5 146
0
~ ~ r I L h I EG R. MILLER AND ICOBERT
I
1G
I
05
15
10
C , gm d I.-’
Fig. 11.-Comparison of cot 2 x 4 and ( T ~ , / C ) - C plots of rabbit actomyosin solution. Data were taken from ref. 26. a high degree of rigidity. It is therefore of interest to test the applicability of the proposed equations to these macromolecular systems. As an example we will choose von Muralt and Edsall’s pioneer work on actomyosin.z6 In Fig. l l are plotted cot 2x against the concentration (assuming the protein contains 16% N) a t two rates of shear (the lowest and the highest measurable RPM). Also included in the figure is the (vsp./t)-t curve, the data of which were measured (26) A . von Rluralt and J. T . Edsall, J. Biol. Ckcm., 89, 3 i 5 , 3.il (1930); J. T. Edsall, Trows. F a r a d a y Soc., 26, 837 (1930).
- _ [COSTRIBUTION FROM
THE
_
~
-1.h L B E R T Y
Yol.
so
in an Ostwald viscometer presumably a t very low rate of shear. The close similarity in the shape of both flow birefringence (at low r .p.m.) and viscosity is self-explanatory. According to the original paper, the lower R.P.M. corresponded to about D = 10 set.? and 01 ( = D / 6 ) was close to 1. Although the actual rate of shear in the viscometer was not mentioned, the listed values seemed also to lie close to the Newtonian region. Thus the two curves were comparable a t about the same range of rates of shear. The apparently straight line obtained a t a high r.p.m. was mostly certainly due to the sharp drop in k ’ [ q ] in eq. >eo or if (eo k 3 ) / k l : A perturbation solution is developed for the caw that kl # ki and the applicability of the steady-state approximation for this case is discussed.
+
Introduction Enzyme kinetic data frequently can be represented by the reversible Michaelis-Menten mechanism E
-+
ka
ki S
s X kz
e
E
+
P
( 1)
k,
(1) This research was supported by grants from the National Science Foundation and from the Research Committee of the Graduate School of the University of Wisconsin from funds supplied by the Wisconsin Alumni Research Foundation. (2) Socony-Mobil Fellow (1957-1958).
where E represents e n ~ y r n e ,S~ and P represent substrates, and X is the intermediate. The total molar concentration of enzyme is represented by eo and the initial concentration of substrate by SO. The concentrations of the intermediate and product (3) The symbol E actually represents the enzymatic site rather than the enzyme but generally the number of sites per molecule is unknown. Letting E represent the enzyme rather than the enzymatic site so that eo represents the total molar Concentration of the enzyme increases the rate constants by a factor equal t o the number of sites per molecule, assuming no site-site interaction. The mechanlsm is not restricted to enzyme catalysis but might be applied to heterogeneous catalysis, with E representing the sites on the surface of the solid.
Oct. 5, 1955
KINETICSOF THE REVERSIBLE MICHAELIS-MENTEN MECHANISM
a t any time are represented by x and p , respectively. The rate equations are therefore
+ (k4 - kt)eop - [kleo + klso + (ka - k J p + kz + k 3 h + kiX2 p (k3 + k4p)x - k4e0p
5147
and
2 = klsoe~
=
(2) (3)
Since equations 7 and 12 involve no approximations in their derivation, they give the concentration of intermediate and product exactly over the entire course of the reaction, including both the (4) transient and steady-state phases. Before discussing the nature of these equations the signifiwhere Ks and K p are the Michaelis constants for cance of the restriction that k1 = kq will be disthe substrate and product, respectively, with Ks cussed. = (k, k3)/kl and Kp = (k2 k3)/k4. Chance5 The condition that k1 = k4 is exactly the restrichas obtained differential analyzer solutions for the case that k4 = 0, and Yang6 has discussed a solu- tion that the Michaelis constants for the subtion for this case using a reversion method. Vari- strate and product must be equal. Enzyme reacous approximate solutions have been discussed tions that can be represented by mechanism 1under for the transient phase of the reaction when kq = given conditions and which have equilibrium conO.'