A new method for the derivation of rate equations in enzyme kinetics using the maximum rate of product formation

BlOCHIlvIICA ET BIOPHYSICA ACTA BBA

I

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A NEW METHOD FOR THE DERIVATION OF RATE EQUATIONS IN ENZYME KINETICS USING THE MAXIMUM RATE OF PRODUCT FORMATION 1. G. DARVEY, S. J. PROKHOVNIK J\.ND J. F. WILLIAMS Department of Biochemistry, School of Biologicai Sciences, and School of 111athematics, The University of New South Wales, Kensington, N.S.W. (Australia) (Received February 25th, 1965)

SUMMA!{¥

A new method for the derivation of the rate equation for the reversible one substrate-one intermediate-one product enzymic mechanism based on the maximum rate of product formation is given. This equation is consistent with the result obtained using the steady state assumption but contains more information than the latter. The equation provides a further method for testing enzyme mechanisms using varying initial enzyme concentrations in addition to varying initial substrate concentrations.

INTRODUCTION

Contemporary theories of enzyme kinetics require the use of a number 'of assumptions in order to obtain rate equations relating the various parameters of an enzymic reaction. The use of assumptions is necessitated by the fact that no analytic solution has yet been found for the differential equations obtained on applying the law of mass action to the simplest proposed mechanisms for enzymic reactions. The most widely used assumption in enzyme kinetics is the BRIGGS AND HALDANEl steady-state assumption, where it is proposed that the enzyme intermediate concentrations remain constant. Mathematically this involves setting the time rates of change of concentration of intermediates equal to zero, which reduces the system of differential equations to a set of algebraic equations. The equations are then simplified by imposing the condition that the initial substrate concentrations are much greater than the initial enzyme concentration. Since experimental measurements are usually made in the initial stages of the reaction, it is further assumed that any changes occurring in the product concentrations are negligible, and terms involving product concentrations in the equations are equated to zero. Using these assumptions, an expression relating the "initial velocity" of the reaction and the initial substrate concentrations is obtained. A precise analysis of enzymic mechanisms tends to be limited by the assumptions and conditions present in the steady-state derivation. The use of the steadyBiochim, Biophys, Acta,

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state assumption does not provided sufficient information for the determination of rate constants of proposed mechanisms, since the rate equations are independent of the number of intermediates present and only lower limits of the rate constants can be obtained 2. Imposing the condition that the initial substrate concentrations are much greater than the initial enzyme concentration restricts the range of concentrations of substrate over which experimental data can be applied to steady-state rate equations. In this paper the rate equation for the reversible one substrate-one intermediate-one product enzymic mechanism will be derived by a new method which does not make use of the steady-state assumption nor imposes any conditions on the concentrations of substrate and enzyme. THEORETICAL

Preliminary theory for the derivation of the rate equation. In the simplest enzymic mechanism one substrate molecule (5) reacts with an enzyme molecule (E) * to form an intermediate (X) which breaks down to form product (P) and regenerate the enzyme. This mechanism is shown in equation (r): (1)

where kv k_lI k-1 ' , and k1 ' are the velocity constants for the steps indicated. The general properties of the product versus time curve for the reversible one substrateone intermediate-one product enzymic mechanism have been discussed by HEARON et at.3 Initially the product concentration P and velocity P (time rate of change of product concentration) are zero. Eventually the system reaches a stable equilibrium when its velocity is again zero. Between the starting point of the reaction and the equilibrium position, the velocity versus time curve must pass through at which time the acceleration 'P (time rate change of velocity) is a maximum zero. When the maximum velocity is reached the product versus time curve passes through a point of inflexion. It is this maximum velocity that is used in deriving the rate equation.

»;

Derivation of maximwm. velocity equaiion From the law of mass action two independent simultaneous differential Eqns. 2 and 3 can be written for the mechanism shown in Eqn. I. Thus the time rate of change of concentration of the intermediate is given by:

x=

klES

+

kl'EP - (k-l

+

k_/)X

and of the product by:

whereE, 5, X, and P represent the concentrations of enzyme, substrate, intermediate, and product respectively at any time. Finally, in order to fully describe the mechanism, the conservation Eqns. 4 and 5 for enzyme and substrate are given by: • Note that this derivation assumes that there

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IS

one active site per molecule of enzyme.

