3368
MILDREDTARASZKA AKD ROBERT A. ALBERTY
Extensions of the Steady-State Ratel Law for the Fumarase Reaction
by Mildred Taraszka and Robert A. Alberty Department of Chemistry, University of Wisconsin, Madison, Wisconsin
(Received J u n e 1 , 2965)
~~
~
~-
When steady-state and relaxation rates for the fumarase reaction are measured with both fumarate and L-malate present, it is found that there is a term in the denominator of the rate law which is proportional to the product of fumarate and L-malate concentrations. The coefficient of this term was measured at three pH values. At the high substrate and product concentrations required to make this denominator term important, numerator terms which represent substrate and product activation must also be taken into account. These experimental data are discussed in terms of a mechanism which provides for the binding of substrate or product molecules a t a neighboring site so that the properties of the enzymatic site are altered. This mechanism may readily be extended to represent other experimental data on the fumarase reaction, namely competitive inhibition and pH effects.
Introduction and Theory The steady-state rate equation2&tbfor a mechanism of the type
E
+ S eXi. . . kl
le,
k-,
kb+1)
Xi d . . . X, k-k*l,
kCr‘+l,
__L k-cn+l,
E
+P
where V S and VP are maximum velocities and Ks and Kp are Michaelis constant,s. Assuming that the time required to reach a steady state is negligible in comparison with the elapsed time,3 eq. 2 may be integrated to yield (P) -= t
+
V d K s VP/KP 1/Ks - 1/KP
tions. This indicates that rate eq. 2 , and therefore mechanism 1, does not adequately represent the steadystate behavior of this enzymatic reaction. The nature of the deviations suggests that there is an additional term of the form of (S)(P)/Kgp in the denominator of the rate equation, written in the form of eq. 2. The simplest mechanism which requires such a term is mechanism 4. The enzyme E’ has to undergo an isomerization t o regenerate the original form E before it can react with another molecule of substrate S.
The steady-state rate equation for mechanism 4 is
where
+
According to this equation, plots of (P)/t us. ( l / t ) In [l - (P)/(P)e,]should be linear. For the fumarase reaction it is found that such plots are not linear, especially at higher substrate concentraT h e Journal of Physical Chemistry
(1) This research was supported by grants from the National Science Foundation and the National Institutes of Health. (2) (a) J. B. S. Haldane, “Enzymes,” Longmans, Green and Co., London, 1930, p. 81; (b) L. Peller and R. A. Alberty, J . Am. Chem. Soc., 81, 5907 (1959). (3) W. Miller and R. A. Alberty, ibid., 80, 5146 (1958).
STEADY-STATE lE(IxErrICs 0 F THE
KP
(IC-1 - __
-
+
k2)(k3
k-2(k-1
KSp
=
FUMARASE REACTION
+
(IC-1 t k2)(k3 -___klk-2
+
k-3)
k3)
t k--3) (6)
If this type of mechanism is to be used to rationalize kinetic data for the fumarase reaction it must be possible to extend it to give a satisfactory representation of the experimental facts about competitive inhibition, substrate activation, the effect of pH, etc. The salts of a number of dicarboxylic acids have been found4-’ to be competitive inhibitors of furmarase, and the same values of the inhibition constant have been obtained whether fumarate or L-malate is used as substrate. If the following steps are added to mechanism 4 KI
E’
+I
inhibition, but the (S)(P)/KSPterm is no longer required in the steady-state rate equation. Thus mechanism 4 does not account for the experimental facts of the fumarase reaction. A second mechanism which leads to a K S P term is one involving substrate and product activation or inhibition. At high concentrations both fumarate and L-malate act as activators of f u m a r a ~ e . ~This ? ~ is in contrast with substrate inhibition which is frequently observed at high substrate concentrations for some other enzymes.1° For the fumarase reaction substrate activation has been attributed to the binding of substrate at sites on the enzyme which have an activating effect upon the enzymatic sites. Mechanism 10 provides for both substrate and product activation.
