Reversible enzyme-catalysed reactions: the quasi steady state as a quasi-equilibrium, and a correction to Haldane’s kinetic constants and to secondary relationships involving kinetic constants

For reversible enzyme-catalysed reactions obeying Henri-Michaelis-Menten kinetics, theoretical solution of the rate equations for the enzyme-substrate intermediate are generally incorrect when the quasi-steady state approximation, equating the rate of change of the concentration of the enzyme-substrate intermediate to zero, is used. For the simplest kinetic model used by Haldane, such a procedure does not reveal that in one direction, that starting with the reactant having the lower binding constant, the quasi-steady state is one of quasiequilibrium, and Haldane’s structure of the Km written in terms of rate constants is incorrect. This is probably also true of more complex mechanisms in which the structure of kcat may also be in error. Modern methods of numerical integration for the solution of rate equations, if applied to reversible reactions to obtain rate constants from measured catalytic constants, will require the correct expressions for kcat and Km. Furthermore, the (now called) Haldane relationship, relating the kinetic constants kcat and Km for the forward and reverse reactions to the equilibrium constant of a reaction, is now seen to be generally incorrect, and in addition another exception for a the theoretical validation of kcat /Km as a specificity constant arises.


Introduction
An equation derived by Henri [1]and by Michaelis and Menten [2] describes the kinetics of many enzyme-catalysed reactions. Their derivations were based on reaction model 1, in which A and B are reactants, E the enzyme and X a reactant-enzyme intermediate, and the small case letters represent their respective concentrations at any instant. In modern form, the equation is written as equation 1 (eo is the total enzyme concentration, e + x), and it contains two kinetic constants: the catalytic constant, kcat, and the Michaelis constant, Km, which were originally defined in terms of rate constants (letters k). Originally, it was assumed that early in the reaction the reverse reaction could be neglected. This equation has one immediate solution for x (the equilibrium state of all components, readily shown by substitution), but otherwise, except by approximation, it has no analytical solutions. The approximation commonly used today was introduced by Briggs and Haldane [3] for an irreversible reaction: dx/dt is equated to zero, and the approximation is the now called quasisteady state assumption. This approximation was later used by Haldane to solve the rate equations for both directions of reversible reactions and to define the kinetic constants pertinent to each direction [4], but his solutions were incorrect. db/dt = kcat aeo/ (a + Km) (1) k1 k2 A + E X B + E Model 1) a e k-1 x k-2 b e dx/dt = (k1a + k-2b)e -(k-1 + k2)x (2) Notwithstanding the early determinations of kcat and Km in terms of rate constants, as more complex kinetic models were examined, particularly for multi-substrate reactions, the common view was adopted around the middle of the last Century that the experimentally derived constants should be viewed as purely empirical ones. This may change as computer aided numerical methods are now used to determine with greater accuracy kcat and Km from primary data (concentrations measured at intervals rather than from secondarily derived velocities) [5]. Furthermore, methods of numerical integration [6,7]

RESULTS
The correct derivation of the Henri-Michaelis-Menten equation for Model 1.
Haldane's analysis of model 1 for an initial velocity (vi), gave equation 3a for a reaction beginning with A, and 3b for that beginning with B (ao and bo are initial concentrations). The capital superscripts identify the reactant. These equations identify the kinetic constants in terms of rate constants: These results are well known, but both of these Michaelis constants are correct only when k1 = k-2. The Km for the forward and reverse reaction are then identical, and the quasi-steady state is one of a quasi-equilibrium of X approaching its ultimate equilibrium concentration.
For an irreversible reaction, x must proceed through a maximum value (when dx/dt is zero) before both x and dx/dt return to zero. For a reversible reaction, experimental evidence for the rôle of quasi-equilibrium (xequ) in the steady state of X was produced about thirty years after Haldane's first use of the steady state approximation. Morales and co-authors [8,9] showed that, for model 1, x could pass through a maximum value greater than its equilibrium concentration only if, in a reaction starting with A, k1 > k-2 (at the maximum dx/dt = 0 and d 2 x/dt 2 is negative). Miller and Alberty [10] observed that there is an exact solution for equation 2 when k1 = k-2 (which occurs without the approximation that ao >>eo), and in this case x rises monotonically by an exponential function to its final equilibrium value, xequ. The only quasi-steady state concentration of x is the one approaching xequ. Using reasonable kinetic data, Miller and Alberty showed that x could rise very rapidly to values close to xequ, and this was the first example of a quasi-stead state being one of quasi-equilibrium, but these authors did not point that out. They extended their observations by numerical approximation to the condition that k1 > k-2, and observed that x proceeded through a maximum before descending to equilibrium. They gave a single progress curve for the condition k-2 = 10k1 in which x rose only to equilibrium. Walter and Morales [11] made computer-aided calculations of progress curves for x, and showed that when k1 = k-2 or k1 < k-2, x rises monotonically to xequ, and more recently, Tzafriri and Edelman [12] gave an example of x rising to xequ when k1 < k-2. In appendix 1, I show by a kinetic analysis that, in a reaction starting with A and when k1 < k-2, it is to be expected that x rises monotonically to xequ.
If the only steady state available for model 1 (in a reaction starting with A and with k1 = k-2 or k1 < k-2) is one in which x is approaching xequ, then the initial quasi-steady state velocity would be written as vi = k2xequ. With xequ = aoeo / ( ao + k-1/k1 + k2/k-2) (appendix 2), this leads to equation 4 which gives the identities kcat = k2 and Km = (k-1/k1 + k2/k-2). The Km is not that derived by Haldane, except when k1 = k-2.
It should be noted that, if the ratio k-2/k1 is too great or the equilibrium constant, (Kequ = bequ/aequ) is too small, Henri-Michaelis-Menten kinetics may not be observed. This is because xequ is approached too slowly so that there is not a steady state such that, in the words of Briggs and Haldane [3] "…x is always negligible compared with b and a, its rate of change must, except during the first instant of the reaction, be negligible compared with theirs." (Emphasis added, but otherwise this quotation is altered only to identifying the concentrations of species with the abbreviations used here). The calculations of Walter and Morales [11] demonstrate this problem, at least with respect to the effect of decreasing Kequ. They presented progress curves of X for reactions in which k1, k-1 and k2 were constant, and only k-2 was varied. As k-2 increases above k1, the slower approach of x to xequ is clearly seen in their computations. Hence reversible reactions must be considered individually in terms of the ratio k1/k-2 and Kequ. For initial velocities, they may display Henri-Michaelis-Menten kinetics in the direction which starts with the reactant with the higher binding constant (and have a Km as derived by Haldane), and in the reverse direction provide a Km which involves a quasi-equilibrium state. But if xequ is approached too slowly it may not be possible to determine an initial velocity (in the usual sense that "initial" is used (see for example the calculation in [10]), and at which the product concentration may be considered negligible compared with the initial reactant concentration).