-lo I n deriving these approximate solutions, stants in the neighborhood of unity have been the product concentration was considered to be studied in both the forward and reverse direcnegligible during the transient phase. However tions.l1Il2 For the fumarase, enolase and phosthe complete analytical solution of equations 2 and phoglucose isomerase reactionsl3 Ks = Kp a t a PH which depends upon the buffer and ionic 3 has not been obtained. In the present paper the exact solution for the strength. For enzyme reactions where the equilibrium conreversible Michaelis-Menten mechanism is derived stant is much greater than unity, i.e., the reaction for the case that k1 = k4 and a perturbation solution goes "to completion," data €or the Michaelis conis developed for the case that k1 # kq. Exact Solutions when kl = kr.-When kl = k4, stants of both the reactant and product are difficult to find in the literature. This in part appears to rate equations 2 and 3 become, respectively be due to a common practice which has developed j: = kleoso - (klso kleo kf k3)x klx2 ( 5 ) among enzyme kineticists that if a reaction goes p = (k3 k4p)x - k4e0p (6) essentially to completion, i t is assumed that kq = Equation 5 is immediately integrable and upon ap- 0. Obviously in this case a Michaelis constant for plying the initial condition t = 0, x = 0 and rear- the product is meaningless. As has been pointed ranging gives out,l4 the condition f o r a reaction following mechanism 1 going to completion i s that klk3 >> k2k4 and this puts no condition on k4. If k1 = kq, k3 >> k2 would be sufficient to cause the reaction to go essenwhere tially to completion. I n many cases product ina = kl (8) hibition could be explained by using mechanism 1 b = -(he0 klso kz ks) (9) rather than letting k4 = 0 and adding a separate c = kleoso (10) reaction to account for the inhibition. Product d = (b2 - 4ac)'In = inhibition constants often have values very close to [(kleo klsa k2 k d 2 - 4k1*e0s0]'/' ( 1 1 ) the Michaelis constant for the substrate,l5 thus sugIf equation 7 is substituted into equation 6, the gesting that if mechanism 1 were used to interpret resulting differential equation is linear in $J but has product inhibition calculated values of k1 and k4 non-constant coefficients. This equation, how- would be nearly equal. The restriction that has ever, can be integrated by the method of variation been used in deriving the equations is thus one that of parameters and upon applying the initial condi- is met under particular conditions in some enzyme tion t = 0, p = 0 and simplifying the following systems and may even be applicable to enzymatic equation is obtained. reactions which essentially BO to comdetion. ~ ' ?st of interest to comkleo(2klso b - d ) [1 - e - d l l ( b - d b - d pare equations 7, 12, 13 (12) and 14 with approximate (kz ka) [d - b (d b)e-dtl equations obtained by others If SO >> eo, d = - b and equations 7 and 12 be- for the case where kq = 0. Equation 7 is identicome cal (if s, the average substrate concentration, kleoso is taken to be so) with the equation derived by [l e ( k l s o f kn + k d t ] x = The steady-state rate equation for this mechanism, derived by Haldane4for the case that SO >> eo, is
+
+
+
+
+ +
+
+
+
+
+ +
+ +
+
+
kiso
_____
+ kz + k3
+ +
(13)
J. B. S. Haldane, "Enzymes," Longmans, Green and Co., London, 1930,p. 81. (5) B. Chance, J . B i d . Chem., 111, 553 (1943). (6) C.Yang, Arch. Biochem, Biophys., 11, 419 (1953). (7) H. Gutfreund. Disc.Faraday Soc., 30, 167 (1955). (8) K.J. Laidler, Can. J . Chem., 99, 1614 (1955). (9) M . F. Morales and D. E. Goldman, THISJOURNAL, 77, 6069 (1955). (10) P. A. T.Swoboda, Biochim. el Biophys. Acta, 38, 70 (1957). (4)
(11) R. M. Bock and R . A . Alberty, THISJOURNAL, 75, 1921 (1953). (12) F . Wold and C. E. Ballou, J . B i d . Chem., 337, 313 (1957). (13) K. K. Tsuboi, J. Estrada and P . B. Hudson, ibid., 391, 19 (1958). (14) R.A. Alberty in P. D. Boyer, H. Lardy and K. Myrback, "The Enzymes," 2nd Ed., Academic Press, New York, N.Y . , 1958. (15) R. J. Foster and C.Niemann, Proc. N a f l . Acad. Sci., U.S . , 99, 999 (1953); H.T. Huang and C. Niemann, THISJOURNAL, 79, 1541 (1951).
