Boris Goldstein and the Derivation ofSteady-State Rate Equations∗

Athel Cornish-BowdenUnité de Bioénergétique et Ingénierie des Protéines, Institut de Microbiologie

de la Méditerranée, Centre National de la Recherche Scientifique,Aix-Marseille Université, 31 chemin Joseph-Aiguier, 13402 Marseille Cedex 20, France

Introduction

Boris Goldstein’s distinguished career in enzymology spanned several decades, buthere I want to discuss the work that he did while still a student with Professor MikhailVladimirovich Volkenstein, in which he extended the graphical method of King andAltman [1] for deriving steady-state rate equations in enzyme kinetics so that it couldbe applied to a wider range of mechanisms with very little additional effort.

Deriving a rate equation for a mechanism as simple as the mechanism for an irre-versible one-substrate reaction (in which E, A and P symbolize enzyme, substrate andproduct respectively) with no inhibition by products,

E + A →← EA→ E + P (1)

leads readily to the familiar Michaelis–Menten equation:

v =k2e0a

k−1 + k2

k1+ a

(2)

in which e0 is the total enzyme concentration and a is the free substrate concentration,k1 and k−1 are the forward and reverse rate constants of the first step, and k2 is the rateconstant of the second step.

However, the ease with which this result can be obtained (as described in any ele-mentary textbook of enzyme kinetics) is deceptive. There is only one intermediate spe-cies in the process, EA, and thus only one steady-state concentration to be set to zero, sosolving is just a matter of solving one simple equation. Even with a three-step reversiblemechanism,

E + A →← EA →← EP →← E + P (3)

the problem becomes significantly more difficult, as now there are two concentrationsof intermediates to be set to zero, and deriving the rate equation requires solution of apair of simultaneous equations. Even the simplest mechanisms for reactions with more

∗This paper is dedicated to the memory of Boris Goldstein.

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E

EAEP

k1ak!1

k−2

k−3 p

k3

k2

k!1

k−2

E

k!1

k3

k3

k2

E E

(k−1k−2 k−1k3 k2k3)

k3

EA EA EA

(k1k−2a

k1a k1ak−3 p

k−2k−2

k1k3a k!2k!3 p)

k1ak−3 p

EPEP EP

k!1k−3 p

k2 k2

(k1k2a k!1k!3 p k2k!3 p)

(a) (b)

(c)

(d)

(e)

(f)

+ +

+ ++ +

Figure 1: The King–Altman method. (a) A closed scheme is drawn to represent themechanism, based here on equation (3), with steps labelled with first-order or pseudo-first order rate constants. (b) This gives rise to a master pattern, from which (c) threeindividual patterns are drawn, each of consists of lines from the master pattern, butwithout containing a closed loop. (d) Arrowheads are drawn on each pattern so thatit terminates at E, and the product of the rate constants gives one of the terms in theexpression for the concentration of E. (e, f) Similar terms are obtained for the concen-trations of EA and EP.

than one substrate typically have four or more intermediates, and although in principlethey are susceptible to the same approach the practice is more difficult and error-prone.As a result, only a few cases had been analysed by the end of the 1950s, as a result ofsubstantial efforts by a small number of groups [2, 3, 4].

The first revolution was brought about by King and Altman [1], but this, with ex-tensions due to Wong and Hanes [5], only resolved part of the problem, and the secondwas brought about by Volkenstein and Goldstein [6]. To explain the importance of theircontribution it will first be necessary to describe the King–Altman method.

The King–Altman method

I shall not set out the principle of the King–Altman method [1] here, or explain whyit works, but interested readers may find this information in my book [7] (available inRussian in a translation of an earlier book [8]). All we need for the present purpose isa description of the procedure, as that will be sufficient to illustrate the advances madeby Volkenstein and Goldstein [6].