Model 2 was introduced by Haldane [4] on the basis of experimental observations, and it
improved model 1 by allowing an isomerization of the intermediate X to Y before a product is released from Y. Although both dx/dt and dy/dt cannot in general be exactly zero at the same instant [11], the analytical methods of Morales and co-authors [8,9] can be applied to this model. The rate equations for x and y are given by equations 5 and 6, and their sum is equation 7a, which on substituting a = (ao -b) becomes equation 7b. If y does proceed through a quasi-steady state maximum, when dy/dt is exactly zero and d 2 y/dt 2 is negative, then dx/dt must also be negligible even if not exactly zero. (That statement might not be true if dx/dt and dy/dt were oscillating out of phase with each other, but in a closed system without a feed-back loop, the possibility of oscillation may be neglected). The second differential of y with respect to t is then given by equation 8. If dx/dt and dy/dt are negligible and zero, respectively, d 2 y/dt 2 can be negative only when k1 > k-3. When this inequality is not obeyed, dy/dt can only be positive, becoming zero only at equilibrium.
dx/dt = k1a(eo -x -y) + k-2y -(k-1 + k2)x (5) dy/dt = k-3b(eo -x -y) + k2x -(k-2 + k3)y (6) dx/dt + dy/dt = (k1a+ k-3b)(eo -x -y) -k1x -k3y (7a) A general evaluation of d 2 y/dt 2 cannot be made as I have done for d 2 x/dt 2 in model 1 (appendix 1) because the relationships between x and y, and dx/dt and dy/dt, are not known except in a particular special case, that when the rate constants for the interconversion of X and Y (k2 and k-2) are large enough to keep X and Y in equilibrium with each other. Then, 8 k2x = k-2y, k2 dx/dt = k-2dy/dt, and this allows (x+y) to be treated as a single unit. Of course, this reduces model 2 to model 1. It is sufficient here to note that, when a reaction starts with A and k-3 > k1, kcat A remains that given by Haldane, but Km A does not.
In view of what has been noted so far about model 1 and the special case of model 2, it seems likely that for model 2 there would be a general problem with Haldane's derivations of the kinetic constants. This view is supported by comparing Haldane's result [4] for the steady state concentration of Y (y* in equation 9) with its equilibrium concentration (yequ), (equation 10, see appendix 2). Equation 9 derives from Haldane's general equation [4] when the concentration b is set to zero. Expanding the inequality y* > yequ and cancellation leads to inequality 11. The right hand side of inequality 11 must be positive, which leads to inequality 12 as a requirement for y* to be greater than yequ. Once again, if there is not a maximum value of y (before equilibrium), y can only rise to its equilibrium value. This observation, however, does not prove that y rises monotonically so that, although dy/dt is always positive (zero at equilibrium), d 2 y/dt 2 is always negative (zero at equilibrium).