S
(7)
E+I+EI K ’I
3369
+Ee
ki
k-1
kz
E X 4 E k-n
+P
ICP
S+E+SE k-i
E’I
ks
P+E+PE
the rate equation may still be written in the form of eq. 5 , but the apparent maximum velocities and Michaelis constants are now given by
vs =
+ SE riSEX =+=SE + P S + PE + P E X + P E +P S
kZk-3(E)O kz[l (I)/K’II
+ + (E) v - _- k3 + k-1[1 + (U/m k--3
IC4
k-lk3
kr
ks
k-4
k-6
k?
ka
k-7
k-a
(10)
Competitive inhibition by I may be provided for by adding the steps
0
KI
E+I+EI KSI
Kp
=
(k-i
+ k z ) { k i [ ] -+ (J)/K’I] + k-3[1 + (I)/Krl] + k-1D + (I)/KI11
KSP=
PI
_.
k-z(k3
(k-i
+I K PE + I =+=
SE
+ kz) fk3[1+ ( ~ ) / K ’ I+] k-s[l + (I)/KI I) klk-2 (9)
The inhibition constants KI and K’I are defined as dissociation constants. In general V Sand 1 7 ~ will be functions of (I) so that the inhibition will not be competitive. If the inhibitor can combine with only one form of the enzyme, either KI or K’1 will be infinite and the corresponding maximum velocity will be independent of (I). However, since the other maximum velocity will be dependent on (I), this mechanism is not consistent with competitive >:> inhibition for both substrate and product. If kz and ka >> k+, that is, if the step involving enzyme isomerization is a rapid equilibrium, V S and V p become independent of (I), and K S and KP show the same dependence on (I) in accordance with competitive
SEI
(11)
PEI
where the equilibrium constants are defined as dissociation constants. The steady-state rate equation for mechanism 10 with the three additional reactions with I can be written as
’
+ + ep(p)l + ( S ) / K s + (P)/Kp + (S)(P)/KSP+ (S)’/Ksz + (P)2/Kpn
I(v~/K~)(s) - (v~/K~)(P)I[~ es(s) =
1
(12) (4) V. Massey, Biochem. J.,5 5 , 172 (1953). (5) C. Frieden, Ph.D. Thesis, University of Wisconsin, 1955, p. 26. (6) G. G. Hammes and R. A. Alberty, J . A m . Chem. Soc., 8 2 , 1564 (1960). (7) P. W. Wigler and R. A. Alberty, {bid , 8 2 , 5482 (1960) ( 8 ) R. A. Alberty and R. M . Bock, PTOCNatl. Acad. Sci. U . S., 39, 895 (1953). (9) R. A. Alberty, V. Massey, C . Frieden, and A. R. Fuhlbrigge, J . Am. Chem. SOC.,7 6 , 2485 (1954). (10) M.Dixon and E. C. Webb, “Enzymes,” Academic Press, Inc., New York, N. Y., 1958, pp. 81-91.