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ENZYME KINETICS: MAXIMUM RATE OF PRODUCT FORMATION

Eo = E

+ X.

So = S

+

+P

X

where Eo and So are the initial concentrations of enzyme and substrate. Elimination of the v ar iables E and 5 from Eqns . 2 and 3 by use of the conservation Eqns. 4 and 5 gives:

X=

k,X' - (h,So

P=

{k_{

+ h,Eo + h_, + k-,')X + h,EoS o + (h,'

+ k{P)X

(6)

- h,) (E o - X )P.

- k,'EoP.

(7)

Differentiation of Eqn. 7 with respect to time gives

P=

(h_,'

+ h,' P)X + h,'(X -

Eorp.

(8)

When the maximum velocity is reached.

p

=

(g)

0.

(10) P = Pm.

(Ill'

X =X m ,

(12)*

X=X m

(13)*

and Eqns. 6, 7 and 8 reduce to a system of three algebraic Eqns. q, IS and I6 in the four unknowns Pm . X m and .Km:

»-:

Xm

= k,X'm -

»; =

+ k_,')Xm + k,EoS o + (k,'

+ k,'Pm)X m - h,'EoP m• + k ,'P m)X m + k,'(X m .-

(1~ -1'

= (tl- 1 '

o

(k,S O + h,ED + h_,

- k, ) (Eo - Xm)P m • (14)

(IS) (16)

Eo)P", .

Substitution of Kmand Pm from Eqns. I4 and IS into Eqn. I6 gives a quadratic Eqo . I7 in X m aX'm - bX

m

+C=

(I7)

0

where a

=

(k-t'

b

= =

(k- I '

C

(L t '

+ k/Pm) (k, + h,')

(I8)

+ kt'P m) [h,So + (k, + ht')E o + k_ t + k_,' + (h,' + kt'P m) [h,SO + {h{ - k,)Pm]E o + k,'2E o2Pm.

- h,)P m]

+ ht' 2E oP m

(19)

(20)

The solution of Eqn. I7 is given by I

Xm = -

(b

2a

±

Vb' - 4acl

I t can be shown that b2

- -

4ac = {(k-t'

+ h,'Pm ) [(k, + k,')Eo -

+ 4(k_ 1 ' + h,' Pm)' (k-, +

k,So - k-, - k_,' - (kt ' - h,)P m] - k,'2E oP m)' k-I') (h, + ht')Eo. (22)

* Pm, X m and X mare th.e respective values of P, X and X at the instant when Note further that Pm. X m and X m are in fact constants, as is Pm.

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Hence the discriminant as expressed in Eqn. 22 is always positive and the roots of Eqn. 17 are real. Further, one of the solutions of Eqn. 21 does not satisfy the physical situation, since, on taking the positive sign in Eqn. 21 and using Eqn. 22, it is seen that

which gives Xm

>

(24)

Eo.

This contradicts the requirement that X m can never be greater than the initial amount of enzyme Eo. Thus the positive sign does not apply to our conditions. The maximum velocity Pm is obtained by substituting Eqn. 21 with the negative sign into Eqn. IS, viz. . Pm =

b - 2k/E o (l11 2

+ hI') Ps« (k 1 + k 1 ' )

Vb" - 4ac

(25)

Eqn. 25 is a general expression for the maximum velocity of product formation as a function of the initial concenfor the mechanism shown in Eqn. 1. It gives trations of enzyme (Eo) and substrate (So) and the concentration of product (Pm) when the maximum velocity is reached. In the next section it will be shown how this expression reduces to a more simple form when a single assumption is invoked.

»;

Simplification of the maximum velocity equation If it is assumed that during the time taken to reach the maximum velocity, a negligible amount of product has been formed. then terms involving Pm in Eqn. 25 may be equated to zero giving . Pm

~

k-1'[k1S0+(k1+k1')Eo+k-1+k-1'] { ( I

-

4k 1(k1+ k1' )SoE o ) 1 }

I -

[k 1S0+(k 1 +1I1')Eo+k-1+k-112

2(k 1+k 1')

•

(26)

Since the expression inside the square root sign is a positive multiple of the discriminant , it is, like the discriminant, positive and hence 0<

4k 1(k\

[k 1S 0

+

(k 1

+k

1

' )S oE o

+ k 1') En + k- 1 + h- 11"

<

(27)

I.

