Vol. 75
ROBERTA. ALBERTY
1928
and obtain 7 p M . Their experiments also indicate that a t 25’ the value of D r e d is greater than a t 4’. Calculations such as described above. show that if DPNH is the only substrate appreciably bound such a value for D r e d is too large to account for the increase in Kappj’Keq which is observed. A value of n r e d of 0.8 p M would be required if this were the case. As pointed out by Neilands6 the low value of the Michaelis constant for pyruvate suggests that this substance is also strongly bound. If it is assumed that the dissociation constant for pyruvate is the same as that for DPNH (that is, i p M ) the v:tlues of Kal,p/f> ks for the forward reaction or as kz >> k4 for the reverse reaction. Four of these kinetic constants are related by equation (5). Since mechanisms (1,l) (rapid equilibrium case), (I,2) and (I,3) yield the same rate equation and four of the kinetic constants are related in the same way to the equilibrium constant for the over-all reaction, it is not possible to distinguish between them with only measurements of over-all kinetics, and so the simplest of the three mechanisms should be used. Mechanism Involving Two Complexes.-If in the preceding mechanism k5 >> k7 and kb >> kZ the same relations are obtained as if
ks
The steady state treatment of this mechanism yields equation (2) for the initial velocity of the forward reaction in which Vf = k , [ E ] o / F , K.4 = G/kaF, KAB = kZG/klk3k5F, F = k4k7/klkrjF, Kg k3(k6 k7)/klks k7/k9 3- 1, and G = krke f k4k7 k~k7. Similarly for the reverse reaction Vr = kd[E]o/H, Kc = G/ksksH, KD = k4k7/kioH, KCD= kgG/ksksk~dIand H = h(k4 4- k5)/ k6k1o f k4/k2 f 1. Four of these kinetic constants are related to the equilibrium constant for the over-all reaction by equation (5). Thus it is seen that the over-all kinetics for this mechanism are indistinguishable from those for the preceding case. Mechanism Involving Three Complexes.-If there is but a single ternary complex mechanism i Ita) may be simplified to
VOl. 75
ALBERTY
+
+ +
I
Ii
li
+
(9)
Mechanism Involving a Single Complex.-The simplest, but probably least likely, mechanism for reaction (I) is ki
E
+ A + B J-EXY kz
k3 J-E kq
+c+D
(13)
The steady state treatment for this mechanism yields the following equation for the initial velocity of the forward reaction.
+
(8)
F h ~ h t n i Haoptilr
38,
where Vf= ks[E]oand K A B= (k2 & ) / k l . Thus the apparent Michaelis constant for A obtained by holding the concentration of B constant for a series of experiments in which the concentration of A is varied will be inversely proportional to the ( 1 4 ) I I Theorell and B Chance i l i l i i Chrm Srnwd
, 5 , 1127 (1051)
April 20, 1953 ENZYME CATALYZED REACTIONS : MICHAELIS CONSTANTS AND MAXIMUM VELOCITIES1931
COMPARISON O F EQUILIBRIUM CONSTANTS FOR
TABLE I ALCOHOLDEHYDROGENASE REACTION AT 20 FROM KINETICCONSTANTS
THE
Keq, d c d .
Keq.“
PH
a
Eq. (7)
Exptl.
Eq. (14)
1 . 0 7 X lo-‘ 39 x 10-4 7.0 1 . 1 1 x 10-1 5 . 5 x 10-3 0 . 4 9 x 10-3 8.0 0.i1 X 9.0 1.06 X 21 x 10-2 4.0 X 10.0 0 . 9 0 x lo-’ 1 . 2 x 10-1 1 . 5 X lo-’ The average value for the lowest enzyme concentration is given.
concentration of B. An equation similar to (10) applies for the reverse reaction where Vr = k z [ E ] ~ and KCD= (kz These kinetic constants are related to the equilibrium constant by equation (5). The same relation would be obtained if there were more intermediate complexes of the same composition as EXY. Mechanisms in which One Substrate Can React with the Enzyme to Form a Product.-In the following mechanism the enzyme might, for example, be alternately oxidized or reduced.
+ kJlP4.
ki
E+A=E’+C
(16)
k2
ks E’+ B = E +
WITH THOSE CALCULATED
Eq. (9)
Vf/Vr
0.030 X .u40 x 10-3 .75 x 10-2 1.8 x 10-1
0,029 0.088 0.19 1.2
The steady state treatment yields the following relation for the initial velocity of the forward reaction. k3k7[E]o/(k3 + KA k7(kz + + and KB + + ki). A similar relation is obtained for the reverse reaction with Vr k,k6[EIO/(k2 + KD kz(k + ki)/ks(kz + and K C h(k2 + f I t is easily verified that these kinetic constants are where Vr
k3)/kl(k3
=
=
k7),
k7),
=
=
k6)
ki)/k5(k3
k6),
=
=
k3)/k4(k2
k6).
