Nov. 20, 1959
ATU1,TIPLE
INTERMEDIATES I N
bands in each compound would tend to become equal as the temperature is raised. This does indeed occur as shown in Table 11, thus lending strong support to the above conclusions. The samples were run as described except that hexachlorobutadiene was used as a solvent. Since the dielectric properties of this solvent are similar to those of carbon tetrachloride, the 0-H spectra obtained from both solvents are nearly identical, but the temperature can be extended well beyond the liquid range of carbon tetrachloride with hexachlorobu tadiene. Finally, as will be shown in a future publication, the association bands of o-halophenols are approximately the Same width as the associated bands of compounds containing O-H-N or O-H-O hydrogen bonds. In these latter compounds, the interacting distances and rotational configurations must be just as restricted as in the o-halophenols.
[CONTRIBUTION FROM THE
STEAD\’ STATE
5907
ENZYME KINRTICS TABLE I1
Halogen substituenls
2
6
F
C1
F
7,
oc.
Absorbance ratiu (peak)
0.61
Br
36 139 33
CI
I
125 35
I3r
I
139 29
.74 2.73 2.08 1.9 1.5
138
.68 I
G8
This
that for these compounds, the A s term in formation Of the hydrogen bond must be nearly zero, as it is for the halophenols. This is the principal for the these types Of bands. PITTSBURGH, CALIF,
DEPARTMENT OF CHEMISTRY, UNIVERSITY O F
WISCONSIN]
Multiple Intermediates in Steady State Enzyme I. The Mechanism Involving a Single Substrate and Product BY LEONARD PELLER AND ROBERTA. ALBERTY RECEIVED JUNE 2, 1959 The method of Christiansen has been employed to derive the general steady-state rate equation for an enzymatically catalyzed reversible reaction involving a single substrate and a single product but with an arbitrary number of intermediates. The observable kinetic parameters (Michaelis constants and maximum velocities) define lower limits to the rate constants for the unimolecular and bimolecular steps. The magnitude of the latter for a number of enzyme systems suggests that a diffusion controlled combination of enzyme and substrate is being observed. The kinetic behavior of the intermediates during the steady-state period is examined. A particular model for the PH variation of the rate as applied to the n-intermediate system is also considered.
Introduction I n studies of the kinetics of enzymatic reactions under steady-state conditions, i t is generally impossible to assign values for the rate constants for specific steps in a mechanism which involves even a reasonable number of intermediates. For a reversible system involving a single substrate and a single product which can be described in terms of one intermediate, the four measurable kinetic parameters (two Michaelis constants and two maximum velocities) suffice to determine the four rate constants. However, such a mechanism if not requiring revision by studies of the influence of other variables, e.g., pH, generally falls far short of providing an adequate description of the chemical steps involved. A number of years ago Christiansena proposed a method for dealing in a straightforward fashion with the steady-state kinetics of complex reaction schemes. Use has been made of this approach in treating multi-barrier kinetic processes by Eyring (1) This research was supported by grants from t h e National Science Foundation and from t h e Research Committee of t h e Graduate School of t h e University of Wisconsin from funds supp!ied by t h e Wisconsin Alumni Research Foundation. (2) Presented in p a r t before the 135th National hleeting of t h e American Chemical Society, Boston, April 5-10, 1959. (3) J . A. Christiansen, 2. p h y s i k . Chcm., 28B,303 (1935); SSB, 145 (1936).