WILMERG. MILLERAND ROBERTA. ALBERTY
5145
Laidlers for the appearance of intermediate during the transient state. This similarity is expected since the initial appearance of intermediate should be independent of the value of k4. Whereas Laidler’s expression is only good for the transient state, equation 7 is exact for the entire course of the reaction if k1 = kd and x asymptotically approaches the steady-state or equilibrium value. For the additional condition that SO>> eo, equations derived by Laidlers and by Swobodalofor the transient state appearance of intermediate are identical with equation 13 which is again to be expected as the value of k4 is unimportant during the transient state. Guffreund? and Swobodalo have also derived an expression for the appearance of product, which may be compared with equation 14. The second term in the equations of Gutfreund and Swoboda is identical with the second term in equation 14 while the first term in their equation can be obtained from our first term by expanding the exponential term in a power series and keeping only the first two terms. An extrapolation of the linear (early steady state) region to obtain kl, as Gutfreund has suggested, must be made with considerable caution. Terms involving e--dt in equations 7 and 12 are responsible for the initial rapid rise in x while terms arising from the first exponential term of equation 12 can be identified with the “steady state.” The applicability of the steady-state approximation depends essentially on the relative values of d and [kle~(2klso b - d ) ] / ( b - d ) , and the larger the relative value of d becomes, the better the degree of approximation of 2 = 0. There are several ways in which the relative values of these quantities can be changed, one of which is to change the ratio of eo t o SO. If so >> eo, equation 12 simplifies to equation 14, and the ratio of klso kZ k3 t o kl(kz ka)eo/(klso kz k3) will always be greater than so/eo. The equation for the concentration of product as a function of time derived using the steady state approximation is given by16
1’01.
so
usually are made. As the enzyme concentration is reduced, the value of - ( l / e o t ) In ( 1 - p/peq) approaches the steady-state value more closely a t smaller and smaller extents of reaction. When the enzyme concentration is 10-4 times the substrate concentration (curve D), the steady-state value is within 1% of the true value when the product concentration has reached only o.3yOof its equilibrium value. But even a t this concentration the fumarase reaction would occur too rapidly for ordinary initial velocity measurements to be made. As a rule experiments to measure “initial velocities” are arranged so that the reaction will be about 5% to its equilibrium value in five minutes or longer. In the case of fumarase this corresponds to an enzyme concentration of about 5 X 10-lo M . Using this concentration of enzyme and equations 13 and 14, a simple calculation shows that a t seconds x is within O . O O l ~ o of its steady-state value while the product concentration has reached only about O . O O l ~ o of its equilibrium value. This completely justifies the use of the steady-state approximation in the study of the fumarase reaction under the condition that the Michaelis constants are equal. Figure 2 illustrates the effect of varying the substrate concentration while holding the enzyme concentration constant. Comparing curve A in Fig. 1 with curve A in Fig. 2, i t is evident that the steady-state assumption becomes a better approximation as both the enzyme and substrate concentrations become much smaller than the Michaelis constant, even if eo 2 so. The values for - ( l / eot) In (1 - $/fie,) in A and C have been multiplied by appropriate constants in order to make their steady-state values coincide with the steady-state value of B. The effect on the appearance of intermediate of changing the initial substrate and enzyme concentrations is shown in Fig. 3. It is thus seen that the steady-state approximation becomes better as eo/so becomes small or as eo SO becomes smaller than (kz k3)/kl. The latter condition is sufficient even though SO = eo or even eo > so. It is of interest that the steady-state approximation can be applicable even though the which is identical with equation 14 except for the concentration of substrate is not much larger than transient term. Equation 15 is that of a pseudo the concentration of enzyme. first-order reaction and a plot of - (l/eot) In (1 Approximate Solution for kl # k4.-The following P/Peq) os. t gives a straight line with zero slope; discussion will be restricted t o the case where therefore any difference between the steady-state so >> eo. In this case so = s p and rate equavalue of product and the actual value would be tions 2 and 3 simplify to seen easily in this type of plot. 2 = klsoeo (k4 - kl)eop - [klso ki + ka (k4 - k i ) P l x Figure 1 illustrates the effect of varying the en(16) zyme concentration while holding the substrate con$= (ka k4p)x - k ~ p (17) centration constant. The values for the rate conEquations 16 and 17 are non-linear and have no stants are those calculated from steady-state kinetic data for the fumarase system a t 2 j 0 , pH known analytical solutions. To investigate the 6.81, 0 . 1 ionic strength in tris-(hydroxymethy1)- nature of x as a function of time equation 16 is first aminomethane acetate buffer containinn 0.09 M differentiated with respect to time, to obtain NaCI.’? The initial substrate concentrahn, jt = (kc - kl)eo$ [klso kz kt M , was chosen to be approximately equal t o the (k4 - k J P l 2 (k4 - k1fXt.l) (18) Michaelis constant, since this is in the range of sub- If ere to go through a maximum = and trate concentrations where initial velocity studies at the Since o, the only way (16) R. A. Alberty, W. G. Miller and H. F. Fisher, THISJOURNAL, that can go through a maximum is for kl > k4.18
+
+
+ +
+ +
+
+
+
+
+
+
+
+
79, 3973 (1957). (17) C. Frieden, R. Wolfe, Jr., and R. A. Albertg, ;bid., 79, 1524 (1957).
