Studies of the Enzyme Fumarase. V.[ Calculation of Minimum and Maximum Values of Constants for the General Fumarase Mechanism' RY
IIOBERT
.'ILBERTY
.WD ~ ' I L L I . u I
H. P E I R C E
RECEIVED ~ O V E M B E Rl R , 1956 Gcnerdly >te;rtly-itatcstudies of enzymatic reactions yield only rate parameters mliich are combinations of individual r a t e clinstantz for the cat'tlytic meclianisin. However, when the rate parameters are determined for both the forward and reverse rcactiony, it beccinies possible to calculate individud rate constants for certain mechanisms. From studies of the effect of P H 011 the initid steady-state velocities of the forward and reverse reactions catalyzed by fumarase, i t is possible t o calculate p apparent acid dissociation constants and 4 pH independent kinetic parameters (2 Michaelis constants and 2 maximum initial velocities). However, t h e simplest mechanism which mill represent the observations involves 12 constants. An investig!.ition has been made of the extent t o which t h e values of t h e desired constants a r e determined b y t h e experimental data. The two acid dissociation constants of the enzymatic site are given directly by t h e experimental data, and it is found that the values of two others are fixed mithin their experiment31 uncertainties. T h e values of 2 of t h e individual r a t e constants are established within their experimental uncertainties, and minimum or maximum values are obtained for t h e remaining constants. It is of special interest t h a t t h e minimuiiz values for t h e second-order rate constants for t h e combination nf enzyme and substrate are of the order of l o 9 set.-' .If-' which is larger than any directly measured second-order r a t e constant for n protein-ion reaction
Introduction e r b l l 1 2 calculated the six specific reaction rate The determination of individual rate constants constants from only rate measurements in the in enzymatic mechanisms by studies of the transient steady state. ki state of the reaction was pioneered by C h a n ~ e . ~ ~ ~ E + ,4 EA I n such studies determination of the time course of kz the concentration of substrate, catalytic intermedika ate, or product make it possible to calculate the E + R s E C + n values of the individual rate constants. The feasikr bility of such experiments is limited by (a) the k5 sensitivity of the experimental method for differE C e E S C entiating the intermediate from the free enzyme or kc detecting low concentrations of the product, (b) the speed of mixing the solutions of enzyme and Calculation of Individual Rate Constants for the substrate, and (c) the speed of response of the Fumarase Mechanism Involving a Single Intermediate Complex measuring instrument. Recently, there has been increased interest in the t e ~ h n i q u eand ~ , ~t h e ~ r y ~ - ~The steady-state treatment of mechanism 1 assuming the substrate and product concentrations of transient state experiments. The information about individual rate constants are large in comparison with the concentration of in enzymatic mechanisms which is obtainable enzymatic sites yields13 from steady-state kinetic studies is usually limited since only certain combinations of the individual rate constants are obtained. However, if the param- where VF and 1 . ' ~ are the maximum steady-state eters in the rate law for the reverse reaction are velocities for fumarate and L-malate and K F and also determined, it is possible to calculate the values K M are the Michaelis constants. When fumarate of the individual rate constants for certain mech- is in the initial substrate the second term in the crnisrns. Two examples of such mechanisms are (1 j numerator is negligible so Iong as the reaction is far from equilibrium, but this is not necessarily true ki k3 of the term involving (ill) in the denominator. E + F ~ E X = E + M (11'0 However, by means of integration of equation 3 ky ki and use of series expansions in time it is possible which is discussed below, and mechanism (a), to show that the steady-state velocities for the forwhich is illustrated by the liver alcohol dehydro- ward, zf, and reverse, z,,reactions obtained by genase reaction for which Theorell and co-work- extrapolation t o t = 0 are given byli (1) T h e preceding article in this series is C. Frieden, R . G. Wolfe and R. A. Alberty, Trim JOURNAL, 7 9 , 1523 (19.57). (2) T h i s research was supported b y t h e National Science Foundation a n d b y t h e Research Committee of t h e University of Wisconsin from f u n d s supplied b y t h e Wisconsin Alumni Research Foundation. ( 3 ) B. Chance, S c i e n c e , 9 2 , 455 (1940). (1) B. Chance, J . B i d . Cheiiz., 151, 553 (1943). 1 3 ) F. J. W. Roughton, F a v a d a y SOC.Disc.. 17, 116 (1954). (0) H. C u t f r e u n d . i b i d . , 2 0 , 167 (1955). 17) l f .T;. hlora!es and D. E . Goidman, THISJ O U R K A L , 77, 0069 (1955). 18) E(. J. Laidler, Cniznifian J . Chem., 33, 1614 (1955). ,9) I,. Ouellet a n d li J . Laidler, i b i d . . 34, 146 (1956). (10) G I3. Iiistiakowsky a n d P. C. Llangelsdorf, Jr., THISJ O U R N A L , 7 8 , 2404 (1956). have mentiorred t h e possibility of calculating all four r a i e conytants in t h i s merhanism for enzymatic ester hydrolysis.
