Rapid-Equilibrium Enzyme Kinetics

Research: Science and Education

Rapid-Equilibrium Enzyme Kinetics Robert A. Alberty Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139; [email protected]

Rapid-equilibrium rate equations for enzyme-catalyzed reactions are especially useful because if experimental data can be fit by these simpler rate equations, the Michaelis constants can be interpreted as equilibrium constants. However, for some reactions it is necessary to use the more complicated steady-state rate equations. Rapid-equilibrium derivations of rate equations have been described by Cha (1) and Segel (2). These equations can be derived for ordered mechanisms by the algebraic method, but there is a problem in deriving rate equations for random mechanisms, that is, mechanisms with more than one pathway. This problem is often solved by using the King–Altman (3) method to derive the complete steady-state rate equation for the random mechanism and then eliminating certain terms, as described by Cleland (4) and Roberts (5). The problem in applying the algebraic method to mechanisms with more than one pathway arises because there is a fundamental difference between kinetics and thermodynamics. Thermodynamics is involved because in the rapid-equilibrium method the assumption is made that all the reactants prior to the rate-determining step are at equilibrium. Random mechanisms have alternate pathways such as

B E A

B EA

EB A

EAB

(1)

products

where A and B are reactants; E is the enzyme; and EA, EB, and EAB are transient complexes. The double arrows, , represent reversible reactions that may be fast or slow, but the single arrow represent an irreversible reaction. Thermodynamics requires that the reactions used in an equilibrium calculation must be independent. The four reactions in the alternate pathway in eq 1 are not independent because the equilibrium constant for any one of them can be calculated from the other three. The following three reactions can be used to calculate the equilibrium composition in the rapid-equilibrium treatment of the mechanism in eq 1:

E A

EA

(2)

E B

EB

(3)

EA B

EAB

products

(4)

The double arrows, , are used in the reactions that are assumed to be equilibrated rapidly. Note that EB + A EAB is not included. Since the issue of the independence of reactions in equilibrium calculations is so important, the next section provides more information. This article describes a new method for obtaining rapidequilibrium rate equations by combining half-reactions. Any forward half-reaction can be combined with any reverse halfreaction. Five half-reactions are described and used to produce 1136

fifteen complete rapid-equilibrium rate equations. More rate equations can be constructed from these half-reactions, and the number of half-reactions can be increased. Relation Between Number of Reactants, Number of Components, and Number of Reactions in Equilibrium Calculations In calculating the equilibrium composition for a biochemical reaction system at a specified pH, apparent equilibrium constants, K', are used. The number of reactants (sums of species) in the system, N', the number of independent reactions R', and the number of components, C', has to satisfy (6) the following relationship:

N b R b C b

(5)

Components are the things that are conserved. In chemical thermodynamics, atoms of elements are conserved, but, even there, if atoms of two elements are always in the same ratio whenever they occur in reacting species, the atoms of the two elements count as one element. Specifying the pH creates pseudoisomer groups (for example, ATP4‒, HATP3‒, and H2ATP2‒) that count as a single reactant. The issue of numbers of components comes up when cycles are postulated, as in eq 1. Alternate pathways are not permitted in thermodynamics (or in rapid-equilibrium kinetics) because the four equilibrium constants in eq 1 are not independent. Therefore, in calculating the equilibrium composition, one of these four reactions has to be omitted so that N' = 6, C' = 3 (E, A, B), and R' = 3. The choice of components is optional, but the number of components C' is not. Equation 5 can be understood in the following way. The basic objective of the thermodynamics of reaction systems is to calculate the equilibrium composition. For biochemical reaction systems, the pH has to be specified; and perhaps pMg needs to be specified. There are equilibrium equations for R' reactions and a conservation equation for each of C' components. This provides just enough information to calculate the equilibrium concentrations of N' reactants. The number of reactants, number of components, and number of independent reactions prior to the rate-determining reaction for five mechanisms of enzymecatalyzed reactions are given in Table 1. These mechanisms are referred to as half-reactions because a forward half-reaction can be combined with a reverse half-reaction to obtain the mechanism of a complete reaction. Notice that in each of these mechanisms for rapid-equilibrium calculations, there is a single reaction for each enzyme–substrate complex that is formed. The half-reactions in Table 1 are similar to the half-reactions of electrochemistry in that any two can be combined. In electrochemistry the two half-reactions have to involve the same number of electrons, and in rapid-equilibrium kinetics the two half-reactions have to have the same denominator terms. The shortest mechanism for the reaction A + B + C → products involves only three reactions and is referred to here as the ordered mechanism. A mechanism may be neither