Consequences of a misidentified Kinetic Constants.
i. The Haldane Relationship. Haldane derived his relationship (equation 13) for both models 1 and 2 [4], but these derivations depended on his identification of the catalytic constants in terms of rate constants. For model 1 and the conditions that k-2 > k1, these are kcat A = k2 and Km A = (k-1/k1 +k2/k-2), but for the reverse reaction kcat B and Km B will then be k-1 and (k-1 + k2)/k-2, respectively. Substitution of these in equation 13 does not give Kequ = k1k2/k-1k-2. Equation 13 is valid only when k1 = k-2, when in both directions the quasi-steady states involve quasi-equilibrium [10]. A similar conclusion may be adduced for model 2 when k1 = k-3 and k2 and k-2 maintain equilibrium between X and Y.
It may be noted that Bock and Alberty [13] made well-controlled kinetic measurements using fumarase, and claimed to have validated the Haldane relationship. Although the reaction catalysed by the enzyme is not one with a single reactant and product in both directions, in the conversion of fumarate to malate the concentration of water can be considered a constant and combined with a second order rate constant. Since the original work, however, fumarase has been shown to be a tetramer with four identical subunits, and extensive kinetic studies by Rose [14] have shown there are several possible kinetic pathways within each subunit. Mescam et al. [6] determined the relative fluxes through three possible kinetic pathways of an 11-state reversible mechanism which they adapted from the work of Rose.
Their results indicate that, except for the binding steps, each pathway had intermediate stages in a state of quasi-equilibrium. This would lead to the prediction that Henri-Michaelis-Menten-kinetics would be displayed by each pathway separately (but the summation for the two such pathways does not do so unless their Michaelis constants were identical). The actual mechanism of fumarase is certainly not that of either model 1 or 2, and so the results of Bock and Alberty [13] are not pertinent to the Haldane relationship.
ii. Enzyme specificity. The ratio kcat / Km is perhaps best known as a measure of enzyme efficiency and specificity. This may be seen intuitively from a consideration of equation 1: when ao <<Km, db/dt approximates to aoeo kcat/Km, and the velocity is maximized for the lowest reactant concentration. It has been shown that kcat/Km is not always useful for comparing different enzymes catalysing the reaction of a single substrate [15], and unfortunately, for the case when the reactions of two substrates are catalysed in a single solution by one enzyme, it must now be stated that kcat/Km is also not always a good measure of specificity. This latter approach to specificity was introduced Fersht [16], and Cornish-Bowden [17], and Cornish-Bowden [18] also extended the use of kcat/Km to reactions more complex than model 1. Unfortunately, none of these avoided the problem which I have pointed out above, the difficulty of simply equating dx/dt to zero in a rate equation and following Haldane to expressions for kcat and Km. I shall not develop proofs here because for model 1 they are elementary, but simply state that if two competing substrates both have binding constants greater than those of their respective products, then the analyses of Fersht and Cornish-Bowden are valid. This is also true in the case when the steady state of both competing reactants involves a quasi-equilibrium, but should the steady state for one reactant be as described by Haldane and for the other a quasi-equilibrium, then the analysis is not correct.

Discussion
When Haldane [4] applied the steady state approximation of Briggs and Haldane [3] to reversible reactions, he did not consider the possibility that the steady state in model 1 could be one in which x was close to and approaching its final equilibrium concentration, a possibility first revealed almost thirty years later [8,9,10,11]. In fairness to Haldane, it should be noted that he may not have been aware of the work of Morales and co-authors [8,9], Miller and Alberty [10], and Walter and Morales [11]. He wrote an Introduction to the second edition of his book [19] in 1964, the year of his death in India at the age of 72, and made clear in the interesting introduction to that edition why his book was deliberately republished unchanged.
I have no explanation for the fact that the consequences resulting from the observations of A recent publication by Johnson [7], has revealed an interesting property of the ratio kcat/Km, although it is to be hoped that the name of specificity constant might be replaced. After the phase of referring to kcat and Km as purely empirical constants fitting the Henri-Michaelis-Menten equation, it is now possible using methods of numerical integration to determine the best set of rate constants which may fit a theoretically derived equation (for kcat and Km) to its experimentally measured value. Should this approach be extended to reversible reactions, this will require the correct deduction of the kinetic constant in terms of rate constants, for which only a start has been made here.

Appendix 1. The steady state in model 1 with a reaction starting with A and k1 < k-2.
The quasi-steady state approximation of Briggs and Haldane [3] stated, "…since x is always negligible compared with b and a, its rate of change must, except during the first instant of the reaction, be negligible compared with theirs." (This quotation is altered only to identifying the concentrations of species with the abbreviations used here). dx/dt must be negligible, but not necessarily zero, and when dx/dt is always positive (except at equilibrium) this condition could be met if dx/dt initially rose rapidly to a phase where it became negligible (and which could be a quasi-steady state), and after which it rose more rapidly before finally declining as x reached its equilibrium concentration. Such a process would require that the sign of d 2 x/dt 2 is initially negative, passes through a positive phase and then a further negative one as the equilibrium concentration is reached. This scenario would lead, if dx/dt were approximated to zero, to the kinetic constants predicted by Haldane, but it cannot occur with model 1 when k-2 > k1. Provided ao >> eo, d 2 x/dt 2 is always negative (zero at equilibrium), there are no inflexions in the progress curve of X, and x rises monotonically to xequ. This is demonstrated as follows.