related to the equilibrium constant for the over-all reaction by the equation Keq = .Vf’KcKn/ Vrr2K.kKn
D
(14)
Application to the Alcohol Dehydrogenase Reaction.-The only reaction of type (I) for which all The initial rate of the forward reaction is given by four Michaelis constants as well as the forward and reverse maximum velocities have been determined appears to be the alcohol dehydrogenase reaction and Theorell where Vf = k&3[E]0,KA = k3 and K B = kl. In studied by Theorell and Bonnichsen lB this case Vf is not the velocity which is approached and Chance.I3 Their best values of the kinetic as the concentrations of A and B are increased. constants a t PH 7, 8, 9 and 10 have been used to If the initial velocity is measured a t concentrations calculate equilibrium constants using equations of A at which KA/[A] >> KB/[B], equation (11) (7), (14) and (9). For this purpose the equilibrium becomes - (d[A]/dt)o = V~[A]/KAso that vf/KA constant is defined by may be determined. The quantities V S / K BVr/& , ICeq = [aldehyde][DPKH]/[alcohol][DPS] (15) and VJKD may also be determined since a similar expression applies to the initial velocity of the These calculated values may be compared with the reverse reaction for which V, = kzkk[Elo, Kc = actual values in Table I . Since equations (7), (14) and (9) differ only in k4 and K D = kz. These kinetic constants are the power of the Vfl V, ratio the differences between related to the equilibrium constant for the overthe results for the various equations will be greatest all reaction by under conditions where this ratio is considerably Keq = VilVr = K ~ K B / K c K I I (12) different from unity. ilt @H 10 this ratio is An interesting feature of this mechanism is that sufficiently close to unity so that it is not possible Since Vf/KA = kl[E]o, Vf/KB = k3[E]o, Vr/Kc = to make a choice between the three equations k%[E]oand V,/KD = k4[E]o, the individual rate without more accurate values for the kinetic conconstants may be determined if the molar concen- stants. At @H 7 and 8 equation (14) gives the tration of the enzyme is known. The usual value which agrees best with the directly deterMichaelis-Menten behavior will be shown by this mined equilibrium constant, while a t fiH 9 equation mechanism if, for example, the initial concentration (9) gives the best agreement. Whether there is of B is held constant for a series of experiments in actually a change in mechanism with pH and which the concentration of A is varied over a wide whether (I, 7) is the mechanism a t pH 7 and 8 will range. In this case equation (3) applies with Vf’ = have to be determined by further experiments. V ~ [ B ] O / Kand B KA’ = Kq[BIO/KB. Mechanisms for Enzymatic Reactions Involving If there is an enzyme-substrate intermediate in One Reactant and Two Products.-LIany enboth steps of the preceding mechanism, it becomes zyme-catalyzed reactions are of the type ki AI_C+D (11) E +A EA I_ E’ + C (I,7)’4 For example, the hydrolysis of esters or peptides kz kd in aqueous solutions could be expressed in this k, k7 way since the concentration of water is high and E’ + B ED If E + D very nearly constant. Since the treatments of ka ks __-~__ ka
e
(14) T h e a u t h o r mechanism
is
indehted t o Dr G W Schwert for suggesting t h i s
115) H Theorell and R
(1951)
Bonnichsen
4 i t t 2 Chem S c n i z i i , 5 , 110;
1932
Vol. 75
ROBERTA. ALBERTY
possible mechanisms for this reaction follow closely those for reaction (I) the rate expressions obtained from the steady-state treatment are simply summarized in Table 11. In all these cases the initial rate of the forward reaction is given by
The steady state treatments for the following two mechanisms have been given by Haldane. ki ka E
+A
E
+
E
EX
ka
ki EX
+C
(II1,l)
kr
kz
If
ks EY
E
+C
(III,2)
k, ke The over-all kinetic behaviors for these mechanisms are indistinguishable, and the initial rates of the forward and reverse reactions are given by equations of the type of (16). The expressions for the over-all kinetic constants in terms of the individual rate constants are given in the preceding paper.2 Haldane pointed out that kz
Mechanism (11, 1) has been treated only for the case that the conversion to EA to ECD is ratedetermining. The kinetic constants for this case are related to the equilibrium constant by Keq =
viKcD/vrKa
(17)
TABLE11 RATEEXPRESSIONS FOR THE INITIAL RATESOF REVERSEREACTION FOR REACTION (11)
- (d[CldOo
Mechanism
(I1,l) E
Keq = T/rKc/vrK~ THE
+ A J _ EA
(19)
This relation is borne out by the data of the pglucoside reaction' and the fumarase reaction.2 The following mechanism involves the reaction of a second molecule of substrate with the enzymesubstrate complex. ki ki ks E + A ~ E A E ; A+A=EC+C; kz kr
E C ~ E + C
ka (IK3)
T I
The initial rate of the forward reaction is given by
+
where Vf = 2ks[E]o, K A = k6(k1 k3)/klk3 and KAZ= k2k~/klka. A similar expression holds for the reverse reaction in which V, = 2k2[E]o, KC = ka(k4 ke)/krke and K C ~= kzks/krks. Thus it
+
a This expression is obtained only for the case that the conversion of EA t o ECD is rate-determining.
If, in addition, the binding of C is not affected by the binding of D, and vice versa K,, = V X C K D / V ~ K A
(18)
This relation is given by Haldane.16 In the case of the other three mechanisms, equation (17) is also found to apply. Mechanisms for Enzymatic Reactions Involving One Reactant and One Product.-The simplest type of enzyme-catalyzed reaction may be represented by __ (16) Reference 1, p. 51,
A
Z
C
(111)
can be seen that the equilibrium constant for the over-all reaction is related to these kinetic constants by K,, = [CI/[Al = ~ V T / ~ K C ~ / V ~(21) KA~
The number of mechanisms which may be invoked in the study of enzymatic reactions is practically limitless but the above mechanisms are suggestive of some general types and the way in which the relation between kinetic constants and equilibrium constants may be used in selecting a suitable mechanism. The author is indebted to Dr. E. L. King for helpful advice. This work was supported by the National Science Foundation and by the Research Committee of the Graduate School of the University of Wisconsin from funds supplied by the Wisconsin Alumni Research Foundation. MADISON, WISCONSIN
