2Y 1 Y[1 0.25 ( p 2n 3q)] Y 1 Yq, a3 Y 1 2Y 1 Yq, a5 Y 1 0.5Y( p 3q)

Stoichiometric parameters

KINETICS, MICROBIAL GROWTH

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biogenic elements forming cellular material. Catabolic substrates include H2 for lithotrophic hydrogen bacteria, NH4 and NO2 for nitrifying bacteria, S0 for sulfuroxidizing bacteria, oxidizable or fermentable organic substances for heterotrophic bacteria and fungi, and so on. Their consumption is accompanied by oxidation and dissipation of chemical substances into extracellular waste products that are no longer reusable as an energy source* (H2O, NO3 , SO2 , CO2, etc.) The anabolic substrates after 4 uptake are incorporated into de novo synthesized cell components, being conserved in biomass (that is why they are called conserved). Contrary to catabolic substrates, they can be reused (e.g., after cell lysis to be taken up by survived cells). The conserved substrates include nearly all the noncarbon sources of biogenic elements (N, P, K, Mg, Fe, and trace elements), CO2 for autotrophs, and the indispensable amino acids and growth factors. Most catabolic substrates are used also as a source of biogenic elements. We can assess both these components separately in terms of respective yields, YE (biomass yield per mass unit of oxidized substrate) and YA (biomass yield per mass unit of assimilated substrate), from the experimentally measured yield Y. For C substrate, equation 4 can be speciﬁed as follows (total carbon consumed equals C incorporated into cell plus C oxidized to CO2 to provide energy plus C incorporated into by-products): dCs dCx dCCO2 dCP (4)

1/YO2

A/Y

B

(7)

where A and B are constants estimated from stoichiometry of their respective combustion reactions (see equation 10 later), for example, the value of A is 33.33 mmol O2 g 1 glucose and B is about 42 mmol O2 g 1 CDW. The relationship between biomass yields on O2 and CO2 is derived from comparison of equations 6 and 7: YCO2 YO2 A rs BY rxY (8)

Now we will go back to the general substrate balance (equation 3) and derive an expression for conserved substrate. Again, we neglect term dEP (because extracellular products are assumed to be reusable) and divide the balance by dx, which is the amount of biomass produced: 1 Y dEx dx rx (9)

Let us neglect the last term dCP (by assuming that extracellular by-products can be reused and functionally are equivalent to C substrate) and divide the substrate balance by dCx, which is the amount of biomass C produced, then: 1 Y where Y mass C g g biomass C g CO2-C 1 Y 1

1

1 YE substrate C and YE

where rx is the intracellular content of element incorporated into biomass from consumed substrate. Sometimes rx is called the cell quota. The values 1/Y and rx are not identical although they have the same dimension (e.g., milligram N per gram biomass) and very close numeric value. The reciprocal 1/Y is characterizing the process (the expenditure of conserved substrate to synthesize biomass unit), whereas rx is an index of cell composition (the content of intracellular N per biomass unit). Formally, 1/Y is equal to the rx value of an inﬁnitely small increment of cell biomass, and rx is the averaged value for entire cell. Notice that although rx is a slow and 1/Y is a rapid variable, their numerical values are exactly the same for balanced steadystate growth and can differ considerably during transients. Yield Variation as Dependent on the Chemical Nature of Organic Substrates

(5)

g bio-

1

rx rs

12 YErs

(6)

where Y g CDW g 1 substrate, YE g CDW mmol 1 CO2, and rx and rs are fractions of carbon in biomass and substrate, respectively. For example, if total measurable yield Y is 0.6 g biomass C g 1 glucose C, it means that from each g of consumed C, 0.6 g is incorporated into biomass (assimilated), and 0.4 g is dissimilated (oxidized to CO2), then YE 0.6/0.4 1.5. To calculate oxygen demand for aerobic growth (or biomass yield on O2) we have a balance (oxygen required to produce 1 g CDW equals oxygen required to burn substrate consumed to produce 1 g CDW minus oxygen required to burn 1 g CDW):

*Fermentation products such as acetate, ethanol, butyrate, and H2 seem to be an exception because they do contain reusable oxidation potential, but it is not available under anaerobic conditions supervising fermentation.

In this section, we will discuss why biomass yield varies when microorganisms are grown on different C substrates. This problem was best solved within the framework of the theory of mass and energy balance (TMEB) (3). Evidently, the fraction of C in dry biomass is almost constant. By contrast, the content of carbon in utilized substrates, rs, and energetic quality of substrate vary over a broad range (e.g., compare methane versus oxalic acid). To characterize substrate and biomass by a single common measure, TMEB uses an index of degree of carbon reduction, c related to the internal energy of organic compounds. The heat liberated by biological or chemical oxidation is proportional to oxygen uptake or equally to the number of electrons gained by oxygen from oxidized substrates, according to Payne’s term available electrons (ae) (4). The heat production from an oxidation reaction averages at 27 kcal per ae equivalent. A carbon reduction degree, c is deﬁned as the number of ae per one carbon atom. Its numeric value can be determined from the stoichiometry of the oxidation reaction: CHpOnNq bO2 CO2 0.5( p 3q)H2O qNH3 (10)

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KINETICS, MICROBIAL GROWTH

c

4b

4

p

2n

3q

(11)

The ae balance for equation 1 can be written as cs b( 4) Ycx YPcp (12)

where cs, cx, and cp are the carbon reduction degree of, respectively, substrate, biomass, and extracellular product. Dividing both sides of equation 1 by cs we obtain the relationship delineating the ae distribution between oxygen (ae used for respiration), biomass, and the intracellular product: 4b cs Ycx cs YPcp cs 1 (13)

Sometimes catabolic machinery is entirely wasteful (respiration without cell growth) and always at least some minor part of energy consumption is diverted from growth. To account for this phenomenon, it was postulated that microbes and cells require energy not only for growth but also for other maintenance purposes. Certain speciﬁc maintenance functions recognized now are turnover of cell material, osmotic work to maintain concentration gradients between the cell and its exterior, and cell motility. According to conventional deﬁnition of maintenance (5), the balance of energy source is total energy source consumed equals consumption for cell growth plus consumption for maintenance: dSE dSG dSM (16)

The second term in this equation is the fraction of ae transferred to biomass from utilized substrate, termed the energetic growth yield. g YCcx /cs (14)

Let us divide it by dx, the amount of biomass produced, then 1 Y dSG dx dSM dx 1 Ymax m l (17)

The third term designates that fraction of total substrate internal energy that is transferred to the product. It is called the energetic product yield f YPcp /cs (15)

Energetic yield g is related to other stoichiometric parameters as follows: g g Yrxcx /(rscs) YCcx /cs

dx/dSG is true growth yield, that is, yield Here, Ymax under imaginary conditions of maintenance being zero. The maintenance coefﬁcient, m, is introduced as the speciﬁc (i.e., expressed per unit of biomass) rate of energy consumption for maintenance functions: m (1/x)(dSM /dt). The ratio m/l on the right side of equation 17 was derived as follows: m/l [(1/x)(dSM /dt)]/[(1/x)(dx/dt)] dSM /dx. If we divide equation 16 by xdt (note that the second term is dSG /(xdt) [dx/(xdt)]/[dx/dSG] l/Ymax), then we have: q l/Ymax m (18)

where Y is g CDW/g substrate and YC is g CDW-C/g substrate C. The advantage of using g is that it varies within a much smaller range than other yield expressions. At one and the same efﬁciency of energy utilization (g), the conventional biomass C yield YC is proportional to substrate reduction degree cs and, for example, it is four times higher on glucose (cs 4) than on oxalate (cs 1), 0.48 and 0.12 g C g 1 C, respectively (assuming g 0.5 and cx 4.2). The energetic growth yield g is more or less constant (0.5 to 0.7) for substrates with cs 4.2 (4.2 corresponds to average reduction degree of microbial biomass), and it declines at higher cs. The attractiveness of macrostoichiometry and TMEB is that all growth coefﬁcients are interrelated and could be measured from any available components of the culture mass balance. For example, if you cannot record microbial growth by conventional routine as dry weight biomass (because of presence of solids in broth liquid), you may still calculate it from N or O2 uptake, CO2 evolution, pH titration rate, and so on. Variations in Yield from Energy Source, Maintenance Requirements To multiply and grow cells requires energy, but the opposite is not true: cells do not require growth to spend energy.

where q is speciﬁc rate of energy source consumption, q (1/x)(dSE /dt). It should be noticed that Ymax is a parameter, but not the yield of a real culture that always has some nonzero maintenance requirements. It is a very common mistake in the application of the maintenance concept to a particular organism: to take the real measured Y value and pick up from literature some average m coefﬁcient. The correct way would be either to borrow concurrently two parameters Ymax and m or to treat actually observed Y as a variable that is altered along with speciﬁc growth rate l according to equation 17: Y lYmax l mYmax (19)

There is another way to formulate maintenance requirements by stating that the net growth of cells l is the difference between true growth (ltrue) and endogenous decay of cellular components (speciﬁc rate, a): l ltrue ltrue l a a

Then, for the rate of energy source uptake, we have

KINETICS, MICROBIAL GROWTH

1517

lx Y or 1 Y

ltruex Ymax

(l

a)x Ymax

1 Ymax

a lYmax

(20) mYmax.

Comparing equations 17 and 20, we see that a Experimental Determination of m

To practically determine the maintenance coefﬁcient, the microorganisms are grown in chemostat culture limited by energy sources at several dilution rates D (numerically D is equal to speciﬁc growth l if steady state is achieved). At each D, we have to measure steady-state biomass x and at ˜ least one of the following quantities (1) residual substrate, s to calculate Y ˜ x /(s0 ˜ s); and (2) the rate of respective ˜ energy-yielding process, such as respiration rate, vresp, from O2 uptake or CO2 production rates to calculate speciﬁc metabolic activity, q vresp /x. These data are ﬁtted to ˜ equations 17 or 18, m and Ymax being found as nonlinear regression parameters. An example is presented in Figure 1. Most available experimental data do obey this relationship. However, considerable deviation occurs at very low growth rates usually attained in chemostat with biomass retention or in dialysis culture. The experimental Y values for slowly growing cells are higher than predicted by equations 17 and 18 (see inset on Figure 1). The explanation is very simple: the maintenance coefﬁcient varies in response to nutritional status and could not be taken as an absolute constant; under substrate deﬁciency, the cells adjust their maintenance requirements to lower values by reducing turnover rate, osmotic work, and motility (2).

The described experimental technique is indirect because it is based on measurements of l-dependent Y variation rather than m itself, and there are some assumptions needed to be conﬁrmed (e.g., that m is constant and that maintenance requirements are the only reason of Y variation). However, some components of maintenance requirements are available for direct estimation. In particular, we can assess the total turnover rate of cellular material a which is one of the main components of maintenance requirements (equation 20). The principal cell constituents that are turned over are proteins, nucleic acids, and cell wall polymers. The turnover rate is very close to endogenous respiration, which is the oxidation of those compounds produced from the turnover (breakdown) of cellular macrocomponents. Accurate measurements of endogenous respiration need to be made under normal growing conditions. It is known that the simple removal of cells from nutrient broth by ﬁltration with subsequent washing and incubation in buffer renders strong stress and may alter the normal turnover rate (6). To avoid artifacts, we can use a label-substitution technique (Fig. 2). The chemostat culture is fed alternately from two bottles containing unlabeled and labeled 14C(U) substrate respectively. The 14CO2 evolution rate is recorded after switching to unlabeled substrate, when the main source of 14CO2 are cell components. The calculated a value was found to be rather high, accounting for the major part of total maintenance determined by the indirect method (2). The endogenous respiration declined at the low growth rate (Fig. 1), indicating that under starving conditions, self-adjustment of the maintenance requirement occurs mainly as a reduction in the turnover rate of macromolecules. Maintenance Requirements and Wasteful Catabolism The described concept of maintenance requirements was the subject of severe criticism (8). One of the strongest arguments against it was an apparent increase in Ymax observed in chemostat cultures limited by P, N, and other conserved substrates under conditions of energy excess. To preserve the constancy of the true yield, Pirt (9) had to modify equation 18 in the following way: q l/Ymax m m (1 l/lm) (21)

0.4 Y, g biomass g –1 glucose

4

0.3

0.4 0.2 0 0

3

0.2

2

0.025 0.05

Endogenous respiration, mmol O2 h–1 g–1

0.1

1

0 0

0.1 0.2 0.3 Specific growth rate, µ (h–1)

0 0.4

Figure 1. Variation of growth yield (circles) and endogenous respiration (squares) as dependent on speciﬁc growth rate in chemostat (open symbols) and continuous dialysis culture (closed symbols). Solid curves were calculated from the synthetic chemostat model (2). The dotted curve was derived from the PirtHerbert model (equations 17 to 20), which predicts quite well intensive growth but fails in the region of extremely low growth rates (see inset).

where m (1 l/lm) is the second l-dependent component of maintenance energy that operates under excess of energy substrate. However, it is better to differentiate maintenance requirements sensu stricto as those more or less a minor component of the cell energy budget that is observed under energy-limitation and wasteful use of catabolic substrate under energy excess. In physiological terms, these two groups of nonproductive catabolic reactions are completely different. The ﬁrst reactions are mainly responsible for compensation of turned-over macromolecules and therefore belong to the category of regular primary catabolism. The catabolic reactions of the second group include excretion into environment of partly oxidized substances (overﬂow metabolism), uncoupling of respiration from ATP generation by metabolic inhibitors, functioning of futile cycles,

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KINETICS, MICROBIAL GROWTH

2

12

1.4

YN, 109 cell mg –1 N

1.6

YN

1.1

CO2

1.2

σN

0.8

0.8 0 15

0.05 0.1 0.15 0.2 Specific growth rate, h–1

0.5 0.25

Figure 3. Relationship between stoichiometric parameters Y and s and speciﬁc growth rates of Chlorella vulgaris grown in chemostat culture limited by nitrogen source (10). The curves are calculated using equations 23 and 24.

CO2, mg CL–1

10 concentration of some conserved limiting substrates that preserve their chemical identity after uptake (K , Mg2 , vitamins). Other conserved substrates (sources of P, N, S, etc.) are incorporated into macromolecular cell constituents (mainly nucleic acids and proteins) whose intracellular content also should be kept high at high growth rate. Both types of changes in cellular composition are manifested as r increase, and both of them require additional maintenance energy (to maintain concentration gradient or compensate turnover of macromolecules). The observed l-dependent variation in r is therefore a compromise between biosynthetic requirements and energy conservation that is attained because of optimal metabolic control of cell performance. However, it would be erroneous to consider l as truly independent variable setting up chemical composition of cells. In fact, both l and r are functions of one common independent variable, the limiting substrate concentration in the medium, s. For steady-state chemostat culture we have: l r q r r0 1 Qs ˜ r Ks s ˜ (rm Kr r0)s ˜ s ˜ (22)

5

14

Total CO2 C 2.5

0

0

0.5

1

1.5 Time (h)

2

Figure 2. Label substitution technique for determination of turnover rate of cell macromolecular constituents. Top, experimental setup including two medium reservoirs containing 14C- and 12Cglucose pumped into a cultivation vessel through a two-way valve. Bottom, example of 14CO2 evolution dynamics before and after (arrow) switching of medium feed from 14C to 12C-glucose, glucoselimited culture of Pseudomonas ﬂuorescens 1472, D 0.08 h 1 (7).

or substrate oxidation through alternative oxidases without ATP generation. These and related phenomena take place in chemostat culture limited by conserved substrates (opposite to limitation by energy source) as well as during lag phase of batch culture started from starving inoculum. We will discuss the mathematical formulation of these phenomena in the section devoted to growth kinetics. Variation in Biomass Yield from Conserved Substrates Yield on conserved substrates varies mainly as a result of alterations in biomass chemical composition expressed by parameter rs, the intracellular content of deﬁcient element or cell quota (see equation 9). For most of known cases, the content rs increases parallel to growth acceleration (Fig. 3). As yield and cell quota are inversely related to each other (equation 9), then Y values decrease with growth rate. The physiological mechanisms of this variation are as follows. The intensive growth requires higher internal

where l is speciﬁc growth rate, q is speciﬁc substrate uptake rates; r0 and rm are, respectively, lower and upper limits of r variation; low limit r r r0 is attained when s r ˜ 0 and upper limits r r rm-when s r . By excluding s from ˜ ˜ both these equations we arrive at following relationship between r and l: l k lm Ks Kr 1 (Ks Kr) we have rm(r r0) r0) r

r[k(rm

r0] (23)

Under realistic assumption k

σN, mg N 10 –9 cell

C

14C

KINETICS, MICROBIAL GROWTH

1519

r Y

1 Ym

(1

r0 r0 /rm)l/lm Y0) l lm (24) Frequency

0.6 0.5 0.4 0.3 0.2 0.1 0 –0.1 5 10 15 YATP, g CDW mol–1 20 Unreliable data Reliable data

(Ym

where Ym and Y0 are, respectively, upper and lower limits of yield variation (Ym 1/r0, Y0 1/rm). As we can see, the linear relationship between Y and l is normally observed in chemostat culture (Fig. 3). Microscopic Approach in Studies of Growth Stoichiometry Equations 1 to 24 exemplify the macroscopic approach in studying of microbial growth stoichiometry. Its typical features are the use of gross formulas for biomass and metabolic products, evaluation of total mass balance for chemical elements (C, N, P), and formal description of microbial growth as a single-step conversion of substrate(s) into biomass. By contrast, the microscopic approach focuses on the much more complex real metabolic reactions and attempts to account for a limited but still quite large number of individual metabolic intermediates. The ﬁnal aim of this approach is to organize the biochemical information into a consistent picture of microbial metabolism at the level of entire cell. The microscopic approach has become possible by virtue of advancements in biochemistry, which has succeeded in establishing a sufﬁciently full picture of metabolic processes in certain microorganisms. The pioneering work in this area was done by Bauchop and Elsden (11), who were able to sum up the balance of ATP for fermenting microorganisms. As a result, a relation was established between the biomass yield (a macroscopic quantity) and the number of generated ATP moles (a stoichiometric characteristic of real catabolic reactions): YATP MYE /n (25)

Figure 4. Frequency distribution of experimentally measured values of YATP at different degrees of creditability. The reliable data refer to studies of anaerobic growth with direct measurements of fermentation products (plotted from data base in Ref. 12. Note that these data are normally distributed with mean value 10.55, whereas all data display considerable skewness.

where n mol of ATP made available to the organism by the metabolism of one mole of energy source, and M molecular weight (g) of energy source. The following example illustrates the YATP calculation: if biomass yield of some organisms aerobically grown on glucose is 0.52 g CDW/g, then YE is 1.49 g CDW per g of oxidized glucose (calculated from equation 6) or YE 1.49 180 268 g CDW/mol (180 is glucose molecular weight); assuming that P/O 2 (that is, 2 mol of ATP produced per atom oxygen taken up) and that 2 ATP mol are produced via glycolysis (substrate phosphorylation) we arrived at n 2 12 2 26 and YATP 268/26 10.3 g CDW/mol ATP. Careful determination of n and YATP is possible only for anaerobic growth of fermenting microorganisms generating ATP via substrate phosphorylation. The mean value tends to be around 10.5 g CDW/mol ATP (Fig. 4). For aerobic growth, we need to make assumptions on the P/O ratio. As soon as the respiratory chain of bacteria differ widely for various organisms and growth conditions, this assumption can never be reliable. To avoid this obstacle, an interesting approach was proposed (1): microbial culture is grown in a chemostat limited by two carboncontaining energy sources, their ratio is varied while the

total carbon feed rate is kept constant; yield measurements should allow one to determine both parameters (P/O and YATP) independently by multiple linear regression. Today, microstoichiometry is quickly progressing as socalled metabolic balancing. Cell growth is viewed as a set of transport and intracellular metabolic reactions known for some particular organisms. As a rule, the produced metabolic networks are composed of a combination of true stoichiometric equations for individual metabolites and empirical gross equations (Table 2). The amount of such equations vary in different models from 20 to 30 to more than 100. For example, van Gulik and Heijnen (13) describe yeast growth by a set of more than 90 reactions including glycolysis and the citric acid cycle (14); PEP phosphotransferase; pentose phosphate pathway (6); glyoxylate shunt (2); oxidative phosphorylation (4); CO2 interaction with THF (3); transport of inorganic P, NH4 , SO2 , ace4 tate, lactate, pyruvate, glucose, gluconate, succinate, and citrate (totally 10 transport reactions); amino acid synthesis (15) and polymerization (2); nucleotide synthesis (9); RNA synthesis; ATP consumption for maintenance; fatty acids synthesis (2); formation of glycogen and polysaccharides; and ﬁnally, the biomass formation from proteins, polysaccharides, RNA, fatty acids, and glycerol. For each compound, i, involved in a metabolic system, a mass balance can be deﬁned: dCi dt rAi Ui (26)

where Ci is concentration of ith compound, rAi, and Ui denote the net rates of, respectively, i chemical conversion and transport over the boundaries of bioreactor (ﬂuxes of

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KINETICS, MICROBIAL GROWTH

Table 2. Metabolic Networks Reaction Glycolysis reaction Equation Glucose ATP r glucose-6-P ADP H Glucose-6-P r fructose-6-P 0.5 Fructose-6-P 0.5 ATP r glyceraldehyde-3-P 0.5 ADP 0.5 H NADH 0.5O2 d1ADP d2Pi (1 d1) H r (1 d1)H2O NAD d1ATP a2polysaccharides a3RNA a4lipids r biomass a1Proteins Stoichiometric Equation x x x x x Empirical Gross Equation

Oxidative phosphorylation Biomass formation

CO2, O2, nutrients, cells, and products). Most metabolic balancing equations are applied to steady-state growth (which means that no intracellular accumulation of metabolites occurs). In such cases, the differential equations like equation 26 are reduced to linear algebraic ones. Besides, an extensive use of matrix calculus is customarily made to obtain a concise notation. The problem of experimental support of such model is especially important (13). The degree of freedom, df, of the resulting system of linear equations is equal to the total number of unknown rates (both intracellular and exchange reactions) minus the total number of linear equations. To resolve the system, df rates have to be measured, and then the system is fully determined. If the number of measured rates is greater than df, then the system is overdetermined, and the redundancy of the data can be used for statistical analysis and error minimization. However, it is much more typical to have an underdetermined system when the sum of measured rates is less than df. In this case, the number of possible solutions is inﬁnite unless additional constraints are applied (e.g., maximization of biomass yield, minimization of energy expenditure) to ﬁnd the one and only one solution by the linear optimization technique. In most studies, ﬂux estimates are obtained using measurements of substrate consumption and product formation. This approach has proved to be efﬁcient in some particular biotechnological cases, such as when only speciﬁc pathways need to be considered (16) or if the contribution of ﬂux for cellular growth is weak, as with mammalian or hybridoma cells (17). The more complex microbial systems are turned out to be seriously underdetermined. In such cases, the application of metabolic balancing requires the use of one or another maneuvers: (1) to lump together several sets of reactions (18); (2) to utilize data from in vitro enzyme assays; (3) to make assumptions on numeric values of some stoichiometric growth parameters, such as YATP, P/O, and H /e ratios, which are the subject of controversial debates (13). However, the best solution would be to get direct experimental data on in vivo ﬂux and resolve the system. Isotopic tracers are one of the best candidates for such a purpose. We will illustrate this point by describing a recently published work (19). This novel approach is based on the analytical power of 1H-detected 13C nuclear magnetic resonance. Corynebacterium glutamicum was grown in chemostat culture continuously fed with [1-13C]-glucose; when steady state was established, the cells were harvested and hydrolyzed and the amino acids were separated by ionexchange chromatography and analyzed by NMR spectros-

copy. NMR provides data on 13C enrichment at each speciﬁed carbon position of amino acid. Because metabolic pathways for amino acid synthesis are exactly deﬁned, then the entire central metabolism can be assessed for in vivo ﬂuxes, including determination of the forward and back rates of bidirectional reactions. In C. glutamicum, the ﬂux through the pentose phosphate pathway turned out to be 66.4% (relative to glucose input ﬂux 1.49 mmol g 1 CDW h 1); the entry into tricarboxylic acid cycle, 62.2%, and the contribution of the succinylase pathway to lysine synthesis, 13.7%. The total net ﬂux of the anaplerotic reactions (carboxylation of PEP/pyruvate into oxaloacetate/ malate) was quantitated as 38%, the true forward ﬂux of C3 r C4 being 68.6% (1.8 times of 38%) and a back ﬂux of C4 r C3 being 30.6% (0.8 times of 38%) (19). The metabolic balancing proved to be very promising and useful to identify metabolic constraints for intensive synthesis (overproduction) of products such as amino acids. On the other hand, this approach still is restricted to steady-state and balanced growth and is not able to cope with complex dynamic behavior of microorganisms (transient growth, changes in biomass composition). BASIC PRINCIPLES OF GROWTH KINETICS Kinetics of Chemical and Enzyme Reactions We need to introduce some basic principles of kinetic analysis of chemical and enzymatic reactions. Quantitative description and understanding of microbial growth dynamics and kinetics are impossible without some elementary knowledge in underlying scientiﬁc disciplines. Enzymatic and chemical reactions play an essential role in biotechnology, which is one of the most important ﬁelds in industrial development. Order and Molecularity of Chemical Reactions. The molecularity of any chemical reaction is deﬁned by the number of molecules that are altered in the reaction (Table 3). The order is a description of the number of concentration terms multipled together in the rate equation (Table 4). Hence, in a ﬁrst-order reaction, the rate is proportional to one concentration of reactant; in a second-order reaction, it is proportional to two concentrations or to the square of one concentration. For a simple single-step reaction, the order is generally the same as the molecularity. For a complex reactions involving a sequence of unimolecular and bimolecular steps, the molecularity is not the same as its order. Reactions of molecularity greater than 2 are com-

KINETICS, MICROBIAL GROWTH Table 3. Molecularity of Chemical Reaction Molecularity of reaction Unimolecular or monomolecular Bimolecular or Trimolecular or termolecular or or Reactants S S S1 S S1 S1 S S2 S S S1 S2 S2 S3 r r r r r r Product P P P P P P

1521

of inspected reactions is different, then we have to equalize the respective rate of expressions; for example, the secondorder rate constant k (time) 1 (concentration) 1 should be multiplied by instant concentration of reactant s to be compared with the ﬁrst-order rate constant having dimension (time) 1. The dimensions of the zero-, ﬁrst-, and secondorder rate constants are shown in Table 4. Reaction Dynamics. If rate constant and so-called initial conditions (concentrations of reactants at zero-time [s01, s02, . . . ]) are known, then it is possible to calculate the time course of reactions either in terms of dynamics of residual reactant concentration, s(t) or product accumulation, p(t). For this purpose, we have to integrate a differential equation under speciﬁed initial conditions (see results of integration in Table 4). The dynamics of s(t) are linear in the case of a zero-order reaction and hyperbolic in the case of the ﬁrst and second order. The difference between the last two dynamic curves can be made visible with a semilogarithmic plot of log(concentration) versus time; it should became linear for the ﬁrst-order reaction and remain to be curvilinear for the reaction of higher order. Reaction Half-Time. The reaction half-time (t0.5) is a very popular kinetic parameter, especially among biologists. It is easily calculated from integral equations by putting s s0 /2 when t t0.5. A unique feature of the ﬁrstorder reaction is the constancy of t0.5 independently of the initial reactant concentration s0. However, the half-time of other reactions does depend on s0; it increases for zeroorder reactions and decreases for the second-order reactions with increase in s0. Thus, it is not recommended to use half-time as a parameter or estimator for reactions other than ﬁrst-order reactions. Determination of the Order of Reaction and Numeric Values of Kinetic Constants. If the reaction has an order n and rate constant k, then the reaction rate v and reactant concentration s are related by the equation

mon, but reactions of order greater than 2 are very rare. For instance, a trimolecular reaction, such as A B C r P, as a rule proceeds through two elementary steps, A B r X and X C r P, each of which are of the second or ﬁrst order. Very often, bimolecular reactions between S1 and S2 occur under the condition that their respective concentrations s2 s1 (e.g., if the second reactant S2 is solvent), then we have a pseudo ﬁrst-order reaction. Some reactions are observed to be of the zero order, that is, the rate appears to be constant, independent of the concentration of reactant. This is a characteristic feature of catalyzed reactions and occurs if reactant is present in such large excess that the full potential of catalyst is realized. Dimensions of Rate Constants. Knowledge of dimensions is very useful to check the correctness of derived kinetic equations: the left- and right-hand sides of an equation must always have the same dimensions. This general rule is applicable to all mathematical models (not only in chemical kinetics). It is incorrect to add or subtract terms of different dimensions, although you may multiply or divide them. For example, if expression “ . . . (1 s)” occurs in an equation, where s has dimension (concentration), then either equation is incorrect, or the 1 is a concentration that happens to have a numerical value of 1 unit. The operation rising to power is allowed for only simple dimensionless numbers, for example, expression e2.5t, where t is time, is correct only if 2.5 has dimension (time) 1. The comparison of velocities of two reactions does make any sense only for kinetic terms of the same dimension. If the kinetic order

Table 4. Kinetic Order of Chemical Reactions Reaction order Zero First Second Second Pseudo-ﬁrst s01 s02 Differential equation ds dt ds dt ds dt dp dt dp dt k ks ks2 ks1s2 ks1s2 Dimension of rate constant (conc.)(time) (time) (conc.) (conc.) (conc.)

1 1

Dynamics of residual reactant s(t) s(t) s0 kt p(t) p(t) p(t) p(t) p

Dynamics of product accumulation p0 s0[1 kt exp( kt)]

Reaction half-timea t0.5 t0.5 t0.5 s0 2k ln2 k 1 b ks0 1/ks01 1/ks02

s0 exp( kt) s0 1 ks0t s01 s02 p(t) p(t)

1

(time) (time) (time)

1

s(t) s1(t) s2(t) s2(t) s1(t)

2 s0kt 2(1 s0kt)

1

1

s01s02(1 exp[(s02 s01 s02 exp[(s02 s02[1

0.

s01)kt]) s01)kt]

t0.5 t0.5

1

1

s02 exp( ks01t) s01

exp( ks01t)]

Note: t time; s reactant concentration; p product concentration; s0 reactant concentration at t a The half-time of the reaction is the time required for half-completion. b The half-time is deﬁned for the reagent that has lower initial concentration and is depleted ﬁrst.

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KINETICS, MICROBIAL GROWTH

v

ds/dt

dp/dt

ksn

(27)

e0 x s

e s s0

x

s0

s

x (29)

The simplest way to ﬁnd both unknown values (n and k) would be to measure reaction rate v at several concentrations of reactant. Then a plot of log(rate) against log(concentration) gives a straight line with a slope equal to n and intercept equal to log(k): log v log k n log s. If there are several reactants, then it is useful to know the order in respect to each one. For this purpose, you need to have several experiments with variation of each reactant concentration while keeping the other concentrations constant. The most frequent goal is the determination of the ﬁrstorder rate constants, ﬁrst because many reactions do obey ﬁrst-order conditions in respect to each reactant and second because it is possible to carry out many reaction under pseudo ﬁrst-order conditions. There are some special methods to determine the values of the ﬁrst-order rate constants from the experimental curves of product formation or substrate depletion dynamics. Some of them were designed to improve the accuracy of k determination when the initial reactant concentration (s0) or ﬁnal value of product concentration ( p ) were not known (methods of Guggenheim, Kezdy-Swinbourne, and others, see details in Ref. 20). All are based on plotting the experimental points in a some sophisticated manner to convert the original curves to a straight lines. Today, these methods are replaced by computer-aided nonlinear regression, which is much more convenient and precise and, contrary to graphic methods, allows for more rigorous estimate of conﬁdence limits of measured quantities. Derivation of Basic Kinetic Equations for Enzymatic Reactions. Contrary to simple chemical reactions, enzymecatalyzed reactions proceed through reversible formation of the dynamic enzyme-substrate complex (ESC). The word reversible is essential because the ESC can be decomposed into free enzyme E and product P or dissociate back to E and substrate S. There are many ways to simulate mathematically the mechanism of the enzymatic reaction. We will consider here equilibrium, steady-state, and general non-steady-state approaches. Equilibrium Approach. This approach was used by Michaelis and Menten (21) to describe the effect of sucrose concentration on invertase activity. They assumed that the ﬁrst step of ESC formation is so rapid that could be represented by an equilibrium constant Ks:

e s Ks x k2 p

The overall reaction rate, v, is a simple ﬁrst-order reaction with rate constant k2: v dp/dt ds/dt k2x (30)

The x value can be found from expression for Ks and massbalance conditions (equation 29): x e0s/(Ks s). Substitution of this expression into equation 30 ﬁnally produces v k2e0s Ks s (31)

Steady-State Approximation. This approach was applied by Briggs and Haldane (22) for the following scheme of enzymatic reaction

e s

E

S i ES r E

k

1

k1

x

k2

p

P

(32)

This scheme implies the reversibility of ESC formation instead of much more restrictive equilibrium postulate. However, still there was steady-state assumption in respect to ESC formation: dx/dt k

1(e0

x)s

k

1x

k

2x

0

(33)

Therefore, x k 1e0s/(k 1s k 1 k 2), and substitution of this expression into equation 30 gives v ds dt k k

1

k

2x

k k

1s

1k

2e0s 1

k Vs s

k

2

2e0s k 2 1

k

s

Km

(34)

E

S } ES r E

P

(28)

where e, s, x, and p denote the concentrations of free enzyme, substrate, FSC, and product, respectively. The equilibrium constant, Ks, is deﬁned as Ks es/x. The instantaneous concentrations s and e are not directly measurable, but they could be expressed in terms of the initial, measured concentrations, e0 and s0, using massbalance relationships:

Equation 34 is what usually called the Michaelis-Menten equation, the fundamental equation of enzymatic kinetics. It contains two parameters, Km, the Michaelis constant, and V, the maximum velocity. V is the rate of reaction that would occur under full substrate saturation of an enzyme’s active sites (s Km). In reality, the V value can be never attained, because extremely high substrate concentrations inhibit enzymes (see later text). It is clear also that V is not a fundamental property of enzyme because it depends on e0, the enzyme concentration. More advantageous as a speciﬁc enzyme characteristic is the catalytic constant or turnover number, kcat, which is V/e0. For equation 32, kcat is identical with k 2, but in general the more noncommittal notation kcat is preferable (e.g., kcat may differ from k 2 if the product formation from ESC is a reversible reaction). Numerically, the Michaelis constant, Km, is the substrate concentration that provides half the maximal reaction rate. Contrary to this simple practical deﬁnition, the mechanistic interpretation of Km is not so lucid. Sometimes

KINETICS, MICROBIAL GROWTH

1523

Km is interpreted as a substrate binding constant, Ks, assuming that k 2 k 1. This is a very dubious assumption. There are very few enzymes for which the individual values of k 2 and k 1 are known. Ironically, the best-studied examples (e.g., horseradish peroxidase) present just opposite case: k 2 k 1. It is important that many speciﬁc mechanisms (not necessarily equations 28 and 32) generate the same steady-state rate equation 34. However, the particular expression for Km should be different for each individual case. Although undeﬁned in mechanistic terms, the parameters Km and V are very useful at the ﬁrst steps of kinetic studies: 1. The use of equation 34 allows the expression of the complex effects with simpler terms. 2. Km and V are helpful as predictive parameters to design a valid enzyme assay. One practical recommendation is to keep substrate concentration in incubation mixture at the level of 10 Km or higher. 3. Equation 34 permits one to obtain at least rough estimate of in vivo enzyme activity provided the internal substrate concentrations are known. General Non-Steady-State Approach. Equation 32 contains four variables (s, p, e, and x) constrained by two massbalance equations (e0 e x and s0 s p x). Therefore, it is enough to integrate just two differential equations (e.g., equations 33 and 34) to characterize the whole system. A typical example of a numeric solution is shown in Figure 5. We can see that enzyme-catalyzed reaction proceeds through three deﬁnite phases well separated on the time scale. Each phase is now safe to analyze with simpler mathematical expressions because any assumptions could be tested against the full exact solution. 1. The ﬁrst transient phase occurs before the steadystate concentration of ESC is reached. It occupies less than

10 3 s for most of enzymes. During this phase, the substrate concentration remains fairly constant (s s0), allowing for the following analytical solution: x(t) A k k

1e0s0

A

1s0

[1 k

1

e

At

],

2

v(t)

Vs [1 Km c

e

At

]

k

(35)

As compared with equation 34, it contains a relaxation term [1 exp( At)] that is very large when t is small, but decays to zero as t increases above s 1/A. Accordingly, the rate of enzymatic reaction is initially zero but increases rapidly to the steady-state value as the exponential term decays. Because the relaxation term contains s0, then the experimentally observed delay depends on substrate concentration. It allows for direct determination of individual rate constants k 1 and (k 1 k 2) by techniques such as in stopped-ﬂow apparatus (23). 2. The second steady-state phase is characterized by the constancy of reaction rate due to the exact balance between the rates of ESC formation and breakdown (dx/dt 0) while the substrate concentration remains close to the initial value s0. This phase proceeds for at least several seconds. As a rule, it is enough to measure the initial velocity of enzymatic reactions unaffected by substrate depletion or product accumulation. Steady-state kinetics is the most popular research domain; it is the most accessible and developed and provides the most kinetic data. However, it fails to determine individual rate constants as fully as the transient and relaxation approaches do. Steadystate equations were derived earlier (equation 34). 3. The third phase is characterized by considerable changes in the concentrations of substrate and product. Thus, we can no longer assume that s s0, and we should integrate equations 32 and 33. However, very good precision provides the quasi-steady-state approximation ds/dt dx/dt 0. Then, we arrive at a relatively simple differential equation that is solved analytically: ds dt For initial conditions s Km ln s0 s Vs(t) Km s(t) s0, p s0 0, t s 0, we have Vt (37) (36)

100 Instant reaction rate, nmol s–1 80 60 40 20

Steady-state phase

Substrate depletion phase

Transient phase

Equation 37 is called the integrated Michaelis-Menten equation. It remains to be valid not only for initial rate measurements but for any point within the reaction progress curve. 1 0.001 Time, s (log scale) 1000 Experimental Determination of Kinetic Parameters of the Michaelis-Menten Equation. Until recently, most enzyme kinetic experiments have been analyzed by means of one of linear plots in Table 5. Linear plots are used to examine an agreement of experimental data with equation 34 as well as to determine numeric values of parameters V and Km from slopes and intercepts. Today, this graphic ap-

0 0.000001

Figure 5. Time course of reaction proceeding by the MichaelisMenten mechanism. Numeric integration of equations 33 and 34: s0 10 4M, e0 10 8M, k 1 106M 1s 1, k 1 103 s 1, k 2 102 s 1.