Volume 68, Number 11
November; 1964
3370
where
~ I I L D RTARASZKA ED AND ROBERT A. ALBERTY
but have different values for the rate constants. The second involves enzymatic sites which are identical but interact in pairs. The first mechanism can be eliminated on the basis of arguments about competitive inhibition which are similar to those given above. The second mechanism is a real possibility but will not be discussed here because it involves the rather special circumstance of pairwise interaction.l2
Experimental Procedure
Both Vs and Vp are functions of (I) instead of being independent of (I) as observed for fumarase, unless the inhibitor dissociation constants K ~ and I Kp1 are much larger than the inhibitor concentrations used. All the competitive inhibition data for fumarase have been obtained under experimental conditions with no apparent product or substrate activation; that is, the squared terms in eq. ,12 were negligible. If insignificant concentrations of the complexes SE and PE are formed under these conditions, it is reasonable to assume that the concentrations of SEI and P E I are also insignificant so that the inhibition steps with dissociation constants KsI and Kp1 may be neglected. Under these conditions V S and V p are independent of (I), and K S and Kp are linear functions of (I) and will yield the same inhibition constant KI. Therefore, mechanism 10 can satisfactorily represent the competitive inhibition data for Mechanism 10 can represent any combination of activation or inhibition by substrate and product. At high concentrations of substrate and product the squared and cross product terms in (S) and (P) will be of significance and, depending on the ratio of their coefficients, either activation or inhibition will result. Two mechanisms suggested by Kistiakowsky and Rosenbergll lead to K S p terms. The first involves two types of enzymatic sites which are independent T h e Journal of Physical Chemistyy
Crystalline fumarase was prepared from pig heart muscle by the method described in ref. 13 and was stored as crystals suspended in 50% saturated ammonium sulfate solution at 0”. Fumaric and L-malic acids were purified by methods described earlier. l4 Tris(hydroxymethyl)aminomethane, “Tris, ” used for “Trls” acetate buffers, was primary grade Sigma 121. All solutions were prepared with conductivity water made with a Barnstead conductivity mater still. A Cary 14 recording spectrophotometer with a scale of 0-0.2 absorbancy unit was used to obtain the initial velocities and the relaxation times. All solutions were thermostated at 25.5”, except the buffered fumarase solution which was kept at 0” in order to keep the decay of enzymatic activity to a minimum. The initial velocity and relaxation time for each reaction mixture were corrected to an enzyme concentration which would give a change of 0.0550 absorbancy unit/niin. at 230 mp when 0.5 ml. of buffered enzyme solution was added to 25 ml. of 204 pLM L-malate, 0.02 ‘14 “Tris” acetate, at pH 7.50 and 25.5’. The initial velocity for each reaction mixture is an average of five determinations. The relaxation times were determined in triplicate for a given substrate concentration and were measured from both sides of the equilibrium, that is, excess fumarate in one experiment and excess malate in the other.
Results of Steady-State Velocity Experiments Initial velocities measured with L-malate and fumarate a t pH 7 are shown in Fig. 1. The maximum velocities and Michaelis constants calculated from the linear parts of these graphs and from siniilar data at other pH values are summarized in Table I. I t is evident that, with both fumarate and L-malate, substrate activation is encountered at the higher substrate con(11) G. B. Kistiakowsky and A. J. Rosenberg, J . Am. Chem. SOC.,74, 5020 (1952). (12) For further details see M .Taraszka, Ph.D. Dissertation, University of Wisconsin, 1962. (13) C. Frieden, R. M. Bock, and R. A . Alberty, J . Am. Chem. Soc., 76, 2482 (1954). (14) C. Frieden, R. G. Wolfe, J r . , and R . A. Alberty, ibid.,7 9 , 1623 (1957).
STEADY-STATE I ' h E T I C S
,,, 0
O F THE
I/(M)
,
0
,
I
0.10
,
0.;5
,
3371
FUMARASE REACTION
,
,
,
0.10
0.30
0.20
,
,
,
0.40
I/(F)
Figure 1. Plot of reciprocal initial velocity us. reciprocal initial fumarate concentration and reciprocal L-malate concentration for p H 7.00, 0.02 M "Tris" acetate, and 25". The initial velocity is expressed in absorbancy units/min. for a 10-cm. cuvette a t 230 mp. The substrate concentrations in both cases are given in pmolar units.