Journal of Chemical Education • Vol. 85 No. 8 August 2008 • www.JCE.DivCHED.org • © Division of Chemical Education

Research: Science and Education Table 1. Number of Reactants N', Number of Components C’, and Number of Reactions R’ for Rapid-Equilibrium Treatments of Half-reactions

completely ordered nor completely random. The most random mechanism involves seven reactions. Between these two extremes, reactions involving EB, EC, EAC, and EBC can be added. When one of these reactants is added, the number of reactants is increased by one and the number of reactions is increased by one; therefore, the number of components is not changed. As a simplification, only the five mechanisms of halfreactions in Table 1 are discussed. In the next three sections the rapid-equilibrium rate equations are derived for (i) ordered A + B P, (ii) random A + B P, and (iii) ordered A + B ordered P + Q, where P and Q are products. Then a shorter way to derive rapid-equilibrium rate equations based on the half-reactions in Table 1 is described and is applied to derive rapid-equilibrium rate equations for fifteen mechanisms of enzyme-catalyzed reactions. Rapid-Equilibrium Rate Equation for Ordered A + B P The ordered mechanism for A + B steps: KIA E A EA KB EA B EAB EAB

EP

EP

P involves four

< A > < EA>

(6)

< A > KIA < EAB>

E P

KP

(7)

< E >t

(9)

< A> KIA

< A> KIA KB

(10)

KP

v kf kr

R’

Simple mechanism E+S ES

3

2 (E, S)

1

Ordered mechanism E+A EA EA + B EAB

5

3 (E, A, B)

2

Random mechanism E+A EA E+B EB EA + B EAB

6

3 (E, A, B)

3

Ordered mechanism E+A EA EA + B EAB EAB + C EABC

7

4 (E, A, B, C)

3

Random mechanism E+A EA E+B EB E+C EC EA + C EAC EB + C EBC EA + B EAB EAB + C EABC

11

4 (E, A, B, C)

7

kf < A > kr
< E > KIA KB KP

(11)

where kf is the rate constant for the forward reaction and kr is the rate constant for the reverse reaction. Replacing [E] in this

Vf < A > V
r KP K IA KB v A> A>< B > < <
1 KIA KIA KB KP

(12)

where Vf = kf[E]t and Vr = kr[E]t. The four terms in the denominator correspond with E, EA, EAB, and EP. The steady-state rate equation for the mechanism in eqs 6–9 has six terms in the denominator (4). If kinetic experiments show that there are only four terms in the denominator, the rapid-equilibrium rate equation in eq 12 represents the experimental data and the Michaelis constants are equilibrium constants. The Haldane equation is given by

The rapid-equilibrium reaction rate v is given by

C’

equation with eq 10 yields the following rapid-equilibrium rate equation,

< EP > 1

N’

(8)

where indicates that the reaction is equilibrated rapidly, and indicates the rate-determining reaction. The nomenclature for Michaelis constants follows Cleland (4). When thermodynamics is used to derive the equilibrium concentrations of E, EA, EAB, and EP, the reactants EAB and EP count as one reactant because they are pseudo-isomers. This was mentioned after eq 5 as applying both to chemical thermodynamics and biochemical thermodynamics at specified pH. The total concentration of enzymatic sites is given by

Half-Reactions

K b A B

P

V f KP Vr K IA KB

eq < A>eq eq

(13)