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KINETICS, MICROBIAL GROWTH

Table 5. Linear Plots 1 v s v v V 1 V Km V V v Km 1 V s s V Km v s Lineweaver-Burk or double-reciprocal plot Langmuir-Hanes plot Eadie-Hofstee plot Direct linear plot (20)

K

p /s

VfKp /VrKs m m

(40)

v Km s

Equation 39 implies that the rate must decrease as the product accumulates, even if the decrease in substrate concentration is negligible. Thus, reversibility is closely associated with and requires the product inhibition. In many essentially irreversible reactions (e.g., invertase-catalyzed hydrolysis of sucrose), product inhibition is also signiﬁcant. It can be explained by equation 38 with only the second step being irreversible. In such a case, the accumulation of product causes the enzyme to be sequestered because the EP complex and rate equation are as follows (compare with equation 39): v

s Vfs/Km s s/Km pKp m

proach seems to be too cumbersome as compared with much more efﬁcient computer routines. The main objection is also that any linear transformation introduces some statistical bias (ironically, the highest bias is attributed to the most popular double-reciprocal plot!). In addition, manual line drawing is very arbitrary, so obtained parameters could not be assessed statistically for conﬁdence limits. However, plotting of original or linearized data is useful as an illustration and the ﬁrst sketchy estimate of enzymatic parameters. For deﬁnitive work, it is advisable to avoid all plots and to use statistical analyses instead. Reversible Enzymatic Reactions. The majority of biochemical reactions are reversible. To account for this feature, the Michaelis-Menten mechanism can be modiﬁed as follows:

k

1

Ks (1 m

Vfs pKp ) m

s

(41)

E

e0 x y

S i ES i EP i

s k

2

1

k x k

2

k y k

3

E

e0 x y

P

p

(38)

2

3

Inhibitors and Activators. Compounds that reduce the rate of enzyme-catalyzed reactions when they are added to the reaction mixture are called inhibitors. Just opposite effects (increase of reaction rate) are caused by activators. Both of them belong to the category of modiﬁers. We will concentrate here on inhibitors as having more practical application and even more speciﬁcally on reversible inhibitors, which form various dynamic complexes with enzymes. The irreversible inhibitors are known as catalytic poisons (many heavy metals, such as mercury); their binding to enzyme molecules reduces activity to zero, leaving no room for quantitative analysis. The reversible inhibitors can form complexes with free enzymes or enzyme-substrate complexes as shown in the scheme of Botts and Morales (20):

Contrary to the basic scheme in equation 32, there are two intermediates, one of which is the normal ESC and another is the enzyme-product complex (EP). The substrate, S, and product, P, can interconvert to each other. Application of the steady-state approach to this scheme results in the following equation: I v Vf Ks m Vr Kp m

s p Vfs/Km Vrp/Km s 1 s/Km p/Kp m

S E P Mixed S ES

I Uncompetitive

Competitive

k k k

2 1k 2

2k 3e0 k 2 k

3 3 2

k

2 2e0 2 1k

1k

k k k

1(k

k k

1k

k k

2k 3)

3

EI P

EIS

k

1 1k 2

k

3(k

2 3 2

k

1

k

k

k k

2k 2)

3

(39)

There are three major simple (or linear) inhibition mechanisms, which can be generated from this scheme by omitting some of the six involved: • Competitive inhibition if EIS is missing (inhibitor binds to the same site on the enzyme as the substrate forming abortive nonproductive complex EI). • Uncompetitive if EI is missing and EIS occurs as a dead-end complex; it implies that the inhibitorbinding site becomes available only after the substrate has bound.

where superscripts f and s denotes parameters of forward reaction, and r and p are indicators of reverse reaction. When a reaction is at equilibrium, the net velocity must be zero and, consequently, if s and p are the equilibrium values of s and p, it follows from equation 39 that equilibrium constant K is expressed via kinetic parameters (the Haldane relationship):

KINETICS, MICROBIAL GROWTH

1525

• Mixed if EI and EIS both occur but are not interconvertible (the complex EIS does not break down to products; this situation frequently occurs when the inhibitor is reaction product). The particular case of mixed inhibition is pure noncompetitive inhibition, which takes place if two inhibitor dissociation constants (for EI and EIS) are exactly matched to each other. For all types of inhibition, the Michaelis-Menten equation remains valid: under constant inhibitor concentration, i, the v-s relationship is of the same hyperbolic type as predicted by equation 34, the only difference is that apparent values of Km and V are now more or less simple functions of i (Table 6). Some parameters (in boldface in Table 6) are not affected by inhibitors, whereas others are changed: competitive inhibitors increase Km, pure noncompetitive inhibitors reduce V, uncompetitive inhibitors decrease at the same degree both V and Km, leaving ﬁrst-order rate constant V/Km to be unchanged. In mixed inhibition, there are no unchanged parameters. One interesting case is so-called substrate inhibition. It occurs when two substrates are bound to the same active site on the enzyme, forming nonproductive triple complex ES2: E S }

Kss

between, respectively, identical and different ligands (e.g., substrate and an allosteric effector). To explain the cooperativity and associated sigmoid kinetics, a number of models have been suggested. The earliest one is the Hill equation, which was originally designed to describe the S-shaped curve of oxygen binding to hemoglobin. It was assumed that each protein molecule E binds n molecules of ligands S in a single step, an amount of other possible forms (ESn 1, ESn 2, . . . ES) being negligible: E nS } ESn

Kh

where Kh is the respective equilibrium constant (Hill and colleagues described equilibrium in terms of association constant, but for the sake of uniformity we will adhere to the previous formalisms, keeping in mind the dissociation constant, Kh [E][S]n /[ESn]). The fractional saturation of protein (enzyme or hemoglobin). H is given as H Number of occupied binding sites Total number of binding sites [ESn] [E] [ESn] [S]n Kh [S]n (43)

Equation 43 can be rearranged into H 1 H [S]n H , log Kh 1 H logKh n log[S] (44)

KSS

ES r E @ ES2

k2

P

then, the enzyme rate (24) is: v Vs s s2 /Kss (42)

Ks

where Kss is the dissociation constant for complex ES2. Cooperativity. Many enzymes respond to changes in metabolite concentrations (substrates, modiﬁers) with much higher sensitivity as compared with predictions from the classic hyperbolic equations 34 to 42. This property is generally known as cooperativity, because it is thought to arise from cooperation between the active sites of the polymeric enzymes. As a rule, such enzymes consist of several subunits and display so-called sigmoid or S-shaped dependence of rate on substrate concentration. Many cooperative enzymes (but not all!) have active sites binding substrate and allosteric sites binding effectors. There are homotropic and heterotropic cooperative effects caused by interactions

A plot of log H/(1 H) against log [S] is known as the Hill plot and should be a straight line of slope n (so). This equation is used to ﬁt the experimental binding and kinetic data displaying a sigmoid shape. When plotting kinetic measurements, it is assumed that H v/V, and maximum velocity V should be known from independent measurements taken at saturation substrate concentration. However, the results of ﬁtting should be interpreted with care. First, the Hill equation is empirical and generally provides good agreement only in the H range 0.1 to 0.9 (the discrepancy at extreme s is probably caused by neglecting of other forms of ESC apart from ESn). Second, parameter n (Hill coefﬁcient) could not be interpreted as the number of subunits in the fully associated protein, rather it is an index of cooperativity. Monod et al. (25) proposed a general model explaining cooperativity and allosteric phenomena within a simple set of postulates: 1. Each subunit of enzyme can exist in two different conformations, designated R (relaxed) and T (tense).

Table 6. Types of Inhibition Type of inhibition Competitive Uncompetitive Mixed Pure noncompetitive V V/(1 V/(1 V/(1 Vapp i/Ki ) i/Ki ) i/Ki ) Vapp /Kapp m (V/Km)/(1 V/Km (V/Km)/(1 (V/Km)/(1 i/Ki)a i/Ki) i/Ki) Km(1 Km(1 Km(1 Km

app km

i/Ki) i/Ki )) i/Ki)/(1

i/Ki )

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KINETICS, MICROBIAL GROWTH

2. All subunits of a molecule must occupy the same conformation at any time (e.g., for tetrameric enzyme only R4 or T4 are permitted, not R3T or R2T2, etc.). 3. The two states are in equilibrium with constant L [R4]/[T4]. 4. The afﬁnity of ligand to subunit depends on the conformation state: KR [R][S]/[RS], KT [T][S]/[TS], c KR /KT. The general equilibrium solution of this scheme is rather bulky, so we demonstrate one particular case of tetrameric protein, if c 0 (i.e., KR KT, S binds only to the R state), then we have H (1 L s/KR)3s/KR (1 s/KR)4 (45)

ESC, whereas V/Km reﬂects the ionization of the free enzyme. The pH effects on Km are more complicated, being affected by both. Generally, the dependence of enzyme activity on pH is described by equation 43 as a bell-shaped curve with maximum at 1/2(pK1 pK2). Effects of Temperature. In chemical kinetics, the dependence of reaction rate on temperature is explained by the transition-state theory developed by Eyring in 1930 to 1935. It is based on the use of thermodynamics and principles of quantum mechanics. The reaction proceeds through a continuum of energy states and must surpass the state of maximum energy, when transient activated complex is formed. Then the dependence of reaction rate constant k on absolute temperature, T, is expressed as follows: d lnk dT DH* RT RT2 (47)

At high s when s/KR L, we can neglect the term L in the denominator, and the entire expression is converted to no more than the simple hyperbolic Langmuir isotherm. At low s, the contribution of L is considerable, so the saturation curve rises slowly from the origin displaying the S-shape. There are other models describing the mechanisms of cooperativity and allosteric effects (20), for example, the sequential model of Koshland and colleagues and the association–dissociation model of Freiden. Effects of pH. Every enzyme contains a large number of acidic and basic groups. Some of them are either fully deprotonated (aspartate, glutamate) or fully protonated (arginine, lysine). However several groups with pKa 5 to 9 (imidazole group of histidine, sulphydryl group of cysteine) do change their ionization state when pH is varied. Assume as a ﬁrst approximation that enzyme is represented as a dibasic acid H2E and only a singly ionized complex, HES , is able to react to give products: H2E H2ES @ KE k 1 @ KES 1 1 HE i HES @ KE k 1 @ KES 2 2 2 E ES2

where R is the gas constant and DH* is the enthalpy of activated complex formation. The classic Arrhenius equation may be obtained from equation 47 under a simpliﬁed condition DH* RT DH Ea (where Ea is activation energy). Most often, the Arrhenius equation is used in its integrated form: lnk or k A exp( Ea /RT) (48) lnA Ea /RT

where A is the integration constant, interpreted as the frequency of collisions of reacting molecules. Apart from mechanistic derivations, there are a number of empirical expressions relating k and Celsius temperature Tc. The most popular is the exponential formula: lnk lnA Tc

k

S

r

2

HE

P

or k A exp( Tc) (49)

Then the reaction rate is dependent on H concentration, h, as follows (20): v V Vs Km s ˜ V h KES 1 h KE 1 1 ˜ ˜ V/Km 1 KE 2 h KES 2 h (46)

V Km

where is the empirical constant related to the widely used temperature coefﬁcient Q10 exp(10 * ). All presented mathematical expressions predict exponential or almost exponential increases of chemical reaction rates with temperature. However, enzymatic reactions deviate from this relationship at high temperature because of thermal denaturation of enzymes. Assume that denaturation is reversible with equilibrium constant KT [E ]/[E], where E represents active enzyme molecules and E represents inactive molecules. Then, the combination of equation 48 with the van’t Hoff relationship for KT ( RT lnKT DGo DHo TDSo) results in v A exp( Ea /RT) exp(DSo /R DHo /RT) (50)

˜ ˜ where V and Km are the pH-corrected constants. It is clear that pH-dependent variation of V reﬂects the ionization of

1

where DGo, DSo, and DHo are the standard Gibbs free en-

KINETICS, MICROBIAL GROWTH

1527

ergy, enthalpy, and entropy of denaturation reaction, respectively, and v is the observed rate of biological process. This equation produces a curve with single maximum and ﬁts to most of the available experimental data on temperature-dependent variations of enzymatic activity. Sometimes denaturation is irreversible, and there are no possibilities for simple mathematical expressions, because temperature effects depends on the exposing time. However, numeric solutions of respective differential equations can still be used. Simple Models of Microbial and Cell Growth This section deals with simple unstructured models. These models mainly ignore any changes in cell quality (biochemical composition, spectrum of enzymatic activity, etc.) induced by environmental factors. Main State Variables and Growth Parameters. There are two types of growth models, deterministic and stochastic. The former describe clear determined and regular processes. The latter deal with random or stochastic processes. The main variables used in deterministic models are the same as in chemical kinetics, concentrations of biomass, substrates, and products, and stochastic models consider instead the probabilities, frequency distributions, variance, and so on. For example, a stochastic model can consider the probability of a single bacteria cell dividing under speciﬁed environmental conditions. Although any real-life biotechnological process has both deterministic and stochastic components, most useful growth models are strictly deterministic. In this section, we will concentrate on this type of model, and the stochastic counterparts will be considered only in “Cell Cycle”. The major state variables of deterministic models describing cell growth are described in Table 7. The concentration of biomass and cell number are related to each other by simple formula: x m N (51)

therefore, it is advisable to make a selection. The biomass, x, has obvious advantage in studies aimed at understanding or control of mass ﬂows, and cell number, N, is preferred in population studies when, for example, mutation or plasmid transfer is an essential factor controlling the efﬁciency of the biotechnological process. The choice of method to determine biomass or cell number depends on many factors (2,5). Today, the preference should be given to those techniques that allow exact and automated measurements (Table 8). The most advanced analytical methodology is now available for automatic recording of gaseous or volatile substrates, intermediates, and end products, such as methane, CO2, O2, H2, volatile fatty acids, alcohols, and other fermentation products (IRanalyzers, mass spectrometry, gas chromatography, NMR, etc.). The primary state variables that are measured directly in cell culture are usually recalculated into the secondary growth characteristics: gross growth rate, dx/dt; speciﬁc growth rate, l (dx/dt)/x; degree of multiplication, x/x0; biomass doubling time, td ln 2/l; growth yield, Y dx/ ds; product yield, Yp /s dp/ds, Yp /x dp/dx; speciﬁc rate of substrate consumption, qs (ds/dt)/x l/Y; speciﬁc rate of product formation, qp (dp/dt)/x lYp /x l(Yp /x /Y). Most of the listed secondary growth characteristics are called speciﬁc rates expressed as a ﬁrst-time derivatives of measured variable per unit of cell mass. The speciﬁc rates are not sensitive to variations in total cell biomass, so they can be considered as analogous to enzyme activity (qs, qp) in most kinetic derivations. The speciﬁc growth rate, l, may be viewed as the activity of an autocatalytic enzyme producing itself. The speciﬁc growth rate, l, measured from biomass dynamic, x, can differ from that estimated from the increase in cell number, N (denoted as lN). It follows from equation 51 that l 1 dx x dt 1 dm m dt 1 d(N m) x dt 1 dN N dt lcell 1 N lN m N dm dt m dN dt (52)

where conversion factor m1 is the average dry mass of a single cell. In some studies, it is save to assume m to be a constant and use both variables x and N as equivalent measures of growth. However, the average size and mass of single cells vary depending on the nature of studied organisms and environmental conditions (see “Cell Cycle”);

0 and l If mass of single cell m is constant, then lcell lN. Otherwise, we have to take into account m variation. Validity of Exponential Growth Low. One of the earliest postulates in microbial kinetics is that under optimal non-

Table 7. Major State Variables of Deterministic Models Variable Concentration of cell biomass Cell numbera Single-cell massa Mycelium lengthb Tips numberb Concentration of limiting substrate Concentration of product

a b

Notation x N m L n s p

Dimension (examples) g CDW L 1 of cultural liquid 109 cell mL 1 of cultural liquid g CDW per cell meter L 1 of cultural liquid 106 tips mL 1 of cultural liquid g L 1 of cultural liquid g L 1 of cultural liquid or g g 1 of CDW

For unicellar organisms (bacteria, yeasts). For ﬁlamentous organisms (fungi, actinomycetes).

1528

KINETICS, MICROBIAL GROWTH

Table 8. Methods Used for Determination of Microbial Biomass Method Dry mass Wet mass Wet biovolume Particulate organic carbon Biuret proteins Measuring principle Mass of separated and dried solids Mass of separated material Linear dimension of pelleted cells or colony CO2 after cell separation and combustion Colorimetric reaction of peptide bonds Colorimetric reaction of tyrosine and tryptophan DL 50 50 100 1.0 Advantage Provides direct unconditional estimate Simplicity, quickness Simplicity, quickness High precision and sensitivity, provides direct estimate of mass High uniformity Disadvantages Interference from dead cells and noncell solids The variation of wet biomass bulk density The variation of wet biomass bulk density Interference from dead cells and noncell solids Variation of protein-to-biomass ratio, possible extracellular accumulation Variation of protein-to-biomass ratio and amino acid composition of cell protein, possible extracellular accumulation Possible extracellular accumulation Variation of the intracellular ATP pool Variation of the intracellular content, possible extracellular accumulation Variation in conversion factor from measured rate to biomass Interference from noncell solids, cell aggregation, wall growth Variation of conversion factor, interference from electrolytes in the medium Time-consuming, low precision

1.0

Folin-Ciocalteu protein

0.1

High sensitivity

DNA ATP Fatty acids

Colorimetric estimation of deoxyribose Bioluminescence assay Gas or liquid chromatography, colorimetric reaction Rate of added substrate uptake or product formation Light scattering Conductivity

1.0 10

5

High speciﬁcity, constancy of the DNA cellular content High sensitivity and speciﬁcity High speciﬁcity, allows identiﬁcation of composition of mixed culture High speciﬁcity, quickness

1.0

Metabolio potential

10

3

–10

Opacity Electrical measurements Manual microscopy

0.1 1

Simplicity, quicknerss, automation Simplicity, quickness, automation Low cost, sensitivity, allows visual assay of biomorphological structure Combined beneﬁt of manual and automated quantiﬁcation, speed, generation of size distribution Quickness, automation, generation of size distribution Low cost, high sensitivity

2% (mg).

Cell count, measuring of hypha length Computer-aided count

10

5

Image analysis

10

5

High cost

Coulter counter Plating

Note: DL

Automated count and sizing Growing of the colonies on Petri dish

10 10

5

Interference from noncell solids Time consuming, low precision

5

detection limit, the minimum CDW required for an estimation with an error

restrictive conditions (nutrient media containing all essential components at nonlimiting concentrations, absence of inhibitors, adequate physicochemical parameters, perfect mixing), the increase in biomass (dx) during an inﬁnitely small time interval (dt) is proportional to this time interval and the instant biomass concentration (x), that is dx or dx/dt lx (53) lxdt

lnx

lnx0

lt ⇒ x

x0 exp(lt)

(54)

where l is the proportionality coefﬁcient. If l is constant, integration of equation 53 gives

where x0 is biomass at zero time or inoculum size. According to early views, true exponential growth occurs only in the case of symmetrically dividing bacteria with equal probability of subsequent division for the mother and daughter cells. Organisms with asymmetrical multiplication (budding yeasts) were thought to obey the exponential low only approximately, whereas the growth of ﬁlamentous organisms (fungi, streptomycetes) was deviated considerably. However, direct measurements performed from 1930 to 1960 (5) revealed that exponential growth does occur in all prokaryotes and eukaryotes independent of their biomorphological features, including protozoa, fungi, and homogeneously cultivated plant and

KINETICS, MICROBIAL GROWTH

1529

animal cells. The deviation is observed only as a result of a growth-associated change in the environment. Wang and Koch (26) made very precise measurements of l dynamics by growing Escherichia coli directly in an aerated cuvette of the computer-linked double-beam spectrophotometer. They found temporary slowdowns in the complex peptone plus beef extract media attributed to diauxie phenomena (e.g., depletion of some preferred oligopeptides) and a gradual increase of the l value during sequential subculture in succinate minimal medium. Long-term cultivation of microorganisms in continuous turbidostat or pH-stat culture (14) revealed that the increase in l is mainly the result of autoselection of mutants having higher maximum speciﬁc growth under speciﬁed cultivation conditions (see “Population Dynamics”). Early Views of Cell Growth. The derivation of the exponential growth equation was done for binary dividing bacteria, based on the geometric progression 2, 22, 23 . . . (27): N N02

n

A similar (but not identical) equation of cell biomass dynamics would be produced if we assumed that the limiting substrate is consumed according to ﬁrst-order kinetics and is converted to biomass with a ﬁxed yield factor: Y then ds dt s x0 (x x0)/(s0 s) (58)

ks Ys0 Y ds Y dt x xm Y kxm 1 x xm x ⇒ dx dt (59)

(55)

where n is the number of divisions, n t/td. The growth dynamics were viewed as a succession of distinct phases approximated with empirical formulas (15): N N0 exp[lF(t)t] (56)

Monod’s Model. This model still is very popular because of its elegant simplicity. It played an important role in the history of microbial growth kinetics as a ﬁrst encouraging example when the theoretical development based on mathematical formalisms came before novel experimental designs. As compared with simple exponential equation 53, this model (29) takes into account the mass-conservation condition linking substrate uptake to biomass formation through constant yield factor Y (equation 2) and the dependence of the speciﬁc growth rate on limiting substrate concentration: dx dt ds dt s Ks s

where function F(t) undergoes stepwise changes as shown in Table 9 (1.56 n 2.7). The alternative way to describe growth dynamics was to use the logistic equation borrowed from demographic studies (28): dN dt or dx dt rx 1 x xm (57) lmt rN 1 N K

l(s)x 1 dx Y dt

lm

x (60)

By substitution of s by x through Y (equation 58), we can reduce equation 60 to a single equation having the analytical solution: (1 P) ln(x/x0) P ln(Q 1) P ln(Q x/x0) (61)

where r is the net growth rate (the difference between true growth and death rates) and K or xm are the upper limits of, respectively, population density or biomass (K is used preferentially in ecological literature, being called the carrying capacity of the ecosystem). This equation simulates S-shaped growth dynamics frequently observed in natural ecosystems and cell cultures.

Table 9. Stepwise Changes of Function F(t) Initial stationary phase Lag phase Logarithmic growth phase Negative growth acceleration phase Maximum stationary phase Accelerating death phase Logarithmic decrease phase F(t) F(t) F(t) F(t) F(t) F(t) F(t) 0 tn 1 tt 0 tn 1

1

1

1

where P YKs /xm, Q xm /x0, xm Ys0 x0, and s0 and x0 are, respectively, s and x at t 0. Equation 61 contains three parameters, Y, lm, and Ks that can be thought of as passport data for a particular organism (e.g., for E. coli grown on glucose at 30 C, Y 0.23, lm 1.35 h 1, and Ks 4 mg L 1) and describe the S-shaped growth dynamics of a batch culture. The initial conditions, s0 and x0, are set by the experimenter, who selects the inoculation dose and medium composition. Thus, knowing the values of all these entities, the growth dynamics can be calculated before the experiment. Moreover, this model was used as a basis for development of the chemostat theory to understand and predict the behavior of microbial cultures in continuous-ﬂow bioreactors before the respective hardware was constructed. For this purpose, original equation 60 was modiﬁed as follows (30):

1530

KINETICS, MICROBIAL GROWTH

dx dt l ds dt D

(l lm

D)x s Ks s) s x 1 lx Y (62)

optimize the productivity by such tools as changing the ﬂow rate and the composition of medium in continuousﬂow bioreactors. Derivation of the Monod Equation from Mechanistic Considerations. The equation relating l and s is known as the Monod equation: l lm s Ks s (65)

D(sr F V

where sr is the concentration of limiting substrate in the fed-fresh medium delivered with pump from reservoir, F is the pumping rate (cm3 /h), V is culture volume (cm3), and D is the dilution rate deﬁned as the ratio D F/V, h 1. The mathematical analysis of equation 62 allows us to make the following conclusions on the growth kinetics of continuous culture: 1. The described open system attains stable steady state when dx/dt 0, ds/dt 0, and variables x and s are not changed with time (denoting steady-state concentrations x and s, respectively). From the ﬁrst ˜ ˜ equation in equation 60, it follows that l D, that is, the speciﬁc rate of microbial growth is equal to the dilution rate, which is under the full control of the experimenter. From the second equation, we ﬁnd x ˜ Y(s0 s) ˜ (63)

It was introduced by the author as a purely empirical relationship resembling adsorption isotherm. However, many microbiologists did view the Monod equation as something having deep inherent meaning rather than as just an empirical formula. Below, we shall outline a number of attempts to deduce this equation logically from the conjectured growth mechanisms. 1. The bottle-neck concept. There is an obvious similarity between the Monod and Michaelis-Menten equations. A theoretical substantiation for this similarity can be the bottleneck postulate originally put forward by Blackman (31). According to this postulate, the growth rate of a cell is determined by a single enzymatic master reaction. Microbial metabolism is symbolized as a unidirectional chain of reactions of substrate S conversion to cell biomass X via some hypothetical intermediates P1, P2 . . . Pn (32): S X r P1 r P2 r . . . r Pn r 2X

k1 k2 kn kn

1

As compared with Y expression for batch culture (equation 58) the formula in equation 63 no longer contains the term, x0, the inoculum size. Thus, a steady-state culture is not dependent on past history, and its properties are determined solely by current cultivation parameters. 2. The dilution rates permitting stable microbial growth (i.e., those D, at which x ˜ 0) are conﬁned between 0 and washout point or critical dilution rate Dc Dc lmsr /(Ks sr) (64)

(66)

Assuming steady state in respect to intermediate concentration pi, dpi /dt 0, we arrive at the Monod equation, composite parameters lm and Ks being expressed via elementary kinetic constants of individual enzymatic reactions: dx dt lm lm s Ks 1

n i 1

s

x

3. The speciﬁc growth rate l does not depend on either s0 or x and is governed solely by the substrate concentration in the cultural liquid. On the other hand, the indirect effect of s0 on l may be evaluated from the dependence of x on D, as soon as x and s are ˜ ˜ ˜ linked by the conservation condition in equation 62. The most important biological implication of the chemostat theory was the discovery of the fact that microorganisms can grow endlessly with any rate between 0 and lm. Such substrate-limited growth is nevertheless exponential, because two subsequent acts of cell division will be separated by a constant time interval. Earlier, substrate limitation had been observed only transiently at the end of the exponential phase of batch growth. The most important biotechnological implication of the chemostat model was the concept of controlled cell biosynthesis, which implies the purposeful manipulation of microbial culture to

1/ki

2

ks k1

1

n i 1

(67) 1/ki

2

where x x p1 . . . pn. The bottleneck idea can now be formulated as follows. If one of the constants kj ki, i 2, . . . , n 1, i j, then lm kj and Ks kj /k1, and so the jth enzymatic reaction is the master reaction. The obvious shortcomings of the reaction scheme in equation 66 are the unjustiﬁed oversimpliﬁcation of the cell metabolism (which is a network, rather than a simple unidirectional sequence of reactions) and the lack of a definite interpretation of variables x and p in real biochemical terms. An evaluation of the characteristic times of major intracellular metabolic reactions showed that there could be two possible bottlenecks: uptake of limiting sub-

KINETICS, MICROBIAL GROWTH

1531

strate (processes of active transport, transphosphorylation of sugars etc.) (33) and the formation of intracellular protein-synthesizing structures such as rRNA and the ribosomal proteins (34). 2. Stochastic considerations. Microbial populations were assumed to consist of active (which utilize the substrate) and inactive cells (35). The probability of a transition from the inactive state to the active one was supposed to be proportional to s, whereas the probability of the reverse transition was s independent. From these assumptions, a relationship between l and s can be inferred, similar to equation 65. 3. Derivations based on mechanistic considerations of protein synthesis (36). The rate of protein synthesis, dp/ dt, is supposed to be determined by rRNA concentration R and by the size of the amino acids pool, A, dp/dt kxRA/ (KA A), k constant. Other conditions are deﬁned as R R0 (Rm R0)l/lm, A bs, and P p/x, where Rm and R0 are, respectively, the upper and lower limits of R variation, and P and b are constants. For a steady-state chemostat culture, dp/dt 0 and dx/dt (1/P)(dp/dt) 0, then l (kRm /P)s/(RmKA /br0 s) lms/(Ks s) (68)

cell density, xm. The residual substrate concentration, s is calculated from mass balance (equation 58); yield is calculated as (xm x0)/s0; the s0 value should be known) and corresponding l(t) values from the slopes d(lnx)/dt. Finally, parameters Ks and lm (equation 65) are to be found graphically or from nonlinear regression as in the case of the Michaelis-Menten equation (see later text). 2. Batch culture, ignoring the substrate uptake. The inoculum size, x0, and the duration of experiment are chosen to minimize the uptake of added substrate s s0 (37). Typically, bacterial cell density should be of the order 10 5 or less, and it is measured by such sensitive instruments as the Coulter counter. The quasi-steady-state growth rate is measured at several s0 values and then ﬁtted to equation 65 as described before. In numerous determinations made at high cell densities (when we can no longer neglect substrate consumption), it was observed that dependence of l during exponential growth phase on s0 is formally described by the Monod equation with s replaced by s0 (38). The explanation of these results could be made only by more complex structured models; however, the described procedure is not appropriate for Ks and lm determination. 3. Batch culture, integral form of equation 61. The Sshaped curve of biomass dynamics is ﬁtted directly to equation 61 through either preliminary linearization or nonlinear regression (preferential). Sometimes (e.g., when cells are grown in opaque media) it is more convenient to follow the dynamics of residual substrate s(t) or product formation p(t), for example, CO2 evolution or dynamics of O2 uptake (respiration). In these cases, we can integrate the set of differential equation 60 in terms of s(t) or p(t) dynamics, taking into account mass–balance relationships in equations 7, 8, and 58 and similar equations. 4. Steady-state chemostat culture. The chemostat provides the opportunity for estimations under steady-state condition, which is commonly believed to be more reliable. By running an experiment at different dilution rates, D, the corresponding s values may be measured and the de˜ pendence of l D on s obtained; in principle, this can be ˜ done as accurately and carefully as desired. Such an approach, however, may and frequently does encounter serious technical problems because of the high afﬁnity to limiting substrate of some microorganisms. There is a need to (1) select highly sensitive analytical techniques to measure extremely low residual concentrations of particular substances; (2) develop instant sampling procedures to minimize substrate loss, and (3) eliminate apparatus-related artifacts such as nonperfect mixing and ﬂuctuations in nutrient medium supply. This can be accomplished by use of radiolabeled substrate, and other tips are covered in speciﬁc experimental works (39,40). However, the most essential objection to this method is that organisms at various steady states do change their kinetic properties, which is not accounted by Monod’s model (see “Structured Models”). 5. Non-steady-state chemostat culture. The measurements are made during short-term experiments started by addition of different amounts of limiting substrate to a steady-state chemostat culture. Then, substrate uptake or respiration rates are recorded in perturbed culture until

Here again, the Monod equation is derived through a consideration of underlying intracellular processes. Needless to say, none of the cited derivations are free from criticism. The Monod equation remains empirical. Any attempts to provide the mechanistic interpretation of this equation inevitably lead to much more complicated mathematical expressions (see late section on structured models). Biological Meaning and Experimental Determination of Growth Parameters Ks and lm. The parameter lm, maximal speciﬁc growth rate, has very lucid biological meaning: it is the upper limit of l variation on speciﬁed nutrient medium. It could not be attained in reality because of its asymptotic nature: l r lm as s r . However, in practice lm is achieved if s Ks. It should be remembered that lm is not absolute maximum of growth rate, because it depends on the nature of the limiting substrate. For example, E. coli has lm above 2.5 h 1 on complex beef-extract medium and below 1.0 h 1 on minimal medium with succinate. Saturation constant Ks could be deﬁned functionally as such concentration of limiting substrate that provides a speciﬁc growth rate equal to 0.5 of lm. We can say also that Ks is a measure cell afﬁnity to substrate: the lower is Ks the better the organism is adapted to consume substrate from diluted solution. Any other deﬁnitions are speculative, e.g., Ks interpretation as the dissociation constant of ESC of the cellular enzyme involved in the ﬁrst step of substrate conversion. There are several ways to determine numeric values of Ks and lm: 1. Batch culture, differential form of Monod equation. The biomass dynamics, x(t), are followed from the start of the exponential phase until the complete consumption of the limiting substrate and the attainment of the maximal

1532

KINETICS, MICROBIAL GROWTH

the new steady state is established (2). The opportunity of continuous culture is that the physiological state of cells just before perturbation is well deﬁned and reproducible. On the other hand, these experiments do provide data on afﬁnity to substrate (Ks) and maximal rates of respiration (e.g., QCO2 or uptake Qs, which are related but are not identical with lm): lm Yp /xQCO2 Ys /xQs (69)

l If we deﬁne lm l s*

lms/(Ks Ym(Qs lm(s

s)

mYm

(75)

m) (2), then s*)/(Ks s) (76)

