Kinetic and Thermodynamic Aspects of Enzyme ... - ACS Publications

16280

J. Phys. Chem. B 2010, 114, 16280–16289

Kinetic and Thermodynamic Aspects of Enzyme Control and Regulation† Johann M. Rohwer*,‡ and Jan-Hendrik S. Hofmeyr*,‡,§ Triple-J Group for Molecular Cell Physiology, Department of Biochemistry, and Centre for Studies in Complexity, Stellenbosch UniVersity, PriVate Bag X1, Stellenbosch 7602, South Africa ReceiVed: September 3, 2010; ReVised Manuscript ReceiVed: October 7, 2010

This paper develops concepts for assessing and quantifying the regulation of the rate of an enzyme-catalyzed reaction. We show how generic reversible rate equations can be recast in two ways, one making the distance from equilibrium explicit, thereby allowing the distinction between kinetic and thermodynamic control of reaction rate, as well as near-equilibrium and far-from-equilibrium reactions. Recasting in the second form separates mass action from rate capacity and quantifies the degree to which intrinsic mass action contributes to reaction rate and how regulation of an enzyme-catalyzed reaction either enhances or counteracts this massaction behavior. The contribution of enzyme binding to regulation is analyzed in detail for a number of enzyme-kinetic rate laws, including cooperative reactions. Introduction The last four decades have seen a massive increase in our understanding of the behavior, control, and regulation of networks of enzyme-catalyzed reactions. This was facilitated by the development of both sophisticated simulation software and theoretical frameworks such as metabolic control analysis1,2 and biochemical systems theory3 for analyzing the control properties of these networks. These developments highlighted both the need for enzyme kinetics of reversible reactions, as opposed to the irreversible rate equations that emerged from studies on enzyme mechanism, and the need for a way of analyzing these rate equations to clarify how phenomena such as saturable binding, cooperativity, and allosterism contribute to the regulation of reaction rate. The central aim of this paper is to develop concepts that allow us to address the latter question of regulation, building on a previous exploration of this matter.4 More specifically, the paper aims to quantify: (1) the degree to which thermodynamics contributes to the rate of a chemical reaction, whether catalyzed or uncatalyzed, thereby distinguishing between near-equilibrium and far-from-equilibrium reactions and, concomitantly, kinetic and thermodynamic control of reaction rate, and (2) the degree to which intrinsic mass action contributes to reaction rate and how regulation of an enzyme-catalyzed reaction either enhances or counteracts this mass-action behavior. The first two sections show how generic reversible rate equations can be recast in two ways, one making the distance from equilibrium explicit, thereby allowing the distinction between kinetic and thermodynamic control of reaction rate, and the other separating mass action from rate capacity. We propose a quantitative measure for distance from equilibrium and compare it with a previously suggested measure.5 The rest of the paper explores the effects of noncooperative and cooperative binding on the sensitivity of enzyme rate to changes in substrate and product concentration, comparing the reversible

Michaelis-Menten, Hill,6 and Monod-Wyman-Changeux7 mechanisms. The importance of using reversible rate equations in metabolic modeling has been stressed8,9 and underlines the need for accurate values for the equilibrium constants of reactions. It therefore gives us great pleasure to be able to acknowledge Bob Alberty’s seminal work in biothermodynamics10,11 as an indispensable cornerstone of the work described in this paper. Without databases of thermodynamic data such as that of Goldberg et al.,12 in which the legacy of Alberty’s work is abundantly clear, metabolic modeling of real pathways would be very difficult, if not impossible.

Contribution of Thermodynamics to Chemical Reaction Rate The net reaction rate V ) Vf - Vr of any reversible reaction (where Vf and Vr symbolize the forward and reverse reaction rate) can be recast in a form where the thermodynamic contribution, i.e., the distance from equilibrium, is made explicit

( )

V ) Vf 1 -

Vr Vf

(1)

In eq 1, Vr/Vf can be seen to be an explicit thermodynamic term that depends only on the distance from equilibrium by considering the general chemical reaction

mA + nB h pC + qD

(2)

†

Part of the “Robert A. Alberty Festschrift”. * To whom correspondence should be addressed. E-mail: [email protected]; [email protected]. Phone: +27 21 8085843. Fax: +27 21 8085863. ‡ Department of Biochemistry. § Centre for Studies in Complexity.

Assuming that the kinetics of this reaction follows the law of mass-action13 and that reaction order equals molecularity, the rate equation can be written as

10.1021/jp108412s  2010 American Chemical Society Published on Web 10/28/2010

Kinetic and Thermodynamic Aspects of Enzyme Control

(

V ) kfambn - krcpdq ) kfambn 1 -

)

cpdq kf / ) ambn kr

(

kfambn 1 -

Γ Keq

)

J. Phys. Chem. B, Vol. 114, No. 49, 2010 16281

(3)

Concentrations (more correctly, activities) of all substances are denoted by the lowercase italic counterpart of the substance; i.e., the concentration of A, B, or C is a, b, or c. The ratio (cpdq)/ (ambn) is called the mass-action ratio, symbolized by Γ (also known as the reaction quotient, Q), while the ratio kf/kr is the equilibrium constant, Keq, of the reaction or, equivalently, the p q m n deq)/(aeq beq) with concentrations that are obtained at ratio (ceq equilibrium when the net reaction rate V ) 0. The ratio Γ/Keq is called the disequilibrium ratio, symbolized by F. Note that whereas the exponents to which concentrations in Γ are raised are always the stoichiometric coefficients with which the corresponding reagents occur in the balanced reaction equation, the exponents in the forward rate term are in general experimentally determined reaction orders that are not necessarily derived from reaction stoichiometry. Given the standard relationship ∆rG ) RT ln F between Gibbs reaction energy and the disequilibrium ratio, the following is therefore one possible formulation of a general rate equation for a chemical reaction.