The mechanism shown as equation (3) can be drawn as a closed scheme, as in Figure1, in which the rate constants for the different steps are shown explicitly. Some arefirst-order rate constants (k−1, k2, k−2, k3). Those that are not (k1, k−3) are converted topseudo-first-order rate constants by incorporating the appropriate concentrations a orp. Then, following the procedure outlined in the legend to Figure 1, the concentrationof each form of enzyme in the steady state is equal to the sum of terms shown under

2

E EA

k1a(a)k!1

k2

(b)

E EAk1a

k!1 k2

(c)

+

Figure 2: Handling parallel steps. (a) The simple Michaelis–Menten mechanism ofequation (1) can be drawn as a closed scheme, and (b) gives a master pattern consistingof two lines, which can be analysed by the method of King and Altman. However, it issimpler to recognize that rate constants for parallel steps can be added, and (c) whenthis is done there is just one line in the master pattern and the solution can be writtendown immediately.

the relevant set of patterns. For example, for [E], we have, from Figure 1d,

[E][E] + [EA] + [EP]

=k−1k−2 + k−1k3 + k2k3

D (4)

in which D is the sum of the numerator terms for all three enzyme forms (i.e. the samedenominator as in the next equation). Combining the three concentration expressionswith an expression for the rate of reaction, v = k2[EA]− k−2[EP], it is then just a matterof straightforward algebra to arrive at the steady-state rate equation:

v =k1k2k3a− k−1k−2k−3 p

k−1k−2 + k−1k3 + k2k3 + k1(k2 + k−2 + k3)a + (k−1 + k2 + k−2)k−3 p(5)

Even this algebra can be avoided by applying the method of Wong and Hanes [5]. Theoriginal method showed only how to obtain the denominator of the rate equation, butthey showed how a simple (and very similar) approach would allow the numerator tobe derived as well, without requiring any cancellation of unwanted terms. However, Ishall not describe this extension here, as it does directly affect the analysis of Volken-stein and Goldstein [6].

Extensions due to Volkenstein and Goldstein

Although the King–Altman method represented a major advance on the purely algeb-raic methods that preceded it, it is extremely cumbersome if one tries to apply it to anybut the simplest mechanisms used in textbooks to illustrate it, even ones that are inprinciple fairly simple, as equation (3) would be if one extended it to take account ofcompetition between two substrates for the same enzyme. This is a serious limitation,as the mechanisms that are difficult to analyse are exactly the ones that a researcher islikely to need to study, as the simpler ones are all readily available from the literature.Volkenstein and Goldstein [6] noted that although the theory of graphs was alreadywidely used in various disciplines, it had not been applied to chemical kinetics:

This theory was developed in topology and is now widely used in calcula-tions to do with electrical networks, in information theory, etc. Graphs areuseful for any problem concerning the branched flow of electrical charges,

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Ek1a

(a)

k−1

k2

A

E

k1ak−1

A

k4k3ak!3

k3a

k!3

k4

k2

E

EAA

EAk1a

(b)

k−1

k4k3ak!3

k2

E

EA

2

2 2

2

EAk1a

(c)

k3a

E

EA

2

2

2(k!3 + k4)

k!1 + k2

Figure 3: Symmetry in a mechanism. The mechanism is symmetrical around the brokenline, because EA and EA have exactly the same properties. (b) Writing them both as EA,and EA

A as EA2 the mechanism can be redrawn in a simpler way, and (c) taking accountof the parallel steps allows a further simplification.

matter or information. Until now, no application of the theory of graphs tochemical kinetics has been made.

They accordingly introduced four ways in which the theory of graphs could be appliedto the King–Altman method, which is, they said, “very cumbersome” if many enzymeforms need to be considered. I shall now describe their four simplifications.

Parallel steps

As described in textbooks the King–Altman method appears to incorporate an assump-tion that the number of reactions connecting any pair of intermediates is one or zero. Inpractice, however, this is by no means always the case. Even the simplest of all mech-anisms of enzyme catalysis, equation (1), contains two connections between E and EA.In fact the King–Altman method can still be applied, if one draws two separate connec-tions, but it is simpler to realize, as Volkenstein and Goldstein [6] explained, that thesetwo connections can be treated as a single one with rate constants equal to the sums ofthose for the parallel steps. This is illustrated in Figure 2.