and associates4 It has been applied to sequences of enzymatic reactions by Christianseq6 Hearons and Lumry.’ Most recently Hearon has pointed out some of the complexities which may have to be taken into consideration in dealing with the effect of inhibitors on the kinetics of complex enzymatic mechanisms.8 It is the purpose of this paper to develop a general steady-state rate equation by the use of Christiansen’s method for an enzymatically catalyzed reversible reaction involving a single substrate and product but with an arbitrary number of intermediates. The resulting expression permits a ready definition of Michaelis constants and maximum velocities. The equations for these parameters are put in such a form that certain general arguments can be made concerning their relation to the actual rate constants in the mechanism. Moreover, such an approach clearly indicates which conclusions commonly drawn from (4) B. J. Zwolinrki, H. Eyring and C. E. Reese, J . P h y s . Colloid Chem., 6 3 , 1426 (1949); R . B. Parlin and H. Eyring in “Ion Transport Across Membranes,’’ ed. by H. T. Clarke, Academic Press, Inc., New York, N. Y., 1954, p. 103. ( 5 ) J. A. Christiansen, Acta Chem. S c a d , 3 , 493 (1919). (6) J . 2. Hearon, Physiol. Rev., 3 2 , 499 (1952). (7) R . Lumry, Disc. Faraday Soc., 20, 257 (1955). (8) J. 2. Hearon, S. A. Bernhard, S. L. Friess, D. F. Botts and M. F. Morale9 in “The Enzymes,” Vol. I , 2nd Ed.. Academic Press, Inc., N e w York, N. Y.,1959, pp. 89-108.
LEOXARII PEI,I,ERA X D ROBERT A..ILBERTT
59Ob
considerations of mechanisms of a simple character are of general significance. 1Yhile the language is that of enzyme kinetics, i t is evident that many of these considerations will apply to heterogeneous reactions which follow Langmuir kinetics and more generally to catalytic reactions which admit the possibility of saturation of the catalytic species. Of alternative methods for developing the general steady-state equations of enzyme kineticsJgmention should be made of the approach of King and Altman.lfl Their schematic method permits the expression of the rate equations in a form which displays the dependence on specific rate constants in a particularly simple fashion. The Steady-state Equations.-Let us consider a reaction involving the reversible interconversion of molecules A and B catalyzed by an enzyme E where the mechanism involves a sequence of n intermecliates. This reaction sequence may be represented as follows k, k,z - r ) “ Z X , F + k-t k-(tAl)
k1 E+A,’X1
k-
1
k ( n + 1) . . . xn4--E --+
+B
(1)
k-(n+1)
where the k,’s are rate constants in the forward path and k-,’s are rate constants in the reverse path The condition of conservation of enzymatic sites must apply so
where the (X$\ is the concentration of intermediates, (E) is the concentration of free enzyme sites, and (E)ois the total concentration of enzyme sites. The condition of conservation of substrate and product can be written as
Vol. s1
The steady-state condition for this sequence is defined by setting d(Xi)/dt equal to zero, i.e., under steady-state conditions = . . . = = . . . = = E,2+1=z al = (5) In more physical terms, the net rate of appearance of each intermediate d(Xi)/dt, where (Xi) is the concentration of a substance in a free energy “valley,” must be small in comparison to the rate of passage of material over the free energy barrier on either side of this “valley” for the steady-state approximation to be applicable. For ld(Xi)/ldt! i > s, KtSis defor n 1 > s > i. For n fined as equal to 1/K,$ while k’,$= 1. 1I7hen v equals zero, the system is a t equilibrium and (Xt)eq/(E)ey(B)ey = K,?z + as must be the case. F o r i = 1, equation 6b becomes
kl
pearance of the ith intermediate can be written as do;’) __- [ k , (X,-1) - k - , d/
where
2 -5:
,
0kb.C’)
(E)
(A)
- (B)Ko“-’
(8)
where
outweighs all the other terms and moreover corresponds to the same value of s = Y, L e .
a
KO"+'= ' + ' k x-r r = l
The conservation of enzymatic sites expressed by equation 2a coupled with equation 6b provides a relation between (E), (E)oand v . v
i
5 2 -!?=
Is
=
i
5
+(E)(B)
&+I)
Kin+'
+
i = l
( E ) = (E)o (9)
Equations 8 and 9 can be combined to give the over-all steady-state rate of reaction ( v ) in terms of (A), (B) , (E)o,and aggregates of the rate constants. (E)o [(A) = 0 &+l)
+(A)[
- (B)Kon+'1
5 5 K,+" k(3
i = l s = i
KA
ka) = Ko?Krh= K o h E (E)eq(A)eq Kh' (Xh)eq G)
>h
Since equation 12d has the same summation in the numerator as equation 12b, the condition that K B become approximately equal to an equilibrium constant yields Kor K B S - 1 k(r+l) = Kan+' Kj' k(r + 1) ~
(E)edB)eqj >
~
(Xj)eq
It follows from the requirement that h 5 r < j that K A and K B can only reduce to equii - 1 librium constants for intermediates (X,) and (Xi), respectively, such that h < j .