KsmYm /lm

where stoichiometric parameters Yp /x dp/dx and Ys /x ds/dx should be determined in independent experiments. Immediate assay of lm in such an experimental setup is possible as follows: the steady state is perturbed by setting the dilution rate by 20 to 100% higher than the critical value Dc (equation 64). The washout dynamics are followed and lm is determined from approximate relationship (5): x x(0) exp[(lm D)t], where x(0) is biomass before perturbation. More rigorous estimate of lm could be provided if such washout experiments were made at several (at least two) input substrate concentrations s0 to account for the fact that sr is still not inﬁnity. Modiﬁcation of the Monod Equation. It was found that not all experimental data could be reasonably well ﬁtted by equation 65. The best-ﬁt hyperbola often passed above experimental points at small s and below them at large s. A better ﬁt was claimed to be provided by using the following, entirely empirical, equations: l l l lm[1

n

exp( Ks)] s ) s)

n

(70) (71) (72)

lms /(Ks lms/(Ksx

Equations 75 and 76 both predict the occurrence of threshold substrate concentration s*, below which growth is impossible. It yields a substantially better ﬁt to experimental data. The difference is that according to equation 75, the parameter lm is the speciﬁc growth rate under imaginary conditions of zero maintenance requirements, whereas equation 76 implies the traditional deﬁnition of lm as the speciﬁc growth rate under substrate excess: l r lm when sr . Account for Substrate Leakage. Mathematically identical to equation 76, a modiﬁcation of the Monod equation was proposed for the case of conserved substrates. However, the biological meaning is entirely different: if the speciﬁc leakage rate is assumed to be constant, then a decrease in s down to some threshold value s* will lead to the counterbalance of the two reverse processes (uptake and leakage), so that the net consumption of the limiting substrate will be zero. Account of Inhibitory Effects. A few valuable reﬁnements of the Monod equation were borrowed from enzymology; most often they were noncompetitive and substrate inhibition. The former inhibition mechanism is by the so-called Monod-Ierusalimsky equation: s Ks s Kp Kp p

The preference of the ﬁrst expression, equation 70 (41), is questionable. An expansion of Monod equation by addition of the third parameter, n in equation 72 (42), or introduction of the second variable, x in equation 71 (43), does improve the approximation capability of kinetic equations. We can even provide the mechanistic basis for this improvement. Thus, the Moser equation (equation 71) is similar to the Hill equation in enzymology (equation 43), indicating the cooperativity effects in performance of some master reaction of cellular metabolisms. The Contois equation (equation 72) could be interpreted in terms of growth autoinhibition by-products, because under the realistic assumptions (x x0, xYp /x /Kp 1 and product p formation coupled with growth), the apparent saturation constant Kapp is almost proportional to accumulated biomass x: s

app Ks

l

lm

(77)

The growth retardation with an excess of such substrates as phenol, methanol, and ethanol is described by an analogue of Haldane’s equation (45): s Ks s s2 /Kss

l

lm

(78)

Ks(1

p/Kp)

Ks[1

Yp /x(x

x0)/Kp]

Ksx (73)

An Account of Maintenance Requirements. Powell (44) assumed that substrate uptake rate q obeys the MichaelisMenten kinetics; then, from mass–balance equation 18 it follows that qs Qs /(Ks s) l/Ym m (74)

If lm is deﬁned as lm

YQs (5), then

Equation 78 simulates a single-peak curve, and so the same l value may be obtained at two different s, one in the substrate-limiting range, dl/dt 0 (stable), and the other in the substrate inhibition range, dl/dt 0 (unstable). A sustainable maintenance of a population under conditions of substrate inhibition is possible either in the second stage of a two-stage chemostat or in the case of plentiful wall growth in a conventional chemostat (46). Account of Diffusion Effects. We will present one example of such models (44). The basic assumption is that substrate is taken up by an enzyme that obeys Michaelis kinetics and is localized on the inner side of cell membrane. The actual substrate concentration around the enzymeactive centers is smaller than in the solution, because of a limited diffusion rate. By applying a simpliﬁed Laplace equation, it was found eventually that

KINETICS, MICROBIAL GROWTH

1533

l

lm(Ks

L 2L s lm Ks L

s)

1

1

(Ks

4Ls L

s)2 (79)

poseful non-steady-state growth may display greater efﬁciency and higher productivity (48). The notions of steady-state and balanced growth are close but not identical. The ﬁrst is more strict; steady-state growth has to be balanced by deﬁnition (otherwise some of the speciﬁc rates responsible for synthesis of the changed cell component should vary). On the other hand, the balanced growth can be for some time nonsteady, perhaps during the late exponential phase of batch culture when l declines while cell composition remains unchanged. During long-term experiments, non-steady-state growth leads inevitably to a change of cell composition; it becomes unbalanced. For heterogeneous populations, the situation may be more complicated. For example, the growth in the second stage of a two-stage chemostat attains a steady state, and the biomass and residual substrate concentration are constant. However, such growth is not balanced, because cells delivered from the ﬁrst stage differ in their properties and composition as compared with the bulk of cells in the second stage. This situation was termed the transient steady state (49). By structured, we mean mathematical models describing growth-associated changes in microbial cell composition. It includes mass–balance equations for all assigned intracellular components. Their concentrations can be expressed either per unit volume of fermenter vessel (c1, c2, . . . , cn), or per unit cell mass (C1, C2, . . . , Cn), and hence Ci ci /x. The mass–balance equations can be written as follows

n

s

where L is a factor determined by membrane permeability and by the maximum rate of the enzymatic reaction. Structured Models The unstructured models (such as Monod’s model or its modiﬁcations) are able to predict and describe only simplest manifestation of growth phenomena. Sometimes it is declared that Monod-type models are able to describe only balanced and steady-state growth. The analysis of more complicated unbalanced and non-steady-state growth requires formulation of structured models. Deﬁnitions. Balanced growth was deﬁned by Campbell (47) as a proportional increase in the amounts of all cell components, in other words, balanced growth produces cells of the same quality without any variation of composition. The terms steady state and non steady state stem from chemical and enzyme kinetics. The ﬁrst one refers to such a regime when the reaction rate remains constant because of an exact balance between formation and breakdown of intermediary products such as the enzyme– substrate complex. In microbial culture, the growth is called steady state if speciﬁc rather than total rates remain constant. In an open system, such as a chemostat, both total (dx/dt, ds/dt) and speciﬁc rates (l (1/x)dx/dt, qs (1/x)ds/dt) tend to have constant steady-state values. A closed system, such as a batch culture, should be considered under steady state only during the exponential phase when l and qs are constant. The linear growth with a constant total rate (dx/dt lx constant) is characterized by monotonously declining l, and is not steady-state growth. However, under some conditions (such as in dialysis culture) it may attain quasi steady state, when ds/dt 0. The non-steady-state kinetic regimes take place before establishment of steady state or after its perturbation. In enzyme kinetics, non-steady-state measurements are taken in the millisecond range of time scale. In microbial cultures, similar non-steady-state transient and perturbation processes advance much more slowly, typically during several hours and days. An example is a transient process in the chemostat induced by changes in D or sr (fed-substrate concentration). In such growth, l, qs, qp, and other metabolic rates exhibit continuous variation in time. The attractiveness of non-steady-state studies for microbiology and biotechnology is obvious: • They allow a wider range of hypotheses to be tested and yield much more data on the studied objects. • They have higher practical value; in biotechnology, steady-state operation is the exception rather than the common routine because of unavoidable disturbances in cultivation conditions. • They provide additional tools for optimal regulation of cell performance in the bioreactor, because pur-

ci

i n 1

x 1 (80)

Ci

i 1

For each variable Ci, a differential equation is written that takes into account all sources, r , and sinks, r , as well as its dilution from cell mass expansion (growth) dCi dt r (s, C1, . . . , Cn) r (s, C1, . . . , Cn) lCi (81) The simplest structured models with n no more than 2 or 3 are called two- or three-compartment models. For example, a model (50) incorporated two compartments: nucleic acids and proteins combined with other active cell components. The model variables also included concentrations of the limiting substrate and the inhibitor. Compared to Monod’s model, the proposed set of four equations was able to account for a much wider range of dynamic patterns. Speciﬁcally, it simulated D-dependent changes in the cell composition (chemostat) and all known growth phases of batch culture from the lag to decline. However, the choice of variables in this model was more or less arbitrary, so it should be regarded more as a bright illustration rather than a research tool. During the past decade, much more realistic structured models based on biochemical data have been developed.

1534

KINETICS, MICROBIAL GROWTH

The simulation model of E. coli growth (51) contains the following dynamic variables: glucose and NH4 , as exosubstrates; CO2 and acetate, as products excreted into the medium; amino acids; ribonucleotides; deoxyribonucleotides; monomeric precursors of cell wall components; rRNA and tRNA; nonprotein polymeric components; glycogen; guanosinetetraphosphate; enzymes transforming ribonucleotides into deoxyribonucleotides; ATP; NAD(H); and protons. Altogether, the dynamic model amounts to a system of 21 differential and 14 algebraic equations. An even more complicated model simulating growth of Bacillus subtilis (52) is the set of 39 nonlinear and coupled differential equations containing nearly 200 parameters! These models are able to simulate particular growth features such as changes in cell sizes, shape, and composition as well as the D-dependent variations in replication time brought about by the shifts in glucose concentration. However, the predictive capability of such an intricate dynamic model should be still estimated as modest as compared with invested modeling efforts; they are still nothing more than a “caricature parody” of microbial biochemistry and are too complex to be studied by conventional mathematical tools (stability analysis, parameters identiﬁcation, etc.) or to be used in biotechnological applications. The best choice of a mathematical model lies, apparently, midway between unstructured and highly structured models outlined here. One of the known compromises has been found through attempts to express quantitatively the cell physiological state. Physiological State of Chemostat Culture. The term was coined by Malek (53) without giving a clear deﬁnition. The impetus for the development of the concept of a physiological state was the evidence on changes in the chemical composition of microorganisms as dependent on dilution rate D and medium composition in chemostat culture (54). It has been found that some of the studied parameters remained constant (content of cell DNA and carbon) while others exhibited regular D-dependent variations, either an increase (RNA content, cell sizes) or a decrease (the content of reserved polysaccharides). Those properties that were D dependent were recognized as components of the vector of the physiological state. Powell (44) combined and put on a quantitative mathematical footing three notions, which were beforehand separated and cloudy: (1) physiological state, (2) past history, and (3) non-steady-state growth kinetics of microbial culture. The speciﬁc rate of substrate uptake, qs, was presented as a product qs(s) qa(s)S(s) (82)

jumps from s(0) to s(1). Immediately, qs will increase from Q(0)S[s(0)] to Q(0)S[s(1)]. If no further changes in s(1) occur, then Q will also eventually attain a new steady-state value equal to qa[s(1)]. In essence, Q is the potential metabolic activity, that is, the speciﬁc rate of a key metabolic process measured just at the moment of relief from substrate limitation. The transient Q dynamics are described as net change equals production minus dilution caused by cell growth: dQ dt r(Q, s) l(Q, s)Q (83)

Substance Q may, in reality, be represented by a single enzyme or multienzymatic complexes as well as by ribosomes or other cell components occupying the bottleneck position. Monod’s model, supplemented by equation 83, made possible at least a qualitative understanding of chemostat-transient processes triggered by a D switch (55). The synthetic chemostat model (SCM) (2) combines Powell’s ideas with the routine of conventional structured models. This model is similar to the cybernetic model (56,57). The basic SCM interprets microbial growth as a conversion of exosubstrate S into cell macromolecules X via a pool of intermediates L:

Cell wall Transport qs

CO2

Synthesis qL

S

Leakage v

L

Turnover a

X'

where S(s) is a simple saturation function (e.g., a Michaelis hyperbola), S(s) s/(Ks s), and qa(s) is a function associated with the microbial physiological state. The instant value of qa(s) is determined by way in which s varies until the given moment (s h ago), effects of later events contributing more than earlier ones. Transient processes are inﬂuenced by the past history of the culture in the following manner. Suppose that steady-state growth of a chemostat culture is upset and the residual substrate concentration

Macromolecular cell components are susceptible to degradation (turnover), and monomeric metabolites can escape into environment because of leakage. Contrary to simple chemical catalysts, the composition of end product X is not uniform and varies in response to a changing environment because of the adaptive nature of microbial metabolism. At the heart of the SCM are the solution of the problem, how to cope with these variations, and how to describe adaptive changes in cell composition by relative simple models. All macromolecular cell constituents are divided into two groups: primary cell constituents necessary for intensive growth (P components), and components needed for cell survival under any kinds of growth restriction (U components). The characteristic examples of P components are ribosomes (rRNA and ribosomal proteins) and enzymes of the primary metabolic pathways. Their intracellular content increases parallel to an increase in l. The contribution of U components to cell biomass decreases with growth acceleration. Examples are enzymes of the secondary metabolism, protective pigments, reserved substances, and transport systems of high afﬁnity. The analysis of available data as well as massconservation conditions allowed the formulation of several rules of variation of cell components taking place because of optimal control of cell biosynthesis: 1. Amounts of P and U components expressed as a fraction of total cell mass (P and U, gram per gram of

KINETICS, MICROBIAL GROWTH

1535

biomass respectively) vary within the upper (Pmax, Umax) and low (Pmin, Umin) limits, the latter being the constitutive part. 2. An increase of one individual P component is accompanied by increase of other P components. 3. The total enlargement of Psum is accompanied by corresponding decrease of Usum and vice versa. 4. The P/U ratio is controlled by the limiting substrate concentration in an environment. These rules are translated into mathematical terms as follows: P1 Pmax 1 U1 Umax 1

min P1 Pmin 1

••• ••• s ˜ Kr

Pn Pmax n Um max Um s ˜

min Pn Pmin n

r 1 r

similar trend was complemented by considerable decrease of YN due to alteration of cell composition in favor of N-rich P components (Fig. 6). SCM adequately describes the transient growth caused by shift-up in chemostat culture. The phenomena of overshoot in substrate concentration and undershoot in biomass during transient growth are explained by slow adjustment of cell composition (RNA content, respiratory activity) to new growth conditions (Fig. 7). Batch culture limited by carbon and energy source was simulated by SCM on the whole from inoculation to death stage; conventional growth phases (lag, exponential, deceleration, stationary, decline) were generated automatically without setting any speciﬁc conditions (Fig. 8). It is important that SCM realistically describes and predicts not only net growth but also the survival dynamics of starving cells after substrate depletion. During this phase,

Umin 1 Umin 1 r ˜

Umin m min Um

Mass fraction of cell components

1.0 RNA 0.8 0.6 Proteins 0.4 0.2 0.0 0.0 Lipids DNA Polysaccharides

(84)

where variable r (index of physiological state) is already scalar (not vector!) function controlled by environmental factors (e.g., concentration of the limiting substrate). The r value in steady-state chemostat culture changes from zero (in culture at almost zero growth rate, when s r 0 and ˜ all P components come down to low limits) to 1.0 (in unlimited culture, when s r and P components attain maxi˜ mum). During transients caused by sudden s changes, an instant r value goes toward new steady state (compare with equation 82): dr dt l s Kr s r (85)

0.2 0.4 0.6 0.8 Chemostat dilution rate, D (h–1)

1.0

Biomass, residual substrate, mg L–1

6

3

2 0.12

qs

r

Qs Ks s

(1

r)

Qs Ks s

(86)

4

0.08

where the ﬁrst and the second terms on the right side stand, respectively, for low (P component) and high (U component) afﬁnity of transport system. Similar r-dependent expressions are derived for other reactions (qL, v, a) and stoichiometric parameters. Predictive and clarifying capabilities of SCM turned out to be higher than more complex structured models. Contrary to all known chemostat models, SCM provides adequate simulation of D-dependent variation of the microbial physiological state. Under energy- and C-limited growth, it was expressed as an increase of apparent Ks, potential uptake and respiration rates, maintenance ration, and turnover parallel to increase of D. Under N limitation, a

2 1 0 –10 –5 0 5

0.04 0 10 15 20 25 30 35 40 Time (h)

Figure 7. SCM simulation of transient growth induced by change of chemostat dilution rate. Residual glycerol concentration (1), biomass (2), and cell RNA content (3). At t 0, dilution rate was shifted from 0.004 to 0.24 h 1. Source: Redrawn from Ref. 2; the original data (59) are for chemostat culture of Aerobacter aerogenes limited by glycerol.

RNA content, fraction of CDW

The introduction of the r variable greatly simpliﬁes the use of structured models because the adaptive variation of cell composition (and metabolic activity, which is determined by the intracellular content of particular enzymes) now could be expressed via one single master variable r. For example, the speciﬁc rate of substrate uptake qs is deﬁned as:

Figure 6. Simulation of D-dependent changes in cell composition of Aerobacter aerogenes grown in NH4 -limited chemostat culture. The curves are calculated from SCM (2), and the original experimental data are from Ref. 58.

8

0.16

1536

KINETICS, MICROBIAL GROWTH

200 Residual glucose (mg L–1) 1 150 2 2 1

160

100

80

50 0 0

40 0

Biomass (mg L–1)

120

mass increase from m0 to m* is inversely related to the speciﬁc growth rate l. It is this difference that yields ldependent variations of cell sizes, because faster-growing cells produce for the same s C D time a larger cell mass than do slower-growing cells. It became clear from the following simple algebra. The steady-state growth of an individual cell proceeds exponentially throughout the entire cell cycle [m m0 • exp(lt)]. At the time of the second part of the cycle, it takes exactly s min for the cell to enlarge from critical mass m* to 2m0. Hence, 2m0 m* exp(ls) ⇒ m0 0.5m* exp(ls) (87)

5

10

0

200 400 600 800 Time (h)

Figure 8. SCM simulation of complete dynamic curve of batch culture; residual glucose (1) and biomass (2) of yeasts Debaryomyces vanrijiae (old name D. formicarius) grown on glucose-mineral medium (2). Note that contrary to old empirical models (equation 56), all growth phases are reproduced automatically without specifying preset time ranges.

surviving bacteria sustain very slow cryptic growth at the expense of cell turnover and L leakage. The rate of decline in biomass gradually decreases as the result of buildup of some U components (parallel to the decrease of such P components as RNA and CN-sensitive respiration enzymes). Batch culture limited by conserved substrate (the source of N, P, Mg, Fe, or other) has an interesting feature: the growth is not stopped after substrate depletion but proceeds at an even higher rate. SCM explains this phenomenon by the partitioning of deﬁcient elements between mother and daughter cells. Cell Cycle The term cell cycle is used to designate the regularly repeated sequence of events that occur between consecutive cell divisions, for example, the formation of two identical daughter cells from one cell, which takes place in most known bacteria. Equivalent cell cycle events include budding (most yeasts and budding bacteria), branching of hypha (ﬁlamentous organisms), fragmentation (some coccoid and corineform bacteria). The majority of available data have been obtained for rod-shaped bacteria (E. coli, Salmonella typhimurium, Bacillus cereus) under steady-state growth conditions when the cell cycle consists of three distinct phases: 1. The growth of the newborn cell without chromosome replication from the initial mass m0 until some critical initiation mass m* 2. DNA replication (C period) 3. Separation of daughter cells (D period) Relationship between Cell Size and Speciﬁc Growth Rate. The periods C and D are constant; in E. coli, they occupy about 40 and 20 min, respectively, independent of environmental conditions. However, the duration of phase 1 depends on cultivation conditions; the time of the single-cell

Equation 87 remains valid for steady-state culture at any l, which can be varied from 0 to lm. Assuming that cell critical mass m* is constant and does not depend on growth rate, we see that equation 87 predicts a l-dependent variation mass of newborn cells m0 and hence the average mass and size of the bacterial population. This ﬁnding is supported by numerous experimental studies from the beginning of this century (60) that displayed the positive correlation between cell size and growth rate. Because the term ls is rather small (ls ltd ln 2), it is difﬁcult to notice the curvature of the experimental curve within the measurement error. Thus, the relationship between the average cell size m (m ¯ ¯ m0 2 ln 2 for rod-shaped bacteria; see derivation in equation 88) and l is given as an empirical linear approximation: m ¯ m(0) ¯ kl (88)

The regression parameters m(0) and k have a meaningful ¯ biological interpretation: y intercept, m(0) ¯ 2 ln 2 0.5m* (ln 2) m* is equal to approximately 69% of the cell critical mass m* initiating chromosome replication, and slope k [exp(lms) 1]/lm s is close to the duration of C D periods. The postulate on the constancy of the critical mass, m* was derived from the observation that the cell has accurate control over its size at division and poorer control over its age at division (61). Recently, accurate measurements with ﬂow cytometry (62) revealed that m* is inversely related to the speciﬁc growth rate; slowly growing cells tend to initiate DNA replication at a slightly higher critical mass as compared with intensively growing cells. However, we may safely assume that m* variation is much smaller than the variation of cell mass during the cell cycle: dm*/dt dm/dt. Simulation of Cell Cycle by Simple Deterministic Structured Model. In biochemical terms, it is difﬁcult to envisage how cell mass per se could determine when to initiate replication. A more likely candidate is some mass-related parameter, such as intracellular concentration of some signal metabolite like guanosine tetraphosphate or an initiator protein, according to the popular model proposed by Helmstetter and Cooper (63,64). It was postulated that the initiation of DNA replication is triggered by a threshold intracellular concentration of this protein V*; this protein is synthesized at a rate proportional to the total growth

KINETICS, MICROBIAL GROWTH

1537

rate and requires exactly one mass doubling time to reach its threshold concentration again. This mechanism is translated into mathematical form of a structured model such as SCM as follows (2):

dV dt dk dt H(l f if V if k a) lV f1 0 else f 0 (initiation) k/2, V 0, m m/2 (division) (89)

V* then f k* then k

Statistical Analysis of the Population Distributions. Equation 87 to 89 were derived for average cells in the population. To embrace the variability of sizes, we have to analyze the frequency distributions. If certain conditions are met— the culture is fully desynchronized, cells grow according to some deterministic low, all divide into only two identical daughter cells, and there is no cell elimination—then age distribution (61) is given by: u(a)da 2le

la

da: 0

a

ln 2/l

td

(90)

where H is the fractional contribution of protein V to total cell synthesis. The V content is an intrinsically transient entity; even during steady-state growth, it continuously changes between zero (Helmstetter and Cooper postulated the annihilation of the initiator protein after every replication cycle) to an upper-threshold value, V*, which is less than the potential steady-state level, H(l a)/l. The second variable, k, imitates the replicating chromosome; it sets up the discontinuity associated with cell division. Analysis of coupled equation set 89 reveals that this model is stable toward perturbations. Suppose that by chance, the content of initiator protein has risen to some abnormally high level. The immediate result would be several more frequent cell divisions, with smooth reversal to a normal multiplication pace. Similar events take place under the opposite situation of V deﬁciency; several divisions are delayed, resulting in production of abnormally long cells, but then steady state is restored. The negative feedback mechanism that brings things back into line is based on the dynamic nature of variable V; it is characterized by a unique, stable steady state that is approached from different initial conditions. It may be easily shown that the described model adequately simulates various morphological effects exhibited during non-steady-state growth (Table 10).

where l is speciﬁc growth rate, td is mean doubling time, a is the age since birth, and u(a) da is the frequency of cells whose ages are between a and a da. Assuming that cells grow exponentially between divisions, then the frequency of mass distribution is u(m)dm 2m0 /m2dm: m0 m 2m0 (91)

where m0 is the mass of newborn cell and u(m) dm is the frequency of cells whose masses are between m and m dm. The mean cell size is 2m0 ln 2 (calculated as an integral of m u(m)dm). If cell growth between two consecutive divisions is linear (65), then u(m)dm 4 ln 2/m0 • exp( m ln 2/m0)dm: m0

m

2m0

(92)

Table 10. Morphological Effects during Non-Steady-State Growth Effects The longer lag phase in batch culture when growth is surveyed by cell count rather than biomass measurements The accumulation of enlarged cells during transition from lag to exponential growth phase The formation of dwarf cells in starving population Explanation The partial synchronization of cell division is delayed as compared with mass growth until attainment of the value 2m ¯ The l–m relationship (equation 87) combined with the transient misbalance in V synthesis (equation 89) The small cells are produced during very slow cryptic growth of surviving organisms This phenomenon is simulated by equation 89 and explained by overproduction of initiator protein in the presence of inhibitor

Equations 90 and 91 are called canonical age or mass distributions to emphasize that they are an idealized form applicable when cell division takes place at a precise size. Assuming that momentary distribution of size at division of individual cells is normal and random (not correlated with other cell cycle events), we can obtain computersimulated curves for any ﬁxed level of noise expressed as the coefﬁcient of variation (CV). As shown in Figure 9, random variations of size at division tend to round the corners of the canonical distribution. Another source of cell size variation can be nonequal separation of mother cells into two daughter cells. It is characterized by the K distribution, which is the distribu-

2.5 2 Frequency 1.5 1 CV = 10% 0.5 0 0.5 1 1.5 2 Cell size, m (10 –9 mg) 2.5 Canonical mass distribution, CV = 0

The accumulation of division potential if division is blocked by inhibitor, then after block release all missing division takes place in quick sequence

Figure 9. Distribution of cell sizes (as single-cell mass) for canonical cases where there is no variation in the size at division and for the case of normal distribution of size at division with CV 10% (see details in Ref. 61).

1538

KINETICS, MICROBIAL GROWTH

tion of the ratio of daughter-cell length to mother-cell length. The average values absolutely necessary equals 0.5, but CV is at best about 4% (e.g., for well-behaved E. coli strains) and can attain rather high values for other organisms. The deviation of observed versus predicted size distribution may be caused also by cell death and different kinds of cell pathology (abnormally long or dwarf cells). Collins and Richmond (66) introduced an entirely different approach based on the use of three distributions:

m m m

Most of the obtained results are in better agreement with the exponential growth model rather than with the linear model. However, there are some serious doubts about whether there is a unique simple mathematical growth low describing bacterial growth during the division cycle. Cooper (67) proposed distinguishing three categories of cell components that are synthesized with a unique pattern: 1. Cytoplasm (proteins, RNA and ribosomes, small molecules) that is accumulated exponentially. 2. Cell DNA that is replicated in a linear fashion as a sequence of constant and zero rates. 3. Cell surface composed of peptiodoglycan and membranes that are synthesized exponentially during most parts of the cell cycle, but immediately before cell division the synthesis accelerates to accomplish new pole formation. Thus, the growth pattern of the whole cell is the sum of these three patterns. Because the cytoplasm is the major constituent (up to 80% of CDW), then the growth of the cell should be approximately exponential. Population Dynamics (Mutations, Autoselection, Plasmid Transfer) Description of Mutation and Autoselection. The continuous culture turned out to be a very efﬁcient tool to study mutation and autoselection (67). Let N be the total cell concentration, M the concentration of mutants, l the speciﬁc growth rate of the main nonmutated part of the cell population, and g the speciﬁc growth rate of neutral mutants, then dM dt dN dt klN lN gM klN DM DN (94)

vm

l{2

0

Wdm

0

u(m)dm

0

k(m)dm}/k(m) (93)

where vx is the growth rate of cells of size m, k(m) is the extent of population distribution, W is the momentary distribution of dividing cells, and u(m) is the momentary distribution of newborn cells. This equation allows the calculation of the mean growth rate of cells of particular size class. Instead of this analytical method, Koch in his work with Shaechter (61) proposed the synthetic approach. Starting from the set of speciﬁc postulates of the cell multiplication mechanism (linear or exponential growth of cell mass between divisions, kind of control, the evenness of the division), they derived the size or age distribution, which then was compared with the observations. The Growth Low. Some believe that there should be a general low of cell growth that can be discovered by sensitive methods of analysis. The rate of biomass growth throughout the cell cycle was hypothesized to be linear, bilinear, exponential, double-exponential, and so on. Two limiting cases have generally been considered: the exponential and linear growth models proposed, respectively, by Cooper and Kubitschek (57). To differentiate between these two mechanisms, three classes of experiments have been used: 1. Size measurements of individual cells growing in the microculture by use of phase-contrast microscopy or recently developed confocal scanning light microscopy combined with image analysis. 2. Pulse-chase labeling of cells with their subsequent separation into different phases of the cell cycle. Most frequently, labeled uracil and leucine are used as precursors of RNA and protein synthesis, respectively. There are two major sources of errors in this approach: poor resolving power of separation methods and artifacts associated with effects of exogenous compounds on intracellular ﬂuxes (feedback inhibition, pool expansion, label dilution, etc.). One of the best options for separation is probably the “baby machine,” based on the membrane elution principle (67). To minimize the second source of error, the mutants blocked in the synthesis of the probe compound can be used. 3. Analysis of the frequency distributions of steadystate populations (see equations 91 and 92). However, the resolving power of this approach is rather low because linear and exponential models produce similar patterns.

where k is the mutation rate expressed as the ratio of the numbers of mutants to total number of cells formed. If k 1 and l g, in the steady state, we obtain l g D, and dM dt or M M0 kDNt (95) kND

If g l, then the original strain will be displaced by the mutant, otherwise, if g l, M will tend to a lower limit M* kN/(1 g/D). Experimental studies of phage-resistant mutants in a tryptophan-limited chemostat culture of E. coli showed that the period of linear M increase in accordance with equation 95 was fairly short. Every 20 to 100 generations there was an abrupt fall in the number of mutants, after which the linear growth resumed at the same rate (67).

KINETICS, MICROBIAL GROWTH

1539

The observed saw-tooth dynamics in M were explained by Moser (43) as a combined effect of mutation and selection. The original wild clone gives not a single but a whole array of mutations with subsequent reversions. Let us denote the total cell population in a chemostat culture as N, which is the sum of all subpopulations, including original and emerging variants, N RNi. All possible transitions between variants are given by the matrix, kirj( j 1, . . . , n; i 1, . . . , n; j i). Then, the chemostat model takes the following form: dN dt l ¯ dNj dt ds dt lj(s)

n j 1

dNj dt

n

l(s)N ¯

DN

1 Nj

ljNj

1 n n

3. Growth of a mutant with a higher resistance to inhibitory metabolic products can be described by equation 77 with Kpk Kp. Under selection pressure r lk l lm(s/Ks s)[Kpk /(Kpk p) Kp /(Kp p)], the original population will be completely displaced, and the product concentration will reach a higher steady-state level, p ¯ lmKpks /(Ks ¯ s)D ¯ Kpk. 4. Growth of mutants resistant to an antibiotic will not be affected by competition if the respective antibiotic is continuously supplied: r lk(s), because l 0 for all other forms. The dynamics of the total population will be governed by the initial density of the mutant. 5. A mutation resulting in enhanced adhesion to fermenter walls will lead to accumulation of slow growing cells (because the adhesion prevents washout); eventually we have r lk D lj. Extrachromosomal Cell Elements. The present-day hot spot in microbial population genetics is the study of extrachromosomal cell elements (ECE) such as plasmids, phages, transposons, and insertion elements. Normally, ECEs do not carry genes, absolutely essential for growth, but they are capable of fast replication, surpassing the chromosome DNA in the number of copies. R plasmids are responsible for bacterial growth in the presence of antibiotics, but under normal conditions (with no antibiotics present), their synthesis becomes too heavy a burden for the host cell, which is manifested in a decreased growth rate. Among the more than 100 R factors studied, about a quarter were found to increase the bacterial generation time by 15% (68). For this reason, plasmid-bearing strains are unable to compete with plasmid-free populations, although there are a few exceptions. Thus, colicin-positive cells carrying respective plasmids are able to withstand the competition with faster-growing plasmid-free strains by virtue of antagonistic inhibition. In recent years, rather intricate and detailed dynamic models of autoselection have been proposed that take into account the ECE-related effects, including the transfer of ECEs within the population, their segregation loss, changes in l arising from the ECEs carriage, and so on. Some of these models have biotechnological and medical applications (69). MICROBIAL GROWTH AS DEPENDENT ON CULTIVATION SYSTEMS The array of laboratory cultivation systems that deﬁne the dynamic patterns of microbial growth is summarized in Table 11. Microbial growth patterns are distinguished by three features: • Regime of substrate supply (1, continuous supply; and 2, single-term addition) • Elimination of growing microorganisms ( , signiﬁcant; b, absent) • Magnitude of spatial gradients (a, homogeneous systems; b, heterogenous systems) Each speciﬁc cultivation procedure can be represented by a point inside a cube with the axes 1 to 2, to b, and a to

lj(s)Nj D(s0 lm s Ksj s)

DNj

i j n

kjriNj

j i

kirjNi

lj(s)Nj /Yj

j 1

s

(96)

Every drop in the saw-tooth dynamics of neutral mutants detected by their phage resistance can be interpreted as the appearance of other types of spontaneous mutant with higher growth capabilities. In a chemostat culture, such mutants overcompete and displace all other cells by virtue of their higher afﬁnity to limiting substrate (a decreased Ks value). Let such a mutant be denoted by the subscript k and l(s) be the average growth rate of all other subpop¯ ulations. The selection pressure for this mutant, r, is given by d(Mk /N) dt r Mk N [lk(s) l(s)] ¯ Mk N (97)

If Ksk Ksj( j k), then r 0 until a new equilibrium is established. In this process, the original cells will be displaced by the mutants and the growth-limiting substrate concentration will decrease from s1 ¯ DKsj /(lm D) to s2 ¯ DKsk /(lm D), and the culture density will rise by Y(s1 ¯ s2). ¯ Autoselection in Turbidostat and pH Auxostat. The afﬁnity to substrate was not always the only driving force of selection outcome. A number of instructive examples were reviewed by Pechurkin (14). 1. In turbidostat and pH auxostat, autoselection is in favor of mutants with higher maximum speciﬁc growth rates, r lmk lmj 0. Because the population density is kept constant instrumentally and the dilution rate is allowed to vary, then autoselection results in the increase in D from lmj to lmk. 2. Mutation toward a higher growth efﬁciency, Yk Yj, will lead to the same result as an increase in lm: r lk lj (lmk lmj)s/(Ksj s), as soon as lk qsYk and lj qsYj.