Reactions of the type A + B P are very common in biochemical kinetics because H2O does not count as a reactant in N'. The concentration of H2O does not change during a reaction, and [H2O] is not used in the rate equation. However, the standard transformed Gibbs energy of formation of water Δf G′°(H2O) is involved in the calculation of the apparent equilibrium constant K', and the hydrogen atoms of H2O do count in calculating the change in binding of hydrogen ions, Δr NH,

© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 8 August 2008 • Journal of Chemical Education

1137


in an enzyme-catalyzed reaction. In a list of 229 reactions for which K' and Δr NH have been calculated at pHs 5, 6, 7, 8, and 9 at 298.25 K and 0.25 M ionic strength, half of the reactions involve H2O as a reactant (6).

The total concentration of enzymatic sites is given by

< E >t

1

< A> K IA

Rapid-Equilibrium Rate Equation for Random A + B P

The random mechanism involves the four steps of the preceding mechanism plus one more step:

E B

EB

KIB

(14)

< E >t

1

< A> K IA

K IB

< A > K IA K B

(15)

KP

Equation 11 applies and substituting eq 15 yields the following rapid-equilibrium rate equation: Vf < A>< B > V
r K IA KB KP v B A A < > < > < >
1 KP KIA K IB KIA KB

(16)

The effect of introducing randomness in the binding of A and B is to add a term in [B] in the denominator. The effects of [A] and [B] are treated in the same way in eq 16, and this indicates they are bound randomly. Since the change is only in the denominator of eq 16, the Haldane equation in eq 13 is not changed.

E A EA B EAB

EPQ

EQ

1138

EA

EAB

(17)

< EA> < A > KB K IA

(18)

EQ P

E Q

KP

(19)

< EQ >

KIQ

K IQ

(22)

K IQ KP

Vf < A> Vr
KIA KB KP KIQ v < A > < A >
1 KIQ KP KIA KIA KB KIQ

(23)

K b A B

P Q

kf KP KIQ kr KIA KB

eq eq < A >eq eq

(24)

Rapid-Equilibrium Rate Equations for Half-reactions There is another way to derive rapid-equilibrium rate equations for reactions and that is to combine rapid-equilibrium rate equations for half-reactions. Any forward half-reaction can be combined with any reverse half-reaction. For example, when the simple mechanism applies to the forward and reverse reactions for S P, the rapid-equilibrium rate equation is

Vf < S > V
r KP KS v < S >
1 KS KP

(25)

This rate equation can be written as

< A > KIA < EA>

EPQ

KIA KB

[A] and [B] play different roles in the denominator, and so it is possible to determine experimentally which reactant is bound first by the enzymatic site. A similar comment applies to the reverse reaction. The Haldane relation is

Rapid-Equilibrium Rate Equation for the Mechanism for Ordered A + B Ordered P + Q This ordered mechanism is given by

< A >

The rapid-equilibrium rate equation (5, 7) is

The step EA + B EAB is not included because it is redundant in the calculation of equilibrium compositions. The total concentration of enzymatic sites is given by

< E >

(20) KIQ (21)

V f Vr
KS KP v S P S < > < > 1 < >
1 KS KP KS KP

(26)

v f vr where vf is the rate of the forward half-reaction and vr is the rate of the reverse half-reaction. The rate equation for the forward reaction must have a denominator term for the reverse reaction because the denominator terms describe the distribution of enzymatic sites between E, ES, and EP at equilibrium. Equation 26 is the key to the use of half-reactions to derive the rate equation for a complete reaction.


Research: Science and Education Table 2. Mechanisms of Enzyme-Catalyzed Reactions and Properties of Their Rapid-Equlibrium Rate Equations Simple mechanism E+S

ES

EP

Ordered A + B E+A

R’

D

Single

Double

Triple

2

3

3

2

0

0

7

3

4

4

2

1

0

8

3

5

5

3

1

0

9

4

5

5

2

2

0

EA EAB

Random A + B

EP

E+P

P

E+A

EA

E+B

EB

EA + B

EAB

Ordered A + B

EP

E+P

Ordered P + Q EA

EA + B EQ

C’

5 E+P

P

EA + B

E+A

N’

EAB

EPQ

EQ + P

E+Q

Note: D is the number of terms in the denominator of the rate equation. Single, double, and triple refer to the numbers of terms in the denominator involving 1, 2, and 3 reactant concentrations. The expanded version of Table 2 is available in the online supplement.