v =
5 K,"
5909
MULTIPLEINTERMEDIATES IN STEADY STATEENZYME KINETICS
Nov. 20: 1959
"I
1)
This result has the obvious consequence that for a single intermediate mechanism only one of the twovillichaelis constants can approximate an equilibrium constant. Lower Limits on the Rate Constants.-Any study of the reaction represented by the scheme in equation 1 conducted under conditions where the steady-state relation of equation 5 applies yields information on only the four kinetic parameters in equation 11, only three of these parameters being independent by virtue of equation 13. However, it is possible to place lower limits on the rate constants for a mechanism like the above. In particular, i t can be seen from equation 12a that retaining just one of the positive terms in the denominator yields the inequality (lo)
Equation 10 can obviously be written in the Michaelis-Menten form for an enzymatically catalyzed reversible reaction, L e . VA -
=
KA
(A) -
VB i~-, (B)
(A)
It-+Ka
(B) KB
(11)
where the maximum velocities (VA and VB) and Rlichaelis constants ( K A and K B ) are defined as
",=
(E:o
= =& n
n
K , ~
(12a)
z = l s =
k(i+,)
2 (E)o "-4
i
# 0
(14a)
inasmuch as Kii = 1. That is, VA/(E)O is the lower limit of any unimolecular rate constant for the forward path. The equality sign holds when there is a single intermediate. Similarly for the reverse path i # n + l
-a relationship between the kinetic parameters and the over-all equilibrium constant usually attributed to Haldane. l 1 The question often has been raised as to the conditions under which the Michaelis constants can be regarded as equilibrium constants, i.e., the existence of a quasi-equilibrium. l 2 From equation 12b i t can be seen that the sums in both numerator and denominator must each contain a term which far ( 1 1 ) J. B. S. Haldane, ''Enzymes," Longmans, Green, London, 1930. (12) Reference 8, p. 57.
as can be seen from equation 12c or from the symmetry of the mechanism in equation 1. The above inequalities simply state that a maximum velocity, e . g . , VA/(E)O,cannot exceed in magnitude any unimolecular rate constant for the conversion of one intermediate to the next in the sequence for the reaction proceeding in a given direction, e . g . , A -+ B. It should be noted that neither VA nor VB contains either K 1 or k - ( % + I), the two bimolecular rate constants in the sequence. This is to be expected physically because the maximum velocities represent rates which are attained only when there is total saturation of the enzymatic sites, and hence when the over-all rate of reaction is independent of the concentrations of A and B. Also V A does not contain k-1 and VB does not contain k ( n + 1) which are the rate constants for
5910
LEONARD PELLER AND ROBERT A. ALBERTY
the dissociation of the first intermediate in the forward (XI) and the reverse paths (X,). In a similar way it is possible to place lower limits on the two bimolecular rate constants, kl and k-(, + 1). From equations 12a, 12b and 12c it can be seen that
Vol. 81
In Table I are gathered estimated lower limits for the bimolecular rate constants for several rather different enzyme systems. As i t has been possible to study few such reactions in the forward and reverse directions, most of the lower limits are based on the inequality
gests for these systems with ( E)&A
5 k”.”
As the numerator of the fraction in braces is greater than or equal to zero, it follows that
with the equality sign holding for the one intermediate mechanism. The symmetry of the mechanism ensures that
(ISb)
volvina a small number of intermvediates the first step may be regarded as a diffusion-controlled combination of substrate and enzyme. It is of considerable interest that such inferences about the rapidity of combination of enzyme and substrate can be made from data obtained during the steady-state period of the reaction.