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KINETICS, MICROBIAL GROWTH

Table 11. Matrix of Cultivation Techniques Substrate input Continuous (1) Spatial organization Homogeneous (a) Cell eliminated ( ) 1a Chemostat Turbidostat pH-auxostat 1b Plug-ﬂow (tubular culture) Colonies No elimination (b) 1ab Fed-batch culture Dialysis culture Retentostat 1bb Column packed with microbe attached Single-term (2) Cell eliminated (a) 2a Continuous culture with substrate pulses Phased culture Forbidden combination No elimination (b) 2ab Simple batch culture

Heterogeneous (b)

b. The 2b combination is logically forbidden because any spatial segregation results in protracted substrate utilization and so transforms a batch process into a continuous one. The dynamics of microbial growth in any type of cultivation system can be described by the following mass– balance equations: ds dt dx dt F V G(s) H(x)x l(s)x/Y l(s)x mx (98)

where F is the substrate input rate; G(s) is the rate of unused substrate removal from cultivation vessel (washout, leaching, evaporation, etc.); V is the rate of microbial biomass input, which may be a single-term inoculation or continuous delivery of cells to the fermenter (specially designed or unintentional, e.g., contamination); and H(x) is the rate of microbial elimination, such as washout, death, grazing, or lysis. The rest of the notation is conventional. To simulate microbial growth dynamics in a particular homogeneous system, one has to make the following selection: F(t) 0, s0 0 for a category 2 (batch culture), F(t) 0 for category 1 (continuous cultivations); H(t) 0 for systems retaining cell biomass (dialysis, fed-batch, simple batch, column with immobilized cells), and H(t) 0 when cell elimination occurs (chemostat, turbidostat, phased culture, etc.) Spatially heterogeneous systems can be simulated either by partial derivatives or compartmental models (e.g., the total biomass x of a microbial colony may be represented as a sum of the peripheral and central components). 1a —Homogeneous Continuous Culture (Continuous-Flow Fermenters with Complete Mixing) There are two subgroups within this type of cultivation technique. In the ﬁrst one, steady-state growth is maintained naturally by the microbial culture itself. Selfregulation is performed through negative feedback that originates from the dependence of the growth rate on substrate concentration (chemostat) or on temperature (caloristat). In the second group, electronic devices are used for the automatic adjustment of dilution rate to the instantaneous growth rate of the microbial culture. Electronic control is based on the sensing of cell density or growth-

linked products, that is, optical density (turbidostat), culture liquid viscosity (viscostat), CO2 concentration in output air, culture pH (pH auxostat), and so on. In theory, the steady-state growth may be established in chemostat between 0 and lm, but in practical terms neither very low nor very high values are attainable because of the long time needed to reach the steady state in the ﬁrst case and the risk of culture washout in the second. The second group of continuous techniques (turbidostat, pH auxostat, viscostat, etc.) are capable of maintaining steady growth at high s when either l r lm or under substrate inhibition, when dl/ds 0. There is also a potpourri mixed technique known as the bistat (70), which combines a chemostat and a pH auxostat. The mass–balance equations for the chemostat and its modiﬁcations have already been given (equations 62 to 64). In a simple, complete mixing cultivator, all cells have an equal probability of being washed out, hence l D. If there is substantial wall growth, biomass retention, or feedback, then l D; this difference increases with the extent of biomass retention in the fermentation vessel. In terms of our scheme, such cultivation systems correspond to points on the edge 1a to 1ab. 1ab—Continuous Cultivation without Cell Washout This group embraces cultures with batch or continuous dialysis, fed-batch culture (FBC), and batch culture with a supply of limiting substrate via the gas phase (gases and volatile compounds). It also includes the chemostat with complete biomass feedback by means of ﬁltration (71). The latter fermenters are less reliable in practical terms as compared with dialysis culture, because the membrane ﬁlters are quickly plugged with cells. The limiting substrate is supplied into the dialysis culture through a semipermeable membrane and, in the case of a gaseous nutrient, through the gas–liquid interface. In both cases, mass transfer is reasonably well described by Fick’s law. The culture volume remains ﬁxed in all systems, with the exception of a FBC. In a FBC, a constant nutrient feed F provides a linear increase of culture volume V during the cultivation span; the dilution rate D F/V is decreased hyperbolically. The great advantage of these cultivation techniques for biotechnology is that they provide the possibility of realizing very slow continuous growth accompanied with derepression of synthesis of many secondary metabolites. With a constant limiting substrate ﬂux, Fs0,

KINETICS, MICROBIAL GROWTH

1541

the absence of cell washout means that at each subsequent moment an equal ration is shared by an increasing microbial biomass, and, as a result, l eventually falls down to negligible values or even to zero (the maintenance state). Unlike the chemostat, no true steady state is established in this case, but when the substrate is virtually depleted, we have ds/dt 0 and the system reaches a quasi steady state (5). If a quasi steady state approximation is found to be sufﬁcient, then extremely slow continuous growth can be obtained after a reasonable period of time, perhaps a few weeks, as compared to the several months needed in a chemostat. 2a —Continuous Cultivation with a Discontinuous Supply of Limiting Substrate Suppose we have a simple chemostat culture fed by nutrient medium lacking just one essential component. This component is added as a small volume of concentrated solution at regular and sufﬁciently large time intervals, Dt. Then ds dt dx dt s0(t) D[s0(t) l(s)x s] Dx iDt, i 0, 1, . . . , n (99) q(s)x

culture liquid, moving along the spatial coordinate z at a linear velocity f F/A (where F is ﬂow rate, cm3 /h, and A is the cross-sectional area, cm3), growth of biomass proceeds as in a simple batch culture. To account for the culture movement per se, we have to pass from ordinary to partial derivatives and replace dx/dt by x/ t f x/ z (along-the ﬂow-growth rate). Allowing for some dispersion of the moving front by diffusion, we can write s t x t f f s z x z Ds Dx s z2

2 2

q(s)x l(s)x (100)

x z2

where Ds and Dx are the diffusion coefﬁcients of the substrate and microbial cells, respectively. 1bb—Continuous-Flow Reactors with Microbes Attached The nutrient solution is pumped through a column ﬁlled with adsorbent material and is utilized as it moves by growing immobilized cells. The mass–balance equations for a packed column are obtained from equation 100 by simplifying the equation for x, s t f s z x t Ds s z2

2

q(s)x(z) Y[q(s) m]x (101)

A 0, when t 0, otherwise

l(s)x

This cultivation method was originally used to obtain synchronized cell division (72). A continuous phased culture (73) is a repeated simple batch culture, that is, at regular intervals, Dt, half of the culture volume is withdrawn and the fermenter is reﬁlled with an equal volume of fresh medium. Under such conditions, repeated batchwise growth proceeds with biomass increasing cyclically from x0 to 2x0. Obviously, the growth dynamics are governed by the time interval between consecutive substrate additions Dt. If Dt ln2/lm, a sawtooth nonlimited growth takes place as in a turbidostat. With Dt ln2/lm the culture should be washed out and, when Dt ln2/lm, the maximal attainable biomass decreases with increasing Dt because of endogenous biomass decomposition and waste respiration during the lag phase. 2ab—Simple Batch Culture Cultivation begins at the initial limiting substrate concentration, s0, and inoculum size, x0. The biomass reaches its maximum, xm, when the limiting substrate is depleted (s 0) and then declines even in the absence of exogenous elimination, so that x r 0 as t r . Description of batch dynamics has been given earlier. Speciﬁed growth phases are described by simple nonstructured models (equations 56, 61, and 73), and entire dynamics are described by SCM and other structured models (equations 84 to 86; Fig. 8). 1b —Plug-Flow (Tubular) Culture The inoculum and the medium are mixed on entry into a long reactor tube, and the culture ﬂows in the tube at a constant velocity without mixing. In each small element of

Bacterial cells accumulate faster on the top of the column because of larger q(s) values, and as a result, a distinctive spatial biomass distribution develops in the form of a hyperbolic decrease of x with column depth. Growth does not reach steady state with respect to x until cell elimination becomes well expressed (the effects of inhibitory products, endogenous cell decomposition, and leaching). Colonies Besides continuous-ﬂow columns, other heterogeneous systems are widely used. Especially popular is plating on solid media made from natural or synthetic gels (agar, PAAG, silica gel, synthetic alumosilicates, etc.) as well as on some porous materials (sand, glass beads, glass ﬁber). Impregnated with nutrient solution, such materials are used to grow microorganisms in the form of colonies or lawns. At ﬁrst glance, it is tempting to consider these techniques as analogies of a simple batch culture with singleterm substrate input (type 2bb in Table 11). However, a closer look at the mechanism of colony growth reveals a greater resemblance to chemostat culture! Here we will outline our considerations. The spread of a colony over a solid substrate, for example, a layer of agar, proceeds by the growth of only a peripheral zone with biomass, xp (Fig. 10). Then ds dt dx dt q(s)x lwxp ax (102)

where a is the speciﬁc decay rate of cell (mycelium) com-

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KINETICS, MICROBIAL GROWTH

w R

Agar

w R

Agar

Figure 10. Schematic illustration of the difference between the growth modes bacterial (top) and fungi (bottom) colonies. Note that unicellular organisms (bacteria) do not penetrate into the agar layer as opposed to ﬁlamentous fungi. Arrows indicate directions of substrate diffusion across concentration gradients. R and w are, respectively, radius and width of the peripheral zone of the colony.

ponents. Straightforward geometrical analysis shows that the linear spread rate for a regularly shaped colony (a cylinder, a sphere, or a strip) of size R is given by the following relation (2,5): dR dt or R R0 Krt (103) lww Kr

parable with the waste cell suspension that is discharged into the product bottle. A steady running of a pump, delivering medium at a rate, F (cm3 /h), corresponds to the active and uniform substrate utilization by the growing mycelium at a rate KrA (cm3 /h, A is the area of colony– medium interface). Finally, both cultivation systems are characterized by the elimination of propagating biomass, that is, by the expulsion of grown cells (mycelium) from further active growth (either by washing out or by the motion of the colony front). In both cases, the elimination rate is equal to biomass growth in the active compartment. The essential deviation from the chemostat lies in properties 3 and 4. Spatial differentiation of hypha and the direction of colony expansion are governed by spatial gradients of limiting substrate and, possibly, some metabolic products. Steepness of gradients is partly diminished by the effects of metabolic translocation along hypha over distances of the order of w. It can be concluded from the previous discussion that colony growth belongs to class 1b , and not to 2bb. In general, it is very likely that the 2b combination is an empty, logically forbidden combination, because a single-term momentary input of substrate may only be realized in a homogeneous system. Any spatial segregation whatever will actually prolong the consumption of substrate and, therefore, transform a batch process into a continuous one. For this reason, growth on any insoluble substrate (lignocellulose and other plant polymers, oil droplets, grains of sulfur, etc.) should always be treated as a continuous process. The solid-phase fermentation can also not be anything but continuous, whether new portions of substrate are added to the reactor or not. (Obviously, this applies to the growth mechanism itself and not to the engineering operation.) BIBLIOGRAPHY

1. A.A. Esener, J.A. Roels, and N.W.F. Kossen, Biotechnol. Bioeng. 25, 2803–2841 (1983). 2. N.S. Panikov, Microbial Growth Kinetics, Chapman & Hall, London, 1995. 3. I.G. Minkevich and V.K. Eroshin, Uspekhi Sovrem. Biol. 82, 103–116 (1976). 4. W.R. Mayberry, G.J. Prochazka, and W.J. Payne, Appl. Microbiol. 15, 1332–1338 (1967). 5. IS.J. Pirt, Principles of Microbe and Cell Cultivation, Blackwell Scientiﬁc Publishers, Oxford, U.K., 1975. 6. A.G. Marr, E.H. Nilson, and D.J. Clark, Ann. N.Y. Acad. Sci. 102, 536–548 (1963). 7. A.G. Dorofeyev and N.S. Panikov, Mikrobiologiya–Engl. Tr. 64, 601–609 (1994). 8. D.W. Tempest and O.M. Neijssel, Annu. Rev. Microbiol. 38, 459–486 (1984). 9. S.J. Pirt, Arch. Microbiol. 133, 300–302 (1982). 10. N. Panikov and J.S. Pirt, J. Gen. Microbiol. 108, 295–303 (1978). 11. T. Bauchop and S.R. Elsden, J. Gen. Microbiol. 23, 457–469 (1960). 12. V.N. Ivanov, Energetics of Microbial Growth, Naukova Dumka, Kiev, 1981 (in Russian). 13. W.M. van Gulik, and J.J. Heijnen, Biotechnol. Bioeng. 48, 681–698 (1996).

where Kr is the colony linear expansion rate, mm/h; lw is the microbial speciﬁc growth rate within the peripheral zone; and w is the zone width. In the case of unicellular organisms (e.g., bacteria, yeasts) that are incapable of penetrating into the gel’s matrix, the substrate is available through passive diffusion to the peripheral zone from the underlying gel layer. As the colony grows, this ﬂux diminishes, Kr decreases, and eventually the growth stops altogether. The ﬁlamentous organisms (fungi and actinomycetes) are able to propagate both on the surface and in the depth of gel. As a result, their growth is not completely dominated by diffusion effects, and the colony front advances at a faster rate than substrate is depleted in the frontier zone. The colony spreads at a constant radial rate, Kr, until the Petri dish is ﬁlled or the agar deteriorates from dryness. (In the case of rich nutrient media, there are also effects of self-inhibition by metabolic products [see equations 76 and 77].) Thus, the colony growth is (1) continuous, (2) substratelimited, (3) directed, and (4) spatially ordered. Properties 1 and 2 suggest a strong similarity between growth of a colony (especially a fungal one) and that of a chemostat culture. If so, a colony’s peripheral zone is analogous to the cell culture in the fermenter, and its central part is com-

KINETICS, MICROBIAL GROWTH 14. N.S. Pechurkin, Populational Microbiology, Novosibirsk, Nauka 1978, p. 277. 15. R.E. Buchanan, J. Infect. Dis. 23, 109–125 (1918). 16. G. Stephanopoulos, and J.J. Vallino, Science 252, 1675–1680 (1991). 17. Y. Kashiwaya, K. Sato, N. Tsuchiya, S. Thomas, D.A. Fell, R.L. Veech, and J.V. Passonneau, J. Biol. Chem. 269, 25502–25514 (1994). 18. J.J. Vallino, and G. Stephanopoulos, Biotechnol. Bioeng. 41, 633–646 (1993). 19. A. Marx, A.A. de Graaf, W. Wiechert, L. Eggeling, and H. Sahm, Biotechnol. Bioeng. 49, 111–119 (1996). 20. A. Cornish-Bowden, Principles of Enzyme Kinetics. Butterworths, London-Boston, 1976. 21. N. Michaelis and M.I. Menten, Biochem. Z. 49, 333–369 (1913). 22. G.E. Briggs and J.B.S. Haldane, Biochem. J. 19, 338–339 (1925). 23. H. Gutfreund, Discuss. Faraday Soc. 20, 167–173 (1955). 24. J.B.S. Haldane, Enzymes. Longmans Green, London, 1930. 25. J. Monod, J. Wyman, and J. Changeux, J. Mol. Biol. 12, 88– 118 (1965). 26. C.H. Wang and A.L. Koch, Bacteriol. 136, 969–975 (1978). 27. O. Rahn, Physiology of Bacteria, P. Blackiston’s, Philadelphia, 1932. 28. P.F. Verhulst, Correspondence Mathematique et Physique 10(1), 113–121 (1838). 29. J. Monod, Recherches sur la Croissance des Cultures Bacteriennes, Paris, Hermann and Co, 1942. 30. J. Monod, Ann. Inst. Pasteur 79, 390–410 (1950). 31. F.F. Blackman, Ann. Bot.–London 19, 281–293 (1905). 32. S.D. Varfolomeyev, in S.D. Varfolomeyev ed., Modern Problems of Biokinetics, Moscow University Publishers, Moscow, 1987, pp. 6–77. 33. J. Monod, Ann. Rev. Microbiol. 3, 371–394 (1949). 34. N.D. Jerusalimsky, in E.O. Powell, C.G.T. Evans, R.E. Strange, and D.W. Tempest eds., Continuous Cultivation of Microorganisms, H.M. Stationery Ofﬁce, Salisbury, U.K., 1967, pp. 23–33. 35. I.A. Shvytov, Zhurnal Obshchei Biologii 35, 904–910 (1974). 36. J.M. Romanovsky, N.V. Stepanova, and D.S. Chemavski, Mathematical Biophysics, Moscow, Nauka, 1984. 37. T.E. Shehata and A.G. Marr, J. Bacteriol. 107, 210–216 (1971). 38. A.F. Guady Jr., A. Obayashi, and E.T. Gaudy, Appl. Microbiol. 22, 1041–1047 (1971). 39. D.K. Button, Microbiol. Rev. 49, 270–297 (1985). 40. B.R. Robertson and D.K. Button, Bacteriology 138, 884–895 (1979). 41. G. Teissier, Ann. Physiol. Physicochem. Biol. 12, 527–586 (1956). 42. D.E. Contois, J. Gen. Microbiol. 21, 40–49 (1959). 43. H. Moser, Carnegie Inst. Wash. Pub. 614 (1958). 44. E.O. Powell, in E.O. Powell, C.G.T. Evans, R.E. Strange, and D.W. Tempest eds., Continuous Cultivation of Microorganisms, H.M. Stationery Ofﬁce, Salisbury, U.K., 1967, pp. 34– 55.

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45. J.F. Andrews, Biotechnol. Bioeng. 10, 707–723 (1968). 46. U. Pawlowsky, S.A. Howell, and C.T. Chi, Biotechnol. Bioeng. 15, 905–916 (1973). 47. A. Campbell, Bacteriol. Rev. 21, 263–272 (1957). 48. C.L. Cooney, Ann. N.Y. Acad. Sci. 326, 295–314 (1979). 49. E.O. Powell, C.G.T. Evans, R.E. Strange, and D.W. Tempest eds., Continuous Cultivation of Microorganisms, H.M. Stationery Ofﬁce, Salisbury, U.K., 1967, pp. 259–261. 50. D. Ramkrishna, A.G. Fredrickson, and H.M. Tsuchiya, Biotech. Bioeng. 9, 129–170 (1967). 51. M.M. Domach, S.K. Leung, R.E. Cahn, G.G. Cacks, and M.L. Shuler, Biotech. Bioeng., 26, 203–216 (1984). 52. J.W. Jeong, J. Snay, and M.M. Ataai, Biotechnol. Bioeng. 35, 160–184 (1990). 53. I. Malek, in I. Malek ed., Continuous Cultivation of Microorganisms: A symposium, Publishing House of Czech. Acad. Sci., 1958, pp. 11–28. 54. D. Herbert, Symp. Soc. Gen. Microbiol. 11, 391–416 (1961). 55. P. Agrawal, H.C. Lim, and D. Ramkrishna, J. Chem. Tech. Biotechnol. 33B, 155–163 (1983). 56. D.S. Kompala, D. Ramkrishna, N.B. Jansen, and G.T. Tsao, Biotechnol. Bioeng. 28, 1044–1055 (1986). 57. B.G. Turner, D. Ramkrishna, N.B. Jansen, Biotechnol. Bioeng. 34, 252–261 (1989). 58. D.W. Tempest, J.R. Hunter, and J. Sykes, J. Gen. Microbiol. 39, 355–366 (1965). 59. D.W. Tempest, D. Herbert, and P.J. Phipps, in E.O. Powell, C.G.T. Evans, R.E. Strange, and D.W. Tempest eds., Continuous Cultivation of Microorganisms, H.M. Stationery Ofﬁce Salisbury, U.K., 1967, pp. 240–253. 60. A.T. Henrici, Morphologic Variation and the Rate of Growth of Bacteria, Charles C. Thomas, Springﬁeld, Ill., 1928, p. 194. 61. A.L. Koch, Crit. Rev. Microbiol. 19, 17–42 (1973). 62. S. Wold, K. Skarstad, H.B. Steen, T. Stokke, E. Boye, EMBO J. 13, 2097–2102 (1994). 63. C.E. Helmstetter, S. Cooper, O. Pierucci, and E. Revela, Cold Springs Harb. Symp. Quant. Biol. 33, 809–822 (1968). 64. S. Cooper, Res. Microbiol. 141, 17–29 (1990). 65. A.L. Koch, J. Gen. Microbiol. 45, 409–417 (1966). 66. J.F. Collins, and M.H. Richmond, J. Gen. Microbiol. 28, 15– 33 (1962). 67. A. Novick, in G. Tunevall ed., Recent Progress in Microbiology, Blackwell Scientiﬁc Publications, Oxford, U.K., 1959, pp. 403– 415. 68. J. Adams, T. Kinney, S. Thompson, L. Rubin, and R.B. Helling, Genetics 91, 627–637 (1979). 69. R. Freter, R.R. Freter, and H. Brickner, Infect. Immunol. 39, 60–84 (1983). 70. I.G. Minkevich, A.Y. Krynitskaya, and V.K. Eroshin, Mikrobiologiya (Engl. Tr.) 56, 254–258 (1987). 71. W.R. Chesbro, T. Evans, and R. Eifert, J. Bacteriol. 139, 625– 638 (1979). 72. B.C. Goodwin, Eur. J. Biochem. 10, 511–514 (1969). 73. P.S.S. Dawson, Can. J. Microbiol. 11, 893–903 (1965).

See also MICROBIAL GROWTH MEASUREMENT, METHODS.

K

KINETICS, ENZYMES

ATHEL CORNISH-BOWDEN

National Center for Scientiﬁc Research Marseilles, France

MICHAELIS–MENTEN KINETICS The starting point for any discussion of enzyme kinetics is the Michaelis–Menten equation, which expresses the initial rate v of a reaction at a concentration a of the substrate transformed in a reaction catalyzed by an enzyme at total concentration e0: v k0e0a Km a Va Km a (1)

KEY WORDS Enzymes Inhibition Kinetics Manipulation of enzyme activity Multienzyme systems OUTLINE Introduction Michaelis–Menten Kinetics Graphical Analysis Two-Substrate Reactions Inhibition and Activation Inhibition Speciﬁcity Activation Irreversible Inhibition Inhibitory Effects in Metabolic Systems Non-Michaelis–Menten Behavior Kinetics of Multienzyme Systems Further Reading Bibliography INTRODUCTION The kinetic behavior of enzymes has been studied in detail for a century, beginning with the classic work of Henri (1) and Michaelis and Menten (2). The objectives have been threefold: to gain an understanding of the mechanisms of enzyme action; to illuminate the physiological roles of enzyme-catalyzed reactions; and, mainly in recent years, to manipulate enzyme properties for biotechnological ends. Experimental practice has been overwhelmingly dominated by the ﬁrst of these aims; most experiments have been designed as if shedding light on the mechanism was the principal, or even the only, objective. However, although much valuable information has been obtained in this way, there are some important aspects of enzyme function that are obscured when working mainly with isolated enzymes in conditions far removed from those that exist in the cell. For this reason the latter part of this chapter will be devoted to a discussion of enzymes as they behave in complex mixtures.

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The parameters are k0, the catalytic constant, and Km, the Michaelis constant. The form shown in the middle is more fundamental than that on the right, but the second form, in which k0e0 is written as the limiting rate V, is often used because the enzyme concentration in meaningful units is often not known. V is the limit that v approaches as the enzyme becomes saturated (that is, when a becomes very large) and Km is the value of a at which v 0.5V (that is, at which the rate is half-maximal). The ratio k0 /Km is called the speciﬁcity constant and given the symbol kA: It is a more fundamental constant than Km in the analysis of enzyme mechanisms (i.e., it has a simpler mechanistic meaning), but the equation is usually written in terms of V (or k0) and Km nonetheless. The principal assumption implied by equation 1 is that the rate of the reverse reaction is negligible: This may be because the reaction is irreversible for practical purposes, but even with reversible reactions the reverse reaction can be made negligible by measuring the rate in the absence of products and by extrapolating the rate back to zero time, that is, by estimating the initial rate at a time when no products have accumulated. Even if there is no signiﬁcant reverse reaction, products can still affect the rate of the forward reaction because product inhibition (discussed later) is a common phenomenon. Another assumption is that the reaction is allowed sufﬁcient time to reach a steady state because all reactions pass through an initial acceleration phase known as the transient state. This phase is normally very brief (a few milliseconds), and in practice enzymes are usually studied under steady-state conditions. Note, however, that this is made experimentally possible by working with extremely small enzyme concentrations compared with those that may exist in the cell. The use of very low enzyme concentrations has two important consequences: First, it normally means that the enzyme concentration can be neglected in comparison with the substrate concentration, and second it makes the steady-state rate sufﬁciently slow to be easily measured and the steady-state phase long enough to be meaningful. If high enzyme concentrations were used, the transient state would be as brief as before, but the steady state would also be very brief, so that there would be no period in which one could adequately treat the rate as constant. Equation 1 may be derived from the following model

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KINETICS, ENZYMES

A

E i EA i EP i E

k

1

k1

k2

k3

P

(2)

v

V

k

2

k

3

which assumes that the reaction passes through an enzyme–substrate complex EA, which undergoes catalytic transformation to an enzyme–product complex EP, which then breaks down to form products. Although real enzyme mechanisms may be more complicated than this, every reaction passes through steps of substrate binding, chemical transformation, and product release. In simple introductory treatments the second and third steps are often treated as a single step, but conceptually they are clearly distinct. In reactions of more than one substrate, the steps do not necessarily occur in the order one might guess; that is, some products may be released before all substrates have bound, but the general principle that any reaction involves the same three kinds of step remains valid. The Michaelis–Menten parameters can be deﬁned as follows in terms of the rate constants shown in equation 2: k0 k2k3 k2 ; kA k k1k2k3 ; k 1k3 k2k3 k 1k3 k2k3 (3) k2 k3)

0.5V

0 0 Km

5 Km

10 Km a

15 Km

20 Km

Figure 1. Michaelis–Menten dependence of rate on substrate concentration. The curve is a rectangular hyperbola through the origin, approaching a limit of v V at saturation. The rate is 0.5 V at a substrate concentration equal to the Michaelis constant, Km, but note that the approach to the limit is slow, so that, for example, even at a 10Km the rate is still nearly 10% less than V.

k

2

k3

k

1k 1k 2

2

Km

k1(k

2

Note that none of the three parameters has a simple transparent meaning. The interpretations commonly attributed to them depend on additional simplifying assumptions that are not always correct. For example, Km is often said to be equal to the equilibrium constant k 1 /k1 for dissociation of A from EA, but the expression in equation 3 does not take this form unless k2 is very small. As there is no good reason for k2 to be small, and indeed ideas of evolutionary optimization of enzyme function lead one to expect the opposite when the enzyme is acting on its natural physiological substrate, it follows that Km should not, in general, be regarded as a measure of the equilibrium dissociation constant. Despite these difﬁculties in providing a detailed mechanistic meaning to Km, it does provide a measure of the tightness of substrate binding in the steady state, as it is quite correct to take a/Km as equal to the ratio of the sum of concentrations of all enzyme complexes (i.e., both EA and EP) over the concentration of free enzyme. Similarly k0 provides a valid measure of the capacity of the enzyme– substrate complex to react to give products, even if it cannot be interpreted as the rate constant for a unique step in the mechanism. The reason for the term speciﬁcity constant for kA, that is, its relationship to enzyme speciﬁcity, will become clear once inhibition has been considered. Graphical Analysis Equation 1 deﬁnes a hyperbolic dependence of rate on substrate concentration, as illustrated in Figure 1. The initial steep rise in v as a increases from zero is rapidly transformed into the phenomenon of saturation, whereby further increases in a produce smaller and smaller increases in v, which approaches, but does not reach or exceed, the limiting rate V. The rectangular hyperbola makes this type

of plot inconvenient for estimating the values of the kinetic parameters (because the line does not approach the saturation limit closely enough at reasonable values of a to allow direct measurement of V). For this purpose, therefore, it is usual to transform equation 1 into one of the following three forms, which underlie the three straight-line plots illustrated in Figures 2 through 4: 1 v a v v 1 V Km V V Km 1 V a 1 a V Km v a (4) (5) (6)

The double-reciprocal plot, illustrated in Figure 2 and based on equation 4, is the mostly widely used, but it is also the least satisfactory because it distorts the effect of experimental error to such an extent that it is difﬁcult to form any visual judgement of where the best line should be drawn. The other two plots are better, and the plot of v

1/v

Km 1 1 1 v = V + V a

Slope = Km/V 1/V

–1/Km

0

1/a

Figure 2. The double-reciprocal plot. This is the most widely used method of plotting the Michaelis–Menten equation as a straight line. However, the severe distortion of any experimental errors in the original data causes it to give a misleading impression.

KINETICS, ENZYMES

1507

a/v

a Km 1 a v= V + V

v v = V – Km V

v a

Slope = 1/V Km /V –Km 0 a

Slope = –Km

0

Km /V

v/a

Figure 3. The plot of a/v against a. This alternative to the plot shown in Figure 2 produces much less distortion of the experimental error.

against v/a (Fig. 4, equation 6) has the particular advantage that the entire observable range of v values, from 0 to V, is mapped onto a ﬁnite range of graph; this makes it easy to judge by eye if an experiment has been well designed. On the other hand, it has the disadvantage that v, normally the less reliable measurement, contributes to both coordinates, and errors in v cause deviations along lines through the origin, rather than parallel with one or the other axis. In modern practice it is usually best to regard these plots as for illustration purposes only, and to use suitable computer programs for the actual parameter estimation. For this purpose it is not sufﬁcient just to apply unweighted linear regression to the straight-line plots, as this suffers from the same statistical distortions as the plots themselves. Full treatment would require more space than is available here, but see Ref. 3. The following two equations for calculating best-ﬁt values of Km and V give satisfactory results if the v values have uniform coefﬁcient of variation (uniform standard deviation expressed as a percentage), as is usually at least approximately correct: Km V Rv2R(v/a) R(v2 /a2)Rv R(v2 /a)Rv R(v2 /a)R(v/a) (7) (8)

Figure 4. The plot of v against v/a. The third way of plotting the Michaelis–Menten equation as a straight line also avoids the error-distorting property of the plot shown in Figure 2, and maps the entire range of observable rates (from 0 to V) onto a ﬁnite range of paper. This is a desirable property because it makes it impossible to disguise deﬁciencies in the experimental design.

equation 1, it can be arranged in the same form if one of the two substrate concentrations, for example b, is treated as a constant: k0b ea KmB b 0 KiAKmB KmB KmAb b a

v

(10)

The two fractions in this equation can be regarded as “apparent” values of the Michaelis–Menten parameters for A, that is, the equation can be written as v with kapp 0 k0b ; kapp KmB b A (k0 /KmA)b ; (KiAKmB /KmA) b KiAKmB KmAb Kapp mA KmB b kappe0a 0 Kapp a mA (11)

R(v2 /a2)Rv2 [R(v2 /a)]2 2 2 R(v /a )Rv R(v2 /a)R(v/a)

(12)

Each summation is made over all observations. Two-Substrate Reactions Enzymes that catalyze reactions of a single substrate are only a small minority of all the enzymes known, but the Michaelis–Menten equation remains useful for examining the kinetics of the more common case of a reaction with two substrates and (often but not necessarily) two products, because such a reaction normally obeys Michaelis– Menten kinetics when only one substrate concentration is varied at a time. This is illustrated by the following typical equation for such a reaction: v k0e0ab KmBa KmAb (9)

Notice that the expressions for the apparent values of k0 and kA are both individually of Michaelis–Menten form with respect to b, whereas that for the apparent value of Km is more complicated: This behavior is quite typical and is one of the reasons why k0 and kA should regarded as more fundamental parameters than Km. More generally, the concept of apparent parameters pervades the analysis of simple cases in steady-state enzyme kinetics, being important for the study of reactions with multiple substrates and inhibition. INHIBITION AND ACTIVATION Inhibition For most enzyme-catalyzed reactions, molecules exist that resemble the substrate closely enough to bind to the en-

KiAKmB

ab

Although at ﬁrst sight this appears quite different from

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KINETICS, ENZYMES

zyme, but not closely enough to undergo a chemical reaction. Such a molecule is known as a competitive inhibitor and causes competitive inhibition, characterized by a rate equation of the following form: v k0e0a i/Kic) (13)

Km(1

a

in which i is the concentration of the inhibitor and Kic is the competitive inhibition constant. (The qualiﬁcation “competitive” and the second subscript c are usually omitted if only this simplest kind of inhibition is being considered.) Inhibitors can interfere with catalysis as well as with substrate binding. In the simplest case, an inhibitory term affects the variable term in the denominator of the Michaelis–Menten equation, instead of the constant term: v k0e0a a(1 i/Kiu) (14)

Km

This is called uncompetitive inhibition, and the inhibition constant Kiu is the uncompetitive inhibition constant. This is important as a limiting case of inhibition, but in its pure form it is not at all common. Much more often one has mixed inhibition, when both competitive and uncompetitive effects occur simultaneously: v k0e0a i/Kic) a(1 (15)

the inhibition is noncompetitive. In general it is simplest to regard kA as the parameter affected by competitive inhibition, negligibly so when the competitive component is negligible, k0 as the parameter affected by uncompetitive inhibition, negligibly so when the uncompetitive component is negligible, and Km just as the ratio of the two, so Km k0 /kA. The effects of the different kinds of inhibition on the common plots as illustrated in Figures 2 through 4 follows naturally from equation 16. Any competitive effect affects the apparent value of kA, hence, it increases the slope of the plot of 1/v against 1/a (Fig. 2), it increases the ordinate intercept of the plot of a/v against a (Fig. 3), and it decreases the abscissa intercept of the plot of v against v/a (Fig. 4). Conversely, any uncompetitive effect increases the ordinate intercept of the plot of 1/v against 1/a, increases the slope of the plot of a/v against a, and decreases the ordinate intercept of the plot of v against v/a. When both components of the inhibition are present, both kinds of effects occur. As an illustration we may consider just one example, the effect of competitive inhibition on the plot of 1/v against 1/a: Plots made at various different inhibitor concentrations produce a family of straight lines intersecting on the ordinate axis, as shown in Figure 5, the lack of effect on the ordinate intercept being a direct consequence of the lack of effect on the apparent value of V. Speciﬁcity Speciﬁcity is the most fundamentally important property of enzymes. Although one is often impressed by the catalytic effectiveness of enzymes, accelerating a reaction is not, in reality, difﬁcult: Heating the reaction mixture in a sealed tube is an efﬁcient way of accelerating virtually any reaction, essentially without limit. What is difﬁcult is to accelerate one selected reaction without at the same time accelerating a great mass of unwanted reactions. What is important about an enzyme, therefore, is not that it is an excellent catalyst for a small set of reactions, but that it is an extremely bad catalyst—virtually without any activity—for all other reactions. In other words, the essential properties of enzymes are that they act under very mild conditions and are highly speciﬁc. In the past, speciﬁcity was often assessed by comparing the kinetic parameters for different reactions measured in

Km(1

i/Kiu)

There is no particular reason for the two inhibition constants Kic and Kiu to be equal, and most of the mechanisms one might propose to account for mixed inhibition lead one to expect them to be different, yet the case where Kic Kiu is often given an undeserved prominence in discussions of inhibition, largely because experiments done many years ago suggested that it was a more common phenomenon than it is. This is called noncompetitive inhibition and its rate equation is the same as equation 15, but with both Kic and Kiu written simply as Ki. All of these kinds of inhibition are conveniently discussed in terms of apparent Michaelis–Menten parameters. In the general case (equation 15), these are as follows: kapp 0 k0 ; kapp i/Kiu A kA ; i/Kic Kapp m Km(1 1 i/Kic) i/Kiu (16)

1

1

1/v Increasing i

Note that the ﬁrst two expressions have the same form, and both simplify to independence of i in the event that one or other inhibition term is negligible. The expression for the apparent value of Km is more complicated, especially when one considers how it varies with the different types of inhibition: It increases with the concentration of a competitive inhibitor, it decreases as the concentration of an uncompetitive inhibitor increases, it may change in either direction as the concentration of a mixed inhibitor increases, or it is independent of inhibitor concentration if

1/V

1 1 Km(1 + i/Kic) 1 v = V + a V 0 1/a

Figure 5. Effect of competitive inhibition on the double-reciprocal plot.