It follows that the square foot term can be expanded as a convergent binomial series so,that Eqn. 26 reduces to Eqn. 28 if quadratic and higher power terms of the series can be considered negligible: .

Pm

~

k 1h- l ' S oE o

h1S 0

+ (hI + hl') Eo + k- l + h-

.

(28)

1'

Eqn. 28 provides a good approximation for Pm if the middle term of Eqn. 27 is very small compared to unity, or equivalently that 11 1 5 0

(k 1

+ hI') Eo +

(hI

+ kl')

Eo

h 1S0

+

2 (k- 1 + k_ 1' ) {hI + k l ' ) Eo

+

2 (11- 1 + k- l ') h1S 0

+

(k- 1 + k- 1')" hI (Ill + k1')SU E •

» 2.

(29)

Inequality 29* will hold provided that anyone of the terms on the left hand • Note that the first two tenlls on the left hand side of Eqn. 29 are reciprocals whose sum is never less than two and so confirms the validity of Eqn, 27·

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side be much greater than two. This depends on the relative initial concentrations of enzyme and substrate and on the relative values for the velocity constants. Consider then the following possible relationships: (i) So

;:j;> Eo

This is the condition generally imposed on the steady-state derivation. For the purposes of this discussion, the condition is required in the form So ( r+hi' ) -»2 Eo

hi

that is, the derivation shows that the condition So » Eo may be necessary but not sufficient to justify Eqn. 28. ~

(ii) Eo

So

I n this case the second term on the left hand side of Eqn. 29 will be large relative to two only if »h l ·

// 1'

(iii) Eo

»

So

This immediately makes the second term on the left hand side of inequality 29 much greater than two and so justifies Eqn. 28. (iv) Further conditions

Even if the first two terms on the left hand side of Eqn. 29 are comparatively small, (though. their sum will never be less than two), Eqn. 28 is still valid if the third or fourth terms are large compared to two, that is,

s., + /1_ 1 ' » k_ + k- l ' » 1

(h l

+

hl')E o,

hlSO'

The magnitude of the last term of inequality 29 is inseparable from the conditions given in Eqns, 32 and 33. Comparison of maximum velocity and steady-state rate equations The maximum velocity Eqn. 28 at a fixed initial errzyme concentration and varying initial substrate concentrations can be rewritten in the form

where

v. = K.=

h_l'E o, (k l

+ kl') Eo + 11-1 + k- l ' . kl

Eqn. 34 is of the "initial velocity" to the the theory if we invoke in the previous section.

(3 6)

same form as the steady-state rate equation relating the initial substrate concentration. This equation emerges from a number of alternative conditions, that is, those discussed In particular, it is seen that if So » Eo in accordance with Bioobim, Biophys. Acta, rro (1965) 1-8

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Eqn. 30, then Eqn. 34 certainly applies. However, in our context, Pm has a slightly different meaning to the one usually assumed and K, is a more complex constant. In the steady-state derivation the "initial velocity" should correspond with the maximum rate of product formation, since the initial rate of product formation is zero. In most experimental studies, the initial lag phase in product formation is rarely seen, and, in fact, it is the maximum rate of product formation that is measurecl. From Eqn. 34 it can be seen that a plot of I/Pm versus I/So should yield a straight line from which the kinetic parameters Va and K s may be estimated. V 6 is the maximum rate of formation of product as So approaches infinity. All experimental data supporting the steady-state rate equation, will also support the maximum velocity rate equation. However the maximum velocity derivation results in a more extensive analysis of the mechanism, oiz., rearrangement of Eqn. 28 for a fixed initial substrate concentration and varying initial enzyme concentrations gives

where kIk_ I' VB' = - - - 5 hI + kt' 0,

Eqn. 37 is of the same form as Eqn. 34 and plots of I/P m uersue IIE o should also yield a straight line. Although it is generally accepted that a plot of "initial velocity" versus initial enzyme concentration is linear-, REINER 5 predicted the theoretical possibility of obtaining a levelling off in such a plot at high enzyme concentrations. REINER 6 showed this by omitting the condition that the initial substrate concentration is much greater than the initial enzyme concentration in the steady-state derivation for the irreversible one substrate-one intermediate-one product enzymic mechanism shown in Eqn. 40: E

+S

kl

k_~'

~

X ---'rE

k_ 1

+ P.