1
TABLE I LOWERLIMITSos BIMOLECULAR RATECONSTANTS Enzyme
Essentially these lower limits have been given for a two intermediate mechanism by a different type of argument.13 The lower limits on the unimolecular rate constants place upper limits on the duration of the transient state period for the production of intermediates in the system. This derives from the fact that the transient terms in the approximate solution of the differential equations for the system are of the form e+’? where r is a measure of the length of the transient state period. For a two intermediate mechanism there are two transient terms of this form for the appearance of product.14-16 At high substrate concentration the more slowly decaying of these exponential terms is given by r =
Of perhaps more interest are the lower limits on the bimolecular rate constants. It has previously been pointed out on the basis of a model for diffusion into a hemispherical sink that the diffusion controlled combination of a small substrate molecule (with a diffusion coefficient D 1 10-6 cm.* sec.-l) with a larger enzyme molecule (Dz lo-’ cm.2 sec.-l> would occur with a bimolecular rate constant of lo8 to 1O’O d4-l sec.-’.17 These values can be regarded as theoretical upper limits to the bimolecular rate constants.
--
(13) R. A. Alberty and W. H . Peirce, THIS J O U R N A L , 79, 1.526 (1957).
(14) H. Gutfreund, Disc. F Q Y O ~ Soc., Q Y 20, IF7 (19%). (15) L. Ouellet and K . J. Laidler, C a n . J . Chcm., 34, 140 (19%). (16) Reference 8, p. 117. (17) R . A. Alberty and G. G. Hammes, J. Phys. Chem., 62, 154 (1958).
Substrate
Minimum value Ref.
( M - 1 sec-1)
(a) @-Amylase Amylose 5 . 8 x 1070 a (b) Urease Urea 5 . 0 X lo6@ (c) Adenosine triphosAdenosine tri- 8 X 10@ phatase phosphate (d) Cytochrome c reductase Cytochrome c 9 X (e) Acetylcholinesterase Acetylcholine -lo9’ e ( f ) Fumarase (1) Phosphate Fumarate 4 . 2 X 107h L-Malate 1 . 6 X 107’ fiH 7.0 (2) “Tris” acetate Fumarate 6 . 5 X loeh L-Malate 1 . 6 X logh pH 7.0 a J. M . Bailey and D. French, J . Biol. Chem., 226, 1 (1957). These authors quote a somewhat lower value for k l than the above limit based on an assumed non-random degradation process. The above cited lower limit applies t o a random degradation process. This problem will be discussed more fully elsewhere by one of us (L. P.). M. c. Wall and K. J . Laidler, Arch. Biochem. Biophys., 43, 299 (1953). L. Ouellet, K . J. Laidler and M . Morales, ibid., 39,37 (1952). T. K. Hogness in “A Symposium on Respiratory Enzymes,” University of Wisconsin Press, Madison, 1942, p. 134. e D. Sachmansohn and I. B. Wilson in “Advances in Enzvmology,” L’ol. X I I , Interscience Publishers, Inc., New Yo&, N. Y . 1951, p. 259. f C. Frieden, R . G. Wolfe and R . -4. Alberty, THIS J O U R N A L , 79, 1523 (1957). Considerably larger values for the lower limits (- 109t o 10’0 M-’ SCC. -1) in the “Tris” buffer system can be given in terms of the PH independent kinetic parameters From (16a). defined below (see ref. 13). 0 From (16c).