The rapid-equilibrium rate equations for the five halfreactions in Table 1 are as follows:

VS KS simple : v mechanism S> < 1 KS

(27)

Vf < A> K IA KB ordered : v mechanism < A> < A> 1 K IA KIA KB

(28)

Vf < A > KIA KB random : v mechanism < A > < A > 1 KIA KIB KIA KB

(29)

Vf < A > KIA KB KC ordered : v A> < A> < < A> (30) mechanism 1 KIA KB

K IA

random : v mechanism

1

< A> K IA

KIA KB KC

Vf < A> KIA KB KC < A> KIB KIC KIA KB

< A > K IA K C

KIB KC

< A >

KIA KB KC

This last equation is given by Cornish-Bowden (7) as the initial rate in the absence of products, but with different symbols for Michaelis constants. Rapid-Equilibrium Rate Equations for Enzyme-Catalyzed Reactions Constructed with Rate Equations of Half-reactions As eq 26 shows, this in not quite as simple as taking the difference between the rate equations for the two half-reactions. Before the difference vf − vr is taken, the denominators of both vf and vr must be augmented to account for all the enzyme– substrate complexes at equilibrium. This has been demonstrated for S P in the previous section. The mechanisms considered in this article are given in Table 2. This table gives N', C', and R', the numbers of terms in the denominator of the rate equation, D, and the number of denominator terms with single concentrations, a product of two concentrations, and a product of three concentrations in the denominator. The numbers N', C', and R' for each rapid-equilibrium mechanism show that the mechanism has the correct number of reactions in the mechanism. The number of components C' is always equal to the number of reactants (A, B, C, P, Q, R) in the enzyme-catalyzed reaction. The way to understand this is that C' is one less than the number of reactants in the enzymecatalyzed reaction plus one for enzymatic sites. The number of terms in the denominator of the rate equation (see column D) is always equal to the number of steps in the mechanism. The sum of the numbers of terms with single, double, and triple products of concentrations of reactants is always one less than D. Ordered A + B P Mechanism The rate of the forward reaction is given by

(31)

vf

Vf < A > K IA KB < A > < A >
1 K IA K IA KB KP


(32)

1139


The denominator has been augmented by the [P]/KP term from the rate equation for the reverse reaction. The rate of the reverse reaction is given by vr

1

K IA

Vr
KP < A >
K IA KB KP

(33)

The denominator has been augmented with [A]/KIA + [A][B]/ KIAKB from the rate equation for the forward reaction. The rate equation for ordered A + B P is given by the difference between eqs 32 and 33. Vf < A > Vr
K IA KB KP v A> A > < <
1 K IA K IA KB KP

(34)

P Mechanism

Vf < A > Vr
KP K IA KB v A> B> A > < < <
1 K IA K IB K IA KB KP

(35)

Ordered P + Q Mechanism

Vf < A > Vr
K IA KB K P K IQ v < A > < A >
1 K IA K IA KB K IQ K P K IQ

(36)

The terms in A and B play different roles, and so it is possible to determine whether A is bound first or second. This applies to the order of dissociation in the reverse reaction as well. The steady-state rate equation has eleven denominator terms (4). (The rate equations for the remaining mechanisms are provided in the online supplement, along with an expanded Table 2.) Haldane Equations Rate equations that include the reverse reaction as well as the forward reaction include the thermodynamics of the catalyzed reaction as well as the kinetics for the two reactions. When v = 0, the relation between the apparent equilibrium constant K' and the kinetic parameters is obtained. These relations are 1140

P

Vf K P Vr K S

(37)

Vf KP Vr KIA KB

(38)

K b A + B

P

K b A + B

P + Q

Vf KP K IQ Vr K IA KB

(39)

K b A + B + C

Vf KP P Vr K IA KB KC

(40)