V ) Vf(1 - F) ) Vf(1 - e∆rG/RT)

(4)

The forward rate (or kinetic) term Vf has rate units, for example, mole amount · time-1 or concentration · time-1; its functional form depends on the mechanism of the reaction and must therefore be determined experimentally. The term (1 - F), on the other hand, depends only on the equilibrium constant and on concentrations raised to the power of stoichiometric coefficients; its form is always the same. When the reaction is in equilibrium (F ) 1) this term is zero; otherwise, when F < 1 it is positive, indicating a net reaction in the forward direction, or when F > 1, it is negative, indicating a net reaction in the reverse direction. The greater the distance from equilibrium (i.e., the smaller F), the faster the rate of reaction. A first approach to assessing the contributions of kinetics and thermodynamics to reaction rate could be to plot V against a in linear rate and concentration space (Figure 1). However, this picture does not suggest a way of assessing in which concentration range the rate is mainly determined by the forward rate (i.e., by kinetics) and in which concentration range it is mainly determined by the distance from equilibrium. The linear representation obscures the discrimination between kinetic and thermodynamic control. The solution to this problem is to take the logarithm of eq 3 to obtain additivity between the forward rate term and the thermodynamic term.

ln V ) ln(kfambn) + ln(1 - F)

Figure 1. Net rate of the chemical reaction in eq 2 as given by eq 3 plotted as a function of the concentration of reactant A. The dashed lines show the contributions of the kinetic and thermodynamic terms to the reaction rate, V. The forward rate constant kf ) 10, c ) 5, while Keq, b, d, m, n, p, and q were all set to 1; equilibrium therefore is obtained at a ) 5.

equilibrium region where the thermodynamic function dominates. We suggest that the degree to which the rate is either kinetically or thermodynamically controlled can be quantified in terms of the fractional difference between the slope of the logarithmic rate function and the slope of either the logarithmic forward rate or thermodynamic function. To obtain these slopes, we rewrite eq 5 as a function of ln a (using the equivalence a ) eln a).

(

ln V ) ln(kfbn) + m ln a + ln 1 - e-m ln a

cpdq bnKeq

)

(6)

and partially differentiate with respect to ln a using the chain rule and the rule [d/(dx)]aebx ) abebx to give

∂ln Vf ∂ln V(1-F) ∂ln V ) + ∂ln a ∂ln a ∂ln a )

∂ (ln(kfbn) + m ln a) + ∂ln a cpdq ∂ ln 1 - e-m ln a n ∂ln a bK

(

)m+ )

(7)

mF 1-F

m 1-F

eq

)

(8) (9) (10)

From here on we shall consistently use the notation εaV, εaVf, and εVa(1-F) for the partial derivatives in this equation. This symbolism stems from metabolic control analysis1,2 (which will play an important part in what follows), where such partial derivatives are called elasticity coefficients or just elasticities. We have therefore established that εaV ) εaVf + εaV(1-F) where

(5)

Figure 2A and B contains plots of V against reactant a or product c with double logarithmic coordinates (the same plots that would be obtained by plotting ln V against ln a or ln c). The right-hand ln V axis is included as an aid to verify the additivity of the forward rate term and the thermodynamic term. It is clear that there is a region far from equilibrium where the rate is effectively determined by the kinetic function and a near-

εVa )

m mF , εVf ) m, εVa(1-F) ) 1-F a 1-F

(11)

Similarly, for product C, the corresponding elasticities are

εVc )

-pF -pF , εVf ) 0, εVc(1-F) ) 1-F c 1-F

(12)

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J. Phys. Chem. B, Vol. 114, No. 49, 2010

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Figure 2. Net rate of the chemical reaction in eq 2 as a function of (A & C) the concentration of reactant A and (B & D) the concentration of product C, plotted with logarithmic coordinates. (A & B) refer to the recasting of the rate equation in eq 3, the dashed lines showing the contributions of the kinetic and thermodynamic terms to the reaction rate, V. (C & D) refer to the recasting of the rate equation in eq 15, the dashed lines showing the contributions of the rate capacity and mass-action terms to the reaction rate, V. The forward rate constant kf ) 10, while Keq, b, d, m, n, p, and q were all set to 1. In (A & C), c ) 5, so that Γ ) 5/a, while in (B & D) a ) 5, so that Γ ) c/5. Equilibrium is therefore approached as a (in A & C) and c (in B & D) tend to 5. As explained in the text, the vertical dotted lines in (A) show the points where the disequilibrium ratio F is 0.1 (right-hand line, a ) 50) and 0.9 (left-hand line, a ) 5/0.9). The shaded region in (A) corresponds to Rolleston’s5 suggested demarcation between near-equilibrium and far-from-equilibrium conditions (discussed in the text).

