April 20, 1953
APPARENT EQUILIBRIUM CONSTANT FOR ENZYME-CATALYZED REACTTON
variation of Vp/Vy or K F / K ~ with PH, temperature or other variable provided the equilibrium constant and one of these ratios is known. For example, Scott and Powell7have observed that the ratio VF/VH varies from 8 t o 0.5 as the pH is varied from 6.0 to 9.0. Since the equilibrium constant does not vary in this PH range, KM/KF must vary markedly. On the other hand, the variation of VF/VM with temperature over the
[CONTRIBUTION FROM
THE
1925
range 8 to 45" nearly parallels the variation of the equilibrium constant so that KM/KFmust remain nearly constant. The authors are indebted to Miss Barbara Greeley for assistance with the measurements and to Mr. Carl Frieden for preparations of the enzyme. This research was supported by a grant from the National Science Foundation. MADISON, WISCONSIN
DEPARTMENT OF CHEMISTRY, UNIVERSITY OF WISCONSIN ]
The Effect of Enzyme Concentration on the Apparent Equilibrium Constant for an Enzyme-catalyzed Reaction BY ROBERT A. ALBERTY RECEIVED NOVEMBER 24, 1952 The change in apparent equilibxium constant for an enzyme-catalyzed reaction which is produced by the use of high concentrations of the enzyme depends upon the values of the dissociation constants for complexes of the various reactants and products with the enzyme. A treatment of this problem is given together with applications to the data of Theorell and Bonnichsen on the alcohol dehydrogenase reaction.
Theorell and Bonnichsen' found that the value of the equilibrium constant Keq = [DPNH ] [CHsCHO][H+]/ [ D P N ][CsHbOH] ( 1 )
for the reaction DPN
+ CIH~OH
DPNH
+ CHsCHO + H +
(2)a
was increased as much as 200-fold a t PH 6.4-7.8 by the addition of large amounts of liver alcohol dehydrogenase (ADH). They concluded that this increase resulted from a stronger binding of DPNH by the enzyme than of any of the other reactants. Since the wave length of maximum absorption of DPNH is shiited from 340 to 328 mp upon formation of a complex with the enzyme, Theorell and Chance3 were able t o measure the dissociation constant for this complex by a spectrophotometric titration. Since the magnitude of the shift of the equilibrium constant in such a case depends upon the values of the dissociation constants for complexes of the various reactants and products with the enzyme, the equilibrium data itself offers information concerning such interactions. Such information is of special value in cases in which a spectral shift does not occur upon formation of the complex. Apparent displacements of equilibria such as observed by Theorell and Bonnichsen are more likely t o be observed in experiments in which high concentrations of the enzyme are used, a situation more closely resembling that actually encountered in the living organism. Such displacements may be caused not only by interactions of reactants with the enzyme but with other substances which are present. The effect of the hydrogen ion concentration upon equilibria is well known and is frequently expressed in terms of the PH dependence of the electromotive force for an oxidation or reduc(1) H. Theorell and R. Bonnichsen, A c f a C h e n . Stand., 6, 1105 (1951). (2) DPN. diphosphopyridinenucleotide; DPNH, reduced diphos-
phopyr'dinenucleotide. (3) H. Theorell and B. Chance, Aclo Chcm. Scand., 6, 1127 (1951).
tion reaction or in terms of the PH dependence of the apparent equilibrium constant for a r e a ~ t i o n . ~ Theory.-The term apparent equilibrium constant (Kspp) will be used to refer to the equilibrium constant calculated without regard for the degree of binding of the reactants by the enzyme. In the case of the reaction A + B Z C + D = [ c IT [D IT/ [A IT [BIT
Kapp
(3) (4)
where the subscript T's represent total concentrations which may be obtained spectrophotometrically, for example. The value of the apparent equilibriumconstant obtained in the presenceof avanishingly small concentration of the enzyme so that the amounts bound are negligible will be represented by Keq. Ke, = [CI[DI/[AI[Bl
(5)
These two equilibrium constants are related by Kspp
Keq
(fAf~/fd~)
(6)
where the f's represent fractions of the indicated reactant not bound by the enzyme. If a product of the reaction is bound by the enzyme, Kappwill be greater than Keq, while if a reactant is bound, Kappwill be smaller. If, for example, only C is bound, equation (6) becomes
where [EC] is the concentration of C in the form of a complex with the enzyme. The following discussion and calculations will be restricted t o the case that the dissociation of this complex may be represented by E C ~ E + C
(8)
where E represents the site on the enzyme responsible for the binding. The dissociation constant is -____
DEC= [EI[CI/[ECl
(9)
(4) R . A. Alberty. R . M. Smith and R . M. Bock, J. B i d Chcm., 193, 425 (1951).