The use of the steady-state assumption was not justified under such conditions, but it should be noted that the maximum velocity and steady-state derivations for the irreversible mechanism (Eqn. 40) are identical since

P=

k_I'X

(4 J )

and

P

=

0

(42 )

implies that ~y = o.

(43)

The derivation for the maximum velocity of product formation for the irreversible Biachim, Biophys, Acta, IIO (1965) 1-8

ENZYME IGNETlCS: MAXIMUM RATE

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mechanism can be simply obtained by putting k1 ' equal to zero in all the above equations. DURE AND CORMIER 6 in kinetic studies on luciferase and horseradish peroxidase obtained a levelling off in plots of "initial velocity" uersus initial enzyme concentration and further showed that double reciprocal plots were linear. This experimental evidence confirms the possibility of obtaining a hyperbolic relationship between the maximum rate of product formation and initial enzyme concentration as predicted by Eqn. 37. DISCUSSION The derivation of the rate Eqn. 28 for the reversible one substrate-one intermediate-one product enzymic mechanism (Eqn. I) using the maximum rate of change of product concentration is theoretically more satisfactory than the steadystate derivation or than the equilibrium approach introduced by MICHAELIS AND MENTEN 7 and developed further by STRAUS AND GOLDSTEIN 8 . The steady-state assumption and the condition that the initial substrate concentration is much greater than tile initial enzyme concentration are not required in the general maximal velocity derivation. However the general equation can be reduced to the usually accepted result on assuming the Eo, So disparity in the more precise form So

k1 '

Eo

1/ 1

->or provided the conditions given in Eqns. 32 and 33 are fulfilled. The maximum velocity derivation contains more information than the steady-state equation, since in addition to predicting the relationship between the maximum velocity and initial substrate concentration, it also describes the behaviour of the maximum velocity as the initial enzyme concentration is varied. This allows a further application of experimental data for testing the mechanism. The rate equation derived using the equilibrium approach 7, which involves more assumptions than the steady-state approach, can also be expressed in the same form as Eqn. 34. Indeed, the relationship between the initial substrate concentration So and the velocity ratio PmJVs where V s is given by Eqn. 35 and Pm by Eqns. 15 and 17 for negligible Pm, is essentially of the same form as that obtained by GOLDSTEIN 9 •

ACKNOWLEDGEMENT One of us (I.G.D.) wishes to acknowledge the support of an Australian Dairy Produce Board Senior Post-Graduate Students hip. REFERENCES I G. E. BRIGGS AND J. B, S. HAT,DANE, Biochem, j., 19 (1925) 338. 2 L. PELLER AND R. A. ALBERTY, j. Am. Chem, Soc., 8x (1959) 5907· 3 J, Z, HEARON, S. A. BERNHARD, S. L. FRIESS, D. J. BOTTS AND M, F. MORALES, in P, D. BOYER, H. LARDY AND K. MYRBACK, The Enzymes, Vol. I, Academic Press, New York, and Ed., 1960, p. 49.

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4 M. DIXON AND E. C. WEBB, En!:)'mes, Longmans, Green and Co., London, 2nd Ed., 1964, P·54· 5 J. M. REINER, Behauiour of Enzyme Systems, Burgess, Minneapolis, 1959, p. 546 L. S. DURE AND M. J. CORMIER, J. et«. Chem., 239 (1964) 235I. 7 L. MICHAELIS AND M. L. MENTEN, Biochem, Z., 49 (19 1 3) 333. 8 O. H. STRAUS AND A. GOLDSTEIN, ]. Gen. Physiol., 26 (1943) 559. 9 A. GOLDSTEIN, J. Gen. Physiol., 27 (1944) 5 29.

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