Near equilibrium, where Ff1, the reactant and product elasticities tend to +∞ and -∞, respectively. Far from equilibrium in the forward direction, where F f 0, εaV tends toward the reaction order m of A, while εVc tends to zero. (Recall that although reaction order is usually defined as the exponent to which a concentration is raised in the rate equation, the alternative definition ∂ln V/∂ln a is more general, as it allows for the reaction order to vary with concentration, as is clearly the case in eq 10. This is another reason for preferring the alternative term “elasticity” to denote the partial derivative form.) All three reactant elasticities depend on m, the stoichiometric coefficient of A, but when the appropriate ratios of elasticities are considered m vanishes, leaving purely thermodynamic discriminators in terms of F. Each ratio is also equivalent to one of the required fractional differences between elasticities, as follows

εVa - εVaf εVa

)

εVa(1-F) εVa

) F (discriminator for kinetic control)

(13) and εaV - εaV(1-F) εaV

)

εaVf

) εaV 1 - F (discriminator for thermodynamic control)

(14)

If we arbitrarily consider the reaction to be controlled by either the forward rate or the distance from equilibrium when the relevant fractional difference in elasticities is 0.1, then eq 13 translates to F e 0.1 (kinetically controlled in the forward direction), while eq 14 translates to F g 0.9 (thermodynamically

controlled); in the range of 0.1 < F < 0.9, the reaction rate depends to a varying degree on both kinetic and thermodynamic functions (see Figure 2A). Traditionally, reactions have been classified as either “equilibrium” or “nonequilibrium”,14 an often-used criterion being that of Rolleston,5 who suggested that: The rate of the reVerse reaction can be regarded as negligible when less than 5% of the rate of forward reaction, and to play a large part in determination of the oVerall rate of flow through the reaction if it is greater than 20% of that of the forward reaction. Application of the relationship Vr/Vf ) Γ/Keq leads to a range of Values of Γ/Keq (0.05-0.2), that separates nonequilibrium from equilibrium reactions. This criterion corresponds to the shaded region in Figure 2A. We suggest that this seemingly arbitrary criterion does not adequately demarcate the situation near-equilibrium from that far-from-equilibrium (our preferred terms). In the F-region proposed by Rolleston, the reaction is still very far from equilibrium; as suggested above, Figure 2A shows that a F-value larger than 0.9 would be a better indication of a reaction near to equilibrium, while F-values less than 0.1 would signify farfrom-equilibrium reactions. In fact, we propose that the alternative concept of regions of kinetic and thermodynamic control of reaction rate is more useful. In the next sections, we quantify this concept as terms in algebraic expressions for elasticity coefficients. Contributions of Rate Capacity and Mass Action to Reaction Rate We have previously4 defined the regulation of a reaction as the modification of its intrinsic mass-action properties; to isolate the contribution of mass action to the rate of a reversible



reaction, its rate equation can be recast in a form that cleanly separates it into a time-dependent, but concentration-independent, component and a mass-action component that depends only on concentrations. Using eq 2 as an example

(

V ) kfambn - krcpdq ) kf ambn -

cpdq Keq

)

(15)

(

)

p ) ln Vcap + ln Θ + Keq ln Vma (18)

ln V ) ln(kse0) + ln Θ + ln s -

At this point, let us return to the elasticities as already briefly introduced in eq 11 and eq 12. In the framework of metabolic control analysis,1,2,16 such an elasticity quantifies the effect of any molecular species on an enzymatic reaction in isolation.

or, in logarithmic form

(

p q

cd ln V ) ln kf + ln a b Keq m n

)

εsV )

Elasticities of Enzymatic Rate Laws The previous section has dealt with splitting a chemical rate equation into components to dissect the relative contributions of mass action and rate capacity. The mass-action term is present in any rate equation; for a simple chemical rate equation such as eq 15, the rate capacity term only consists of a single rate constant. In biological systems, however, enzymes bring two additional properties into the mix, i.e., binding and catalysis. These allow for additional levels of regulation. Any reversible rate equation for an enzyme-catalyzed reaction S h P can always be cast in the following form15 (for the rest of the paper, we use a simple uni-uni reaction as an example, so as not to complicate the binding terms unnecessarily).

(

)

( )

) p,r, ...

( ∂V/V ∂s/s )

) p,r, ...

V ( ∂ln ∂ln s )

p,r, ...

(19)

(16)

The bracketed term on the right contains only the equilibrium constant and concentrations, which are, as before, raised to exponents that represent the stoichiometry with which each species participates in the reaction; it describes the contribution of mass action alone to reaction rate. For this simple reaction, the kinetic contribution to rate (the left-hand term) is simply the rate constant kf with units of concentration-(m+n-1) · time-1 (further on we shall see that in enzyme rate equations this term equates to the specificity constant multiplied by the enzyme concentration). We refer to this term as the rate capacity of the reaction. Figure 2C and D show the contributions of mass action and rate capacity to reaction rate with varying concentrations of reactant A and product C. The mass-action contribution varies with concentration, while the rate capacity term kf augments mass action by a constant factor (here 10). The elasticities with respect to the mass action term are of course equal to the elasticities with respect to reactant or product concentration, i.e., those given in eqs 11 and 12.

p V ) kse0Θ s Keq

s ∂V V ∂s

where s is the concentration of any molecular species (substrate, product, or effector) that affects the enzyme directly. The subscript p, r, ... indicates that the concentrations of all other substrates, products, and effectors are kept constant at their prevailing values while s is varied. As mentioned before, elasticities are apparent kinetic orders, which derive directly from the kinetic properties of the enzymes. With the definition of eq 19, the elasticity expression for the substrate of the enzymatic rate equation in eq 18 is given by

∂ln(kse0) ∂ln(s - p/Keq) ∂ln Θ + + ∂ln s ∂ln s ∂ln s ∂ln Θ 1 + ) 0+ ∂ln s 1-F ) εsVcap + εsΘ + εsVma

εsV )

(20)