KINETICS, ENZYMES

1509

isolation from one another, which led to sterile arguments as to whether speciﬁcity was best measured in terms of k0, Km, or some combination of the two. This type of argument was resolved once it was realized that the only meaningful way of deﬁning speciﬁcity is as a property of an enzyme that allows it to discriminate between substrates that are mixed together. The simplest way to consider this is with a model similar to that for competitive inhibition, except that one assumes that both molecules are capable of reacting. The equation for reaction of one substrate A in the presence of a competing substrate A follows an equation similar to that for competitive inhibition (equation 13): v k0e0a a /Km) (17)

Km(1

a

with the inhibitor concentration replaced by a , the concentration of A , and the inhibition constant by Km, the Michaelis constant for the reaction of A considered in isolation. The rate of reaction of A is given by the same equation with an obvious transposition of symbols: v k0e0a a/Km) (18)

Km(1

a

NAD, but although consideration of the metabolic pathway in which the reaction occurs may make it convenient to regard ethanol as the substrate and NAD as the coenzyme, this is meaningless when the reaction is considered in isolation. So far as alcohol dehydrogenase is concerned, it catalyzes a reaction that requires two substrates, ethanol and oxidized NAD; the reaction will not proceed unless both are present, and neither has any more reason to be called the substrate than the other. When such improper uses of the term are excluded, there remain a number of enzymes for which the true inverse of inhibition occurs. In the simplest cases the equations are just the inverse of inhibition equations, with terms of the form i/Ki replaced by ones of the form Kx /x (for an activator X with activation constant Kx). However, the simplest cases constitute a smaller proportion of the whole than they do for inhibition. This is because whereas most inhibitors inhibit completely, in the sense that enzyme species with inhibitor bound retain no activity as long as the inhibitor remains bound, many enzymes subject to activation retain some activity in the absence of the activator. As a result, full analysis of activation is often more complicated than it is for inhibition, but this will not be discussed further here. Irreversible Inhibition The types of inhibition considered so far are examples of reversible inhibition; the inhibitor binds reversibly and catalytic activity returns when the inhibitor is released. Irreversible inhibition also occurs, in which the inhibitor either binds so tightly that for practical purposes it cannot be removed, or reacts with the enzyme and converts it irreversibly to a form that has no catalytic activity. These two cases are conceptually different, and the former is more correctly called tight-binding inhibition, rather than irreversible inhibition. However, they are not easy to distinguish in practice, and have similar practical effects and, hence, similar practical uses. Although irreversible inhibition has played a smaller part than reversible inhibition in the academic study of enzyme mechanisms, it has far greater industrial and pharmacological importance. This is because competitive inhibitors, the most common kind of reversible inhibitors, are almost completely ineffective in complete physiological systems, for reasons to be considered shortly. By contrast, whenever irreversible (or tight-binding) inhibition occurs in a physiological system, it can be expected to have profound effects. Many toxic and pharmacologically active substances owe their effects to irreversible inhibition. Both tight-binding and irreversible inhibition manifest themselves in ways that allow them to be confused with noncompetitive inhibition, as in equation 15 with the two inhibition constants equal. This is because the practical effect of irreversible inhibition is not on any of the kinetic parameters in the Michaelis–Menten equation, but on e0, the total enzyme concentration. However, a decrease in e0 can be confused with a decrease in the apparent value of k0, as they occur as a product in equation 1. Although uncompetitive inhibition affects k0, it does so without affecting kA, and thus also changes Km. Decreasing k0 without

It can then be seen that the ratio of rates is the ratio of substrate concentrations multiplied by the ratio of speciﬁcity constants: v v k0 /Km a k0 /Km a kAa kAa (19)

This result, which is still less well known than its importance merits, is the reason for the term speciﬁcity constant. Note that although inspection of equation 1 suggests that kA is no more than the parameter that deﬁnes the rate at very low substrate concentrations, no assumption about the magnitudes of the concentrations was made in arriving at equation 19. It is thus valid at all concentrations, and the speciﬁcity constant measures speciﬁcity at all concentrations, not just low ones. Activation Activation is the opposite from inhibition, in which a reaction proceeds more rapidly in the presence of a particular molecule than in its absence. It is less common than inhibition, and discussion is complicated by the fact that a variety of quite different phenomena have been termed “activation.” The most important of these is a confusion between true activation and the case where the activator is really a component of the substrate. Numerous ATPhandling enzymes are said to be activated by magnesium ions, when in reality the complex MgATP is the true substrate, that is, the species that reacts with the enzyme. Other metal ions, such as the zinc in a number of enzymes, may be true activators as they bind to the enzyme itself and confer catalytic properties on it. Another misuse of the term activation relates to coenzymes such as NAD in many dehydrogenases: Alcohol dehydrogenase, for example, may be said to be activated by

1510

KINETICS, ENZYMES

affecting Km, similar to what one would observe if e0 decreases, is a deﬁnition of noncompetitive inhibition. Inhibitory Effects in Metabolic Systems Competitive and uncompetitive inhibition are sufﬁciently similar in their effects in artiﬁcial experiments on isolated enzymes that they are often not distinguished, and an uncompetitive component in mixed inhibition often passes unnoticed. Many inhibitors are described in the literature as competitive inhibitors in the absence of any real evidence of the type of inhibition. This sort of confusion can easily lead to the entirely false idea that they are similar in their effects in systems where the inhibited enzyme is mixed with other enzymes and catalyzes a step in the middle of a pathway. In a typical experiment in vitro, one decides the concentrations of the various components in advance and measures the rate that results; however, this is very different from what happens in the cell. To a ﬁrst approximation, an enzyme catalyzing a step in the middle of a pathway must transform its substrate at the rate at which it arrives, that is, within certain limits it has little or no effect on the rate of its reaction, but instead determines the concentrations of the metabolites around it. (This is an oversimpliﬁcation, but is useful for discussion.) It is useful therefore to transform equations 13 and 14 into expressions for a in terms of i: vKm(1 k0e0 k0e0 vKm v(1 i/Kic) v a/Km

6 4 2 0 0 1 i/Kic (a) 6 a/Km 4 2 0 0 1 i/Kiu (b)

Figure 6. Effects of (a) competitive and (b) uncompetitive inhibition on the concentration of substrate in a constant-rate system. Both curves are drawn for the case of a Km in the absence of inhibitor. Note that both kinds of inhibitor have quantitatively equal effects at very low concentrations, but the initial slope is maintained indeﬁnitely if the inhibition is competitive, whereas it rapidly becomes inﬁnite if the inhibition is uncompetitive.

2

2

a

(20) NON-MICHAELIS–MENTEN BEHAVIOR All of the cases considered so far can be regarded as generalizations of the Michaelis–Menten equation (equation 1). However, although many enzymes do behave in this way, at least as a ﬁrst approximation, there are some important exceptions. It is simple to calculate from equation 1 that if a Km /9 then v 0.1V and if a 9Km then v 0.9V; in other words, spanning the 10–90% range of available rates requires an 81-fold range of substrate concentrations, almost two orders of magnitude. Similar calculations may be done with any of the equations of the Michaelis–Menten type for additional substrates, inhibitors, or activators. Their implication is that as long as enzymes follow Michaelis–Menten kinetics, their rates cannot be adequately varied by manipulating concentrations of substrates, for example, because effective regulation will often require sensitivity to small changes in signals— certainly changes much smaller than two orders of magnitude. A second difﬁculty arises from the fact that inhibition of the types considered commonly derives from structural similarities between inhibitors and substrates or products, whereas there is no reason to expect the molecules needed for metabolic signals to resemble the substrates or products of the enzymes that need to respond to the signals. In reality, the concentration of the end product of a pathway often serves as such a signal: too low, and the pathway

a

i/Kiu)

(21)

However similar equations 13 and 14 may seem, their transformed versions are drastically different. Equation 20 shows a linear dependence of a on i, which means that increasing i can never result in uncontrolled increases in a. This is illustrated in Fig. 6a. Even at an inhibition concentration equal to the inhibition constant, the substrate concentration is only doubled. By contrast, the curve deﬁned by equation 21 is a rectangular hyperbola (Fig. 6b) that produces a steep and uncontrolled rise in substrate concentration at quite moderate inhibitor concentrations (4). The point is that in competitive inhibition, rises in substrate and inhibitor concentrations oppose one another— not only does the inhibitor compete with the substrate, but equally, the substrate competes with the inhibitor. In uncompetitive inhibition, however, these effects potentiate one another. It follows that although it is relatively easy to ﬁnd molecules that will act as competitive inhibitors, it is also largely useless as a strategy for designing pesticides or drugs because it is correspondingly easy for the organism to counteract the effect of the inhibition. To produce major metabolic effects one needs uncompetitive inhibitors, irreversible inhibitors, or tight-binding inhibitors: None of these are as easy to produce as weakly binding competitive inhibitors, but they are far more effective.

KINETICS, ENZYMES

1511

needs to be activated; too high, and it needs to be inhibited. It is often found, therefore, that the enzyme that catalyzes the ﬁrst committed step of a pathway, that is, the ﬁrst step after a branch point, in the branch that leads to the end product in question, is inhibited by that end product. For inhibition of this kind to be possible, the enzyme must have a speciﬁc binding site for the end product, independent of the binding sites for substrates and products. Such a site is called an allosteric site, and the phenomenon is called allosteric inhibition. Because the need for it often coincides with the need for higher sensitivity than is provided by Michaelis–Menten kinetics and the common kinds of inhibition, allosteric inhibition is often cooperative: This means that the equations that deﬁne it are more complicated than those considered above, allowing, for example, a change from 10% to 90% inhibited over a concentration range much smaller than 81-fold, and typically less than 10-fold (though rarely, if ever, less than 3-fold) (Fig. 7). The fact that allosteric and cooperative behavior are often found together has led many authors to treat the two terms as synonymous, a tendency encouraged by the fact that in one of the most widely accepted models of nonMichaelis–Menten behavior, that of Monod, Wyman, and Changeux (5), both result from the same structural properties attributed to the enzyme. Nonetheless, most careful authors consider the two properties to be conceptually distinct and not necessarily occurring together, so the two terms should be considered distinct as well. KINETICS OF MULTIENZYME SYSTEMS As noted in the introduction, nearly all kinetic studies of enzymes have been carried out using isolated enzymes, and although this has been very valuable for arriving at a good understanding of the nature of enzyme catalysis, it is quite inadequate as a guide to how systems of enzymes will behave. One cannot assume that the ﬂux through a met-

v V 0.9V

0.1V 0 81× 9×

abolic pathway is a property of a unique enzyme catalyzing the rate-limiting step, and that the properties of the pathway as a whole can be deduced from studies of the kinetics of this one enzyme in isolation. Space does not permit a full analysis of this subject, which has developed from a classic paper by Kacser and Burns (6), but it should sufﬁce to examine an example of a pathway in which biotechnologists have attempted to increase the ﬂux by identifying the rate-limiting enzyme and using genetic manipulation to increase its activity. Tryptophan biosynthesis in yeast provides such an example, tryptophan being a commercially valuable metabolite for which increased production would be very desirable. The tryptophan pathway consists of ﬁve enzymes, and in the classical model any one of these could be the “key” enzyme catalyzing the rate-limiting step. However, when the activity of each of these enzymes was increased in turn, either singly or at the same time as others in the pathway, the results were trivial: Increases of enzyme activity of 20fold or greater produced ﬂux increases of perhaps 30%. Only when all ﬁve enzyme activities were increased (or all but one, apparently unimportant, activity) was there a substantial increase in ﬂux, which even then was much smaller than the increase in enzyme activity (7). The object here is not to analyze in detail why manipulation of tryptophan biosynthesis did not produce the desired results, but to use it to illustrate the point that the whole approach is misconceived. One cannot treat the kinetic behavior of systems of enzymes as if it were determined by the properties of a single component. Moreover, abundant evidence exists to show that all organisms have evolved regulatory mechanisms to control metabolic ﬂuxes and concentrations to satisfy their own requirements. Artiﬁcially trying to force more activity in a pathway by increasing the activities of certain enzymes simply stimulates the regulatory mechanisms to resist. Tryptophan biosynthesis is just one example of a general result, and similar efforts in other pathways—for example, increased alcohol production in yeast and increased starch production in potatoes—have produced similar results. The essential point is that ﬂux control is not a property of a single enzyme in a pathway, but is shared among all the enzymes in the pathway. Strictly speaking, it is shared among all the enzymes in the cell, but it is usually safe to assume that enzymes catalyzing reactions remote from the pathway of interest have very little effect on the ﬂux through it, so to a ﬁrst approximation one can consider ﬂux control to be shared among the enzymes of the pathway. To express this idea in quantitative terms, one can deﬁne a ﬂux control coefﬁcient for any enzyme by the following equation: lnJ p lnvi p

a CJ i (22)

Figure 7. Non-Michaelis–Menten kinetics. For an enzyme obeying the Michaelis–Menten equation (cf. Fig. 1), an 81-fold increase in substrate concentration is needed to bring the rate from 10% to 90% of V. If the enzyme shows positive cooperativity the curve typically becomes sigmoid (S-shaped), and this range of substrate concentrations is decreased (to nine-fold in the example, but in strongly cooperative cases it can be as small as three-fold).

This deﬁnition compares the effect on the ﬂux J through a pathway of some perturbation of the activity of the ith enzyme, represented by a change in the parameter p, with the effect the same perturbation would have on the rate vi of the same reaction if it were considered in isolation. The

1512

KINETICS, ENZYMES

identity of the parameter p does not have to be speciﬁed, because as long as it affects only one enzyme, the control coefﬁcient deﬁned by equation 22 is independent of the manner in which the ﬂux and isolated rate are perturbed. However, to make the deﬁnition more concrete, consider the case where p is the logarithm of the enzyme concentration, ei. As most reactions are considered under conditions where the rate is proportional to the enzyme concentration, it will often be true to write dlnvi dlnei, so that the denominator in equation 22 has a value of unity and the whole equation simpliﬁes to the following: lnJ lnei

FURTHER READING The topic of this article is referred to in all textbooks of biochemistry, but unfortunately the treatment is nearly always very superﬁcial and potentially misleading. Since Mahler and Cordes (8), there has been no serious attempt to cover enzyme kinetics in a general biochemistry text. More specialized textbooks were abundant in the past, but have become much less so. There are, however, two recent ones (9,10) that may be consulted for more information about most of the points discussed here, as well as topics such as the pH- and temperature-dependence of enzymecatalyzed reactions, which require more detail than can be covered adequately in a short article. Another source is the book by Segel (11), which was originally published in 1975, but has been reissued recently. Fast reactions are not dealt with in this article, and as they require both techniques and methods of analysis different from those of steady-state kinetics, it is best to refer to a specialized source, such as Hiromi (12), who provides a thorough treatment. Two recent books deal with the kinetics of multienzyme systems, one (13) at a very readable level, the other (14) much more advanced. This is also covered much more brieﬂy in two of the books mentioned earlier (9,10).

CJ i

(23)

This equation is less abstract and simpler to understand than equation 22, and in the past was often used as a primary deﬁnition of a control coefﬁcient. However, this is not recommended, ﬁrst because it is not always true that the isolated rate is proportional to the enzyme concentration, and second because equation 23 can give the false impression that control coefﬁcients are concerned only with effects brought about by changes in enzyme concentration. Flux control coefﬁcients are measures of how much the ﬂux through a pathway is dependent on the activities of the individual enzymes. Mathematical analysis shows that they satisfy a property called the summation relationship: RC J i 1 (24)

BIBLIOGRAPHY

1. V. Henri, Lois Generales de l’Action des Diastases, Hermann, ´ ´ Paris, 1903. 2. L. Michaelis and M.L. Menten, Biochem. Z. 49, 333–369 (1913). 3. A. Cornish-Bowden, Analysis of Enzyme Kinetic Data, Oxford University Press, Oxford, 1995. 4. A. Cornish-Bowden, FEBS Lett. 203, 3–6 (1986). 5. J. Monod, J. Wyman and J.-P. Changeux, J. Molec. Biol. 12, 88–118 (1965). 6. H. Kacser and J.A. Burns, Symp. Soc. Exp. Biol. 27, 65–104 (1973). 7. P. Niederberger, R. Prasad, G. Miozzari, and H. Kacser, Biochem. J. 287, 473–479 (1992). 8. H.R. Mahler and E.H. Cordes, Biological Chemistry, 2nd ed., Harper & Row, New York, 1972. 9. A.R. Schulz, Enzyme Kinetics: From Diastase to MultiEnzyme Systems, Cambridge University Press, Cambridge, 1994. 10. A. Cornish-Bowden, Fundamentals of Enzyme Kinetics, 2nd ed., Portland Press, London, 1995. 11. I.H. Segel, Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems, Wiley, New York, 1993. 12. K. Hiromi, Kinetics of Fast Enzyme Reactions: Theory and Practice, Kodansha, Tokyo 1979. 13. D. Fell, Understanding the Control of Metabolism, Portland Press, London, 1997. 14. R. Heinrich and S. Schuster, The Regulation of Cellular Systems, Chapman & Hall, New York, 1996.

The interpretation of this relationship would be straightforward if all ﬂux control coefﬁcients were always positive, but in reality negative ﬂux control coefﬁcients also occur; for example, if a pathway contains a branchpoint, the enzymes in one branch normally have negative ﬂux control coefﬁcients for ﬂux through the other. Nonetheless, in practice, negative ﬂux control coefﬁcients are nearly always small in magnitude, so although the sum of positive ﬂux control coefﬁcients may be greater than 1, it is rarely much greater than 1, so that the idea of sharing ﬂux control among all the enzymes of a system is reasonably accurate. It follows that we should not expect any enzyme to have complete control over ﬂux, and the closer to complete control any enzyme approaches, the less control all of the others have, taken together. Moreover, a ﬂux control coefﬁcient is not a constant, but tends to decrease when the activity of an enzyme is increased. In other words, even if one can identify one enzyme that has a large proportion of the total ﬂux control, increasing its activity will tend to decrease that proportion, so that the amount the ﬂux can be increased by increasing the activity of one enzyme will always be small. Once this is understood, the failure to increase the ﬂux to tryptophan (in the example mentioned) or to achieve signiﬁcant ﬂux increases (in other similar examples), ceases to be a mysterious result, but is simply what was to be expected.

KINETICS, MICROBIAL GROWTH

1513

KINETICS, MICROBIAL GROWTH

NICOLAI S. PANIKOV

Institute of Microbiology, Russian Academy of Sciences Moscow, Russian Federation

Colonies Bibliography INTRODUCTION Kinetics (Greek jimesijor, forcing to move) is a branch of natural science that deals with the rates and mechanisms of any processes—physical, chemical, or biological. Kinetic studies in microbiology cover all dynamic manifestations of microbial life: growth itself, survival and death, product formation, adaptations, mutations, cell cycles, environmental effects, and biological interactions. Kinetics provides a theoretical framework for optimal design in biotechnologies based on fermentation and enzyme catalysis, as well as on employment of outdoor activity of natural microbial populations (wastewater treatment, soil bioremediation, etc.) Contrary to simple rates measurements, kinetic studies require the perception of the underlying basic mechanisms of studied processes. We will deﬁne mechanistic studies as those that interpret some complex process as an interplay of several simpler reactions, for example, cell growth can be explained through activity of enzymes and microbial community dynamics can be interpreted through behavior of individual cells and populations. Ideally, mechanistic studies infer the coupling of experimental measurements with analysis of simulating mathematical models. The models formalize postulated mechanisms, so that the comparison of observations and the model’s predictions allows one to discard an incorrect hypotheses. The quantitative studies in microbiology often involve the assessment of growth stoichiometry. Stoichiometry [Greek rsoijgeiom, element] is the quantitative relationship between reactants and products in a chemical reaction. In microbiology, stoichiometry stands for a quantitative relationship between substrates and products of microbial processes, including biomass formation (the consequence of complying with mass and energy conservation laws). In practical terms, kinetic and stoichiometry are tightly linked to each other, but stoichiometry mainly addresses problems of a static nature (how much? in what proportion?), whereas kinetics considers the dynamics questions (at what rate? by which mechanism?). GROWTH STOICHIOMETRY Macrostoichiometry of Microbial Growth By analogy to simple chemical reactions, we can represent growth as a conversion of a number of substrates (medium components) into cell mass and products. Growth of aerobic heterotrophic microorganisms can be approximated by the following stoichiometric equation (substrates biomass products) (1,2): CHmO1

2 a1NH3 a2HPO4 YCHpOnNqPoKv . . .

KEY WORDS Cell size distribution Colonies Energy and conserved substrates Growth models Macrostoichiometry Maintenance Microstoichiometry Physiological state Steady-state and transient dynamics Yield OUTLINE Introduction Growth Stoichiometry Macrostoichiometry of Microbial Growth Growth Yield: Catabolic and Conserved Substrates Yield Variation as Dependent on the Chemical Nature of Organic Substrates Variations in Yield from Energy Source, Maintenance Requirements Experimental Determination of m Maintenance Requirements and Wasteful Catabolism Variation in Biomass Yield from Conserved Substrates Microscopic Approach in Studies of Growth Stoichiometry Basic Principles of Growth Kinetics Kinetics of Chemical and Enzyme Reactions Simple Models of Microbial and Cell Growth Structured Models Cell Cycle Population Dynamics (Mutations, Autoselection, Plasmid Transfer) Microbial Growth as Dependent on Cultivation Systems 1a —Homogeneous Continuous Culture (Continuous-Flow Fermenters with Complete Mixing) 1ab—Continuous Cultivation without Cell Washout 2a —Continuous Cultivation with a Discontinuous Supply of Limiting Substrate 2ab—Simple Batch Culture 1b —Plug-Flow (Tubular) Culture 1bb—Continuous-Flow Reactors with Microbes Attached

a3K a4CO2

... bO2 a5H2O (1)

Here, microbial biomass is empirically expressed by the gross formula CHpOnNqPoKv . . . , for example, if some av-

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KINETICS, MICROBIAL GROWTH D A B C

Table 1. Selected Macrostoichiometric Equations Describing Growth of Microorganisms with Different Types of Energy Generation

Growth Yield: Catabolic and Conserved Substrates The growth yield is one of the main stoichiometric parameters. It is deﬁned as follows Y dx ds Dx Ds (2)

Heterotrophic growth and by-product formation

Microbial process

Phototrophic growth (algae or plant cells)

where Dx is the increase in microbial biomass consequent on utilization of the amount Ds of substrate, and dx and ds are respective inﬁnitely small increments. Rigorous deﬁnition of Y as derivative dx/ds stems from the fact that Y can vary in time, the negative sign being introduced because x and s vary in opposite senses. Sometimes, it is used as the reciprocal of Y: 1/Y, which is called the economic coefﬁcient. It expresses explicitly the nutrient requirements for growth: how many mass units of a particular substrate should be consumed to produce one unit mass of cell material. The growth efﬁciency depends generally on the partitioning of consumed element between new cell biomass and extracellular products. The mass balance (total element consumed amount incorporated into cell plus amount incorporated into extracellular products) is as follows: dEs dEx dEp (3)

Methanogenesisa

There are two groups of substrates for microbial growth: (1) catabolic substrates, which are sources of energy; and (2) anabolic or conserved substrates, which are sources of

Nitriﬁcationb

a

b

Particular example of H2-utilizing methanogenic bacteria. The total growth balance for two phases of nitriﬁcation: oxidation of ammonium to nitrite and subsequent oxidation of nitrite to nitrate. Y is growth yield of bacterial mass per mass unit of oxidized N.

YPCHrOsNt

a3CHrOsNt

YCHpOnNq

YCHpOnNq

YCHpOnNq

a2NH3

a2NH3

Substrates

a1O2

a1H2O

a2NH3

CHmOl

a1H2

YCO2

CO2

CO2

a1O2

a2NH4

YCHpOnNq

Biomass

a3NO3

a3CH4

erage microbial cell contains per dry cell weight (%) C 46, H 7.5, O 31, N 11; and P, 1.3, then the biomass formula is CH1.9O0.5N0.2P0.01. The stoichiometric quotients a1–a5 . . . , b and Y (biomass yield) specify quantities of substrate and products of microbial growth. If we know biomass yield and gross formulas of all substrates and products, then quotients a1–a6 . . . are easily calculated from conservation conditions. There are at least two such conditions. First, the mass of each element (C, H, O, N, P, K, . . . ) on the left side of equation 1 should be equal to that on the right side (mass balance). Second, if ionized substances are involved, we should take into account the balance of charges to satisfy the condition of electroneutrality. Table 1 demonstrates some examples of stoichiometric growth reactions relevant to biotechnology. Described formalism is useful as a ﬁrst step in biotechnological studies aimed at planning and optimizing microbial growth. It estimates how much nutrient should be supplied to the fermenter to obtain the required amount of biomass or target product. However, it should be absolutely clear that stoichiometric equations like equation 1 are no more than an approximation to reality. The most severe deviation stems from the fact that unlike chemical reagents, microbial cells are characterized by changeable composition, and stoichiometric coefﬁcients are not true constants. One task of contemporary microbial stoichiometry is to ﬁnd out the functional relationships between stoichiometric parameters and internal (physiological) and external (environmental) factors.

a4H2O

Products

a3CO2

a4O2

a4H2O

a4H

a5H2O

a1 a2 a4 a1 a2 a4 a1 a2 a4 a1 a2 a4

0.5(Yn Ys l a4) a3 Yq Y t, a3 1 Y Y 0.5[m Y(3q p) Y (3t r)] 0.5[Y( p 3n) (r 3t)(1 Y)] Y(n t) t, a3 1 Y 0.5(2 a1 Yn a3s) 4 0.5Y(4 2n 3q p) Yq, a3 1 Y 2 Yn 2Y 1 Y[1 0.25 ( p 2n 3q)] Y 1 Yq, a3 Y 1 2Y 1 Yq, a5 Y 1 0.5Y( p 3q)

Stoichiometric parameters

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biogenic elements forming cellular material. Catabolic substrates include H2 for lithotrophic hydrogen bacteria, NH4 and NO2 for nitrifying bacteria, S0 for sulfuroxidizing bacteria, oxidizable or fermentable organic substances for heterotrophic bacteria and fungi, and so on. Their consumption is accompanied by oxidation and dissipation of chemical substances into extracellular waste products that are no longer reusable as an energy source* (H2O, NO3 , SO2 , CO2, etc.) The anabolic substrates after 4 uptake are incorporated into de novo synthesized cell components, being conserved in biomass (that is why they are called conserved). Contrary to catabolic substrates, they can be reused (e.g., after cell lysis to be taken up by survived cells). The conserved substrates include nearly all the noncarbon sources of biogenic elements (N, P, K, Mg, Fe, and trace elements), CO2 for autotrophs, and the indispensable amino acids and growth factors. Most catabolic substrates are used also as a source of biogenic elements. We can assess both these components separately in terms of respective yields, YE (biomass yield per mass unit of oxidized substrate) and YA (biomass yield per mass unit of assimilated substrate), from the experimentally measured yield Y. For C substrate, equation 4 can be speciﬁed as follows (total carbon consumed equals C incorporated into cell plus C oxidized to CO2 to provide energy plus C incorporated into by-products): dCs dCx dCCO2 dCP (4)

1/YO2

A/Y

B

(7)

where A and B are constants estimated from stoichiometry of their respective combustion reactions (see equation 10 later), for example, the value of A is 33.33 mmol O2 g 1 glucose and B is about 42 mmol O2 g 1 CDW. The relationship between biomass yields on O2 and CO2 is derived from comparison of equations 6 and 7: YCO2 YO2 A rs BY rxY (8)

Now we will go back to the general substrate balance (equation 3) and derive an expression for conserved substrate. Again, we neglect term dEP (because extracellular products are assumed to be reusable) and divide the balance by dx, which is the amount of biomass produced: 1 Y dEx dx rx (9)

Let us neglect the last term dCP (by assuming that extracellular by-products can be reused and functionally are equivalent to C substrate) and divide the substrate balance by dCx, which is the amount of biomass C produced, then: 1 Y where Y mass C g g biomass C g CO2-C 1 Y 1

1

1 YE substrate C and YE

where rx is the intracellular content of element incorporated into biomass from consumed substrate. Sometimes rx is called the cell quota. The values 1/Y and rx are not identical although they have the same dimension (e.g., milligram N per gram biomass) and very close numeric value. The reciprocal 1/Y is characterizing the process (the expenditure of conserved substrate to synthesize biomass unit), whereas rx is an index of cell composition (the content of intracellular N per biomass unit). Formally, 1/Y is equal to the rx value of an inﬁnitely small increment of cell biomass, and rx is the averaged value for entire cell. Notice that although rx is a slow and 1/Y is a rapid variable, their numerical values are exactly the same for balanced steadystate growth and can differ considerably during transients. Yield Variation as Dependent on the Chemical Nature of Organic Substrates

(5)

g bio-

1

rx rs

12 YErs

(6)

where Y g CDW g 1 substrate, YE g CDW mmol 1 CO2, and rx and rs are fractions of carbon in biomass and substrate, respectively. For example, if total measurable yield Y is 0.6 g biomass C g 1 glucose C, it means that from each g of consumed C, 0.6 g is incorporated into biomass (assimilated), and 0.4 g is dissimilated (oxidized to CO2), then YE 0.6/0.4 1.5. To calculate oxygen demand for aerobic growth (or biomass yield on O2) we have a balance (oxygen required to produce 1 g CDW equals oxygen required to burn substrate consumed to produce 1 g CDW minus oxygen required to burn 1 g CDW):

*Fermentation products such as acetate, ethanol, butyrate, and H2 seem to be an exception because they do contain reusable oxidation potential, but it is not available under anaerobic conditions supervising fermentation.

In this section, we will discuss why biomass yield varies when microorganisms are grown on different C substrates. This problem was best solved within the framework of the theory of mass and energy balance (TMEB) (3). Evidently, the fraction of C in dry biomass is almost constant. By contrast, the content of carbon in utilized substrates, rs, and energetic quality of substrate vary over a broad range (e.g., compare methane versus oxalic acid). To characterize substrate and biomass by a single common measure, TMEB uses an index of degree of carbon reduction, c related to the internal energy of organic compounds. The heat liberated by biological or chemical oxidation is proportional to oxygen uptake or equally to the number of electrons gained by oxygen from oxidized substrates, according to Payne’s term available electrons (ae) (4). The heat production from an oxidation reaction averages at 27 kcal per ae equivalent. A carbon reduction degree, c is deﬁned as the number of ae per one carbon atom. Its numeric value can be determined from the stoichiometry of the oxidation reaction: CHpOnNq bO2 CO2 0.5( p 3q)H2O qNH3 (10)

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KINETICS, MICROBIAL GROWTH

c

4b

4

p

2n

3q

(11)

The ae balance for equation 1 can be written as cs b( 4) Ycx YPcp (12)

where cs, cx, and cp are the carbon reduction degree of, respectively, substrate, biomass, and extracellular product. Dividing both sides of equation 1 by cs we obtain the relationship delineating the ae distribution between oxygen (ae used for respiration), biomass, and the intracellular product: 4b cs Ycx cs YPcp cs 1 (13)

Sometimes catabolic machinery is entirely wasteful (respiration without cell growth) and always at least some minor part of energy consumption is diverted from growth. To account for this phenomenon, it was postulated that microbes and cells require energy not only for growth but also for other maintenance purposes. Certain speciﬁc maintenance functions recognized now are turnover of cell material, osmotic work to maintain concentration gradients between the cell and its exterior, and cell motility. According to conventional deﬁnition of maintenance (5), the balance of energy source is total energy source consumed equals consumption for cell growth plus consumption for maintenance: dSE dSG dSM (16)

The second term in this equation is the fraction of ae transferred to biomass from utilized substrate, termed the energetic growth yield. g YCcx /cs (14)

Let us divide it by dx, the amount of biomass produced, then 1 Y dSG dx dSM dx 1 Ymax m l (17)

The third term designates that fraction of total substrate internal energy that is transferred to the product. It is called the energetic product yield f YPcp /cs (15)

Energetic yield g is related to other stoichiometric parameters as follows: g g Yrxcx /(rscs) YCcx /cs

dx/dSG is true growth yield, that is, yield Here, Ymax under imaginary conditions of maintenance being zero. The maintenance coefﬁcient, m, is introduced as the speciﬁc (i.e., expressed per unit of biomass) rate of energy consumption for maintenance functions: m (1/x)(dSM /dt). The ratio m/l on the right side of equation 17 was derived as follows: m/l [(1/x)(dSM /dt)]/[(1/x)(dx/dt)] dSM /dx. If we divide equation 16 by xdt (note that the second term is dSG /(xdt) [dx/(xdt)]/[dx/dSG] l/Ymax), then we have: q l/Ymax m (18)

where Y is g CDW/g substrate and YC is g CDW-C/g substrate C. The advantage of using g is that it varies within a much smaller range than other yield expressions. At one and the same efﬁciency of energy utilization (g), the conventional biomass C yield YC is proportional to substrate reduction degree cs and, for example, it is four times higher on glucose (cs 4) than on oxalate (cs 1), 0.48 and 0.12 g C g 1 C, respectively (assuming g 0.5 and cx 4.2). The energetic growth yield g is more or less constant (0.5 to 0.7) for substrates with cs 4.2 (4.2 corresponds to average reduction degree of microbial biomass), and it declines at higher cs. The attractiveness of macrostoichiometry and TMEB is that all growth coefﬁcients are interrelated and could be measured from any available components of the culture mass balance. For example, if you cannot record microbial growth by conventional routine as dry weight biomass (because of presence of solids in broth liquid), you may still calculate it from N or O2 uptake, CO2 evolution, pH titration rate, and so on. Variations in Yield from Energy Source, Maintenance Requirements To multiply and grow cells requires energy, but the opposite is not true: cells do not require growth to spend energy.

where q is speciﬁc rate of energy source consumption, q (1/x)(dSE /dt). It should be noticed that Ymax is a parameter, but not the yield of a real culture that always has some nonzero maintenance requirements. It is a very common mistake in the application of the maintenance concept to a particular organism: to take the real measured Y value and pick up from literature some average m coefﬁcient. The correct way would be either to borrow concurrently two parameters Ymax and m or to treat actually observed Y as a variable that is altered along with speciﬁc growth rate l according to equation 17: Y lYmax l mYmax (19)

There is another way to formulate maintenance requirements by stating that the net growth of cells l is the difference between true growth (ltrue) and endogenous decay of cellular components (speciﬁc rate, a): l ltrue ltrue l a a

Then, for the rate of energy source uptake, we have

KINETICS, MICROBIAL GROWTH

1517

lx Y or 1 Y

ltruex Ymax

(l

a)x Ymax

1 Ymax

a lYmax

(20) mYmax.

Comparing equations 17 and 20, we see that a Experimental Determination of m

To practically determine the maintenance coefﬁcient, the microorganisms are grown in chemostat culture limited by energy sources at several dilution rates D (numerically D is equal to speciﬁc growth l if steady state is achieved). At each D, we have to measure steady-state biomass x and at ˜ least one of the following quantities (1) residual substrate, s to calculate Y ˜ x /(s0 ˜ s); and (2) the rate of respective ˜ energy-yielding process, such as respiration rate, vresp, from O2 uptake or CO2 production rates to calculate speciﬁc metabolic activity, q vresp /x. These data are ﬁtted to ˜ equations 17 or 18, m and Ymax being found as nonlinear regression parameters. An example is presented in Figure 1. Most available experimental data do obey this relationship. However, considerable deviation occurs at very low growth rates usually attained in chemostat with biomass retention or in dialysis culture. The experimental Y values for slowly growing cells are higher than predicted by equations 17 and 18 (see inset on Figure 1). The explanation is very simple: the maintenance coefﬁcient varies in response to nutritional status and could not be taken as an absolute constant; under substrate deﬁciency, the cells adjust their maintenance requirements to lower values by reducing turnover rate, osmotic work, and motility (2).