where for mechanism 1 V F = ka(E)o
VM = kn(E)o
(11) H. Theorell, A . P. Nygaard a n d R. Bonnichsen, A c f a C h i n . .Scaizd.. 8 , 1490 (1954). (12) H. Theorell, A . P. I'Jygaard a n d R . Bonnichsen, i b i d , 9 , 11-i8 (1955). (13) J. B. S. Haldane, "Enzymes," Longmans. Green and C o . , L o n d o n , 1930. (14) 11. .4. d l h e r t ? and B E. k f e y r r s , unpuhlishrrl
April 5 , 1957
CONSTANTS
FOR
THE
GENERAL FUhLIRASE
These four kinetic parameters are not independent but are related through the equilibrium constant for the over-all reaction13 as has been shown.’j9l6 However, for present purposes i t is simpler not to eliminate one of these equations by introduction of the equilibrium constant. All the rate constants calculated in this article will be consistent with the equilibrium expression because the experimental data are consistent. Equation 3 is suitable for representing kinetic data for the fumarase reaction only a t sufficiently low substrate concentrations that the effect of substrate activation or inhibition is avoided.l6 Since the two Michaelis constants and two maximum initial velocities have been determined over a range of p H values it is of interest to calculate the corresponding values of k1, k2, ks and kq. To show how these constants vary with p H for 0.01 ionic strength “tris”17 acetate buffers, the values have been calculated for DH 6, 7 and 8 using:
One of the striking things about the values of the specific rate constants which are summarized in Table I is that kl and kq are very large, of the order of lo8 set.-' M-l. These are very large secondorder constants for a reaction of a protein. Unfortunately, mechanism 1 is too simple to represent the effect of p H on the kinetics since according to this mechanism it would be expected that k2 and ks would vary in the same way with pH. It is evident from Table I that this is not the case. However, it is of interest to inquire as to the duration of the transient phase of the reaction using the values of the rate constants of Table I. Since the transient phase is short the product concentration term in the equation for d(EX)/dt may be ignored and the initial substrate concentration (F)o may be considered constant during the transient phase. It is shown8readily that the concentration of EX increases in a first-order manner with a halflife of tl/, = 0.693/[kl(F)0 (k2 k3)]. At low substrate concentrations the half-lives are independent of substrate concentration and would be 4.0 X 3.3 X and 5.4 X sec. a t $H 6, 7 and 8, respectively. Thus it does not appear that the transient state of this enzymatic reaction could be studied by direct methods using presently available techniques. It is of interest to note that in the case of mechanism 1 the concentration of E X may either increase or decrease during the steady-state phase of the reaction. Provided the substrate and product concentrations during the steady state are large in comparison with the concentration of enzymatic sites
+
+
1.527
~IECHANISJI
According to the values of the rate constants in Table I when fumarate is the initial reactant, the concentration of EX increases in the steady state so that d(EX)/dt does not actually equal zero until equilibrium is reached. Jf L-malate is the initial reactant, the concentration of E X goes through a maximum very early in the reaction. TABLE I KINETICPARAMETERS FOR THE FUMARASE REACTIONFOR 25’ AND 0.01 IOSICSTREXGTH “TRIS”ACETATE BUFFER 9H
7
6
8
Experimental parameters (seC.-’) 1.45 1.20 [ ~ F / ( E ) ox ] [ V M / ( E ) Ox ] (seC.-’) 0.26 0.93 KF X lo6 ( M ) 5.7 4.7 K M X lo6 (,liT) 4.6 15.9 Derived rate constants 0.30 0.26 kz X (sec.-I) k3 X 10-3 (sec.-l) 1.45 k4 X 10-9 (sec.-lJR-l) 0.38
kl X lo-@(sec.-lAR-l)
0.30 0.98 7.3 103
0.45 0.93 1.20 0.13
0.18 .98 .30 ,012
Calculation of Individual Rate Constants for the General Fumarase Mechanism In order to provide a basis for the interpretation of the effect of p H on the fumarase reaction it is necessary to extend mechanism 1 to include two intermediate enzyme-substrate complexes and to provide for two proton dissociations of each complex and of the free enzymatic site.18 The following mechanism is believed to be the simplest one which may be used to represent the effect of hydrogen ion concentration on the reaction. E
ki
F f E H
J_
kt
EHF
kz KbEti
EHz
EM
EF
If
k5
EHM s E H + M
k4 KbEFTi EHzF
E
ke
KLEMTJ EHnM
KtJZji
(11)
EHz
In this mechanism k1 to ke are the individual rate constants for the interconversions of EH, E H F and EHM; while K s E , K b E l K a E F , K b E F , K a E M and K ~ E M are the first and second acid dissociation constants for the ionization of the enzymatic site. As expected for the steady-state treatment of a mechanism of this type i t is found that only the equilibrium constants for the acid dissociations are involved and not the individual rate constants for the proton dissociations and associations. Furthermore the steady-state rate equation may be arranged in the form of equation 3. The Michaelis constants and maximum velocities are given by
(15) R. M. Bock a n d R. A. Alberty, THISJOURNAL, 7 6 , 1921 (1953). (16) R. A. Alberty, V. Massey, C. Frieden a n d A. R. Fuhlbrigge,
ibid., 76, 2485 (1954). (17) T h e abbreviation “tris” will be used f o r tris-(hydroxymethy1)aminomethane cation.