The rate capacity term, being constant, never contributes toward elasticity, whether toward substrates, products, or effectors. Hofmeyr4 defined regulation as the “counteraction or augmentation of the intrinsic mass-action trend in open reaction networks”; eq 20 neatly splits the contributions of mass action and binding to the value of the elasticity. Comparison of this equation with eq 11, which was written for the pure mass-action case, shows that the contribution of enzyme binding to the elasticity is given by the term εsΘ. For further discussion of the bounds between which both the binding and the mass-action terms vary, as well as the conditions under which either dominates, the reader is referred to Hofmeyr.4 Equation 20 also illustrates that the only distinguishing feature between different enzyme mechanisms is given by different expressions for Θ; the mass-action term is always the same. The effect of various mechanisms, including cooperativity, is exemplified in the next section. Kinetic Regulation in Different Enzymatic Rate Laws

(17)

where ks ) kcat/KM is the specificity constant of the enzyme for S and e0 is the total enzyme concentration. The term kse0 is equivalent to the rate capacity term kf in eq 15 and has units of time-1; we denote it by the shorthand Vcap. (The limiting forward rate, Vf, is given by kcate0). The presence of the enzyme now adds an additional binding term to the rate equation: Θ quantifies the effect of saturation on the reaction rate. Denoting the final mass-action term by Vma, the logarithmic form of eq 17 reads

Enzyme catalysis has a profound effect on chemical reactions, not only in terms of a vastly increased speed of catalysis but also by opening up additional levels of regulation. Here, we explore this in greater detail, first with the reversible MichaelisMenten equation, and subsequently with two rate laws for enzymes with cooperative kinetics, viz., the reversible Hill and Monod-Wyman-Changeux equations. Reversible Michaelis-Menten Kinetics. The reversible Michaelis-Menten equation for an enzyme-catalyzed reaction S h P can be rewritten in the form of eq 17 to separate the rate capacity, binding, and mass-action terms.

16284


V)

Vf × Ks

(

1 p × sKeq s p 1+ + Ks Kp

)

Rohwer and Hofmeyr

where Ks and Kp are the Michaelis constants for S and P, respectively, and Vf is the limiting rate in the forward direction. The substrate and product dependence of the reversible Michaelis-Menten equation is shown in Figure 3A and C, respectively. Both graphs indicate how the rate capacity, binding, and mass-action terms contribute to the overall rate. In particular, comparison with Figure 2C and D shows that the rate capacity and mass-action terms are the same as for an ordinary uncatalyzed chemical reaction, except of course that the magnitude of the rate capacity may be vastly increased by the enzyme. As s increases, the binding term becomes smaller (more negative in logarithmic space) and precisely counteracts the mass-action term, so that the overall reaction rate tends to a constant value as the enzyme saturates with S (Figure 3A). By contrast, when p is increased, the binding and mass-action term augment each other, and the reaction rate starts to decrease at a lower value of p as would have been the case for an uncatalyzed chemical reaction (Figure 3C). As shown previously,4 the value of the elasticities toward substrate (εsV) and product (εpV) are given by

εsV )

1+

s Ks

s p + Ks Kp

+

1 ) εsΘ + εsVma 1-F

(

)

1 + (σ + π)h

(25)

1 (1 + σ + π)(σ + π)h-1 × 1+σ+π 1 + (σ + π)h ) Θ1 × Θh

Θ )

(26) Here, Θ1 is identical to the Θ-term in eq 21, while Θh quantifies how the contribution of the binding term to the rate is modified through cooperativity and subunit interactions; naturally, this term depends on h. The elasticity of the reversible Hill equation toward S is obtained by symbolic differentiation (see Hofmeyr et al.17) of eq 24. Importantly, as in eq 26 above, we now split εsΘ to separate the contribution of a single subunit (εsΘ1) from that of subunit interaction and cooperativity (εsΘh).

∂ln V ) εsΘ + εsVma ∂ln s (h - 1)σ 1 hσ(σ + π)h-1 + ) h σ+π 1 F 1 + (σ + π)

εsV )

(22)

(

)

σ -σ (h - 1)σ + + 1+σ+π 1+σ+π σ+π

(

)

1 hσ(σ + π)h-1 + h 1 F 1 + (σ + π) ) εsΘ1 + εsΘh + εsVma

(23)

(27)

The variation of εsV with s is plotted in Figure 3B and that of εVp with p in Figure 3D. The split between the binding and massaction contributions to the elasticities is also shown in both cases. The binding term elasticity (εΘ) varies between zero (at low s or p) and -1 (at high s or p). The mass-action term elasticity becomes numerically very large as the reaction approaches equilibrium (small s in Figure 3B; large p in Figure 3D). Both εVs and εVp are thus dominated by thermodynamics close to equilibrium, whereas further from equilibrium the binding term contributes significantly to their value. Reversible Hill Kinetics. While the reversible MichaelisMenten equation takes into account thermodynamic reversibility and the effect of binding, it cannot account for cooperative binding of substrate or product nor for allosteric interactions. A possible kinetic rate law that takes these properties into account is the reversible Hill equation, which is given by6

Vf p s(σ + π)h-1 s0.5 Keq

(σ + π)h-1 1 + (σ + π)h

To separate the contributions of a single subunit in isolation from those of subunit interaction and cooperativity, we rewrite eq 25 as

)

p K p -F εVp ) + ) εΘp + εVpma 1-F s p 1+ + Ks Kp

V)

Θ)

(21)

(24)

with σ ) s/s0.5 and π ) p/p0.5; s0.5 and p0.5 denote the halfsaturation constants for S and P, respectively, and h is the Hill coefficient. The binding term is now given by