Taking account of symmetry

Sometimes a mechanism may be drawn in a more complicated way than in necessary,for example if it contains multiple enzyme intermediates with identical properties. Forexample, the mechanism shown as Figure 3a contains four enzyme forms, of which EA

and EA are equivalent and the reactions in which they participate have identical rateconstants. Analysis by the full method of King and Altman is tedious and error-prone,as there are 32 patterns to be considered. Moreover, even once the rate equation isderived a considerable amount of further algebraic effort is needed to recognize thesymmetrical aspects of the result, which allows various terms to be combined.

However, when the symmetry is taken into account, as in Figure 3b, the schemebecomes much simpler; and then the addition of parallel steps as in the previous section

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E

EAEP

k1ak!1

k!2

k−3 p

k2

k3

EA!EP′

k!1a!k!"1

k!"2

k′−3 p′k!3

k!2 k!2

k!"1 k!"1k!3

k!"2

k!3

k!2

k!1 k!1k3 k3

F ! = (k!"1k!"2 + k!"1k

!3 + k!2k

!3)

F = (k!1k!2 + k!1k3 + k2k3)

k2

Figure 4: Competition between substrates for the same free enzyme. The two halves ofthe mechanism share just one enzyme form in common, the free enzyme E. As a result,the two halves have their own sets of patterns that can be analysed separately

produces a scheme sufficiently simple to allow the rate equation to be written down byinspection.

Depending on the relationships between the rate constants that may exist, the mech-anism of Figure 3 can exhibit cooperativity, negative cooperativity or no cooperativity.In the simplest case, if k1 = k3, k−1 = k−3 and k2 = k4 it just collapses to the case of non-interacting sites and leads to an equation equivalent to the Michaelis–Menten equation.If k2 = k4 but k3/k−3 > k1/k−1 it generates cooperativity, or negative cooperativity ifk3/k−3 < k1/k−1. More complicated behaviour, such as cooperativity in one range ofsubstrate concentrations, but negative cooperativity in another, is possible if k2 6= k4.In a companion paper, Volkenstein and Goldstein [9] analysed these ideas further, anddiscussed their relationship to the well known allosteric model of Monod, Wyman andChangeux [10] for cooperativity.

Partial mechanisms with just one intermediate in common

When two substrates compete for the same free enzyme the mechanism may consistof two partial mechanisms that are quite separate apart from sharing the free enzyme.Other mechanisms can also exist in which two parts touch at a unique point, but thecase of competition between substrates is the example most often encountered. It isillustrated in Figure 4. In such a case the expression for each form of the enzyme issimply the product of the separate sums for the two halves of the mechanism. Forexample, for [E] the top (primed) half of the mechanism gives F ′ = (k′−1k′−2 + k′−1k′3 +k′2k′3), and the lower (unprimed) half gives the same without the primes: F = (k−1k−2 +k−1k3 + k2k3). The complete expression for [E] is then the product of the two sums:

[E][E]total

=(k′−1k′−2 + k′−1k′3 + k′2k′3)(k−1k−2 + k−1k3 + k2k3)

D =F ′FD (6)

where [E]total is the sum of the concentrations of the five enzyme forms, and D is againthe sum of the numerator terms for all of the enzyme forms. A further simplicationcomes from the fact that F ′ occurs again as a factor in the expressions for [EA] and[EP], and similarly F occurs again as a factor in the expressions for [EA′] and [EP′]. For