K b A + B + C

P + Q

(41)

K b A + B + C

Vf K P K IQ K R P + Q + R Vr KIA KB KC

Vf KP KIQ Vr KIA KB KC

(42)

Note that KIB, KIC, KIP, and KQ do not appear in these Haldane equations. The Haldane equation gives a property of the reaction that is catalyzed, and it is independent of the mechanism. Discussion

The denominator for the forward reaction has been augmented with [P]/KP and the denominator of the reverse reaction has been augmented with 3 terms proportional to [A], [B], and [A][B]. Note that A and B play the same roles in this rate equation. Ordered A + B

K b S

This rate equation has been given in eq 12, but it is repeated here so that it can be compared with other rate equations. Random A + B

called Haldane equations because he pointed out the first of these relations in 1930. The mechanisms in Table 2 involve only the following six Haldane equations:

Rapid-equilibrium rate equations are the ones to start with in determining the mechanism of an enzyme-catalyzed reaction. If one of these rate equations represents all the experimental data, the Michaelis constants are equilibrium constants. If rapid-equilibrium rate equations do not have enough terms to represent the experimental data, the complete steady-state rate equation has to be used. The values of the number of reactants N', the number of components C', and the number of independent reactions R' for enzyme-catalyzed reactions can be calculated by adding the values in Table 1 for half-reactions. The number of reactants in an enzyme-catalyzed reaction is equal to the sum of the N' of the two half-reactions minus one so that E is not counted twice. The number of components C' is determined by the catalyzed reaction and is one less than the number of reactants in the catalyzed reaction. The number of independent reactions R' is one greater than the sum of the number of reactions in the two half-reactions because the isomer interconversion is counted. The sum of the number of single, double, and triple terms in the denominator is one less than the number of denominator terms because the enzyme site contributes the 1 in the denominator. Note that in identifying N', C', and R', counting ES EP, EAB EPQ, and EABC EPQR as one reactant (because they are isomers) or as two reactants is optional. When both members of the pair are counted as reactants (as is done in Table 2), there is one more reaction and one more reactant so that C' is not affected. In this article both reactants and the isomerization reaction are counted. This has the additional advantage that



when pH effects are discussed, the pH dependencies of Vf and Vr can be different (8). Complete rate equations provide a connection to thermodynamics through the Haldane equation, and they can also be used in calculating the way that systems of enzyme-catalyzed reaction go from an initial state to equilibrium. However, kinetic data discussed here cannot be used to unambiguously describe the mechanism of a reaction. It is desirable to have additional experimental results, isotope labeling, spectroscopic data, and so forth to sort out alternative mechanisms. Acknowledgments I am indebted to Robert N. Goldberg (NIST), Carl Frieden (Washington University), and Athel Cornish-Bowden (France) for many helpful discussions. Grateful thanks to NIH for support of this research by award number 5-RO1GM48348-11. Literature Cited 1. Cha, S. J. Biol. Chem. 1968, 243, 820–825.

2. Segel, I. H. Enzyme Kinetics: Behavoir and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems; Wiley: Hoboken, NJ, 1975. 3. King, E. L.; Altman, C. J. Phys. Chem. 1956, 60, 1375–1378. 4. Cleland, W. W. Biochim. Biophys. Acta 1963, 67, 104–137. 5. Roberts, D. V. Enzyme Kinetics; Cambridge University Press: Cambridge, 1977. 6. Alberty, R. A. Biochemical Thermodynamics: Applications of Mathematica; Wiley: Hoboken, NJ, 2006. 7. Cornish-Bowden, A. Fundamentals of Enzyme Kinetics; Portland Press Ltd.: London, 2004. 8. Alberty, R. A. Biophys. Chem. 2006, 124, 11–17.

Supporting JCE Online Material

http://www.jce.divched.org/Journal/Issues/2008/Aug/abs1136.html Abstract and keywords Full text (PDF) Supplement

The remaining mechanisms

Extended Table 2


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Rapid-Equilibrium Enzyme Kinetics

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