Likewise, εpV for the reversible Hill equation is given by εVp )

-π ∂ln V ) + ∂ln p 1+σ+π

(

)

(h - 1)π hπ(σ + π)h-1 -F π + + 1+σ+π σ+π 1-F 1 + (σ + π)h

Θh Vma 1 ) εΘ p + εp + εp

(28) The contribution of all the terms in the reversible Hill equation (eq 24) to the reaction rate and their effect on the elasticity are graphically presented in Figure 4. The rate capacity, mass-action, and single-subunit binding terms (and their elasticities) are identical to the reversible Michaelis-Menten case (compare Figure 4A with Figure 3A and Figure 4C with Figure 3C) and will not be further discussed here. By contrast, the origin of cooperativity becomes very apparent: the term Θh contributes negatively (in logarithmic space) to the rate and “pulls down” the rate curve at s values around 100 (Figure 4A). This yields the cooperative response (an elasticity >1 away from equilibh rium); at these values of s the εΘ s component has a large positive value and contributes significantly to the overall elasticity (Figure 4B). In theory, the maximal value of εsΘ (i.e., εsΘ1 + εsΘh) is h - 1 (at low s and p, see eq 27), and the maximal value of the overall elasticity is h in the absence of P (see eq



Figure 3. Substrate (A) and product dependence (C) of the net rate of an enzyme-catalyzed reaction obeying reversible Michaelis-Menten kinetics as given by eq 21, plotted with logarithmic coordinates. The dashed lines show the contributions to the overall rate V of the rate capacity (Vcap), binding (Θ), and mass-action (Vma) terms, as defined in eq 18. (B) The elasticity of the reaction with respect to S as given by eq 22. The dashed lines show the contributions to the overall elasticity εsV of the binding (εsΘ) and mass-action (εsVma) terms. (D) The elasticity of the reaction with Vma respect to P as given by eq 23. The dashed lines show the contributions to the overall elasticity (εVp) of εΘ p and εp . The parameters were: Vf ) 100, Keq ) 10, Ks ) 5, Kp ) 1, p ) 1 (A,B), and s ) 1 (C,D).

Figure 4. Substrate (A) and product dependence (C) of the net rate of an enzyme-catalyzed reaction obeying reversible Hill kinetics as given by eq 24, plotted with logarithmic coordinates. The dashed lines show the contributions to the overall rate V of the rate capacity (Vcap), single-subunit binding (Θ1), subunit interaction (Θh), and mass-action (Vma) terms, as defined in eqs 18 and 26. (B) The elasticity of the reaction with respect to S as given by eq 27. The dashed lines show the contributions to the overall elasticity εsV of the single-subunit binding (εsΘ1), subunit interaction (εsΘh), and mass-action (εsVma) terms. (D) The elasticity of the reaction with respect to P as given by eq 28. The dashed lines show the contributions to the Θh Vma 1 overall elasticity (εVp) of εΘ p , εp , and εp . The parameters were: Vf ) 100, Keq ) 10, and h ) 4. In (A,C), s0.5 ) 5, p0.5 ) 10, and p ) 1. In (B,D), s0.5 ) 10, p0.5 ) 1, and s ) 1.

33 below); the fact that this value is not reached in Figure 4B is the result of P being present in significant concentration. Figure 4C illustrates another defining feature of the reversible Hill equation, i.e., product activation when p ≈ p0.5, and this concentration of P is insufficiently high to cause thermodynamic inhibition of the reaction rate. The product activation stems from the σ + π term in the numerator of eq 24 and can lead to interesting phenomena such as bistability and multistationarity.18 Product activation is synonymous with a positive value for εpV;

εpΘh is the only positive term in the product elasticity and dominates at these concentrations of P, making the overall elasticity positive (Figure 4D). The reversible Hill equation also describes the effects of allosteric modifiers (inhibitors or activators) on the reaction rate. However, as these modifiers do not take part in the reaction and hence do not contribute to the thermodynamics, their contribution is purely kinetic. Since this paper deals with separating kinetic and thermodynamic aspects of metabolic

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Rohwer and Hofmeyr

Figure 5. Substrate (A) and product dependence (C) of the net rate of an enzyme-catalyzed reaction obeying reversible MWC kinetics as given by eq 29, plotted with logarithmic coordinates. The dashed lines show the contributions to the overall rate V of the rate capacity (Vcap), singlesubunit binding (Θ1), subunit interaction (Θn), and mass-action (Vma) terms, as defined in eqs 18 and 30. (B) The elasticity of the reaction with respect to S as given by eq 31. The dashed lines show the contributions to the overall elasticity εsV of the single-subunit binding (εsΘ1), subunit interaction (εsΘn), and mass-action (εsVma) terms. (D) The elasticity of the reaction with respect to P as given by eq 32. The dashed lines show the Vma 1 Θn contributions to the overall elasticity (εVp) of εΘ p ,εp , and εp . The parameters were: Vf ) 100, Keq ) 10, L ) 200, and n ) 4. In (A,B) Ks ) 5, Kp ) 10, and p ) 1. In (B,D), Ks ) 10, Kp ) 1, and s ) 1.

regulation, allosteric modifiers will not be discussed here (for an analysis of their elasticity coefficients, refer to ref 17). Reversible Monod-Wyman-Changeux Kinetics. A different model for cooperativity and allosterism is given by the Monod-Wyman-Changeux (MWC) mechanism, whereby an enzyme can exist in one of two forms, the tense (T) or relaxed (R) state. Assuming that S and P only bind to the R state, and furthermore that only the R state is active, the reversible MWC equation is given by7,19