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EA

EABE

k1a k!1

k−2

k!8 p

k2b

k3b

EQ

EB

EP

k−4

k4a

k8 k6

k−6q

k−5 pk5k7

k−7q

k−3

EAB

EAB

EAB

EAB

E

E

E

E

EA

EB

EP

EQ

k3b

k−5 p+ + +() () )(

k1a k−2

k4a k8

k−6qk7

Figure 5: Compression of patterns. This is the mechanism for a reaction with two sub-strates A and B and two products P and Q that can bind or be released in either or-der. If we consider King–Altman patterns for EAB, it is clear that each contains oneand only one of the sequences shown at the right, E→ EA→ EAB, E→ EB→ EAB,E→ EP→ EAB or E→ EQ→ EAB. In addition it must contain three further ar-rows, one for each of the three intermediates that are missing from the sequenceE→ · · · → EAB. The are shown for the first sequence, E→ EA→ EAB, in the bottomline of the figure, and leads to the expression k1k3ab(k−2 + k4a)(k−6q + k8)(k−5 p + k7).Similar expressions must then be derived for the other three sequences E→ · · · → EAB.

example, the expression for [EA] is

[EA]

[E]total=F ′(k1k−2a + k1k3a + k−2k−3 p)

D (7)

Compression of patterns

The last of Volkenstein and Goldstein’s modifications to the King–Altman method (ac-tually the third in their enumeration) is more complicated than the other three, andcannot usefully be illustrated in terms of mechanisms as simple as those considereduntil now. In my first kinetics book [8] I described it in the following terms:

The fourth modification . . . is the most useful and important, because itprovides the only practical method of analysing mechanisms with six ormore enzyme forms. Unfortunately, it is also the most difficult to under-stand and use, because it is not purely mechanical but requires careful thoughtif it is to be used profitably. In essence, it provides a means of recognizingand using repetitive features of the master pattern, so that one can writedown terms for several patterns simultaneously.

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Here I shall discuss it in relation to the mechanism shown in Figure 5, which refersto a reaction with two substrates A and B and two products P and Q that can bind orbe released in either order. If we consider the unmodified King–Altman method wesee that it requires 32 patterns to be analysed, so it is barely usable, and has a highprobability of errors. However, many sums occur multiple times in the full expansion,and the analysis in the figure shows how these are obtained for patterns terminating atEAB.

The patterns terminating at E can be obtained by just transposing those for EAB, asE→ EA→ EAB becomes E← EA← EAB, and k1k3ab needs to be replaced by k−1k−3,and so on. The terms for EA require more thought, because it occupies a position in themaster pattern that is topologically dissimilar from those occupied by E and EAB. Forexample, the fragment E — EB — EAB is E→ EB→ EAB, represented by k2k4ab, if it isconnected to EA by EAB→ EA, but it is E← EB← EAB, represented by k−2k−4, if it isconnected by E→ EA. Similar considerations apply to EB, EP and EQ.

Apart from decreasing the labour involved in deriving a rate equation for a mech-anism like the one considered, the technique discussed in this section has the additionalimportant advantage of providing a equation in which repetitive and symmetrical fea-tures are obvious, and thus it is very helpful for locating and eliminating errors.

Discussion: are all these methods obsolete?

40 years ago [8] I described the fourth of Volkenstein and Goldstein’s modifications tothe King–Altman method as “the most useful and important”, but I omitted it frommy most recent book [7], while retaining the description of the other three. Why thisinconsistency? The main reason is that it is the most difficult to understand and ap-ply, and thus less suitable for a more elementary book. However, there has been animportant change in biochemical practice, in that rather than deriving and using fullrate equations for complex mechanisms it has become usual to use computer analysis.Computer methods for deriving enzyme rate equations were described long ago [11],but were initially not very practical to use. However, today very powerful methodshave been developed [12] that allow a great deal of the analysis to be done by com-puter. Given, therefore, that the modern biochemist rarely needs to derive and presenta full rate equation for the mechanism of an enzyme-catalysed reaction, do the variousmethods for doing it manually [6, 1, 5] still have any importance?