*
’
In order to obtain values for the lower limits on the rate constants the “value” of (E)o must be known. Most often the only estimate of (E)o available is the molar concentration of the enzyme. Use of such an estimate ignores the possibility of multiple sites in the enzyme and tends to give a spuriously high value for the lower limit. This is possibly the case in systems (b), (c) and (f) in Table I. Determinations of the number of active sites are available for a number of enzyme
5911
MULTIPLE INTERMEDIATES IN STEADYSTATE ENZYME KINETICS
Nov. 20, 1959
systems, however, and additional estimates should become available as these systems are more thoroughly studied. It should be realized that the above treatment applies to enzymes with multiple sites only if they are independent. Use of a value for (E)o expressed in terms of the molar concentration of sites still involves the assumption that the stoichiometric concentration of sites represents active enzyme. This assumption is difficult to circumvent in enzymatic assays and may lead to a spuriously low limit if an appreciable fraction of the enzyme is unreactive. Lastly the influence of solution environment on the kinetic parameters may be quite significant. As can be seen from Table I, the lower limits for the two bimolecular rate constants in the fumarase system are more than an order of magnitude smaller in sodium phosphate than in “tris” acetate a t the same pH. This difference can reasonably be accounted for if the phosphate with a double negative charge clings to the enzymatic site more tenaciously than the acetate anion with a single negative charge. Such an interaction manifests itself by increasing the Michaelis constant in phosphate over “tris.” It seems likely from studies of the effect of the variation of ionic strength on the kinetic parameters that there is an important electrostatic contribution to the interaction of the substrate anions with the e n ~ y m e . ~ ’ JThe ~ nature of the buffer ions may thus greatly influence the apparent rate constants. The Kinetic Behavior of Intermediates.-During the steady-state period of a reversible enzymatic reaction, the concentration of an intermediate may obviously either decline or rise to its equilibrium value. The first possibility corresponds to the existence of a maximum in the concentration of an intermediate during the transient period of the reaction. Before examining the temporal behavior of the intermediates, i t is of interest to consider the kinetic behavior of the free enzyme concentration in the post transient state period. From equations 8, 11, 12a and 12b, we have the familiar expression for the free enzyme concentration (E) =
(E)O
1+
P !+ L?J KA KB
(17)
It follows from the conservation equations 2a and 2b and the above expression that
latter situation corresponds to the existence of a minimum in the time course of its concentration. The behavior of the free enzyme concentration with increasing time for the reaction proceeding in the direction of A 4 B is schematically illustrated in Fig. 1. If K A = K B ,then d(E)/d(B) = 0 A-B
----_--__________________
a t. Fig. 1.-Schematic representation of the time course of the free enzyme concentration.
and the steady-state value of (E) is identical to its equilibrium value as can be seen from equations 2a, 2b and 17 (E) = ~ [ K A + ( A ) ~ - ( E ) ~ I ~ + ~ ( E ) ~[KA+(A)O-(E)O] KA]”~2
-
where the second relation is obtained when (A)o >> (E)o by binomial expansion of the square root. This latter condition coupled with the equality of the Michaelis constants makes equation 11 pseudofirst order. It is obvious from equation 2a that the time course of (E) determines the time course of the n
total
Concentration
of
(Xi).
intermediates i = l
For example, a minimum in (E)implies a maximum n
(Xi). Also it follows from the fact that
in
i P 1
d(E)/d(A) is opposite in sign to d(E)/d(B) that a minimum for (E) in the forward direction (A + B) implies the absence of a minimum in the reverse direction (B 4A). For a mechanism involving a single intermediate, equality of K A and K B requires that the two 1 (E)o bimolecular rate constants be equal, ;.e., kl = k--2. In this case the differential equations for the ap(A) ( B ) I 2 (18) [l pearance of intermediate and product are integrable where the second equality holds when (E)o > (E)o is sufficient to ensure maximum inasmuch as initially it was equal to that the intermediate and free enzyme concentraeffectively attain their equilibrium values (E)o, and ultimately it must approach a value tions early in the reaction, and the steady-state approx(E)es < (E)o. However, i t may decline to this value with increasing time (or (B)) if K A > K B imation can be employed with confidence after or rise to its equilibrium value if K B > KA. The this brief transient period.lg
(z; k) + +
(18) (1959).
G. G. Hamrnes and R. A. Alberty, J . Phrr. Chcm., 63, 274
(19) (1958).