ν)

(

)

Vf p s(1 + σ + π)n-1 Ks Keq L + (1 + σ + π)n

(29)

MWC equation (eq 29) to the reaction rate, and their effect on the elasticity, are graphically presented in Figure 5. As previously, we obtain the elasticities of the reversible MWC equation toward S and P by symbolic differentiation17 of eq 29. Again, the term εΘ is split to separate the contribution of a single subunit (εΘ1) from that of subunit interaction and cooperativity (εΘn). εsV )

∂ln V -σ ) + ∂ln s 1+σ+π

(

)

1 (n - 1)σ nσ(1 + σ + π)n-1 σ + + 1+σ+π 1+σ+π 1-F L + (1 + σ + π)n

) εsΘ1 + εsΘn + εsVma

(31)

with σ ) s/Ks and π ) p/Kp; Ks and Kp denote the intrinsic dissociation constants of S and P, respectively, with a single R-state subunit. L ) t0/r0 is the equilibrium ratio of T and R state enzyme concentrations in the absence of substrate or product, and n is the number of enzyme subunits. As was done in eq 26 for the reversible Hill equation, we separate the contributions of a single subunit in isolation from those of subunit interaction and cooperativity by rewriting the binding term as

1 (1 + σ + π)(1 + σ + π)n-1 × 1+σ+π L + (1 + σ + π)n × Θ Θ ) 1 n

Θ )

(30) Note that Θ1 is the same in eqs 26 and 30 and identical to Θ in eq 21. Again, Θn quantifies how the contribution of the binding term to the rate is modified through cooperativity and subunit interactions. The contributions of all the terms in the reversible

εpV )

∂ln V -π ) + ∂ln p 1+σ+π

( )

)

-F (n - 1)π nπ(1 + σ + π)n-1 π + + 1+σ+π 1+σ+π 1-F L + (1 + σ + π)n

εpΘ1

+

εpΘn

+

εpVma

(32) Despite the fact that the reversible Hill and MWC equations were derived with vastly different assumptions, their kinetic profiles are quite similar, and the same arguments that were made for the reversible Hill equation in the previous section hold here equally well. For example, the subunit-interaction term Θn and its elasticity εsΘn are the sole determinants of the cooperative response (Figure 5A,B), and the product activation and its associated positive product elasticity, which is observed with the MWC equation just as with the reversible Hill equation, is due to Θn and εpΘn (Figure 5C,D). In spite of the similarities between the kinetic profiles of the Hill and MWC mechanisms, their elasticity profiles differ to some extent, especially at low substrate concentrations (compare



Figure 6. Substrate and product dependence of the reversible Hill (A and C) and MWC (B and D) equations at different levels of saturation with product and substrate, respectively. Common parameters: Vf ) 100, Keq ) 10, p ) 1 (when varying s), and s ) 1 (when varying p). Reversible Hill parameters (A,C): h ) 4; when varying s (A): s0.5 ) 5, p0.5 as indicated on the graph; when varying p (C): p0.5 ) 1, s0.5 as indicated on the graph. MWC parameters (B,D): n ) 4, L ) 200; when varying s (B): Ks ) 5, Kp as indicated on the graph; when varying p (D): Kp ) 1, Ks as indicated on the graph.

Figure 4B and Figure 5B). The elasticity of the Hill enzyme is greater than that of the MWC enzyme. In the limit, when sf0, and in the absence of product, the Hill elasticity tends to h, whereas the MWC elasticity tends to unity, as can be derived from the elasticity expressions

Hill: lim εsV ) lim p)0,sf0

σf0

MWC: lim εsV ) lim p)0,sf0

σf0

(

(

)

(h - 1)σ hσh +1 )h σ 1 + σh (33) (n - 1)σ 1+σ

)

nσ(1 + σ) + 1 ) 1 (34) L + (1 + σ)n n-1

Mechanistically, the different responses can be related to the fact that the derivation of the Hill mechanism assumes infinite cooperativity; i.e., only the free and fully bound enzyme forms, and no intermediate forms, are considered to exist. The MWC mechanism, with its transition between the T and R states, does not have this restriction. Abolishment of Cooperativity When Both Substrate and Product Are Bound. An interesting feature of both the reversible Hill and MWC equations is that cooperative binding in substrate (or product) is completely abolished when the product (or substrate) is also bound to the enzyme in significant amounts. We have investigated this by plotting the substrate and product dependence for both mechanisms (Figure 6). To increase the binding to the enzyme of the second binding partner (product or substrate) that is not being varied, without changing the thermodynamics of the reaction, we decrease its halfsaturation constant (p0.5 or Kp in the former case; s0.5 or Ks in the latter), thus increasing the affinity of the enzyme and the fractional saturation with the second binding partner. In the limit, when the half-saturation constant of the binding partner that is not being varied is in the same range as its concentration, the

response toward the varying binding partner resembles that of the ordinary reversible Michaelis-Menten equation (for substrate dependence, compare Figure 6A (p0.5 ) 1) and B (Kp ) 0.2) with Figure 3A; for response to changes in product concentration, compare Figure 6C (s0.5 ) 1) and D (Ks ) 0.2) with Figure 3C). In all cases, cooperative binding is lost. The abolishment of cooperativity can be understood from the rate equations, specifically from the split of the term Θ into a single-subunit and a subunit interaction term. In the case of the reversible Hill equation, it follows from eq 26 that h-1 Θh ) (1 + (σ + π))(σ + π) h 1 + (σ + π) (σ + π)h-1 + (σ + π)h ) 1 + (σ + π)h