The implied criticism misses the point, however. The importance of these methodstoday is not that they are tools for deriving rate equations, but that they provide insightand understanding. Once one is thoroughly conversant with them one can understandwhere every term in a rate equation comes from, and why it is there, and one can de-duce important conclusions about the steady-state properties without having to do anyexplicit algebra. Wong [13] called this process topological reasoning, and it provides thekey to understanding why the methods of deriving rate equations continue to be im-portant. The last of Volkenstein and Goldstein’s modifications is particularly importantin this respect: with just the King–Altman method the full equation for the mechanismof Figure 5 is just an impenetrable thicket of rate constants with no obvious regular-ities, but the analysis of the preceding sub-section makes the regularities visible andallows understanding of why they are there. After nearly half a century, therefore, thework that Boris Goldstein did for his doctorate remains a major advance in biochemical

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science.

Remembering Boris Goldstein

I only met Boris Goldstein a few times, and would like to have known him better thanI did, because I knew and admired his work for many years. I am thinking especiallyof the work that forms the principal topic of this paper, which I learned about while Iwas writing my first book on enzyme kinetics [8]. When I read his paper [6] — prob-ably some time in 1972 — I realized immediately that it constituted a major advance inknowledge of enzyme rate equations. It was not until a considerable time afterwardsthat I met Boris for the first time, when I invited him to the NATO Advanced ResearchWorkshop on “Control of Metabolic Processes” that I organized at Il Ciocco (Italy) in1989. At that time it was still difficult to obtain permission to invite participants fromthe USSR to attend NATO meetings, and still difficult for them to obtain permission tocome, so I was very happy that Boris was able to participate, and to present a posteron “Models of Symmetry-driven Kinetics for Enzymatic Assemblies”, presenting workapplying the ideas of graph theory that I have been discussing. Later in the same year Imet him again at a meeting at Holzhau that took place in the dying days of the GermanDemocratic Republic, just three weeks after the opening of the Berlin Wall. Apart fromsome other research meetings that we both attended, I am happy to say that Boris alsomade a brief visit to me at home in Marseilles in 1990, and we have a decorated Russianspoon that he brought us as a gift on that occasion.

References

[1] King E L & Altman C (1956) A schematic method of deriving the rate laws forenzyme-catalyzed reactions, J. Phys. Chem. 60, 1375–1378.

[2] Segal H L, Kachmar J F & Boyer P D (1952) Kinetic analysis of enzyme reactions.I. Further considerations of enzyme inhibition and analysis of enzyme activation,Enzymologia 15, 187–198.

[3] Dalziel K (1957) Initial steady state velocities in the evaluation of enzyme-coenzyme-substrate reaction mechanisms, Acta Chem. Scand. 11, 1706–1723.

[4] Alberty R A (1958) On the determination of rate constants for coenzyme mechan-isms, J. Am. Chem. Soc. 80, 1777–1782.

[5] Wong J T & Hanes C S (1962) Kinetic formulations for enzymic reactions involvingtwo substrates, Canad. J. Biochem. Physiol. 40, 763–804.

[6] Volkenstein M V & Goldstein B N (1966) A new method for solving the problemsof the stationary kinetics of enzymological reactions, Biochim. Biophys. Acta 115,471–477.

[7] Cornish-Bowden A (2012) Fundamentals of Enzyme Kinetics, Wiley–Blackwell,Weinheim, 4th edition.

[8] Cornish-Bowden A (1976) Principles of Enzyme Kinetics, Butterworths, London,Russian translation by V. I. Kurganov, Mir, Moscow, 1979.

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[9] Volkenstein M V & Goldstein B N (1966) Allosteric enzyme models and their ana-lysis by the theory of graphs, Biochim. Biophys. Acta 115, 478–485.

[10] Monod J, Wyman J & Changeux J P (1965) On the nature of allosteric transitions— a plausible model, J. Mol. Biol. 12, 88–118.

[11] Hurst R O (1967) A simplified approach to the use of determinants in the calcu-lation of the rate equation for a complex enzyme system, Canad. J. Biochem. 45,2015–2038.

[12] Qi F, Dash R K, Han Y & Beard D A (2009) Generating rate equations for complexenzyme systems by a computer-assisted systematic method, BMC Bioinformatics10, 238.

[13] Wong J T (1975) Kinetics of Enzyme Mechanisms, Academic Press, London.

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