The described experimental technique is indirect because it is based on measurements of l-dependent Y variation rather than m itself, and there are some assumptions needed to be conﬁrmed (e.g., that m is constant and that maintenance requirements are the only reason of Y variation). However, some components of maintenance requirements are available for direct estimation. In particular, we can assess the total turnover rate of cellular material a which is one of the main components of maintenance requirements (equation 20). The principal cell constituents that are turned over are proteins, nucleic acids, and cell wall polymers. The turnover rate is very close to endogenous respiration, which is the oxidation of those compounds produced from the turnover (breakdown) of cellular macrocomponents. Accurate measurements of endogenous respiration need to be made under normal growing conditions. It is known that the simple removal of cells from nutrient broth by ﬁltration with subsequent washing and incubation in buffer renders strong stress and may alter the normal turnover rate (6). To avoid artifacts, we can use a label-substitution technique (Fig. 2). The chemostat culture is fed alternately from two bottles containing unlabeled and labeled 14C(U) substrate respectively. The 14CO2 evolution rate is recorded after switching to unlabeled substrate, when the main source of 14CO2 are cell components. The calculated a value was found to be rather high, accounting for the major part of total maintenance determined by the indirect method (2). The endogenous respiration declined at the low growth rate (Fig. 1), indicating that under starving conditions, self-adjustment of the maintenance requirement occurs mainly as a reduction in the turnover rate of macromolecules. Maintenance Requirements and Wasteful Catabolism The described concept of maintenance requirements was the subject of severe criticism (8). One of the strongest arguments against it was an apparent increase in Ymax observed in chemostat cultures limited by P, N, and other conserved substrates under conditions of energy excess. To preserve the constancy of the true yield, Pirt (9) had to modify equation 18 in the following way: q l/Ymax m m (1 l/lm) (21)

0.4 Y, g biomass g –1 glucose

4

0.3

0.4 0.2 0 0

3

0.2

2

0.025 0.05

Endogenous respiration, mmol O2 h–1 g–1

0.1

1

0 0

0.1 0.2 0.3 Specific growth rate, µ (h–1)

0 0.4

Figure 1. Variation of growth yield (circles) and endogenous respiration (squares) as dependent on speciﬁc growth rate in chemostat (open symbols) and continuous dialysis culture (closed symbols). Solid curves were calculated from the synthetic chemostat model (2). The dotted curve was derived from the PirtHerbert model (equations 17 to 20), which predicts quite well intensive growth but fails in the region of extremely low growth rates (see inset).

where m (1 l/lm) is the second l-dependent component of maintenance energy that operates under excess of energy substrate. However, it is better to differentiate maintenance requirements sensu stricto as those more or less a minor component of the cell energy budget that is observed under energy-limitation and wasteful use of catabolic substrate under energy excess. In physiological terms, these two groups of nonproductive catabolic reactions are completely different. The ﬁrst reactions are mainly responsible for compensation of turned-over macromolecules and therefore belong to the category of regular primary catabolism. The catabolic reactions of the second group include excretion into environment of partly oxidized substances (overﬂow metabolism), uncoupling of respiration from ATP generation by metabolic inhibitors, functioning of futile cycles,

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KINETICS, MICROBIAL GROWTH

2

12

1.4

YN, 109 cell mg –1 N

1.6

YN

1.1

CO2

1.2

σN

0.8

0.8 0 15

0.05 0.1 0.15 0.2 Specific growth rate, h–1

0.5 0.25

Figure 3. Relationship between stoichiometric parameters Y and s and speciﬁc growth rates of Chlorella vulgaris grown in chemostat culture limited by nitrogen source (10). The curves are calculated using equations 23 and 24.

CO2, mg CL–1

10 concentration of some conserved limiting substrates that preserve their chemical identity after uptake (K , Mg2 , vitamins). Other conserved substrates (sources of P, N, S, etc.) are incorporated into macromolecular cell constituents (mainly nucleic acids and proteins) whose intracellular content also should be kept high at high growth rate. Both types of changes in cellular composition are manifested as r increase, and both of them require additional maintenance energy (to maintain concentration gradient or compensate turnover of macromolecules). The observed l-dependent variation in r is therefore a compromise between biosynthetic requirements and energy conservation that is attained because of optimal metabolic control of cell performance. However, it would be erroneous to consider l as truly independent variable setting up chemical composition of cells. In fact, both l and r are functions of one common independent variable, the limiting substrate concentration in the medium, s. For steady-state chemostat culture we have: l r q r r0 1 Qs ˜ r Ks s ˜ (rm Kr r0)s ˜ s ˜ (22)

5

14

Total CO2 C 2.5

0

0

0.5

1

1.5 Time (h)

2

Figure 2. Label substitution technique for determination of turnover rate of cell macromolecular constituents. Top, experimental setup including two medium reservoirs containing 14C- and 12Cglucose pumped into a cultivation vessel through a two-way valve. Bottom, example of 14CO2 evolution dynamics before and after (arrow) switching of medium feed from 14C to 12C-glucose, glucoselimited culture of Pseudomonas ﬂuorescens 1472, D 0.08 h 1 (7).

or substrate oxidation through alternative oxidases without ATP generation. These and related phenomena take place in chemostat culture limited by conserved substrates (opposite to limitation by energy source) as well as during lag phase of batch culture started from starving inoculum. We will discuss the mathematical formulation of these phenomena in the section devoted to growth kinetics. Variation in Biomass Yield from Conserved Substrates Yield on conserved substrates varies mainly as a result of alterations in biomass chemical composition expressed by parameter rs, the intracellular content of deﬁcient element or cell quota (see equation 9). For most of known cases, the content rs increases parallel to growth acceleration (Fig. 3). As yield and cell quota are inversely related to each other (equation 9), then Y values decrease with growth rate. The physiological mechanisms of this variation are as follows. The intensive growth requires higher internal

where l is speciﬁc growth rate, q is speciﬁc substrate uptake rates; r0 and rm are, respectively, lower and upper limits of r variation; low limit r r r0 is attained when s r ˜ 0 and upper limits r r rm-when s r . By excluding s from ˜ ˜ both these equations we arrive at following relationship between r and l: l k lm Ks Kr 1 (Ks Kr) we have rm(r r0) r0) r

r[k(rm

r0] (23)

Under realistic assumption k

σN, mg N 10 –9 cell

C

14C

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r Y

1 Ym

(1

r0 r0 /rm)l/lm Y0) l lm (24) Frequency

0.6 0.5 0.4 0.3 0.2 0.1 0 –0.1 5 10 15 YATP, g CDW mol–1 20 Unreliable data Reliable data

(Ym

where Ym and Y0 are, respectively, upper and lower limits of yield variation (Ym 1/r0, Y0 1/rm). As we can see, the linear relationship between Y and l is normally observed in chemostat culture (Fig. 3). Microscopic Approach in Studies of Growth Stoichiometry Equations 1 to 24 exemplify the macroscopic approach in studying of microbial growth stoichiometry. Its typical features are the use of gross formulas for biomass and metabolic products, evaluation of total mass balance for chemical elements (C, N, P), and formal description of microbial growth as a single-step conversion of substrate(s) into biomass. By contrast, the microscopic approach focuses on the much more complex real metabolic reactions and attempts to account for a limited but still quite large number of individual metabolic intermediates. The ﬁnal aim of this approach is to organize the biochemical information into a consistent picture of microbial metabolism at the level of entire cell. The microscopic approach has become possible by virtue of advancements in biochemistry, which has succeeded in establishing a sufﬁciently full picture of metabolic processes in certain microorganisms. The pioneering work in this area was done by Bauchop and Elsden (11), who were able to sum up the balance of ATP for fermenting microorganisms. As a result, a relation was established between the biomass yield (a macroscopic quantity) and the number of generated ATP moles (a stoichiometric characteristic of real catabolic reactions): YATP MYE /n (25)

Figure 4. Frequency distribution of experimentally measured values of YATP at different degrees of creditability. The reliable data refer to studies of anaerobic growth with direct measurements of fermentation products (plotted from data base in Ref. 12. Note that these data are normally distributed with mean value 10.55, whereas all data display considerable skewness.

where n mol of ATP made available to the organism by the metabolism of one mole of energy source, and M molecular weight (g) of energy source. The following example illustrates the YATP calculation: if biomass yield of some organisms aerobically grown on glucose is 0.52 g CDW/g, then YE is 1.49 g CDW per g of oxidized glucose (calculated from equation 6) or YE 1.49 180 268 g CDW/mol (180 is glucose molecular weight); assuming that P/O 2 (that is, 2 mol of ATP produced per atom oxygen taken up) and that 2 ATP mol are produced via glycolysis (substrate phosphorylation) we arrived at n 2 12 2 26 and YATP 268/26 10.3 g CDW/mol ATP. Careful determination of n and YATP is possible only for anaerobic growth of fermenting microorganisms generating ATP via substrate phosphorylation. The mean value tends to be around 10.5 g CDW/mol ATP (Fig. 4). For aerobic growth, we need to make assumptions on the P/O ratio. As soon as the respiratory chain of bacteria differ widely for various organisms and growth conditions, this assumption can never be reliable. To avoid this obstacle, an interesting approach was proposed (1): microbial culture is grown in a chemostat limited by two carboncontaining energy sources, their ratio is varied while the

total carbon feed rate is kept constant; yield measurements should allow one to determine both parameters (P/O and YATP) independently by multiple linear regression. Today, microstoichiometry is quickly progressing as socalled metabolic balancing. Cell growth is viewed as a set of transport and intracellular metabolic reactions known for some particular organisms. As a rule, the produced metabolic networks are composed of a combination of true stoichiometric equations for individual metabolites and empirical gross equations (Table 2). The amount of such equations vary in different models from 20 to 30 to more than 100. For example, van Gulik and Heijnen (13) describe yeast growth by a set of more than 90 reactions including glycolysis and the citric acid cycle (14); PEP phosphotransferase; pentose phosphate pathway (6); glyoxylate shunt (2); oxidative phosphorylation (4); CO2 interaction with THF (3); transport of inorganic P, NH4 , SO2 , ace4 tate, lactate, pyruvate, glucose, gluconate, succinate, and citrate (totally 10 transport reactions); amino acid synthesis (15) and polymerization (2); nucleotide synthesis (9); RNA synthesis; ATP consumption for maintenance; fatty acids synthesis (2); formation of glycogen and polysaccharides; and ﬁnally, the biomass formation from proteins, polysaccharides, RNA, fatty acids, and glycerol. For each compound, i, involved in a metabolic system, a mass balance can be deﬁned: dCi dt rAi Ui (26)

where Ci is concentration of ith compound, rAi, and Ui denote the net rates of, respectively, i chemical conversion and transport over the boundaries of bioreactor (ﬂuxes of

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KINETICS, MICROBIAL GROWTH

Table 2. Metabolic Networks Reaction Glycolysis reaction Equation Glucose ATP r glucose-6-P ADP H Glucose-6-P r fructose-6-P 0.5 Fructose-6-P 0.5 ATP r glyceraldehyde-3-P 0.5 ADP 0.5 H NADH 0.5O2 d1ADP d2Pi (1 d1) H r (1 d1)H2O NAD d1ATP a2polysaccharides a3RNA a4lipids r biomass a1Proteins Stoichiometric Equation x x x x x Empirical Gross Equation

Oxidative phosphorylation Biomass formation

CO2, O2, nutrients, cells, and products). Most metabolic balancing equations are applied to steady-state growth (which means that no intracellular accumulation of metabolites occurs). In such cases, the differential equations like equation 26 are reduced to linear algebraic ones. Besides, an extensive use of matrix calculus is customarily made to obtain a concise notation. The problem of experimental support of such model is especially important (13). The degree of freedom, df, of the resulting system of linear equations is equal to the total number of unknown rates (both intracellular and exchange reactions) minus the total number of linear equations. To resolve the system, df rates have to be measured, and then the system is fully determined. If the number of measured rates is greater than df, then the system is overdetermined, and the redundancy of the data can be used for statistical analysis and error minimization. However, it is much more typical to have an underdetermined system when the sum of measured rates is less than df. In this case, the number of possible solutions is inﬁnite unless additional constraints are applied (e.g., maximization of biomass yield, minimization of energy expenditure) to ﬁnd the one and only one solution by the linear optimization technique. In most studies, ﬂux estimates are obtained using measurements of substrate consumption and product formation. This approach has proved to be efﬁcient in some particular biotechnological cases, such as when only speciﬁc pathways need to be considered (16) or if the contribution of ﬂux for cellular growth is weak, as with mammalian or hybridoma cells (17). The more complex microbial systems are turned out to be seriously underdetermined. In such cases, the application of metabolic balancing requires the use of one or another maneuvers: (1) to lump together several sets of reactions (18); (2) to utilize data from in vitro enzyme assays; (3) to make assumptions on numeric values of some stoichiometric growth parameters, such as YATP, P/O, and H /e ratios, which are the subject of controversial debates (13). However, the best solution would be to get direct experimental data on in vivo ﬂux and resolve the system. Isotopic tracers are one of the best candidates for such a purpose. We will illustrate this point by describing a recently published work (19). This novel approach is based on the analytical power of 1H-detected 13C nuclear magnetic resonance. Corynebacterium glutamicum was grown in chemostat culture continuously fed with [1-13C]-glucose; when steady state was established, the cells were harvested and hydrolyzed and the amino acids were separated by ionexchange chromatography and analyzed by NMR spectros-

copy. NMR provides data on 13C enrichment at each speciﬁed carbon position of amino acid. Because metabolic pathways for amino acid synthesis are exactly deﬁned, then the entire central metabolism can be assessed for in vivo ﬂuxes, including determination of the forward and back rates of bidirectional reactions. In C. glutamicum, the ﬂux through the pentose phosphate pathway turned out to be 66.4% (relative to glucose input ﬂux 1.49 mmol g 1 CDW h 1); the entry into tricarboxylic acid cycle, 62.2%, and the contribution of the succinylase pathway to lysine synthesis, 13.7%. The total net ﬂux of the anaplerotic reactions (carboxylation of PEP/pyruvate into oxaloacetate/ malate) was quantitated as 38%, the true forward ﬂux of C3 r C4 being 68.6% (1.8 times of 38%) and a back ﬂux of C4 r C3 being 30.6% (0.8 times of 38%) (19). The metabolic balancing proved to be very promising and useful to identify metabolic constraints for intensive synthesis (overproduction) of products such as amino acids. On the other hand, this approach still is restricted to steady-state and balanced growth and is not able to cope with complex dynamic behavior of microorganisms (transient growth, changes in biomass composition). BASIC PRINCIPLES OF GROWTH KINETICS Kinetics of Chemical and Enzyme Reactions We need to introduce some basic principles of kinetic analysis of chemical and enzymatic reactions. Quantitative description and understanding of microbial growth dynamics and kinetics are impossible without some elementary knowledge in underlying scientiﬁc disciplines. Enzymatic and chemical reactions play an essential role in biotechnology, which is one of the most important ﬁelds in industrial development. Order and Molecularity of Chemical Reactions. The molecularity of any chemical reaction is deﬁned by the number of molecules that are altered in the reaction (Table 3). The order is a description of the number of concentration terms multipled together in the rate equation (Table 4). Hence, in a ﬁrst-order reaction, the rate is proportional to one concentration of reactant; in a second-order reaction, it is proportional to two concentrations or to the square of one concentration. For a simple single-step reaction, the order is generally the same as the molecularity. For a complex reactions involving a sequence of unimolecular and bimolecular steps, the molecularity is not the same as its order. Reactions of molecularity greater than 2 are com-

KINETICS, MICROBIAL GROWTH Table 3. Molecularity of Chemical Reaction Molecularity of reaction Unimolecular or monomolecular Bimolecular or Trimolecular or termolecular or or Reactants S S S1 S S1 S1 S S2 S S S1 S2 S2 S3 r r r r r r Product P P P P P P

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of inspected reactions is different, then we have to equalize the respective rate of expressions; for example, the secondorder rate constant k (time) 1 (concentration) 1 should be multiplied by instant concentration of reactant s to be compared with the ﬁrst-order rate constant having dimension (time) 1. The dimensions of the zero-, ﬁrst-, and secondorder rate constants are shown in Table 4. Reaction Dynamics. If rate constant and so-called initial conditions (concentrations of reactants at zero-time [s01, s02, . . . ]) are known, then it is possible to calculate the time course of reactions either in terms of dynamics of residual reactant concentration, s(t) or product accumulation, p(t). For this purpose, we have to integrate a differential equation under speciﬁed initial conditions (see results of integration in Table 4). The dynamics of s(t) are linear in the case of a zero-order reaction and hyperbolic in the case of the ﬁrst and second order. The difference between the last two dynamic curves can be made visible with a semilogarithmic plot of log(concentration) versus time; it should became linear for the ﬁrst-order reaction and remain to be curvilinear for the reaction of higher order. Reaction Half-Time. The reaction half-time (t0.5) is a very popular kinetic parameter, especially among biologists. It is easily calculated from integral equations by putting s s0 /2 when t t0.5. A unique feature of the ﬁrstorder reaction is the constancy of t0.5 independently of the initial reactant concentration s0. However, the half-time of other reactions does depend on s0; it increases for zeroorder reactions and decreases for the second-order reactions with increase in s0. Thus, it is not recommended to use half-time as a parameter or estimator for reactions other than ﬁrst-order reactions. Determination of the Order of Reaction and Numeric Values of Kinetic Constants. If the reaction has an order n and rate constant k, then the reaction rate v and reactant concentration s are related by the equation

mon, but reactions of order greater than 2 are very rare. For instance, a trimolecular reaction, such as A B C r P, as a rule proceeds through two elementary steps, A B r X and X C r P, each of which are of the second or ﬁrst order. Very often, bimolecular reactions between S1 and S2 occur under the condition that their respective concentrations s2 s1 (e.g., if the second reactant S2 is solvent), then we have a pseudo ﬁrst-order reaction. Some reactions are observed to be of the zero order, that is, the rate appears to be constant, independent of the concentration of reactant. This is a characteristic feature of catalyzed reactions and occurs if reactant is present in such large excess that the full potential of catalyst is realized. Dimensions of Rate Constants. Knowledge of dimensions is very useful to check the correctness of derived kinetic equations: the left- and right-hand sides of an equation must always have the same dimensions. This general rule is applicable to all mathematical models (not only in chemical kinetics). It is incorrect to add or subtract terms of different dimensions, although you may multiply or divide them. For example, if expression “ . . . (1 s)” occurs in an equation, where s has dimension (concentration), then either equation is incorrect, or the 1 is a concentration that happens to have a numerical value of 1 unit. The operation rising to power is allowed for only simple dimensionless numbers, for example, expression e2.5t, where t is time, is correct only if 2.5 has dimension (time) 1. The comparison of velocities of two reactions does make any sense only for kinetic terms of the same dimension. If the kinetic order

Table 4. Kinetic Order of Chemical Reactions Reaction order Zero First Second Second Pseudo-ﬁrst s01 s02 Differential equation ds dt ds dt ds dt dp dt dp dt k ks ks2 ks1s2 ks1s2 Dimension of rate constant (conc.)(time) (time) (conc.) (conc.) (conc.)

1 1

Dynamics of residual reactant s(t) s(t) s0 kt p(t) p(t) p(t) p(t) p

Dynamics of product accumulation p0 s0[1 kt exp( kt)]

Reaction half-timea t0.5 t0.5 t0.5 s0 2k ln2 k 1 b ks0 1/ks01 1/ks02

s0 exp( kt) s0 1 ks0t s01 s02 p(t) p(t)

1

(time) (time) (time)

1

s(t) s1(t) s2(t) s2(t) s1(t)

2 s0kt 2(1 s0kt)

1

1

s01s02(1 exp[(s02 s01 s02 exp[(s02 s02[1

0.

s01)kt]) s01)kt]

t0.5 t0.5

1

1

s02 exp( ks01t) s01

exp( ks01t)]

Note: t time; s reactant concentration; p product concentration; s0 reactant concentration at t a The half-time of the reaction is the time required for half-completion. b The half-time is deﬁned for the reagent that has lower initial concentration and is depleted ﬁrst.

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KINETICS, MICROBIAL GROWTH

v

ds/dt

dp/dt

ksn

(27)

e0 x s

e s s0

x

s0

s

x (29)

The simplest way to ﬁnd both unknown values (n and k) would be to measure reaction rate v at several concentrations of reactant. Then a plot of log(rate) against log(concentration) gives a straight line with a slope equal to n and intercept equal to log(k): log v log k n log s. If there are several reactants, then it is useful to know the order in respect to each one. For this purpose, you need to have several experiments with variation of each reactant concentration while keeping the other concentrations constant. The most frequent goal is the determination of the ﬁrstorder rate constants, ﬁrst because many reactions do obey ﬁrst-order conditions in respect to each reactant and second because it is possible to carry out many reaction under pseudo ﬁrst-order conditions. There are some special methods to determine the values of the ﬁrst-order rate constants from the experimental curves of product formation or substrate depletion dynamics. Some of them were designed to improve the accuracy of k determination when the initial reactant concentration (s0) or ﬁnal value of product concentration ( p ) were not known (methods of Guggenheim, Kezdy-Swinbourne, and others, see details in Ref. 20). All are based on plotting the experimental points in a some sophisticated manner to convert the original curves to a straight lines. Today, these methods are replaced by computer-aided nonlinear regression, which is much more convenient and precise and, contrary to graphic methods, allows for more rigorous estimate of conﬁdence limits of measured quantities. Derivation of Basic Kinetic Equations for Enzymatic Reactions. Contrary to simple chemical reactions, enzymecatalyzed reactions proceed through reversible formation of the dynamic enzyme-substrate complex (ESC). The word reversible is essential because the ESC can be decomposed into free enzyme E and product P or dissociate back to E and substrate S. There are many ways to simulate mathematically the mechanism of the enzymatic reaction. We will consider here equilibrium, steady-state, and general non-steady-state approaches. Equilibrium Approach. This approach was used by Michaelis and Menten (21) to describe the effect of sucrose concentration on invertase activity. They assumed that the ﬁrst step of ESC formation is so rapid that could be represented by an equilibrium constant Ks:

e s Ks x k2 p

The overall reaction rate, v, is a simple ﬁrst-order reaction with rate constant k2: v dp/dt ds/dt k2x (30)

The x value can be found from expression for Ks and massbalance conditions (equation 29): x e0s/(Ks s). Substitution of this expression into equation 30 ﬁnally produces v k2e0s Ks s (31)

Steady-State Approximation. This approach was applied by Briggs and Haldane (22) for the following scheme of enzymatic reaction

e s

E

S i ES r E

k

1

k1

x

k2

p

P

(32)

This scheme implies the reversibility of ESC formation instead of much more restrictive equilibrium postulate. However, still there was steady-state assumption in respect to ESC formation: dx/dt k

1(e0

x)s

k

1x

k

2x

0

(33)

Therefore, x k 1e0s/(k 1s k 1 k 2), and substitution of this expression into equation 30 gives v ds dt k k

1

k

2x

k k

1s

1k

2e0s 1

k Vs s

k

2

2e0s k 2 1

k

s

Km

(34)

E

S } ES r E

P

(28)

where e, s, x, and p denote the concentrations of free enzyme, substrate, FSC, and product, respectively. The equilibrium constant, Ks, is deﬁned as Ks es/x. The instantaneous concentrations s and e are not directly measurable, but they could be expressed in terms of the initial, measured concentrations, e0 and s0, using massbalance relationships:

Equation 34 is what usually called the Michaelis-Menten equation, the fundamental equation of enzymatic kinetics. It contains two parameters, Km, the Michaelis constant, and V, the maximum velocity. V is the rate of reaction that would occur under full substrate saturation of an enzyme’s active sites (s Km). In reality, the V value can be never attained, because extremely high substrate concentrations inhibit enzymes (see later text). It is clear also that V is not a fundamental property of enzyme because it depends on e0, the enzyme concentration. More advantageous as a speciﬁc enzyme characteristic is the catalytic constant or turnover number, kcat, which is V/e0. For equation 32, kcat is identical with k 2, but in general the more noncommittal notation kcat is preferable (e.g., kcat may differ from k 2 if the product formation from ESC is a reversible reaction). Numerically, the Michaelis constant, Km, is the substrate concentration that provides half the maximal reaction rate. Contrary to this simple practical deﬁnition, the mechanistic interpretation of Km is not so lucid. Sometimes

KINETICS, MICROBIAL GROWTH

1523

Km is interpreted as a substrate binding constant, Ks, assuming that k 2 k 1. This is a very dubious assumption. There are very few enzymes for which the individual values of k 2 and k 1 are known. Ironically, the best-studied examples (e.g., horseradish peroxidase) present just opposite case: k 2 k 1. It is important that many speciﬁc mechanisms (not necessarily equations 28 and 32) generate the same steady-state rate equation 34. However, the particular expression for Km should be different for each individual case. Although undeﬁned in mechanistic terms, the parameters Km and V are very useful at the ﬁrst steps of kinetic studies: 1. The use of equation 34 allows the expression of the complex effects with simpler terms. 2. Km and V are helpful as predictive parameters to design a valid enzyme assay. One practical recommendation is to keep substrate concentration in incubation mixture at the level of 10 Km or higher. 3. Equation 34 permits one to obtain at least rough estimate of in vivo enzyme activity provided the internal substrate concentrations are known. General Non-Steady-State Approach. Equation 32 contains four variables (s, p, e, and x) constrained by two massbalance equations (e0 e x and s0 s p x). Therefore, it is enough to integrate just two differential equations (e.g., equations 33 and 34) to characterize the whole system. A typical example of a numeric solution is shown in Figure 5. We can see that enzyme-catalyzed reaction proceeds through three deﬁnite phases well separated on the time scale. Each phase is now safe to analyze with simpler mathematical expressions because any assumptions could be tested against the full exact solution. 1. The ﬁrst transient phase occurs before the steadystate concentration of ESC is reached. It occupies less than

10 3 s for most of enzymes. During this phase, the substrate concentration remains fairly constant (s s0), allowing for the following analytical solution: x(t) A k k

1e0s0

A

1s0

[1 k

1

e

At

],

2

v(t)

Vs [1 Km c

e

At

]

k

(35)

As compared with equation 34, it contains a relaxation term [1 exp( At)] that is very large when t is small, but decays to zero as t increases above s 1/A. Accordingly, the rate of enzymatic reaction is initially zero but increases rapidly to the steady-state value as the exponential term decays. Because the relaxation term contains s0, then the experimentally observed delay depends on substrate concentration. It allows for direct determination of individual rate constants k 1 and (k 1 k 2) by techniques such as in stopped-ﬂow apparatus (23). 2. The second steady-state phase is characterized by the constancy of reaction rate due to the exact balance between the rates of ESC formation and breakdown (dx/dt 0) while the substrate concentration remains close to the initial value s0. This phase proceeds for at least several seconds. As a rule, it is enough to measure the initial velocity of enzymatic reactions unaffected by substrate depletion or product accumulation. Steady-state kinetics is the most popular research domain; it is the most accessible and developed and provides the most kinetic data. However, it fails to determine individual rate constants as fully as the transient and relaxation approaches do. Steadystate equations were derived earlier (equation 34). 3. The third phase is characterized by considerable changes in the concentrations of substrate and product. Thus, we can no longer assume that s s0, and we should integrate equations 32 and 33. However, very good precision provides the quasi-steady-state approximation ds/dt dx/dt 0. Then, we arrive at a relatively simple differential equation that is solved analytically: ds dt For initial conditions s Km ln s0 s Vs(t) Km s(t) s0, p s0 0, t s 0, we have Vt (37) (36)

100 Instant reaction rate, nmol s–1 80 60 40 20

Steady-state phase

Substrate depletion phase

Transient phase

Equation 37 is called the integrated Michaelis-Menten equation. It remains to be valid not only for initial rate measurements but for any point within the reaction progress curve. 1 0.001 Time, s (log scale) 1000 Experimental Determination of Kinetic Parameters of the Michaelis-Menten Equation. Until recently, most enzyme kinetic experiments have been analyzed by means of one of linear plots in Table 5. Linear plots are used to examine an agreement of experimental data with equation 34 as well as to determine numeric values of parameters V and Km from slopes and intercepts. Today, this graphic ap-

0 0.000001

Figure 5. Time course of reaction proceeding by the MichaelisMenten mechanism. Numeric integration of equations 33 and 34: s0 10 4M, e0 10 8M, k 1 106M 1s 1, k 1 103 s 1, k 2 102 s 1.

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KINETICS, MICROBIAL GROWTH

Table 5. Linear Plots 1 v s v v V 1 V Km V V v Km 1 V s s V Km v s Lineweaver-Burk or double-reciprocal plot Langmuir-Hanes plot Eadie-Hofstee plot Direct linear plot (20)

K

p /s

VfKp /VrKs m m

(40)

v Km s

Equation 39 implies that the rate must decrease as the product accumulates, even if the decrease in substrate concentration is negligible. Thus, reversibility is closely associated with and requires the product inhibition. In many essentially irreversible reactions (e.g., invertase-catalyzed hydrolysis of sucrose), product inhibition is also signiﬁcant. It can be explained by equation 38 with only the second step being irreversible. In such a case, the accumulation of product causes the enzyme to be sequestered because the EP complex and rate equation are as follows (compare with equation 39): v

s Vfs/Km s s/Km pKp m

proach seems to be too cumbersome as compared with much more efﬁcient computer routines. The main objection is also that any linear transformation introduces some statistical bias (ironically, the highest bias is attributed to the most popular double-reciprocal plot!). In addition, manual line drawing is very arbitrary, so obtained parameters could not be assessed statistically for conﬁdence limits. However, plotting of original or linearized data is useful as an illustration and the ﬁrst sketchy estimate of enzymatic parameters. For deﬁnitive work, it is advisable to avoid all plots and to use statistical analyses instead. Reversible Enzymatic Reactions. The majority of biochemical reactions are reversible. To account for this feature, the Michaelis-Menten mechanism can be modiﬁed as follows:

k

1

Ks (1 m

Vfs pKp ) m

s

(41)

E

e0 x y

S i ES i EP i

s k

2

1

k x k

2

k y k

3

E

e0 x y

P

p

(38)

2

3

Inhibitors and Activators. Compounds that reduce the rate of enzyme-catalyzed reactions when they are added to the reaction mixture are called inhibitors. Just opposite effects (increase of reaction rate) are caused by activators. Both of them belong to the category of modiﬁers. We will concentrate here on inhibitors as having more practical application and even more speciﬁcally on reversible inhibitors, which form various dynamic complexes with enzymes. The irreversible inhibitors are known as catalytic poisons (many heavy metals, such as mercury); their binding to enzyme molecules reduces activity to zero, leaving no room for quantitative analysis. The reversible inhibitors can form complexes with free enzymes or enzyme-substrate complexes as shown in the scheme of Botts and Morales (20):

Contrary to the basic scheme in equation 32, there are two intermediates, one of which is the normal ESC and another is the enzyme-product complex (EP). The substrate, S, and product, P, can interconvert to each other. Application of the steady-state approach to this scheme results in the following equation: I v Vf Ks m Vr Kp m

s p Vfs/Km Vrp/Km s 1 s/Km p/Kp m

S E P Mixed S ES

I Uncompetitive

Competitive

k k k

2 1k 2

2k 3e0 k 2 k

3 3 2

k

2 2e0 2 1k

1k

k k k

1(k

k k

1k

k k

2k 3)

3

EI P

EIS

k

1 1k 2

k

3(k

2 3 2

k

1

k

k

k k

2k 2)

3

(39)

There are three major simple (or linear) inhibition mechanisms, which can be generated from this scheme by omitting some of the six involved: • Competitive inhibition if EIS is missing (inhibitor binds to the same site on the enzyme as the substrate forming abortive nonproductive complex EI). • Uncompetitive if EI is missing and EIS occurs as a dead-end complex; it implies that the inhibitorbinding site becomes available only after the substrate has bound.

where superscripts f and s denotes parameters of forward reaction, and r and p are indicators of reverse reaction. When a reaction is at equilibrium, the net velocity must be zero and, consequently, if s and p are the equilibrium values of s and p, it follows from equation 39 that equilibrium constant K is expressed via kinetic parameters (the Haldane relationship):

KINETICS, MICROBIAL GROWTH

1525

• Mixed if EI and EIS both occur but are not interconvertible (the complex EIS does not break down to products; this situation frequently occurs when the inhibitor is reaction product). The particular case of mixed inhibition is pure noncompetitive inhibition, which takes place if two inhibitor dissociation constants (for EI and EIS) are exactly matched to each other. For all types of inhibition, the Michaelis-Menten equation remains valid: under constant inhibitor concentration, i, the v-s relationship is of the same hyperbolic type as predicted by equation 34, the only difference is that apparent values of Km and V are now more or less simple functions of i (Table 6). Some parameters (in boldface in Table 6) are not affected by inhibitors, whereas others are changed: competitive inhibitors increase Km, pure noncompetitive inhibitors reduce V, uncompetitive inhibitors decrease at the same degree both V and Km, leaving ﬁrst-order rate constant V/Km to be unchanged. In mixed inhibition, there are no unchanged parameters. One interesting case is so-called substrate inhibition. It occurs when two substrates are bound to the same active site on the enzyme, forming nonproductive triple complex ES2: E S }

Kss

between, respectively, identical and different ligands (e.g., substrate and an allosteric effector). To explain the cooperativity and associated sigmoid kinetics, a number of models have been suggested. The earliest one is the Hill equation, which was originally designed to describe the S-shaped curve of oxygen binding to hemoglobin. It was assumed that each protein molecule E binds n molecules of ligands S in a single step, an amount of other possible forms (ESn 1, ESn 2, . . . ES) being negligible: E nS } ESn

Kh

where Kh is the respective equilibrium constant (Hill and colleagues described equilibrium in terms of association constant, but for the sake of uniformity we will adhere to the previous formalisms, keeping in mind the dissociation constant, Kh [E][S]n /[ESn]). The fractional saturation of protein (enzyme or hemoglobin). H is given as H Number of occupied binding sites Total number of binding sites [ESn] [E] [ESn] [S]n Kh [S]n (43)

Equation 43 can be rearranged into H 1 H [S]n H , log Kh 1 H logKh n log[S] (44)

KSS

ES r E @ ES2

k2

P

then, the enzyme rate (24) is: v Vs s s2 /Kss (42)

Ks

where Kss is the dissociation constant for complex ES2. Cooperativity. Many enzymes respond to changes in metabolite concentrations (substrates, modiﬁers) with much higher sensitivity as compared with predictions from the classic hyperbolic equations 34 to 42. This property is generally known as cooperativity, because it is thought to arise from cooperation between the active sites of the polymeric enzymes. As a rule, such enzymes consist of several subunits and display so-called sigmoid or S-shaped dependence of rate on substrate concentration. Many cooperative enzymes (but not all!) have active sites binding substrate and allosteric sites binding effectors. There are homotropic and heterotropic cooperative effects caused by interactions

A plot of log H/(1 H) against log [S] is known as the Hill plot and should be a straight line of slope n (so). This equation is used to ﬁt the experimental binding and kinetic data displaying a sigmoid shape. When plotting kinetic measurements, it is assumed that H v/V, and maximum velocity V should be known from independent measurements taken at saturation substrate concentration. However, the results of ﬁtting should be interpreted with care. First, the Hill equation is empirical and generally provides good agreement only in the H range 0.1 to 0.9 (the discrepancy at extreme s is probably caused by neglecting of other forms of ESC apart from ESn). Second, parameter n (Hill coefﬁcient) could not be interpreted as the number of subunits in the fully associated protein, rather it is an index of cooperativity. Monod et al. (25) proposed a general model explaining cooperativity and allosteric phenomena within a simple set of postulates: 1. Each subunit of enzyme can exist in two different conformations, designated R (relaxed) and T (tense).