(18) C. Frieden and R. A. Alberty, J . Biol. Chem., 212, 859 (1955).
The evaluation of the parameters in the righthand side members of these equations has been discussed in the preceding article.’ It is of interest to note that two kinetic constants (&E and K b E ) for mechanism 11 are obtained directly from the experimental data. The PH-independent maximum initial velocities and Michaelis constants are related to the rate constants of mechanism 11 by
(30)
k,,
The apparent acid dissociation constants (primed) are related to those in mechanism 11 by19
=
b17’t
+
(1-~ - ci)I-’,, R’,l( 1 - ( I ) ~
(33)
These equations are obtained by first eliminating k j between equations 16 and 24 and k2 between equations 18 and 25. The two simultaneous equations in ks, and kq are then solved to obtain equations 32 and 33. It remains now to determine how a and b must be restricted so that the 10 unknown constants will all be positive. From the definition of a and b we immediately have that 0 < a < 1 and 0 < b < 1 , and from equations 32 and 33 we have that (a b - 1) > 0. With these restrictions on a and b it follows that kl, kz, kB,kq, k j and k6 will all be positive. It is apparent from equations 33-29 that the acid dissociation constants will be positive if
+
The values of the ten experimental parameters for the fumarase reaction in “tris” acetate buffers a t 2.5’ are given in Table I of the preceding article. Although the 8 non-linear relationships (equations 16-23) between the ten unknown quantities are insufficient to determine the values of these constants, they do place certain restrictions on the values of these constants. The 8 non-linear relationships are such that it is possible to write the 10 unknown constants in terms of the 8 measurable quantities and two combinations of specific rate constants given by
bKfaEiI - (1 - ~ ) K ’ * E> F0 uK‘~EF - (1 - b ) K ’ * ~ > v 0 bK’t,EF - (1 - a)K’bcM > 0 aK’bE\{ - (1 - b ) K ’ h E F > 0
(36)
(37) (38) (39)
If the inequality signs are replaced with equal signs the plots in a - b plane of these four equations are shown in Fig. 1. This figure is for the special case, always observed for fumarase, that K ’ a >~ K~a E S I and K ’ b E F > K ’ b E M . Since the inequality signs require that we use values of a and b to the right and above the corresponding straight lines, the shaded area in this figure gives the range of permitted values of a and h. kr ks a = _____ (24) In considering the ranges of values for the ten ks ka ks unknown constants we can interpret these ten unknowns geometrically as surfaces above the a - B plane, and we can consider the heights of the surwhich are the only combinations of constants ap- faces over the shaded area in the a - b plane. Expearing in equations 20-23. The first equation cept for kz and ks,which can be handled quite simgiven below is obtained by eliminating K a E M be- ply, it turns out that the surfaces for the remaining tween equations 20 and 22 after introducing 24 eight unknowns have a constant height above either and 25, and the subsequent equations are obtained the boundary imposed by equation 30 or the boundin a similar fashion. ary imposed by equation 39. It turns out further a f b - 1 that for each of the remaining eight unknowns, the KsEF = (F) partial derivatives with respect to a and b are both _ _b_ - ____ (26) K’*EF K‘,cir positive, or both negative. If they are positive the constant height above the boundary due to equan f b - 1 ILru = n tion 36 or 39 will be the minimum value of the un(1 _-_ b ) __ - (27) known, and the maximum value is obtained a t c K’*EM K’*EF = b = 1. If the partial derivatives are both negative, the constant height above the boundary due to equation 36 or 39 will be the maximum value of the unknown, and the minimum value is obtained a t a = b = 1. For example, k l has a constant The individual reaction rate constants kl - kg height above the boundary due to equation 39 and are expressed in terms of a and b and the experi- substituting a / ( l - b ) = K’bEF/K’bEH in equation mental parameters by 30 yields k l = [(K’bET;/K’bEM) I/’RIf V’F]/KF. Since b K1 ’bn > 0 a n d a k j lb b > 0 this is the minii i ~ i )11 h Alrtrrt> .I f,il Cc7 /> r h r c i o , 4 7 , 245 ,185(>)
+ + +
COKST.ZNTS FOR
April 5, 1937
SPECIFIC
p = 0.003
= 0001 hlaui-
fi
Minimum
mum
17 30 1.2 1.2 6
25
m
45 2.0
m
2.0 1.3
1.8
m
pKbEM
8.6
m
m m
2.2 2.0
+
K’F
v’F