T'ol. 75
-9I, R ERTY Thus if there is more than one binding site on the enzyme molecule, i t is assumed that these sites are independent arid have the same intrinsic dissociation constant.5 If there are n independent sites per enzyme molecule, [E]Tin the following equations is n times the molar concentration of the enzyme. The concentration of bound C may be expressed in terms of D I ~ c , [ C ] , and [E]T. Two special cases may be considered. If [ E]T >> [e1 I'
DPNH) and roughly 3 p i ! a t $H 10 (based on the fact that ,411H binds one DPNH). Since the ratio of the dissociation constants is 200 a t #H 7 and approximately unity a t pH 10, the dissociation constant for DPX (Dox)would be 20 p M a t pH 7 and about 3 p M a t PH 10. The values of the ratio Kap,,,'Keqdetermined by Theorell and Bonnichsen' a t PH 7 over a range of enzyme concentrations from 0.3 to 79 piV! are giren in Table I along with the total concentrations of D P S :tnd DPYH a t equilibrium. The 1
TABLE
( 1 1)
while in the general case [EC] must be obtained as the solution of a quadratic equation. The value of the ratio K a p p l K e q a t high concentrations of the enzyme may be obtained by introducing equation (10) into (7). As the enzyme concentration is increased the limiting equation becomes ( K L , ) , , / K ~ Q J ( E=) ~[-E~[ ] DFT
which in the limit of increasing enzyme concentration reduces to I -+m
= DFR/UF~
0.1 @.\I
AUD
no,
= 20
@ h ' Km~/Keo
2[ADExpt. no
HIT,
irZI
fl&-
[DPNH]T,
9
39.4
,'3 4 .j
Exptl
(1) 1.08
p M
6
2
(DPNIT,
p M
0.6 31.8 0 . 6 16.0 1 . 4 21.7 5.022.7 10.0 21.1 10.8 20.0
1
fl'i
Thus if a single product is bound the apparent equilibrium constant obtained in the presence of high concentrations of enzyme will be directly proportional to the enzyme concentration. If two product molecules are bound the apparent equilibrium constant will be proportional to the square of the total enzyme concentration a t high en7yme concentrations, etc. The fact that Theorell and Bonnichsen' find definite indications that K,,, reaches a maximum value for reaction (2) indicates that it is necessary to consider, in addition, the binding of a reactant. If B and C are both bound by the enzyme to an appreciable extent, either a t the same or different sites, and the enzytne concentration is high
:
COMPARISOY O F EXPERIMENT.4L VALUES FOR Kapp/Keqa 4 T PH 7 .\'ID 20" TVITH THOSE CALCULATED WITH D r p d =
8.2 24.0 23.3 22.3
18.0
23.9 24.1 82 0 27.0
13.0 12.4
3 9 27 0 22.6
0.92 1.16
1.32 1.118 6.04 11.7 20.7 35.1 72 1 87.3 163 133 17:i
10
13 14
79.0 158 0
Compet. compet. calcd. calcd. (2) (3) 1.02 1.04 1.12 1.28 1.78 7.73 36.8 32.3 74.1 96.7 111 136 141 173
DPNH only (4)
0.998 1.02 1.025 1.04 1.06 1.12 1.14 1.28 150 I89 6.61 10.3 75.0 108 36.1 70 96.9 216 13.5 306 119 377 143 .?SO 168 66 1 184 1.160
The value of K,, was taken t o be 1.11 p M , the average of the two experimental d u e s at the lowest enzyme concentration.
values of the ratio Kappl'Keq a t pH 7 t o be expected from the values of D r e d and Do,obtained by Theore11 and Chance have been calculated for the case that there is no competition between DPX and DPNH for the binding sites (column 3 ) , that there is competition for the same site (column 2) and that only DPNH is bound (column 4). If there is n o competition the following equations may be used for the calculation A T
(11)
-
- ..( .[-A D _ _H - D P S I / P P S l r ) - ( [ADH-I)PSH]/[DP?;H]T)
1
.-
1
~
~
(15)
--(AIIH-DPSH] I(_ [L)PSH]r - [ADH-DPNH]) Thus a t sufficiently high enzyme Dred = (L'JADHJI .. _ ____-__ ~ - (16) ( AUH- 11P Y H concentrations the apparent equilibrium constant will be independ( ~ / A I ) H ]-I ~ A I l H'AIIII - I ) P S ]i,p j[( I I P N-~-T . IADH-DPSI) - -~. _ _ = (17) ent of the concentration of the enzyme and of the concentrations of the reactants. This is an important relation be- where [;ZDHj,p is the molar concentration of the cxuse i t offers a means for calculating the ratio enzyme which is multiplied by 2 since there are two of the dissociation constants for the reactant and binding sites. I n case DPN and DPNH do comproduct which are bound. Thus the ratio of the pete for the same site, a supposition which is dissociation constants for the complex of ADH supported by certain observations," it is necessary with DPN and DPNH is 200 a t $H 7.' to write the concentration of free enzyme as ( 2 The Binding of DPN and DPNH by Liver Alcohol [ADH] - [-4DH-!JPN] - [ADH-DPNH]) Dehydrogenase.--In the case of horse liver alcohol in equations ( lri i arid (1 7 ) . Values of Kapp/Keq caldehydrogenase Theorell and Chance'j found the culated for the case that there is competition are dissociation constant (D,,.,i) for the ADH-DPNH also plotted in Fig. 1 for comparison with the complex a t low enzyme concentration to be 0.1 p experimental values o f Theorell and Bonnichsen. at pH 7 (based on the fact that .4DH binds 2 The values of Kal,l,ICeq are in good agreement with the experimental values and offer an independent ( 5 ) T h e relationship between the intrinsic constant and the usual equilibrium constant has heen discussed by H. S. Simms, THIS test of the dissociation constants obtained by J O U R K A I . . 48, 1239 (1026). spectrophotometric titration. ~-
~
ncl\
, ;-
April 20, 11353
APPARENTEQUILIBRIUM
CONSTANT FOR
ENZYME-CATALYZED REACTION
1927
TABLE I1 VALUES OF Dred A N D Dox FOR ALCOHOLDEHYDROGENASE AT 2.0
_200 _
t-
Dred in p M
DLa
I
.