Table 6. Types of Inhibition Type of inhibition Competitive Uncompetitive Mixed Pure noncompetitive V V/(1 V/(1 V/(1 Vapp i/Ki ) i/Ki ) i/Ki ) Vapp /Kapp m (V/Km)/(1 V/Km (V/Km)/(1 (V/Km)/(1 i/Ki)a i/Ki) i/Ki) Km(1 Km(1 Km(1 Km

app km

i/Ki) i/Ki )) i/Ki)/(1

i/Ki )

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KINETICS, MICROBIAL GROWTH

2. All subunits of a molecule must occupy the same conformation at any time (e.g., for tetrameric enzyme only R4 or T4 are permitted, not R3T or R2T2, etc.). 3. The two states are in equilibrium with constant L [R4]/[T4]. 4. The afﬁnity of ligand to subunit depends on the conformation state: KR [R][S]/[RS], KT [T][S]/[TS], c KR /KT. The general equilibrium solution of this scheme is rather bulky, so we demonstrate one particular case of tetrameric protein, if c 0 (i.e., KR KT, S binds only to the R state), then we have H (1 L s/KR)3s/KR (1 s/KR)4 (45)

ESC, whereas V/Km reﬂects the ionization of the free enzyme. The pH effects on Km are more complicated, being affected by both. Generally, the dependence of enzyme activity on pH is described by equation 43 as a bell-shaped curve with maximum at 1/2(pK1 pK2). Effects of Temperature. In chemical kinetics, the dependence of reaction rate on temperature is explained by the transition-state theory developed by Eyring in 1930 to 1935. It is based on the use of thermodynamics and principles of quantum mechanics. The reaction proceeds through a continuum of energy states and must surpass the state of maximum energy, when transient activated complex is formed. Then the dependence of reaction rate constant k on absolute temperature, T, is expressed as follows: d lnk dT DH* RT RT2 (47)

At high s when s/KR L, we can neglect the term L in the denominator, and the entire expression is converted to no more than the simple hyperbolic Langmuir isotherm. At low s, the contribution of L is considerable, so the saturation curve rises slowly from the origin displaying the S-shape. There are other models describing the mechanisms of cooperativity and allosteric effects (20), for example, the sequential model of Koshland and colleagues and the association–dissociation model of Freiden. Effects of pH. Every enzyme contains a large number of acidic and basic groups. Some of them are either fully deprotonated (aspartate, glutamate) or fully protonated (arginine, lysine). However several groups with pKa 5 to 9 (imidazole group of histidine, sulphydryl group of cysteine) do change their ionization state when pH is varied. Assume as a ﬁrst approximation that enzyme is represented as a dibasic acid H2E and only a singly ionized complex, HES , is able to react to give products: H2E H2ES @ KE k 1 @ KES 1 1 HE i HES @ KE k 1 @ KES 2 2 2 E ES2

where R is the gas constant and DH* is the enthalpy of activated complex formation. The classic Arrhenius equation may be obtained from equation 47 under a simpliﬁed condition DH* RT DH Ea (where Ea is activation energy). Most often, the Arrhenius equation is used in its integrated form: lnk or k A exp( Ea /RT) (48) lnA Ea /RT

where A is the integration constant, interpreted as the frequency of collisions of reacting molecules. Apart from mechanistic derivations, there are a number of empirical expressions relating k and Celsius temperature Tc. The most popular is the exponential formula: lnk lnA Tc

k

S

r

2

HE

P

or k A exp( Tc) (49)

Then the reaction rate is dependent on H concentration, h, as follows (20): v V Vs Km s ˜ V h KES 1 h KE 1 1 ˜ ˜ V/Km 1 KE 2 h KES 2 h (46)

V Km

where is the empirical constant related to the widely used temperature coefﬁcient Q10 exp(10 * ). All presented mathematical expressions predict exponential or almost exponential increases of chemical reaction rates with temperature. However, enzymatic reactions deviate from this relationship at high temperature because of thermal denaturation of enzymes. Assume that denaturation is reversible with equilibrium constant KT [E ]/[E], where E represents active enzyme molecules and E represents inactive molecules. Then, the combination of equation 48 with the van’t Hoff relationship for KT ( RT lnKT DGo DHo TDSo) results in v A exp( Ea /RT) exp(DSo /R DHo /RT) (50)

˜ ˜ where V and Km are the pH-corrected constants. It is clear that pH-dependent variation of V reﬂects the ionization of

1

where DGo, DSo, and DHo are the standard Gibbs free en-

KINETICS, MICROBIAL GROWTH

1527

ergy, enthalpy, and entropy of denaturation reaction, respectively, and v is the observed rate of biological process. This equation produces a curve with single maximum and ﬁts to most of the available experimental data on temperature-dependent variations of enzymatic activity. Sometimes denaturation is irreversible, and there are no possibilities for simple mathematical expressions, because temperature effects depends on the exposing time. However, numeric solutions of respective differential equations can still be used. Simple Models of Microbial and Cell Growth This section deals with simple unstructured models. These models mainly ignore any changes in cell quality (biochemical composition, spectrum of enzymatic activity, etc.) induced by environmental factors. Main State Variables and Growth Parameters. There are two types of growth models, deterministic and stochastic. The former describe clear determined and regular processes. The latter deal with random or stochastic processes. The main variables used in deterministic models are the same as in chemical kinetics, concentrations of biomass, substrates, and products, and stochastic models consider instead the probabilities, frequency distributions, variance, and so on. For example, a stochastic model can consider the probability of a single bacteria cell dividing under speciﬁed environmental conditions. Although any real-life biotechnological process has both deterministic and stochastic components, most useful growth models are strictly deterministic. In this section, we will concentrate on this type of model, and the stochastic counterparts will be considered only in “Cell Cycle”. The major state variables of deterministic models describing cell growth are described in Table 7. The concentration of biomass and cell number are related to each other by simple formula: x m N (51)

therefore, it is advisable to make a selection. The biomass, x, has obvious advantage in studies aimed at understanding or control of mass ﬂows, and cell number, N, is preferred in population studies when, for example, mutation or plasmid transfer is an essential factor controlling the efﬁciency of the biotechnological process. The choice of method to determine biomass or cell number depends on many factors (2,5). Today, the preference should be given to those techniques that allow exact and automated measurements (Table 8). The most advanced analytical methodology is now available for automatic recording of gaseous or volatile substrates, intermediates, and end products, such as methane, CO2, O2, H2, volatile fatty acids, alcohols, and other fermentation products (IRanalyzers, mass spectrometry, gas chromatography, NMR, etc.). The primary state variables that are measured directly in cell culture are usually recalculated into the secondary growth characteristics: gross growth rate, dx/dt; speciﬁc growth rate, l (dx/dt)/x; degree of multiplication, x/x0; biomass doubling time, td ln 2/l; growth yield, Y dx/ ds; product yield, Yp /s dp/ds, Yp /x dp/dx; speciﬁc rate of substrate consumption, qs (ds/dt)/x l/Y; speciﬁc rate of product formation, qp (dp/dt)/x lYp /x l(Yp /x /Y). Most of the listed secondary growth characteristics are called speciﬁc rates expressed as a ﬁrst-time derivatives of measured variable per unit of cell mass. The speciﬁc rates are not sensitive to variations in total cell biomass, so they can be considered as analogous to enzyme activity (qs, qp) in most kinetic derivations. The speciﬁc growth rate, l, may be viewed as the activity of an autocatalytic enzyme producing itself. The speciﬁc growth rate, l, measured from biomass dynamic, x, can differ from that estimated from the increase in cell number, N (denoted as lN). It follows from equation 51 that l 1 dx x dt 1 dm m dt 1 d(N m) x dt 1 dN N dt lcell 1 N lN m N dm dt m dN dt (52)

where conversion factor m1 is the average dry mass of a single cell. In some studies, it is save to assume m to be a constant and use both variables x and N as equivalent measures of growth. However, the average size and mass of single cells vary depending on the nature of studied organisms and environmental conditions (see “Cell Cycle”);

0 and l If mass of single cell m is constant, then lcell lN. Otherwise, we have to take into account m variation. Validity of Exponential Growth Low. One of the earliest postulates in microbial kinetics is that under optimal non-

Table 7. Major State Variables of Deterministic Models Variable Concentration of cell biomass Cell numbera Single-cell massa Mycelium lengthb Tips numberb Concentration of limiting substrate Concentration of product

a b

Notation x N m L n s p

Dimension (examples) g CDW L 1 of cultural liquid 109 cell mL 1 of cultural liquid g CDW per cell meter L 1 of cultural liquid 106 tips mL 1 of cultural liquid g L 1 of cultural liquid g L 1 of cultural liquid or g g 1 of CDW

For unicellar organisms (bacteria, yeasts). For ﬁlamentous organisms (fungi, actinomycetes).

1528

KINETICS, MICROBIAL GROWTH

Table 8. Methods Used for Determination of Microbial Biomass Method Dry mass Wet mass Wet biovolume Particulate organic carbon Biuret proteins Measuring principle Mass of separated and dried solids Mass of separated material Linear dimension of pelleted cells or colony CO2 after cell separation and combustion Colorimetric reaction of peptide bonds Colorimetric reaction of tyrosine and tryptophan DL 50 50 100 1.0 Advantage Provides direct unconditional estimate Simplicity, quickness Simplicity, quickness High precision and sensitivity, provides direct estimate of mass High uniformity Disadvantages Interference from dead cells and noncell solids The variation of wet biomass bulk density The variation of wet biomass bulk density Interference from dead cells and noncell solids Variation of protein-to-biomass ratio, possible extracellular accumulation Variation of protein-to-biomass ratio and amino acid composition of cell protein, possible extracellular accumulation Possible extracellular accumulation Variation of the intracellular ATP pool Variation of the intracellular content, possible extracellular accumulation Variation in conversion factor from measured rate to biomass Interference from noncell solids, cell aggregation, wall growth Variation of conversion factor, interference from electrolytes in the medium Time-consuming, low precision

1.0

Folin-Ciocalteu protein

0.1

High sensitivity

DNA ATP Fatty acids

Colorimetric estimation of deoxyribose Bioluminescence assay Gas or liquid chromatography, colorimetric reaction Rate of added substrate uptake or product formation Light scattering Conductivity

1.0 10

5

High speciﬁcity, constancy of the DNA cellular content High sensitivity and speciﬁcity High speciﬁcity, allows identiﬁcation of composition of mixed culture High speciﬁcity, quickness

1.0

Metabolio potential

10

3

–10

Opacity Electrical measurements Manual microscopy

0.1 1

Simplicity, quicknerss, automation Simplicity, quickness, automation Low cost, sensitivity, allows visual assay of biomorphological structure Combined beneﬁt of manual and automated quantiﬁcation, speed, generation of size distribution Quickness, automation, generation of size distribution Low cost, high sensitivity

2% (mg).

Cell count, measuring of hypha length Computer-aided count

10

5

Image analysis

10

5

High cost

Coulter counter Plating

Note: DL

Automated count and sizing Growing of the colonies on Petri dish

10 10

5

Interference from noncell solids Time consuming, low precision

5

detection limit, the minimum CDW required for an estimation with an error

restrictive conditions (nutrient media containing all essential components at nonlimiting concentrations, absence of inhibitors, adequate physicochemical parameters, perfect mixing), the increase in biomass (dx) during an inﬁnitely small time interval (dt) is proportional to this time interval and the instant biomass concentration (x), that is dx or dx/dt lx (53) lxdt

lnx

lnx0

lt ⇒ x

x0 exp(lt)

(54)

where l is the proportionality coefﬁcient. If l is constant, integration of equation 53 gives

where x0 is biomass at zero time or inoculum size. According to early views, true exponential growth occurs only in the case of symmetrically dividing bacteria with equal probability of subsequent division for the mother and daughter cells. Organisms with asymmetrical multiplication (budding yeasts) were thought to obey the exponential low only approximately, whereas the growth of ﬁlamentous organisms (fungi, streptomycetes) was deviated considerably. However, direct measurements performed from 1930 to 1960 (5) revealed that exponential growth does occur in all prokaryotes and eukaryotes independent of their biomorphological features, including protozoa, fungi, and homogeneously cultivated plant and

KINETICS, MICROBIAL GROWTH

1529

animal cells. The deviation is observed only as a result of a growth-associated change in the environment. Wang and Koch (26) made very precise measurements of l dynamics by growing Escherichia coli directly in an aerated cuvette of the computer-linked double-beam spectrophotometer. They found temporary slowdowns in the complex peptone plus beef extract media attributed to diauxie phenomena (e.g., depletion of some preferred oligopeptides) and a gradual increase of the l value during sequential subculture in succinate minimal medium. Long-term cultivation of microorganisms in continuous turbidostat or pH-stat culture (14) revealed that the increase in l is mainly the result of autoselection of mutants having higher maximum speciﬁc growth under speciﬁed cultivation conditions (see “Population Dynamics”). Early Views of Cell Growth. The derivation of the exponential growth equation was done for binary dividing bacteria, based on the geometric progression 2, 22, 23 . . . (27): N N02

n

A similar (but not identical) equation of cell biomass dynamics would be produced if we assumed that the limiting substrate is consumed according to ﬁrst-order kinetics and is converted to biomass with a ﬁxed yield factor: Y then ds dt s x0 (x x0)/(s0 s) (58)

ks Ys0 Y ds Y dt x xm Y kxm 1 x xm x ⇒ dx dt (59)

(55)

where n is the number of divisions, n t/td. The growth dynamics were viewed as a succession of distinct phases approximated with empirical formulas (15): N N0 exp[lF(t)t] (56)

Monod’s Model. This model still is very popular because of its elegant simplicity. It played an important role in the history of microbial growth kinetics as a ﬁrst encouraging example when the theoretical development based on mathematical formalisms came before novel experimental designs. As compared with simple exponential equation 53, this model (29) takes into account the mass-conservation condition linking substrate uptake to biomass formation through constant yield factor Y (equation 2) and the dependence of the speciﬁc growth rate on limiting substrate concentration: dx dt ds dt s Ks s

where function F(t) undergoes stepwise changes as shown in Table 9 (1.56 n 2.7). The alternative way to describe growth dynamics was to use the logistic equation borrowed from demographic studies (28): dN dt or dx dt rx 1 x xm (57) lmt rN 1 N K

l(s)x 1 dx Y dt

lm

x (60)

By substitution of s by x through Y (equation 58), we can reduce equation 60 to a single equation having the analytical solution: (1 P) ln(x/x0) P ln(Q 1) P ln(Q x/x0) (61)

where r is the net growth rate (the difference between true growth and death rates) and K or xm are the upper limits of, respectively, population density or biomass (K is used preferentially in ecological literature, being called the carrying capacity of the ecosystem). This equation simulates S-shaped growth dynamics frequently observed in natural ecosystems and cell cultures.

Table 9. Stepwise Changes of Function F(t) Initial stationary phase Lag phase Logarithmic growth phase Negative growth acceleration phase Maximum stationary phase Accelerating death phase Logarithmic decrease phase F(t) F(t) F(t) F(t) F(t) F(t) F(t) 0 tn 1 tt 0 tn 1

1

1

1

where P YKs /xm, Q xm /x0, xm Ys0 x0, and s0 and x0 are, respectively, s and x at t 0. Equation 61 contains three parameters, Y, lm, and Ks that can be thought of as passport data for a particular organism (e.g., for E. coli grown on glucose at 30 C, Y 0.23, lm 1.35 h 1, and Ks 4 mg L 1) and describe the S-shaped growth dynamics of a batch culture. The initial conditions, s0 and x0, are set by the experimenter, who selects the inoculation dose and medium composition. Thus, knowing the values of all these entities, the growth dynamics can be calculated before the experiment. Moreover, this model was used as a basis for development of the chemostat theory to understand and predict the behavior of microbial cultures in continuous-ﬂow bioreactors before the respective hardware was constructed. For this purpose, original equation 60 was modiﬁed as follows (30):

1530

KINETICS, MICROBIAL GROWTH

dx dt l ds dt D

(l lm

D)x s Ks s) s x 1 lx Y (62)

optimize the productivity by such tools as changing the ﬂow rate and the composition of medium in continuousﬂow bioreactors. Derivation of the Monod Equation from Mechanistic Considerations. The equation relating l and s is known as the Monod equation: l lm s Ks s (65)

D(sr F V

where sr is the concentration of limiting substrate in the fed-fresh medium delivered with pump from reservoir, F is the pumping rate (cm3 /h), V is culture volume (cm3), and D is the dilution rate deﬁned as the ratio D F/V, h 1. The mathematical analysis of equation 62 allows us to make the following conclusions on the growth kinetics of continuous culture: 1. The described open system attains stable steady state when dx/dt 0, ds/dt 0, and variables x and s are not changed with time (denoting steady-state concentrations x and s, respectively). From the ﬁrst ˜ ˜ equation in equation 60, it follows that l D, that is, the speciﬁc rate of microbial growth is equal to the dilution rate, which is under the full control of the experimenter. From the second equation, we ﬁnd x ˜ Y(s0 s) ˜ (63)

It was introduced by the author as a purely empirical relationship resembling adsorption isotherm. However, many microbiologists did view the Monod equation as something having deep inherent meaning rather than as just an empirical formula. Below, we shall outline a number of attempts to deduce this equation logically from the conjectured growth mechanisms. 1. The bottle-neck concept. There is an obvious similarity between the Monod and Michaelis-Menten equations. A theoretical substantiation for this similarity can be the bottleneck postulate originally put forward by Blackman (31). According to this postulate, the growth rate of a cell is determined by a single enzymatic master reaction. Microbial metabolism is symbolized as a unidirectional chain of reactions of substrate S conversion to cell biomass X via some hypothetical intermediates P1, P2 . . . Pn (32): S X r P1 r P2 r . . . r Pn r 2X

k1 k2 kn kn

1

As compared with Y expression for batch culture (equation 58) the formula in equation 63 no longer contains the term, x0, the inoculum size. Thus, a steady-state culture is not dependent on past history, and its properties are determined solely by current cultivation parameters. 2. The dilution rates permitting stable microbial growth (i.e., those D, at which x ˜ 0) are conﬁned between 0 and washout point or critical dilution rate Dc Dc lmsr /(Ks sr) (64)

(66)

Assuming steady state in respect to intermediate concentration pi, dpi /dt 0, we arrive at the Monod equation, composite parameters lm and Ks being expressed via elementary kinetic constants of individual enzymatic reactions: dx dt lm lm s Ks 1

n i 1

s

x

3. The speciﬁc growth rate l does not depend on either s0 or x and is governed solely by the substrate concentration in the cultural liquid. On the other hand, the indirect effect of s0 on l may be evaluated from the dependence of x on D, as soon as x and s are ˜ ˜ ˜ linked by the conservation condition in equation 62. The most important biological implication of the chemostat theory was the discovery of the fact that microorganisms can grow endlessly with any rate between 0 and lm. Such substrate-limited growth is nevertheless exponential, because two subsequent acts of cell division will be separated by a constant time interval. Earlier, substrate limitation had been observed only transiently at the end of the exponential phase of batch growth. The most important biotechnological implication of the chemostat model was the concept of controlled cell biosynthesis, which implies the purposeful manipulation of microbial culture to

1/ki

2

ks k1

1

n i 1

(67) 1/ki

2

where x x p1 . . . pn. The bottleneck idea can now be formulated as follows. If one of the constants kj ki, i 2, . . . , n 1, i j, then lm kj and Ks kj /k1, and so the jth enzymatic reaction is the master reaction. The obvious shortcomings of the reaction scheme in equation 66 are the unjustiﬁed oversimpliﬁcation of the cell metabolism (which is a network, rather than a simple unidirectional sequence of reactions) and the lack of a definite interpretation of variables x and p in real biochemical terms. An evaluation of the characteristic times of major intracellular metabolic reactions showed that there could be two possible bottlenecks: uptake of limiting sub-

KINETICS, MICROBIAL GROWTH

1531

strate (processes of active transport, transphosphorylation of sugars etc.) (33) and the formation of intracellular protein-synthesizing structures such as rRNA and the ribosomal proteins (34). 2. Stochastic considerations. Microbial populations were assumed to consist of active (which utilize the substrate) and inactive cells (35). The probability of a transition from the inactive state to the active one was supposed to be proportional to s, whereas the probability of the reverse transition was s independent. From these assumptions, a relationship between l and s can be inferred, similar to equation 65. 3. Derivations based on mechanistic considerations of protein synthesis (36). The rate of protein synthesis, dp/ dt, is supposed to be determined by rRNA concentration R and by the size of the amino acids pool, A, dp/dt kxRA/ (KA A), k constant. Other conditions are deﬁned as R R0 (Rm R0)l/lm, A bs, and P p/x, where Rm and R0 are, respectively, the upper and lower limits of R variation, and P and b are constants. For a steady-state chemostat culture, dp/dt 0 and dx/dt (1/P)(dp/dt) 0, then l (kRm /P)s/(RmKA /br0 s) lms/(Ks s) (68)

cell density, xm. The residual substrate concentration, s is calculated from mass balance (equation 58); yield is calculated as (xm x0)/s0; the s0 value should be known) and corresponding l(t) values from the slopes d(lnx)/dt. Finally, parameters Ks and lm (equation 65) are to be found graphically or from nonlinear regression as in the case of the Michaelis-Menten equation (see later text). 2. Batch culture, ignoring the substrate uptake. The inoculum size, x0, and the duration of experiment are chosen to minimize the uptake of added substrate s s0 (37). Typically, bacterial cell density should be of the order 10 5 or less, and it is measured by such sensitive instruments as the Coulter counter. The quasi-steady-state growth rate is measured at several s0 values and then ﬁtted to equation 65 as described before. In numerous determinations made at high cell densities (when we can no longer neglect substrate consumption), it was observed that dependence of l during exponential growth phase on s0 is formally described by the Monod equation with s replaced by s0 (38). The explanation of these results could be made only by more complex structured models; however, the described procedure is not appropriate for Ks and lm determination. 3. Batch culture, integral form of equation 61. The Sshaped curve of biomass dynamics is ﬁtted directly to equation 61 through either preliminary linearization or nonlinear regression (preferential). Sometimes (e.g., when cells are grown in opaque media) it is more convenient to follow the dynamics of residual substrate s(t) or product formation p(t), for example, CO2 evolution or dynamics of O2 uptake (respiration). In these cases, we can integrate the set of differential equation 60 in terms of s(t) or p(t) dynamics, taking into account mass–balance relationships in equations 7, 8, and 58 and similar equations. 4. Steady-state chemostat culture. The chemostat provides the opportunity for estimations under steady-state condition, which is commonly believed to be more reliable. By running an experiment at different dilution rates, D, the corresponding s values may be measured and the de˜ pendence of l D on s obtained; in principle, this can be ˜ done as accurately and carefully as desired. Such an approach, however, may and frequently does encounter serious technical problems because of the high afﬁnity to limiting substrate of some microorganisms. There is a need to (1) select highly sensitive analytical techniques to measure extremely low residual concentrations of particular substances; (2) develop instant sampling procedures to minimize substrate loss, and (3) eliminate apparatus-related artifacts such as nonperfect mixing and ﬂuctuations in nutrient medium supply. This can be accomplished by use of radiolabeled substrate, and other tips are covered in speciﬁc experimental works (39,40). However, the most essential objection to this method is that organisms at various steady states do change their kinetic properties, which is not accounted by Monod’s model (see “Structured Models”). 5. Non-steady-state chemostat culture. The measurements are made during short-term experiments started by addition of different amounts of limiting substrate to a steady-state chemostat culture. Then, substrate uptake or respiration rates are recorded in perturbed culture until

Here again, the Monod equation is derived through a consideration of underlying intracellular processes. Needless to say, none of the cited derivations are free from criticism. The Monod equation remains empirical. Any attempts to provide the mechanistic interpretation of this equation inevitably lead to much more complicated mathematical expressions (see late section on structured models). Biological Meaning and Experimental Determination of Growth Parameters Ks and lm. The parameter lm, maximal speciﬁc growth rate, has very lucid biological meaning: it is the upper limit of l variation on speciﬁed nutrient medium. It could not be attained in reality because of its asymptotic nature: l r lm as s r . However, in practice lm is achieved if s Ks. It should be remembered that lm is not absolute maximum of growth rate, because it depends on the nature of the limiting substrate. For example, E. coli has lm above 2.5 h 1 on complex beef-extract medium and below 1.0 h 1 on minimal medium with succinate. Saturation constant Ks could be deﬁned functionally as such concentration of limiting substrate that provides a speciﬁc growth rate equal to 0.5 of lm. We can say also that Ks is a measure cell afﬁnity to substrate: the lower is Ks the better the organism is adapted to consume substrate from diluted solution. Any other deﬁnitions are speculative, e.g., Ks interpretation as the dissociation constant of ESC of the cellular enzyme involved in the ﬁrst step of substrate conversion. There are several ways to determine numeric values of Ks and lm: 1. Batch culture, differential form of Monod equation. The biomass dynamics, x(t), are followed from the start of the exponential phase until the complete consumption of the limiting substrate and the attainment of the maximal

1532

KINETICS, MICROBIAL GROWTH

the new steady state is established (2). The opportunity of continuous culture is that the physiological state of cells just before perturbation is well deﬁned and reproducible. On the other hand, these experiments do provide data on afﬁnity to substrate (Ks) and maximal rates of respiration (e.g., QCO2 or uptake Qs, which are related but are not identical with lm): lm Yp /xQCO2 Ys /xQs (69)

l If we deﬁne lm l s*

lms/(Ks Ym(Qs lm(s

s)

mYm

(75)

m) (2), then s*)/(Ks s) (76)

KsmYm /lm

where stoichiometric parameters Yp /x dp/dx and Ys /x ds/dx should be determined in independent experiments. Immediate assay of lm in such an experimental setup is possible as follows: the steady state is perturbed by setting the dilution rate by 20 to 100% higher than the critical value Dc (equation 64). The washout dynamics are followed and lm is determined from approximate relationship (5): x x(0) exp[(lm D)t], where x(0) is biomass before perturbation. More rigorous estimate of lm could be provided if such washout experiments were made at several (at least two) input substrate concentrations s0 to account for the fact that sr is still not inﬁnity. Modiﬁcation of the Monod Equation. It was found that not all experimental data could be reasonably well ﬁtted by equation 65. The best-ﬁt hyperbola often passed above experimental points at small s and below them at large s. A better ﬁt was claimed to be provided by using the following, entirely empirical, equations: l l l lm[1

n

exp( Ks)] s ) s)

n

(70) (71) (72)

lms /(Ks lms/(Ksx

Equations 75 and 76 both predict the occurrence of threshold substrate concentration s*, below which growth is impossible. It yields a substantially better ﬁt to experimental data. The difference is that according to equation 75, the parameter lm is the speciﬁc growth rate under imaginary conditions of zero maintenance requirements, whereas equation 76 implies the traditional deﬁnition of lm as the speciﬁc growth rate under substrate excess: l r lm when sr . Account for Substrate Leakage. Mathematically identical to equation 76, a modiﬁcation of the Monod equation was proposed for the case of conserved substrates. However, the biological meaning is entirely different: if the speciﬁc leakage rate is assumed to be constant, then a decrease in s down to some threshold value s* will lead to the counterbalance of the two reverse processes (uptake and leakage), so that the net consumption of the limiting substrate will be zero. Account of Inhibitory Effects. A few valuable reﬁnements of the Monod equation were borrowed from enzymology; most often they were noncompetitive and substrate inhibition. The former inhibition mechanism is by the so-called Monod-Ierusalimsky equation: s Ks s Kp Kp p

The preference of the ﬁrst expression, equation 70 (41), is questionable. An expansion of Monod equation by addition of the third parameter, n in equation 72 (42), or introduction of the second variable, x in equation 71 (43), does improve the approximation capability of kinetic equations. We can even provide the mechanistic basis for this improvement. Thus, the Moser equation (equation 71) is similar to the Hill equation in enzymology (equation 43), indicating the cooperativity effects in performance of some master reaction of cellular metabolisms. The Contois equation (equation 72) could be interpreted in terms of growth autoinhibition by-products, because under the realistic assumptions (x x0, xYp /x /Kp 1 and product p formation coupled with growth), the apparent saturation constant Kapp is almost proportional to accumulated biomass x: s

app Ks

l

lm

(77)

The growth retardation with an excess of such substrates as phenol, methanol, and ethanol is described by an analogue of Haldane’s equation (45): s Ks s s2 /Kss

l

lm

(78)

Ks(1

p/Kp)

Ks[1

Yp /x(x

x0)/Kp]

Ksx (73)

An Account of Maintenance Requirements. Powell (44) assumed that substrate uptake rate q obeys the MichaelisMenten kinetics; then, from mass–balance equation 18 it follows that qs Qs /(Ks s) l/Ym m (74)

If lm is deﬁned as lm

YQs (5), then

Equation 78 simulates a single-peak curve, and so the same l value may be obtained at two different s, one in the substrate-limiting range, dl/dt 0 (stable), and the other in the substrate inhibition range, dl/dt 0 (unstable). A sustainable maintenance of a population under conditions of substrate inhibition is possible either in the second stage of a two-stage chemostat or in the case of plentiful wall growth in a conventional chemostat (46). Account of Diffusion Effects. We will present one example of such models (44). The basic assumption is that substrate is taken up by an enzyme that obeys Michaelis kinetics and is localized on the inner side of cell membrane. The actual substrate concentration around the enzymeactive centers is smaller than in the solution, because of a limited diffusion rate. By applying a simpliﬁed Laplace equation, it was found eventually that

KINETICS, MICROBIAL GROWTH

1533

l

lm(Ks

L 2L s lm Ks L

s)

1

1

(Ks

4Ls L

s)2 (79)

poseful non-steady-state growth may display greater efﬁciency and higher productivity (48). The notions of steady-state and balanced growth are close but not identical. The ﬁrst is more strict; steady-state growth has to be balanced by deﬁnition (otherwise some of the speciﬁc rates responsible for synthesis of the changed cell component should vary). On the other hand, the balanced growth can be for some time nonsteady, perhaps during the late exponential phase of batch culture when l declines while cell composition remains unchanged. During long-term experiments, non-steady-state growth leads inevitably to a change of cell composition; it becomes unbalanced. For heterogeneous populations, the situation may be more complicated. For example, the growth in the second stage of a two-stage chemostat attains a steady state, and the biomass and residual substrate concentration are constant. However, such growth is not balanced, because cells delivered from the ﬁrst stage differ in their properties and composition as compared with the bulk of cells in the second stage. This situation was termed the transient steady state (49). By structured, we mean mathematical models describing growth-associated changes in microbial cell composition. It includes mass–balance equations for all assigned intracellular components. Their concentrations can be expressed either per unit volume of fermenter vessel (c1, c2, . . . , cn), or per unit cell mass (C1, C2, . . . , Cn), and hence Ci ci /x. The mass–balance equations can be written as follows

n

s

where L is a factor determined by membrane permeability and by the maximum rate of the enzymatic reaction. Structured Models The unstructured models (such as Monod’s model or its modiﬁcations) are able to predict and describe only simplest manifestation of growth phenomena. Sometimes it is declared that Monod-type models are able to describe only balanced and steady-state growth. The analysis of more complicated unbalanced and non-steady-state growth requires formulation of structured models. Deﬁnitions. Balanced growth was deﬁned by Campbell (47) as a proportional increase in the amounts of all cell components, in other words, balanced growth produces cells of the same quality without any variation of composition. The terms steady state and non steady state stem from chemical and enzyme kinetics. The ﬁrst one refers to such a regime when the reaction rate remains constant because of an exact balance between formation and breakdown of intermediary products such as the enzyme– substrate complex. In microbial culture, the growth is called steady state if speciﬁc rather than total rates remain constant. In an open system, such as a chemostat, both total (dx/dt, ds/dt) and speciﬁc rates (l (1/x)dx/dt, qs (1/x)ds/dt) tend to have constant steady-state values. A closed system, such as a batch culture, should be considered under steady state only during the exponential phase when l and qs are constant. The linear growth with a constant total rate (dx/dt lx constant) is characterized by monotonously declining l, and is not steady-state growth. However, under some conditions (such as in dialysis culture) it may attain quasi steady state, when ds/dt 0. The non-steady-state kinetic regimes take place before establishment of steady state or after its perturbation. In enzyme kinetics, non-steady-state measurements are taken in the millisecond range of time scale. In microbial cultures, similar non-steady-state transient and perturbation processes advance much more slowly, typically during several hours and days. An example is a transient process in the chemostat induced by changes in D or sr (fed-substrate concentration). In such growth, l, qs, qp, and other metabolic rates exhibit continuous variation in time. The attractiveness of non-steady-state studies for microbiology and biotechnology is obvious: • They allow a wider range of hypotheses to be tested and yield much more data on the studied objects. • They have higher practical value; in biotechnology, steady-state operation is the exception rather than the common routine because of unavoidable disturbances in cultivation conditions. • They provide additional tools for optimal regulation of cell performance in the bioreactor, because pur-

ci

i n 1

x 1 (80)

Ci

i 1

For each variable Ci, a differential equation is written that takes into account all sources, r , and sinks, r , as well as its dilution from cell mass expansion (growth) dCi dt r (s, C1, . . . , Cn) r (s, C1, . . . , Cn) lCi (81) The simplest structured models with n no more than 2 or 3 are called two- or three-compartment models. For example, a model (50) incorporated two compartments: nucleic acids and proteins combined with other active cell components. The model variables also included concentrations of the limiting substrate and the inhibitor. Compared to Monod’s model, the proposed set of four equations was able to account for a much wider range of dynamic patterns. Speciﬁcally, it simulated D-dependent changes in the cell composition (chemostat) and all known growth phases of batch culture from the lag to decline. However, the choice of variables in this model was more or less arbitrary, so it should be regarded more as a bright illustration rather than a research tool. During the past decade, much more realistic structured models based on biochemical data have been developed.