Table I : Kinetic Parameters for the Fumarase Reaction in 0.02 M "Tris" Acetate Buffer a t 25"" Parameter
pH 6.60
pH 7.00
1.63 f 0.07 1.26 =t0.04 4 . 6 =t0 . 5 16.3 f 1 . 6 4.6 250 f 100 10-3 10-4
'S
=
0.94 f 0.05 1.38 f O . 0 9 4.8 f O . 0 7 29.4 f 2 . 3 4.2 400 f 150
fumarate; P = r,-malate.
centrations. As required by eq. 2 and 12 the quantity VFKM/VMKF is equal to the directly determined equilibrium ratio of ],-malate to fumarate, which is 4.53 in this buffer at 26' and pH 7. When the product concentration is negligible eq. 12 becomes
+
=
(VS/KS)(S)[1 1 lT(S)/Ks (S)2/Ksz
+
(F)
pH 7.50
(14)
The fact that substrake activation is observed (Fig. 1) indicates that the numerator term in substrate concentration squared is more important than the corresponding denominator term. The values of V S and Ks were calculated from the linear portions of plots of the type of Fig. 1. Then the value of 0s required to give the best fit of the nonlinear portion was determined. The values es = pM-' and
pM
Figure 2. Grid of K M Fvalues for pH 7.00, 0.02 M "tris" acetate, and 25'. The K M Fvalues were calculated using eq. 12, es = 10-3pM-1, ep = lo-' pM-', K s 2 = K M =~ 03, and the kinetic parameters V S ,Vp, Ks, and Kp given in Table I.
ep
= p M - l are found to represent the data a t pH 7. For the fumarase reaction we will let S represent fumarate and P represent L-malate. The steady-state velocities for various mixtures of fumarate and malate were then used to calculate K S P using eq. 12, the values of all of the other parameters being known. The values of KSPobtained at pH 7 are shown in a grid in Fig. 2. These values increase from approximately 150 to 300 p M 2 as the total concentration of fumarate and malate is increased from 80 to 300 p M , but it would not have been expected that a very accurate determination of this parameter could be made, especially in view of the approximate values of Bs, BP, Ks,, and Kpa. As can be seen from Table I nothing can be said about the pH dependence of KSPdue to the large experimental error. Only the initial velocities for solutions in which the sum of the concentrations of fumarate and L-malate was less than 300 p M were considered so that the ionic strength contribution of fumarate plus L-malate was less than 5% of the total ionic strength. Experimental data require an (S)(P) term rather than an (S)2(P)
Volume 68, Number 1 1
November, 1964
MILDRED TARASZKA AND ROBERT A. ALBERTY
3372
term or (S)(P)zterm, and this is in agreement with the proposed mechanism. If the term were (S)z(P), the observed value of K S p would equal K s p , / ( S ) and the value of KSPwould decrease to half its value when (S) is doubled.
Steady-State Relaxation Theory and Results The form of the complete steady-state rate law may also be investigated by studying the rate of approach to equilibrium starting with a mixture which differs only slightly from the equilibrium mixture. This relaxation method6 has already been used to study the fumarase reaction. If the initial displacement of the substrate concentration from equilibrium, [A(S) lo, is sufficiently small, the difference between the substrate concentration at any time and its equilibrium concentration is given by
A(S)
=
[A@) ~ o e - ~ ’ ‘ ~ ~
(15)
where 7-98 is the steady-state relaxation time. Using procedures described earlier it may be shown that for rate eq. 12, corresponding with mechanism 10, the expression for the relaxation time is 0
1
100
200
300
(SIo pM
=
TSS
+ (S)/Ks + (F)/KP + (S)(P)/KSP-f (S)Z/Ks*+ (P)2/KP* (vS/ixs+ v ~ / K [i~ +) es(S) + ep(P)I (16)
Figure 3. Plot of steady-state relaxation time us. total substrate concentration, (S)O = (S) (P), for pH 7.00, 0.02 M “Tris” acetate, and 2 5 ” . The points are experimental. Curves A, B, C, 8, and y are all obtained by using V B ,Ks, VP, Kp, and K,, determined by initial velocity experiments. Curve A is a plot of eq. 18; curves B and C are plots of eq. 17 with K S Pequal to 300 and 200 p M 2 , respectively; curves p and are plots of eq. 17 with es = ~ L M -Op ~ ,= I M - ~ ,and Ksp = 300 and 200 p M 2 , respectively.