(35)

For large values of π, the values of the numerator and denominator in eq 35 are approximately equal for all values of σ, implying that cooperativity no longer has an appreciable effect on the reaction rate (i.e., Θh f 1). This effect becomes particularly marked as π g 1 (i.e., p g p0.5); for example, with h ) 4, at positive σ the maximal value of Θh at π ) 1 is 1.42; at π ) 10 it is 1.10. Similarly, for the MWC mechanism, it follows from eq 30 that

Θn )

(1 + σ + π)n L + (1 + σ + π)n

(36)

As soon as (π + 1)n . L, Θn f 1 irrespective of the value of σ, leading to the abolishment of cooperativity in S. The abolishment of cooperativity in substrate binding by saturating product concentrations and vice versa described above is similar to the abolishment of modifier effects by saturating

16288


substrate concentrations found by Hofmeyr and CornishBowden6 in their derivation and study of the reversible Hill equation. Discussion This paper has dealt with recasting a generic reversible rate equation for a chemical reaction in two ways. The first split makes the contribution of thermodynamics and kinetics explicit and allows one to distinguish between near-equilibrium and farfrom-equilibrium reactions. The kinetic contribution is given by the forward rate term and the thermodynamic contribution by the term 1 - F. On the basis of the fractional difference between the elasticities (i.e., the dependence on reactant concentration) of each of these terms, we propose a new criterion for distinguishing between near-equilibrium, thermodynamically controlled reactions (F > 0.9) and far-from-equilibrium, kinetically controlled reactions (F < 0.1). We also show that the frequently used criterion of Rolleston5 to classify reactions with F > 0.2 as equilibrium reactions is flawed; in fact, such reactions are closer to being kinetically controlled than thermodynamically. The second split separates a rate equation into a rate capacity term, a mass-action term, andsin the case of an enzymecatalyzed reactionsa binding term. In the case of a generic chemical reaction, the rate capacity term is merely given by its forward rate constant kf; when the reaction is enzyme-catalyzed, it is given by the product of the specificity constant and total enzyme concentration, kse0. This term does not depend on substrate or product concentrations. The mass-action term quantifies the contribution of mass action to the reaction rate; it is the same for all chemical reactions, uncatalyzed or catalyzed with different enzyme mechanisms. The contribution of enzyme binding and catalysis to the reaction rate can be quantified with the binding term Θ. By defining regulation as the trend to counteract or augment mass action, the same split of the reaction elasticity between mass action and binding allows us to quantify the contribution of enzyme binding to the regulation of reaction rate. The only term in the rate equation that differs between enzyme mechanisms is this binding term, and if a reaction follows cooperative kinetics, Θ can be further split to separate the contribution of a single enzyme subunit from that of subunit interaction and cooperativity. We have previously proposed to split enzyme elasticities into a binding and a mass-action term4 to quantify the contribution of enzyme catalysis to regulation. Note, however, that in the previous paper4 a slightly different terminology was used: the mass-action term was also called “thermodynamic”, and the binding term was referred to as “regulatory kinetic”. As shown in Figure 2, the rate equation needs to be recast in two different ways to distinguish between thermodynamic and kinetic regulation on the one hand and the contribution of mass action and binding to reaction rate on the other. We prefer to use massaction for the term Vma in eq 18 as it describes precisely this contribution; it does not answer the question of whether the reaction is thermodynamically controlled or not. The novel contributions of this paper compared to the previous one4 are: first, that we introduce the two ways of recasting the general chemical rate equation; second, that we make a finer distinction between different aspects that affect reaction rate and elasticity (by splitting the term Θ); and third, that we plot and analyze the contributions of the individual terms to reaction rate over a range of substrate and product concentrations for different mechanisms. Recently, Yuan et al.20 published a detailed kinetic model of central nitrogen metabolism in Escherichia coli. As part of their

Rohwer and Hofmeyr analysis, they evaluated according to Hofmeyr4 the contribution of mass action and kinetics (enzyme binding in the terminology of this paper) to the elasticity coefficients of the enzymes GOGAT and aspartate aminotransferase both before and after a nitrogen upshift. The elasticity of GOGAT for glutamine was 0.37 and 0.18 pre- and postshift, respectively, reflecting a substantial negative contribution of enzyme binding to the overall elasticity. This was even more pronounced for the substrate R-ketoglutarate, where the overall elasticity was close to zero and the mass-action and binding terms canceled each other out. By contrast, aspartate aminotransferase was close to equilibrium, resulting in huge elasticities (>10) for all substrates and products. N upshift resulted in a shift away from equilibrium with a concomitant decrease in the numerical value of the elasticities and a diminished role of the mass action term. This analysis20 gives a good example of how the different contributions of mass action and binding to metabolic regulation can be dissected; using the tools presented in this paper, it is possible to extend such analyses by, first, separating subunit interactions from single-subunit effects in the case of cooperative enzymes and, second, quantifying the different contributions over a whole range of conditions including metabolic transients. A central issue in the construction of kinetic models of large cellular networks is the choice of kinetic rate laws for the enzyme-catalyzed reactions. In a recent paper, Liebermeister et al.21 introduced a number of thermodynamically consistent, modular rate laws for enzymatic reactions. The rate laws were split into terms accounting for enzyme levels, enzyme regulation, turnover number (Tr), and a denominator. The elasticity expressions were split along the same lines and analytical formulas given for the denominator terms, which differ between the different rate laws. This largely corresponds with our analysis in the present paper, with the denominator term equating to our binding term Θ (however, they did not consider cooperative kinetics). The only difference is that Liebermeister et al.21 include the forward and reverse kcat with the turnover number term Tr (which would equate to our mass-action term), keeping only the enzyme level separate. By contrast, we include kcat in the rate capacity term, together with the enzyme concentration, so that the mass-action term is only a function of substrate and product concentrations and the equilibrium constant of the reaction. In our view, such a split makes more sense to address the questions we asked in this paper because the rate constant for an uncatalyzed reaction (or the turnover number in the case of an enzyme-catalyzed reaction) is really an indicator of rate capacity (eqs 15 and 17). In addition, forward and reverse turnover numbers of an enzyme are not independent parameters, and their effects cannot be split when analyzing the Tr term.21 Our formulation of the rate law, which makes the Keq explicit, takes this into account automatically. In closing, we believe that the analytical methods presented in this paper can be usefully applied to two areas of biochemical research. First, in enzyme-kinetic analyses, our generalized method of splitting an enzyme-kinetic rate equation into rate capacity, mass action, and binding terms can dissect the contribution of each of these terms toward regulation of the enzyme activity. Although we give only three examples of enzyme-kinetic rate laws, it must be mentioned that eq 17 is completely general, and the binding term Θ can be derived for any rate equation. A second area of application is the analysis of kinetic models in computational systems biology. An inspection of the online repositories JWS Online22 and BioModels23 reveals that the number of available models, and the number of cellular pathways that they cover, increases monthly.