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KINETICS, MICROBIAL GROWTH

The simulation model of E. coli growth (51) contains the following dynamic variables: glucose and NH4 , as exosubstrates; CO2 and acetate, as products excreted into the medium; amino acids; ribonucleotides; deoxyribonucleotides; monomeric precursors of cell wall components; rRNA and tRNA; nonprotein polymeric components; glycogen; guanosinetetraphosphate; enzymes transforming ribonucleotides into deoxyribonucleotides; ATP; NAD(H); and protons. Altogether, the dynamic model amounts to a system of 21 differential and 14 algebraic equations. An even more complicated model simulating growth of Bacillus subtilis (52) is the set of 39 nonlinear and coupled differential equations containing nearly 200 parameters! These models are able to simulate particular growth features such as changes in cell sizes, shape, and composition as well as the D-dependent variations in replication time brought about by the shifts in glucose concentration. However, the predictive capability of such an intricate dynamic model should be still estimated as modest as compared with invested modeling efforts; they are still nothing more than a “caricature parody” of microbial biochemistry and are too complex to be studied by conventional mathematical tools (stability analysis, parameters identiﬁcation, etc.) or to be used in biotechnological applications. The best choice of a mathematical model lies, apparently, midway between unstructured and highly structured models outlined here. One of the known compromises has been found through attempts to express quantitatively the cell physiological state. Physiological State of Chemostat Culture. The term was coined by Malek (53) without giving a clear deﬁnition. The impetus for the development of the concept of a physiological state was the evidence on changes in the chemical composition of microorganisms as dependent on dilution rate D and medium composition in chemostat culture (54). It has been found that some of the studied parameters remained constant (content of cell DNA and carbon) while others exhibited regular D-dependent variations, either an increase (RNA content, cell sizes) or a decrease (the content of reserved polysaccharides). Those properties that were D dependent were recognized as components of the vector of the physiological state. Powell (44) combined and put on a quantitative mathematical footing three notions, which were beforehand separated and cloudy: (1) physiological state, (2) past history, and (3) non-steady-state growth kinetics of microbial culture. The speciﬁc rate of substrate uptake, qs, was presented as a product qs(s) qa(s)S(s) (82)

jumps from s(0) to s(1). Immediately, qs will increase from Q(0)S[s(0)] to Q(0)S[s(1)]. If no further changes in s(1) occur, then Q will also eventually attain a new steady-state value equal to qa[s(1)]. In essence, Q is the potential metabolic activity, that is, the speciﬁc rate of a key metabolic process measured just at the moment of relief from substrate limitation. The transient Q dynamics are described as net change equals production minus dilution caused by cell growth: dQ dt r(Q, s) l(Q, s)Q (83)

Substance Q may, in reality, be represented by a single enzyme or multienzymatic complexes as well as by ribosomes or other cell components occupying the bottleneck position. Monod’s model, supplemented by equation 83, made possible at least a qualitative understanding of chemostat-transient processes triggered by a D switch (55). The synthetic chemostat model (SCM) (2) combines Powell’s ideas with the routine of conventional structured models. This model is similar to the cybernetic model (56,57). The basic SCM interprets microbial growth as a conversion of exosubstrate S into cell macromolecules X via a pool of intermediates L:

Cell wall Transport qs

CO2

Synthesis qL

S

Leakage v

L

Turnover a

X'

where S(s) is a simple saturation function (e.g., a Michaelis hyperbola), S(s) s/(Ks s), and qa(s) is a function associated with the microbial physiological state. The instant value of qa(s) is determined by way in which s varies until the given moment (s h ago), effects of later events contributing more than earlier ones. Transient processes are inﬂuenced by the past history of the culture in the following manner. Suppose that steady-state growth of a chemostat culture is upset and the residual substrate concentration

Macromolecular cell components are susceptible to degradation (turnover), and monomeric metabolites can escape into environment because of leakage. Contrary to simple chemical catalysts, the composition of end product X is not uniform and varies in response to a changing environment because of the adaptive nature of microbial metabolism. At the heart of the SCM are the solution of the problem, how to cope with these variations, and how to describe adaptive changes in cell composition by relative simple models. All macromolecular cell constituents are divided into two groups: primary cell constituents necessary for intensive growth (P components), and components needed for cell survival under any kinds of growth restriction (U components). The characteristic examples of P components are ribosomes (rRNA and ribosomal proteins) and enzymes of the primary metabolic pathways. Their intracellular content increases parallel to an increase in l. The contribution of U components to cell biomass decreases with growth acceleration. Examples are enzymes of the secondary metabolism, protective pigments, reserved substances, and transport systems of high afﬁnity. The analysis of available data as well as massconservation conditions allowed the formulation of several rules of variation of cell components taking place because of optimal control of cell biosynthesis: 1. Amounts of P and U components expressed as a fraction of total cell mass (P and U, gram per gram of

KINETICS, MICROBIAL GROWTH

1535

biomass respectively) vary within the upper (Pmax, Umax) and low (Pmin, Umin) limits, the latter being the constitutive part. 2. An increase of one individual P component is accompanied by increase of other P components. 3. The total enlargement of Psum is accompanied by corresponding decrease of Usum and vice versa. 4. The P/U ratio is controlled by the limiting substrate concentration in an environment. These rules are translated into mathematical terms as follows: P1 Pmax 1 U1 Umax 1

min P1 Pmin 1

••• ••• s ˜ Kr

Pn Pmax n Um max Um s ˜

min Pn Pmin n

r 1 r

similar trend was complemented by considerable decrease of YN due to alteration of cell composition in favor of N-rich P components (Fig. 6). SCM adequately describes the transient growth caused by shift-up in chemostat culture. The phenomena of overshoot in substrate concentration and undershoot in biomass during transient growth are explained by slow adjustment of cell composition (RNA content, respiratory activity) to new growth conditions (Fig. 7). Batch culture limited by carbon and energy source was simulated by SCM on the whole from inoculation to death stage; conventional growth phases (lag, exponential, deceleration, stationary, decline) were generated automatically without setting any speciﬁc conditions (Fig. 8). It is important that SCM realistically describes and predicts not only net growth but also the survival dynamics of starving cells after substrate depletion. During this phase,

Umin 1 Umin 1 r ˜

Umin m min Um

Mass fraction of cell components

1.0 RNA 0.8 0.6 Proteins 0.4 0.2 0.0 0.0 Lipids DNA Polysaccharides

(84)

where variable r (index of physiological state) is already scalar (not vector!) function controlled by environmental factors (e.g., concentration of the limiting substrate). The r value in steady-state chemostat culture changes from zero (in culture at almost zero growth rate, when s r 0 and ˜ all P components come down to low limits) to 1.0 (in unlimited culture, when s r and P components attain maxi˜ mum). During transients caused by sudden s changes, an instant r value goes toward new steady state (compare with equation 82): dr dt l s Kr s r (85)

0.2 0.4 0.6 0.8 Chemostat dilution rate, D (h–1)

1.0

Biomass, residual substrate, mg L–1

6

3

2 0.12

qs

r

Qs Ks s

(1

r)

Qs Ks s

(86)

4

0.08

where the ﬁrst and the second terms on the right side stand, respectively, for low (P component) and high (U component) afﬁnity of transport system. Similar r-dependent expressions are derived for other reactions (qL, v, a) and stoichiometric parameters. Predictive and clarifying capabilities of SCM turned out to be higher than more complex structured models. Contrary to all known chemostat models, SCM provides adequate simulation of D-dependent variation of the microbial physiological state. Under energy- and C-limited growth, it was expressed as an increase of apparent Ks, potential uptake and respiration rates, maintenance ration, and turnover parallel to increase of D. Under N limitation, a

2 1 0 –10 –5 0 5

0.04 0 10 15 20 25 30 35 40 Time (h)

Figure 7. SCM simulation of transient growth induced by change of chemostat dilution rate. Residual glycerol concentration (1), biomass (2), and cell RNA content (3). At t 0, dilution rate was shifted from 0.004 to 0.24 h 1. Source: Redrawn from Ref. 2; the original data (59) are for chemostat culture of Aerobacter aerogenes limited by glycerol.

RNA content, fraction of CDW

The introduction of the r variable greatly simpliﬁes the use of structured models because the adaptive variation of cell composition (and metabolic activity, which is determined by the intracellular content of particular enzymes) now could be expressed via one single master variable r. For example, the speciﬁc rate of substrate uptake qs is deﬁned as:

Figure 6. Simulation of D-dependent changes in cell composition of Aerobacter aerogenes grown in NH4 -limited chemostat culture. The curves are calculated from SCM (2), and the original experimental data are from Ref. 58.

8

0.16

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KINETICS, MICROBIAL GROWTH

200 Residual glucose (mg L–1) 1 150 2 2 1

160

100

80

50 0 0

40 0

Biomass (mg L–1)

120

mass increase from m0 to m* is inversely related to the speciﬁc growth rate l. It is this difference that yields ldependent variations of cell sizes, because faster-growing cells produce for the same s C D time a larger cell mass than do slower-growing cells. It became clear from the following simple algebra. The steady-state growth of an individual cell proceeds exponentially throughout the entire cell cycle [m m0 • exp(lt)]. At the time of the second part of the cycle, it takes exactly s min for the cell to enlarge from critical mass m* to 2m0. Hence, 2m0 m* exp(ls) ⇒ m0 0.5m* exp(ls) (87)

5

10

0

200 400 600 800 Time (h)

Figure 8. SCM simulation of complete dynamic curve of batch culture; residual glucose (1) and biomass (2) of yeasts Debaryomyces vanrijiae (old name D. formicarius) grown on glucose-mineral medium (2). Note that contrary to old empirical models (equation 56), all growth phases are reproduced automatically without specifying preset time ranges.

surviving bacteria sustain very slow cryptic growth at the expense of cell turnover and L leakage. The rate of decline in biomass gradually decreases as the result of buildup of some U components (parallel to the decrease of such P components as RNA and CN-sensitive respiration enzymes). Batch culture limited by conserved substrate (the source of N, P, Mg, Fe, or other) has an interesting feature: the growth is not stopped after substrate depletion but proceeds at an even higher rate. SCM explains this phenomenon by the partitioning of deﬁcient elements between mother and daughter cells. Cell Cycle The term cell cycle is used to designate the regularly repeated sequence of events that occur between consecutive cell divisions, for example, the formation of two identical daughter cells from one cell, which takes place in most known bacteria. Equivalent cell cycle events include budding (most yeasts and budding bacteria), branching of hypha (ﬁlamentous organisms), fragmentation (some coccoid and corineform bacteria). The majority of available data have been obtained for rod-shaped bacteria (E. coli, Salmonella typhimurium, Bacillus cereus) under steady-state growth conditions when the cell cycle consists of three distinct phases: 1. The growth of the newborn cell without chromosome replication from the initial mass m0 until some critical initiation mass m* 2. DNA replication (C period) 3. Separation of daughter cells (D period) Relationship between Cell Size and Speciﬁc Growth Rate. The periods C and D are constant; in E. coli, they occupy about 40 and 20 min, respectively, independent of environmental conditions. However, the duration of phase 1 depends on cultivation conditions; the time of the single-cell

Equation 87 remains valid for steady-state culture at any l, which can be varied from 0 to lm. Assuming that cell critical mass m* is constant and does not depend on growth rate, we see that equation 87 predicts a l-dependent variation mass of newborn cells m0 and hence the average mass and size of the bacterial population. This ﬁnding is supported by numerous experimental studies from the beginning of this century (60) that displayed the positive correlation between cell size and growth rate. Because the term ls is rather small (ls ltd ln 2), it is difﬁcult to notice the curvature of the experimental curve within the measurement error. Thus, the relationship between the average cell size m (m ¯ ¯ m0 2 ln 2 for rod-shaped bacteria; see derivation in equation 88) and l is given as an empirical linear approximation: m ¯ m(0) ¯ kl (88)

The regression parameters m(0) and k have a meaningful ¯ biological interpretation: y intercept, m(0) ¯ 2 ln 2 0.5m* (ln 2) m* is equal to approximately 69% of the cell critical mass m* initiating chromosome replication, and slope k [exp(lms) 1]/lm s is close to the duration of C D periods. The postulate on the constancy of the critical mass, m* was derived from the observation that the cell has accurate control over its size at division and poorer control over its age at division (61). Recently, accurate measurements with ﬂow cytometry (62) revealed that m* is inversely related to the speciﬁc growth rate; slowly growing cells tend to initiate DNA replication at a slightly higher critical mass as compared with intensively growing cells. However, we may safely assume that m* variation is much smaller than the variation of cell mass during the cell cycle: dm*/dt dm/dt. Simulation of Cell Cycle by Simple Deterministic Structured Model. In biochemical terms, it is difﬁcult to envisage how cell mass per se could determine when to initiate replication. A more likely candidate is some mass-related parameter, such as intracellular concentration of some signal metabolite like guanosine tetraphosphate or an initiator protein, according to the popular model proposed by Helmstetter and Cooper (63,64). It was postulated that the initiation of DNA replication is triggered by a threshold intracellular concentration of this protein V*; this protein is synthesized at a rate proportional to the total growth

KINETICS, MICROBIAL GROWTH

1537

rate and requires exactly one mass doubling time to reach its threshold concentration again. This mechanism is translated into mathematical form of a structured model such as SCM as follows (2):

dV dt dk dt H(l f if V if k a) lV f1 0 else f 0 (initiation) k/2, V 0, m m/2 (division) (89)

V* then f k* then k

Statistical Analysis of the Population Distributions. Equation 87 to 89 were derived for average cells in the population. To embrace the variability of sizes, we have to analyze the frequency distributions. If certain conditions are met— the culture is fully desynchronized, cells grow according to some deterministic low, all divide into only two identical daughter cells, and there is no cell elimination—then age distribution (61) is given by: u(a)da 2le

la

da: 0

a

ln 2/l

td

(90)

where H is the fractional contribution of protein V to total cell synthesis. The V content is an intrinsically transient entity; even during steady-state growth, it continuously changes between zero (Helmstetter and Cooper postulated the annihilation of the initiator protein after every replication cycle) to an upper-threshold value, V*, which is less than the potential steady-state level, H(l a)/l. The second variable, k, imitates the replicating chromosome; it sets up the discontinuity associated with cell division. Analysis of coupled equation set 89 reveals that this model is stable toward perturbations. Suppose that by chance, the content of initiator protein has risen to some abnormally high level. The immediate result would be several more frequent cell divisions, with smooth reversal to a normal multiplication pace. Similar events take place under the opposite situation of V deﬁciency; several divisions are delayed, resulting in production of abnormally long cells, but then steady state is restored. The negative feedback mechanism that brings things back into line is based on the dynamic nature of variable V; it is characterized by a unique, stable steady state that is approached from different initial conditions. It may be easily shown that the described model adequately simulates various morphological effects exhibited during non-steady-state growth (Table 10).

where l is speciﬁc growth rate, td is mean doubling time, a is the age since birth, and u(a) da is the frequency of cells whose ages are between a and a da. Assuming that cells grow exponentially between divisions, then the frequency of mass distribution is u(m)dm 2m0 /m2dm: m0 m 2m0 (91)

where m0 is the mass of newborn cell and u(m) dm is the frequency of cells whose masses are between m and m dm. The mean cell size is 2m0 ln 2 (calculated as an integral of m u(m)dm). If cell growth between two consecutive divisions is linear (65), then u(m)dm 4 ln 2/m0 • exp( m ln 2/m0)dm: m0

m

2m0

(92)

Table 10. Morphological Effects during Non-Steady-State Growth Effects The longer lag phase in batch culture when growth is surveyed by cell count rather than biomass measurements The accumulation of enlarged cells during transition from lag to exponential growth phase The formation of dwarf cells in starving population Explanation The partial synchronization of cell division is delayed as compared with mass growth until attainment of the value 2m ¯ The l–m relationship (equation 87) combined with the transient misbalance in V synthesis (equation 89) The small cells are produced during very slow cryptic growth of surviving organisms This phenomenon is simulated by equation 89 and explained by overproduction of initiator protein in the presence of inhibitor

Equations 90 and 91 are called canonical age or mass distributions to emphasize that they are an idealized form applicable when cell division takes place at a precise size. Assuming that momentary distribution of size at division of individual cells is normal and random (not correlated with other cell cycle events), we can obtain computersimulated curves for any ﬁxed level of noise expressed as the coefﬁcient of variation (CV). As shown in Figure 9, random variations of size at division tend to round the corners of the canonical distribution. Another source of cell size variation can be nonequal separation of mother cells into two daughter cells. It is characterized by the K distribution, which is the distribu-

2.5 2 Frequency 1.5 1 CV = 10% 0.5 0 0.5 1 1.5 2 Cell size, m (10 –9 mg) 2.5 Canonical mass distribution, CV = 0

The accumulation of division potential if division is blocked by inhibitor, then after block release all missing division takes place in quick sequence

Figure 9. Distribution of cell sizes (as single-cell mass) for canonical cases where there is no variation in the size at division and for the case of normal distribution of size at division with CV 10% (see details in Ref. 61).

1538

KINETICS, MICROBIAL GROWTH

tion of the ratio of daughter-cell length to mother-cell length. The average values absolutely necessary equals 0.5, but CV is at best about 4% (e.g., for well-behaved E. coli strains) and can attain rather high values for other organisms. The deviation of observed versus predicted size distribution may be caused also by cell death and different kinds of cell pathology (abnormally long or dwarf cells). Collins and Richmond (66) introduced an entirely different approach based on the use of three distributions:

m m m

Most of the obtained results are in better agreement with the exponential growth model rather than with the linear model. However, there are some serious doubts about whether there is a unique simple mathematical growth low describing bacterial growth during the division cycle. Cooper (67) proposed distinguishing three categories of cell components that are synthesized with a unique pattern: 1. Cytoplasm (proteins, RNA and ribosomes, small molecules) that is accumulated exponentially. 2. Cell DNA that is replicated in a linear fashion as a sequence of constant and zero rates. 3. Cell surface composed of peptiodoglycan and membranes that are synthesized exponentially during most parts of the cell cycle, but immediately before cell division the synthesis accelerates to accomplish new pole formation. Thus, the growth pattern of the whole cell is the sum of these three patterns. Because the cytoplasm is the major constituent (up to 80% of CDW), then the growth of the cell should be approximately exponential. Population Dynamics (Mutations, Autoselection, Plasmid Transfer) Description of Mutation and Autoselection. The continuous culture turned out to be a very efﬁcient tool to study mutation and autoselection (67). Let N be the total cell concentration, M the concentration of mutants, l the speciﬁc growth rate of the main nonmutated part of the cell population, and g the speciﬁc growth rate of neutral mutants, then dM dt dN dt klN lN gM klN DM DN (94)

vm

l{2

0

Wdm

0

u(m)dm

0

k(m)dm}/k(m) (93)

where vx is the growth rate of cells of size m, k(m) is the extent of population distribution, W is the momentary distribution of dividing cells, and u(m) is the momentary distribution of newborn cells. This equation allows the calculation of the mean growth rate of cells of particular size class. Instead of this analytical method, Koch in his work with Shaechter (61) proposed the synthetic approach. Starting from the set of speciﬁc postulates of the cell multiplication mechanism (linear or exponential growth of cell mass between divisions, kind of control, the evenness of the division), they derived the size or age distribution, which then was compared with the observations. The Growth Low. Some believe that there should be a general low of cell growth that can be discovered by sensitive methods of analysis. The rate of biomass growth throughout the cell cycle was hypothesized to be linear, bilinear, exponential, double-exponential, and so on. Two limiting cases have generally been considered: the exponential and linear growth models proposed, respectively, by Cooper and Kubitschek (57). To differentiate between these two mechanisms, three classes of experiments have been used: 1. Size measurements of individual cells growing in the microculture by use of phase-contrast microscopy or recently developed confocal scanning light microscopy combined with image analysis. 2. Pulse-chase labeling of cells with their subsequent separation into different phases of the cell cycle. Most frequently, labeled uracil and leucine are used as precursors of RNA and protein synthesis, respectively. There are two major sources of errors in this approach: poor resolving power of separation methods and artifacts associated with effects of exogenous compounds on intracellular ﬂuxes (feedback inhibition, pool expansion, label dilution, etc.). One of the best options for separation is probably the “baby machine,” based on the membrane elution principle (67). To minimize the second source of error, the mutants blocked in the synthesis of the probe compound can be used. 3. Analysis of the frequency distributions of steadystate populations (see equations 91 and 92). However, the resolving power of this approach is rather low because linear and exponential models produce similar patterns.

where k is the mutation rate expressed as the ratio of the numbers of mutants to total number of cells formed. If k 1 and l g, in the steady state, we obtain l g D, and dM dt or M M0 kDNt (95) kND

If g l, then the original strain will be displaced by the mutant, otherwise, if g l, M will tend to a lower limit M* kN/(1 g/D). Experimental studies of phage-resistant mutants in a tryptophan-limited chemostat culture of E. coli showed that the period of linear M increase in accordance with equation 95 was fairly short. Every 20 to 100 generations there was an abrupt fall in the number of mutants, after which the linear growth resumed at the same rate (67).

KINETICS, MICROBIAL GROWTH

1539

The observed saw-tooth dynamics in M were explained by Moser (43) as a combined effect of mutation and selection. The original wild clone gives not a single but a whole array of mutations with subsequent reversions. Let us denote the total cell population in a chemostat culture as N, which is the sum of all subpopulations, including original and emerging variants, N RNi. All possible transitions between variants are given by the matrix, kirj( j 1, . . . , n; i 1, . . . , n; j i). Then, the chemostat model takes the following form: dN dt l ¯ dNj dt ds dt lj(s)

n j 1

dNj dt

n

l(s)N ¯

DN

1 Nj

ljNj

1 n n

3. Growth of a mutant with a higher resistance to inhibitory metabolic products can be described by equation 77 with Kpk Kp. Under selection pressure r lk l lm(s/Ks s)[Kpk /(Kpk p) Kp /(Kp p)], the original population will be completely displaced, and the product concentration will reach a higher steady-state level, p ¯ lmKpks /(Ks ¯ s)D ¯ Kpk. 4. Growth of mutants resistant to an antibiotic will not be affected by competition if the respective antibiotic is continuously supplied: r lk(s), because l 0 for all other forms. The dynamics of the total population will be governed by the initial density of the mutant. 5. A mutation resulting in enhanced adhesion to fermenter walls will lead to accumulation of slow growing cells (because the adhesion prevents washout); eventually we have r lk D lj. Extrachromosomal Cell Elements. The present-day hot spot in microbial population genetics is the study of extrachromosomal cell elements (ECE) such as plasmids, phages, transposons, and insertion elements. Normally, ECEs do not carry genes, absolutely essential for growth, but they are capable of fast replication, surpassing the chromosome DNA in the number of copies. R plasmids are responsible for bacterial growth in the presence of antibiotics, but under normal conditions (with no antibiotics present), their synthesis becomes too heavy a burden for the host cell, which is manifested in a decreased growth rate. Among the more than 100 R factors studied, about a quarter were found to increase the bacterial generation time by 15% (68). For this reason, plasmid-bearing strains are unable to compete with plasmid-free populations, although there are a few exceptions. Thus, colicin-positive cells carrying respective plasmids are able to withstand the competition with faster-growing plasmid-free strains by virtue of antagonistic inhibition. In recent years, rather intricate and detailed dynamic models of autoselection have been proposed that take into account the ECE-related effects, including the transfer of ECEs within the population, their segregation loss, changes in l arising from the ECEs carriage, and so on. Some of these models have biotechnological and medical applications (69). MICROBIAL GROWTH AS DEPENDENT ON CULTIVATION SYSTEMS The array of laboratory cultivation systems that deﬁne the dynamic patterns of microbial growth is summarized in Table 11. Microbial growth patterns are distinguished by three features: • Regime of substrate supply (1, continuous supply; and 2, single-term addition) • Elimination of growing microorganisms ( , signiﬁcant; b, absent) • Magnitude of spatial gradients (a, homogeneous systems; b, heterogenous systems) Each speciﬁc cultivation procedure can be represented by a point inside a cube with the axes 1 to 2, to b, and a to

lj(s)Nj D(s0 lm s Ksj s)

DNj

i j n

kjriNj

j i

kirjNi

lj(s)Nj /Yj

j 1

s

(96)

Every drop in the saw-tooth dynamics of neutral mutants detected by their phage resistance can be interpreted as the appearance of other types of spontaneous mutant with higher growth capabilities. In a chemostat culture, such mutants overcompete and displace all other cells by virtue of their higher afﬁnity to limiting substrate (a decreased Ks value). Let such a mutant be denoted by the subscript k and l(s) be the average growth rate of all other subpop¯ ulations. The selection pressure for this mutant, r, is given by d(Mk /N) dt r Mk N [lk(s) l(s)] ¯ Mk N (97)

If Ksk Ksj( j k), then r 0 until a new equilibrium is established. In this process, the original cells will be displaced by the mutants and the growth-limiting substrate concentration will decrease from s1 ¯ DKsj /(lm D) to s2 ¯ DKsk /(lm D), and the culture density will rise by Y(s1 ¯ s2). ¯ Autoselection in Turbidostat and pH Auxostat. The afﬁnity to substrate was not always the only driving force of selection outcome. A number of instructive examples were reviewed by Pechurkin (14). 1. In turbidostat and pH auxostat, autoselection is in favor of mutants with higher maximum speciﬁc growth rates, r lmk lmj 0. Because the population density is kept constant instrumentally and the dilution rate is allowed to vary, then autoselection results in the increase in D from lmj to lmk. 2. Mutation toward a higher growth efﬁciency, Yk Yj, will lead to the same result as an increase in lm: r lk lj (lmk lmj)s/(Ksj s), as soon as lk qsYk and lj qsYj.

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KINETICS, MICROBIAL GROWTH

Table 11. Matrix of Cultivation Techniques Substrate input Continuous (1) Spatial organization Homogeneous (a) Cell eliminated ( ) 1a Chemostat Turbidostat pH-auxostat 1b Plug-ﬂow (tubular culture) Colonies No elimination (b) 1ab Fed-batch culture Dialysis culture Retentostat 1bb Column packed with microbe attached Single-term (2) Cell eliminated (a) 2a Continuous culture with substrate pulses Phased culture Forbidden combination No elimination (b) 2ab Simple batch culture

Heterogeneous (b)

b. The 2b combination is logically forbidden because any spatial segregation results in protracted substrate utilization and so transforms a batch process into a continuous one. The dynamics of microbial growth in any type of cultivation system can be described by the following mass– balance equations: ds dt dx dt F V G(s) H(x)x l(s)x/Y l(s)x mx (98)

where F is the substrate input rate; G(s) is the rate of unused substrate removal from cultivation vessel (washout, leaching, evaporation, etc.); V is the rate of microbial biomass input, which may be a single-term inoculation or continuous delivery of cells to the fermenter (specially designed or unintentional, e.g., contamination); and H(x) is the rate of microbial elimination, such as washout, death, grazing, or lysis. The rest of the notation is conventional. To simulate microbial growth dynamics in a particular homogeneous system, one has to make the following selection: F(t) 0, s0 0 for a category 2 (batch culture), F(t) 0 for category 1 (continuous cultivations); H(t) 0 for systems retaining cell biomass (dialysis, fed-batch, simple batch, column with immobilized cells), and H(t) 0 when cell elimination occurs (chemostat, turbidostat, phased culture, etc.) Spatially heterogeneous systems can be simulated either by partial derivatives or compartmental models (e.g., the total biomass x of a microbial colony may be represented as a sum of the peripheral and central components). 1a —Homogeneous Continuous Culture (Continuous-Flow Fermenters with Complete Mixing) There are two subgroups within this type of cultivation technique. In the ﬁrst one, steady-state growth is maintained naturally by the microbial culture itself. Selfregulation is performed through negative feedback that originates from the dependence of the growth rate on substrate concentration (chemostat) or on temperature (caloristat). In the second group, electronic devices are used for the automatic adjustment of dilution rate to the instantaneous growth rate of the microbial culture. Electronic control is based on the sensing of cell density or growth-

linked products, that is, optical density (turbidostat), culture liquid viscosity (viscostat), CO2 concentration in output air, culture pH (pH auxostat), and so on. In theory, the steady-state growth may be established in chemostat between 0 and lm, but in practical terms neither very low nor very high values are attainable because of the long time needed to reach the steady state in the ﬁrst case and the risk of culture washout in the second. The second group of continuous techniques (turbidostat, pH auxostat, viscostat, etc.) are capable of maintaining steady growth at high s when either l r lm or under substrate inhibition, when dl/ds 0. There is also a potpourri mixed technique known as the bistat (70), which combines a chemostat and a pH auxostat. The mass–balance equations for the chemostat and its modiﬁcations have already been given (equations 62 to 64). In a simple, complete mixing cultivator, all cells have an equal probability of being washed out, hence l D. If there is substantial wall growth, biomass retention, or feedback, then l D; this difference increases with the extent of biomass retention in the fermentation vessel. In terms of our scheme, such cultivation systems correspond to points on the edge 1a to 1ab. 1ab—Continuous Cultivation without Cell Washout This group embraces cultures with batch or continuous dialysis, fed-batch culture (FBC), and batch culture with a supply of limiting substrate via the gas phase (gases and volatile compounds). It also includes the chemostat with complete biomass feedback by means of ﬁltration (71). The latter fermenters are less reliable in practical terms as compared with dialysis culture, because the membrane ﬁlters are quickly plugged with cells. The limiting substrate is supplied into the dialysis culture through a semipermeable membrane and, in the case of a gaseous nutrient, through the gas–liquid interface. In both cases, mass transfer is reasonably well described by Fick’s law. The culture volume remains ﬁxed in all systems, with the exception of a FBC. In a FBC, a constant nutrient feed F provides a linear increase of culture volume V during the cultivation span; the dilution rate D F/V is decreased hyperbolically. The great advantage of these cultivation techniques for biotechnology is that they provide the possibility of realizing very slow continuous growth accompanied with derepression of synthesis of many secondary metabolites. With a constant limiting substrate ﬂux, Fs0,

KINETICS, MICROBIAL GROWTH

1541

the absence of cell washout means that at each subsequent moment an equal ration is shared by an increasing microbial biomass, and, as a result, l eventually falls down to negligible values or even to zero (the maintenance state). Unlike the chemostat, no true steady state is established in this case, but when the substrate is virtually depleted, we have ds/dt 0 and the system reaches a quasi steady state (5). If a quasi steady state approximation is found to be sufﬁcient, then extremely slow continuous growth can be obtained after a reasonable period of time, perhaps a few weeks, as compared to the several months needed in a chemostat. 2a —Continuous Cultivation with a Discontinuous Supply of Limiting Substrate Suppose we have a simple chemostat culture fed by nutrient medium lacking just one essential component. This component is added as a small volume of concentrated solution at regular and sufﬁciently large time intervals, Dt. Then ds dt dx dt s0(t) D[s0(t) l(s)x s] Dx iDt, i 0, 1, . . . , n (99) q(s)x

culture liquid, moving along the spatial coordinate z at a linear velocity f F/A (where F is ﬂow rate, cm3 /h, and A is the cross-sectional area, cm3), growth of biomass proceeds as in a simple batch culture. To account for the culture movement per se, we have to pass from ordinary to partial derivatives and replace dx/dt by x/ t f x/ z (along-the ﬂow-growth rate). Allowing for some dispersion of the moving front by diffusion, we can write s t x t f f s z x z Ds Dx s z2

2 2

q(s)x l(s)x (100)

x z2

where Ds and Dx are the diffusion coefﬁcients of the substrate and microbial cells, respectively. 1bb—Continuous-Flow Reactors with Microbes Attached The nutrient solution is pumped through a column ﬁlled with adsorbent material and is utilized as it moves by growing immobilized cells. The mass–balance equations for a packed column are obtained from equation 100 by simplifying the equation for x, s t f s z x t Ds s z2

2

q(s)x(z) Y[q(s) m]x (101)

A 0, when t 0, otherwise

l(s)x

This cultivation method was originally used to obtain synchronized cell division (72). A continuous phased culture (73) is a repeated simple batch culture, that is, at regular intervals, Dt, half of the culture volume is withdrawn and the fermenter is reﬁlled with an equal volume of fresh medium. Under such conditions, repeated batchwise growth proceeds with biomass increasing cyclically from x0 to 2x0. Obviously, the growth dynamics are governed by the time interval between consecutive substrate additions Dt. If Dt ln2/lm, a sawtooth nonlimited growth takes place as in a turbidostat. With Dt ln2/lm the culture should be washed out and, when Dt ln2/lm, the maximal attainable biomass decreases with increasing Dt because of endogenous biomass decomposition and waste respiration during the lag phase. 2ab—Simple Batch Culture Cultivation begins at the initial limiting substrate concentration, s0, and inoculum size, x0. The biomass reaches its maximum, xm, when the limiting substrate is depleted (s 0) and then declines even in the absence of exogenous elimination, so that x r 0 as t r . Description of batch dynamics has been given earlier. Speciﬁed growth phases are described by simple nonstructured models (equations 56, 61, and 73), and entire dynamics are described by SCM and other structured models (equations 84 to 86; Fig. 8). 1b —Plug-Flow (Tubular) Culture The inoculum and the medium are mixed on entry into a long reactor tube, and the culture ﬂows in the tube at a constant velocity without mixing. In each small element of

Bacterial cells accumulate faster on the top of the column because of larger q(s) values, and as a result, a distinctive spatial biomass distribution develops in the form of a hyperbolic decrease of x with column depth. Growth does not reach steady state with respect to x until cell elimination becomes well expressed (the effects of inhibitory products, endogenous cell decomposition, and leaching). Colonies Besides continuous-ﬂow columns, other heterogeneous systems are widely used. Especially popular is plating on solid media made from natural or synthetic gels (agar, PAAG, silica gel, synthetic alumosilicates, etc.) as well as on some porous materials (sand, glass beads, glass ﬁber). Impregnated with nutrient solution, such materials are used to grow microorganisms in the form of colonies or lawns. At ﬁrst glance, it is tempting to consider these techniques as analogies of a simple batch culture with singleterm substrate input (type 2bb in Table 11). However, a closer look at the mechanism of colony growth reveals a greater resemblance to chemostat culture! Here we will outline our considerations. The spread of a colony over a solid substrate, for example, a layer of agar, proceeds by the growth of only a peripheral zone with biomass, xp (Fig. 10). Then ds dt dx dt q(s)x lwxp ax (102)

where a is the speciﬁc decay rate of cell (mycelium) com-

1542

KINETICS, MICROBIAL GROWTH

w R

Agar

w R

Agar

Figure 10. Schematic illustration of the difference between the growth modes bacterial (top) and fungi (bottom) colonies. Note that unicellular organisms (bacteria) do not penetrate into the agar layer as opposed to ﬁlamentous fungi. Arrows indicate directions of substrate diffusion across concentration gradients. R and w are, respectively, radius and width of the peripheral zone of the colony.

ponents. Straightforward geometrical analysis shows that the linear spread rate for a regularly shaped colony (a cylinder, a sphere, or a strip) of size R is given by the following relation (2,5): dR dt or R R0 Krt (103) lww Kr

parable with the waste cell suspension that is discharged into the product bottle. A steady running of a pump, delivering medium at a rate, F (cm3 /h), corresponds to the active and uniform substrate utilization by the growing mycelium at a rate KrA (cm3 /h, A is the area of colony– medium interface). Finally, both cultivation systems are characterized by the elimination of propagating biomass, that is, by the expulsion of grown cells (mycelium) from further active growth (either by washing out or by the motion of the colony front). In both cases, the elimination rate is equal to biomass growth in the active compartment. The essential deviation from the chemostat lies in properties 3 and 4. Spatial differentiation of hypha and the direction of colony expansion are governed by spatial gradients of limiting substrate and, possibly, some metabolic products. Steepness of gradients is partly diminished by the effects of metabolic translocation along hypha over distances of the order of w. It can be concluded from the previous discussion that colony growth belongs to class 1b , and not to 2bb. In general, it is very likely that the 2b combination is an empty, logically forbidden combination, because a single-term momentary input of substrate may only be realized in a homogeneous system. Any spatial segregation whatever will actually prolong the consumption of substrate and, therefore, transform a batch process into a continuous one. For this reason, growth on any insoluble substrate (lignocellulose and other plant polymers, oil droplets, grains of sulfur, etc.) should always be treated as a continuous process. The solid-phase fermentation can also not be anything but continuous, whether new portions of substrate are added to the reactor or not. (Obviously, this applies to the growth mechanism itself and not to the engineering operation.) BIBLIOGRAPHY

1. A.A. Esener, J.A. Roels, and N.W.F. Kossen, Biotechnol. Bioeng. 25, 2803–2841 (1983). 2. N.S. Panikov, Microbial Growth Kinetics, Chapman & Hall, London, 1995. 3. I.G. Minkevich and V.K. Eroshin, Uspekhi Sovrem. Biol. 82, 103–116 (1976). 4. W.R. Mayberry, G.J. Prochazka, and W.J. Payne, Appl. Microbiol. 15, 1332–1338 (1967). 5. IS.J. Pirt, Principles of Microbe and Cell Cultivation, Blackwell Scientiﬁc Publishers, Oxford, U.K., 1975. 6. A.G. Marr, E.H. Nilson, and D.J. Clark, Ann. N.Y. Acad. Sci. 102, 536–548 (1963). 7. A.G. Dorofeyev and N.S. Panikov, Mikrobiologiya–Engl. Tr. 64, 601–609 (1994). 8. D.W. Tempest and O.M. Neijssel, Annu. Rev. Microbiol. 38, 459–486 (1984). 9. S.J. Pirt, Arch. Microbiol. 133, 300–302 (1982). 10. N. Panikov and J.S. Pirt, J. Gen. Microbiol. 108, 295–303 (1978). 11. T. Bauchop and S.R. Elsden, J. Gen. Microbiol. 23, 457–469 (1960). 12. V.N. Ivanov, Energetics of Microbial Growth, Naukova Dumka, Kiev, 1981 (in Russian). 13. W.M. van Gulik, and J.J. Heijnen, Biotechnol. Bioeng. 48, 681–698 (1996).

where Kr is the colony linear expansion rate, mm/h; lw is the microbial speciﬁc growth rate within the peripheral zone; and w is the zone width. In the case of unicellular organisms (e.g., bacteria, yeasts) that are incapable of penetrating into the gel’s matrix, the substrate is available through passive diffusion to the peripheral zone from the underlying gel layer. As the colony grows, this ﬂux diminishes, Kr decreases, and eventually the growth stops altogether. The ﬁlamentous organisms (fungi and actinomycetes) are able to propagate both on the surface and in the depth of gel. As a result, their growth is not completely dominated by diffusion effects, and the colony front advances at a faster rate than substrate is depleted in the frontier zone. The colony spreads at a constant radial rate, Kr, until the Petri dish is ﬁlled or the agar deteriorates from dryness. (In the case of rich nutrient media, there are also effects of self-inhibition by metabolic products [see equations 76 and 77].) Thus, the colony growth is (1) continuous, (2) substratelimited, (3) directed, and (4) spatially ordered. Properties 1 and 2 suggest a strong similarity between growth of a colony (especially a fungal one) and that of a chemostat culture. If so, a colony’s peripheral zone is analogous to the cell culture in the fermenter, and its central part is com-

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