i
~
7.
i
-I
0.5
-
06
6.
c
7 8
.
Dred
From spect. titr.3
Calcd. from equil. data
200
0.1
140
...
0.1 0. 2 6
9 30 ... 10 1 3 ... a From the value of Kapp/Keq approached a t the highest enzyme concentrations.
and the ratios K a p p / K e q calculated from these values may be compared with the experimental results in Fig. 2 .
1
I
-
1
I
1 1
2.0
0
1
2
3
Log(ADH) t 7, Fig. l.-Plot of log (&,/&q) versus log [ADHIT a t pH 7 and 20": 0 , experimental data of Theorell and Bonnichsenx; 0, values calculated from equations (15)-( 17) using Dred = 0.1 p M and Do= = 20 p M .
The ratio K a p p / K e q is actually not a continuous function of [El0 since the concentrations of DPN and DPNH a t equilibrium vary from experiment to experiment. There are three pairs of equilibrium experiments in which the total enzyme concentration is constant but the concentrations of DPN and DPNH are varied (experiments 6 and 7, 9 and 10, 12 and 13). In each case there are quite appreciable differences between the experimentally determined apparent equilibrium constants. It is interesting to see the extent to which the theory predicts these differences. In the first pair of experiments (6 and 7) the concentration of DPNH is less in experiment 7 so that it would be expected that a larger fraction would be bound. If a larger fraction of DPNH is bound it would be expected that K a p p l K e q would be larger than in experiment 6, as is actually observed. In the second pair of experiments (9 and 10) the total concentration of DPNH is lower in experiment 10, so that the larger value of K a p p / K e q in this experiment is in agreement with the theory. In the third pair of experiments (12 and 13) the theory predicts relatively little effect of the change of the [DPNH]T/[DPN]T ratio, while an effect in the opposite direction is observed. The values of D r e d a t PH 8 and 9 have not beefi determined spectrophotometrically, but since the apparent equilibrium constants have been measured over a range of enzyme concentration under these conditions, the values for D r e d may be calculated from the equilibrium data. A method of successive approximations has been used to obtain Dred for the case that the two forms of the coenzyme compete for the same site. The values obtained in this way are given in the last column of Table I1
.
c
0 0
I
a
1
0.5
3
I
2
3
Log(ADH) + 7 . Fig. 2.-Plot of log (Kapp/Keq) versus log [ADHIT a t pH 8 and 9 and 20": 0 , experimental data of Theorell and Bonnichsenl; 0, values calculated from equations (15)-( 17) using a t PH 8, Dred = 0.2 p M , Do, = 28 p M and at PH 9, Dred = 6 p M , Do, = 180 p h l . The points at pH 9 are represented by 0 - (exptl.) and 0- (theoret.), and only the points a t the three highest enzyme concentrations are included since a t the lower enzyme concentrations the ratio is unity within the experimental error.
The Binding of DPN and DPNH by Heart Lactic Dehydrogenase.-Neilandss finds that the apparent equilibrium constant for the reaction Lactate
+ DPN
DPNH
+ Pyruvate + H +
(18)
is dependent upon the concentration of heart lactic dehydrogenase and values of Kapp/Keqof 50 and 36 were reached a t PH 7 and 8, respectively, a t 25'. The values of K a p p / K e q did not appear to approach a maximum for the enzyme concentrations used as was true in the case of alcohol dehydrogenase. Chance and Neilands' have measured the dissociation constant for the complex of DPNH with the enzyme spectrophotometrically a t PH 7.15 and 5' ( 6 ) J. Neilands, J . B i d . Chcm., 199, 373 (1962). (7) B. Chance and J. Neilands, i b i d . , 199, 383 (1952).
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