Kinetic and Thermodynamic Aspects of Enzyme Control Such models are useful tools for analyzing pathway behavior and regulation; in particular, our approach can be used as a model interrogation tool. This can identify, first, which reactions are thermodynamically and which ones are kinetically controlled and, second, what is the relative contribution of mass action and enzyme binding to their regulation under various environmental or physiological conditions. Acknowledgment. We thank Dr. Athel Cornish-Bowden for helpful comments on the manuscript. The authors acknowledge financial support from the South African National Research Foundation (NRF). Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors, and therefore the NRF does not accept any liability in regard thereto. References and Notes (1) Kacser, H.; Burns, J. A. Symp. Soc. Exp. Biol. 1973, 27, 65–104. (2) Heinrich, R.; Rapoport, T. A. Eur. J. Biochem. 1974, 42, 89–95. (3) Savageau, M. A. Biochemical systems analysis; Addison-Wesley: London,1976. (4) Hofmeyr, J.-H. S. J. Bioenerg. Biomembr. 1995, 27, 479–489. (5) Rolleston, F. S. Curr. Top. Cell. Regul. 1972, 5, 47–75. (6) Hofmeyr, J.-H. S.; Cornish-Bowden, A. Comput. Appl. Biosci. 1997, 13, 377–385. (7) Monod, J.; Wyman, J.; Changeux, J.-P. J. Mol. Biol. 1965, 12, 88– 118. (8) Cornish-Bowden, A.; Ca´rdenas, M. L. Eur. J. Biochem. 2001, 268, 6616–6624.

J. Phys. Chem. B, Vol. 114, No. 49, 2010 16289 (9) Cornish-Bowden, A.; Hofmeyr, J.-H. S. The Biochemist 2005, 27, 11–14. (10) Alberty, R. A. Thermodynamics of Biochemical Reactions; Wiley: Hoboken, NJ, 2003. (11) Alberty, R. A. Biochemical Thermodynamics: Applications of Mathematica; Wiley: Hoboken, NJ, 2006. (12) Goldberg, R. N.; Tewari, Y. B.; Bhat, T. N. Bioinformatics 2004, 20, 2874–2877. (13) Waage, P.; Guldberg, C. J. Chem. Educ. 1962, 63, 1044–1047 (translated by Abrash, H. I. from the original publication: Forhandlinger: Videnskabs-Selskabet i Christiana, 1864, 35). (14) Newsholme, E. A.; Start, C. Regulation in Metabolism; John Wiley: London, 1973. (15) Reich, J. G.; Sel’kov, E. E. Energy metabolism of the cell; AcademicPress: London, 1981. (16) Kacser, H.; Burns, J. A.; Fell, D. A. Biochem. Soc. Trans. 1995, 23, 341–366. (17) Hofmeyr, J.-H. S.; Rohwer, J. M.; Snoep, J. L. IEE Proc.: Syst. Biol. 2006, 153, 327–331. (18) Olivier, B. G.; Rohwer, J. M.; Snoep, J. L.; Hofmeyr, J.-H. S. IEE Proc.: Syst. Biol. 2006, 153, 335–337. (19) Popova, S. V.; Sel’kov, E. E. FEBS Lett. 1975, 53, 269–273. (20) Yuan, J.; Doucette, C. D.; Fowler, W. U.; Feng, X.-J.; Piazza, M.; Rabitz, H. A.; Wingreen, N. S.; Rabinowitz, J. D. Mol. Syst. Biol. 2009, 5, 302. (21) Liebermeister, W.; Uhlendorf, J.; Klipp, E. Bioinformatics 2010, 26, 1528–1534. (22) Olivier, B. G.; Snoep, J. L. Bioinformatics 2004, 20, 2143–2144. (23) le Nove`re, N.; Bornstein, B.; Broicher, A.; Courtot, M.; Donizelli, M.; Dharuri, H.; Li, L.; Sauro, H.; Schilstra, M.; Shapiro, B.; Snoep, J. L.; Hucka, M. Nucleic Acids Res. 2006, 34, D689–D691.

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