W. G. Miller and R. A. Alberty, THISJOURNAL, 80, 5146
LEONARD PELLER A N D ROBERT A.XLBERTY
5912
The time dependence of the concentration of intermediates during the steady-state period is somewhat more complicated. From equations 6b, 11 and 17 we have an expression for the concentration of the i t h intermediate (Xi) =
(20)
The behavior of (Xi) during the steady-state period can be most easily examined by considering the sign of the derivative d(Xi) d(B)
- -
t =
Ok!R+l)
Vol. 81
t = 0k--
!S+l)
where h < i. Hence, if the concentration of the ith intermediate goes through a maximum as the reaction proceeds in the direction of A + B, the concentration of the hth intermediate will go through a maximum also. In other words, the inequality (22a) will be sati$ed by all intermediates preceding the i t h in the reaction sequence and all will exhibit concentration maxima in the direction A+B. If the inequality (22a) applies to the nth intermediate, then all n intermediates go through concentration maxima. This is a suficient but not a necessary condition (save for a single intermediate mechanism) for K A < Kp, and for the concentration of the free enzyme (E) to exhibit a concentration minimum. I n this extreme case, inequality (22a) becomes
where it has been assumed that (A)o >> (E)o. For (Xi) t o decrease during the steady-state period d(Xi)/d(B) < 0 or the quantity in braces in equation 21 must be greater than zero, ;.e.
The condition that the derivative in equation 21 be less than zero corresponds to the concentration of the ith intermediate having gone through a maximum in the transient period. For d(Xi)/ d(B) > 0, (Xi) increases monotonically from its initial value of zero to its final equilibrium value. These two possible modes of behavior of the concentration of an intermediate with increasing time for the reaction proceeding in the direction of A B are schematically represented in Fig. 2.
-
I
A-cB
For a system involving a single intermediate (i = n = 1) the inequality (22a) reduces to the requirement that kl > k-2 or K A < K B which is to be expected from the earlier remarks concerning the temporal behavior of (E). Reversing the inequality 22a defines the condition that d(X,)/d(B) > 0 in which case (X,) increases monotonically to its equilibrium value. By the same argument as presented above i t can be seen readily that if (Xi) does not go through a maximum then the (Xj) will not exhibit maxima for a l l j > i in the direction A += B. For several enzymes reaction conditions may be found where K A = K B usually by operating a t a particular pH.l9 As has been remarked, the equality of the Michaelis constants is a necessary and sufficient condition for the free enzyme concentration (E) to be effectively a t its equilibrium value after the transient period. For a single intermediate sequence this also obviously applies to the concentration of the intermediate (Xt). When K A = K B ,inequality 22a becomes
Lj== 0 !Y+l)
t. Fig. 2.-Schematic representation of the time course of the concentration of an intermediate.
From the definition of the sums appearing in the above inequality, i t is immediately apparent that
;.e., the condition for the existence of a maximum in the concentration of the ith intermediate is independent of the total substrate concentration. When d(X,)/d(B) = 0,the concentration of the ith intermediate must be effectively a t its equilibrium value in the steady-state period of the reaction, i.e., equation (23a) therefore is derivable. In this circumstance inequality (22a) becomes an equality. From the definition of the sums appearing in (22a) it is apparent that if d(Xj)/d(B) =
s o v . 20, 1959
MULTIPLE INTERMEDIATES IN STEADY STATE E
1
+ KO"+' + (A)o
Kjn+l j = 1
0, then d(Xh)/d(B) < 0 for h < i and d(Xj)/ d(B) > 0 for j > i. Hence the concentrations of all intermediates preceding the ith exhibit maxima while none succeeding it do. The condition that d(Xi)/d (B) be identically zero is of course artificial. However, it leads to the physically obvious conclusion by consideration of the sums appearing in (22a) that if the ith intermediate is a t its equilibrium concentration, then the intermediates close to it in the sequence may be correspondingly close to their equilibrium concentrations as well. It is of some interest to point out that if Kin + = 0for all i then from equation 6b
Hence, the ratio of the concentrations of any two intermediates during the steady-state period is constant. One way of achieving this condition is for k - ( n + 1 ) = 0. This is a common assumption in treating the kinetics of so-called irreversible enzyme catalyzed reactions. Irreversibility of course requires only that KO" +