+
where the bars indicate equilibrium concentrations. K S (S)o = Introducing K,, = (P)/(S) = V ~ K P / V P and (S) (F) this equation may be rewritten
+ =
TSS
+
+
+
+ +
1 [(S)o/(l Keq)](l/Ks Keq/Kp) [(S)02/(1 Re, ) z l ( K e q / K ~ ~ 1/Ksa Keq2/Kp,) ( ~ s / ~ VP/KP) s { 1 [(s)o/(l K e g ) ](os Ke,ep)
+
+
+
+ +
+
1
(17) For the simple reversible Michaelis-Menten mechanism this reduces to 7-88
=
1
+ [(S)O/(1 + Ked1 + Ke,/KP) VSIKS + VP/KP
(18)
which has been confirmed experimentally for the fumarase reaction at low total substrate concentrations. As may be seen from eq. 17, a nonlinear dependence of T S S on (S)o would be expected at higher concentrations. This nonlinear dependence is illustrated in Fig. 3. To evaluate 7-98 a smooth line was drawn through the spectrophotometer tracing and Guggenheimlb plots of In ( A , + A, - A , ) us. time were used to obtain relaxation times since the rate of reThe Journal of Physical Chemistry
action is proportional to the displacement from equilibrium. A , is the absorbancy at time t and A , + A , is the absorbancy of t plus At, where At is a constant time interval. The relaxation time is equal to the negative reciprocal of the slope of such a plot. All experimental ~ g values g were corrected to the enzyme activity used for the initial velocity data so they can be compared with the values calculated from the parameters determined in initial velocity experiments. The corrected experimental TSS values are plbtted 11s. (S),, in Fig. 3 where (S)c,is the total concentration of fumarate plus L-malate. The graph shows that at low substrate concentrations, (S)o < 50 I M , the data may be represented by eq. 18, which applies to the simple reversible Michaelis-Menten mechanism. At high substrate concentrations the experimental T S S values are in agreement with eq. 17 with the initial (15) E. A. Guggenheim, Phil. Mag., 2, 538 (1926).
STEADY-STATE KINE:rrIcs O F THE
FUMARASE REACTION
velocity parameters from Table I and 6 s pi+P1 and 6p = pL14-1.
=
1OVa
Discussion This paper illustrates the fact that more thorough studies of the kinetics of an enzymatic reaction generally disclose further terms in the rate law which in turn require extension of the mechanism. There are at least two ways of explaining the present results, but mechanism 10 has the advantage that it represents a simple extension of the mechanism which is generally used to account for substrate activation and substrate inhibition effects. The extension consists of providing for effects of both substrate and product. Actually, mechanism 10 has been extended to include more intermediates to provide for the effect of pH on the steady-state rates of the fumarase reaction.12 Since the effect of pH on various kinetic parameters is not discussed in this article, this extension is not necessary here and would only increase the complexity of the rate equation.
3373
There are several difficulties in the determination of further kinetic parameters of the type discussed here. First, since the substrate is an electrolyte, the range of substrate concentration which can be used must be well below the buffer concentration. As the substrate concentration approaches the buffer concentration, the medium effects on the kinetics change appreciably as the substrate concentration is varied. Second, since the rate equation is not linear, it is difficult to devise objective methods for evaluating all of the kinetic parameters a t one time. I n the method used here, errors in the values of the Michaelis constants and maximum velocities determined at low concentrations lead to larger errors in kinetic parameters such as KSP. An analysis of the effect of these errors indicates that KSPis uncertain by about 3 ~ 4 0 % . An important conclusion from this study is that initial velocity studies with only substrate or product present are not sufficient to reveal all of the terms that may he present in the complete rate law for an enzymatic reaction.
Volume 68, J'umber 11
.Vowmber, 1964