+ + +
+
(18) M. F . Morales, in P. D. Boyer, H. Lardy and K. MyrbBck, “The Enzymes,’’ 2nd Ed., Academic P r e s , New York. h’. Y..1958.
Oct. 5, 1958
KINETICS OF
THE
5149
REVERSIBLE MICHAELIS-MENTEN MECHANISM
eo t x IO~~M-SEC).
Fig. 1.-Kinetic plots a t various enzyme concentrations with SO = lo-' M : A, eo = 10-4 M ; B, eo = 10-6 M; C, eo = lo-' M ; D, eo = iM. Values used for the rate constants are kl = k4 = 7.9 X 108 fi1-l rnin.-', k2 = 5.5 X lo4 mix^.-^ and kl = 1.24 X 10' min.-'. The steady-state value is labelled S.S. The value of ed a t which the product concentration is one per cent. of its equilibrium value is marked by a short vertical line.
Thus as the ratio of k4 to k1 changes, an envelope of curves for x vs. t is obtained, with all curves for kl > k4 going through a maximum and with no curves for kg 2 kl having a maximum, but instead approaching their maximum value asymptotically a t infinite time.lS If x goes through a maximum for the forward reaction, i t cannot for the reverse reaction. A logical way to obtain solutions for k1 # kq is to consider the desired solutions as perturbationsZ0of the solution for kl = k4. The desired solutions are then x = xo
+ (k4 -
+ (ka - k J k z + ....
ki)~l
m
X"(k4
- kl)"
(19)
0
0.006
0.004
0.002
0.008
t (sec.). Fig. 3.-Fractional approach to equilibrium for both intermediate ( x ) and product ( p ) for the values of the rate constants given for Fig. 1. Curves marked 1A and 1B were calculated using the same enzyme and substrate concentrations as for curves A and B in Fig. 1; similarly curves 2A correspond with curve A in Fig. 2. The steady-state value is labelled S.S. Equations 7 and 12 were used in making the calculations.
P
=
Po
+ (ka - k i ) p i + (k4 - ki)'Pz + m
n=O
Pdk4 n-0
=
- k3"
(20)
with xo and PObeing the solutions of x and p , respectively, when kl = k4. The quantities x1 and are the first-order perturbation terms, x2 and p z the second-order perturbation terms, etc. The expression for xo is given by equation 13
where P = klso
klk3S0 [I k4k2 ka) where a = [k4(kz
+
I
I
I
I
I
2
3
e , t
I
4
x IO8 (M SEC.)
Fig. 2.-Kinetic plots a t various substrate concentrations with ea = 10-6 M : A, SO = 10-6 M; B, SO = M; C, M. The values of rate constants are the same as SO = for Fig. 1. The steady-state value is labelled S.S. The value of ed at which the product concentration is one per cent. of its equilibrium value is marked by a short vertical line. (19) The stme condition applies t o general equation 2 where there is no restriction on the relative concentrations of enzyme and substrate. (20) R . Courant and D. Hilbert, "Methods of Mathematical Physics, I," Interscionce Publishers, New York, N. Y.,1953.
+ kz + ka and the equation for - e-at
] -
+
klkssoeo [1 P2
$0
is
(22)
+ +
k3)eo]/(klso kz k 3 ) . Equation 22 differs from equation 14 in that in the derivation of equation 12, k4 is allowed t o keep its identity wherever possible. T o obtain x and P as functions of time equations 19 and 20 and their first derivatives are substituted into equations 16 and 17 and the coefficient of each power of (k4 - kl) is equated to zero. Thus, considering equation 19, the coefficient of the (kd k1) term will be a first-order differential equation in XI,the coefficient of (kl - k1)2involves x2, etc. The coefficient of (k4 - kJn gives &
+ Bx,
n-1
=
eopn-l
pn-~-m xm (n f 0) (23)
m=O
5150 Pn
+ k4e0 - xo)Pn = kax, + k4
WILRIER
G. hTILLER
AND ROBERT
Vol. 80
B. XLBERTY
n
Pn-mmxm ( n f 0) m=l
(2Jj
The homogeneous parts of both equations are the same as the corresponding homogeneous equations for xo and Po, which have been integrated earlier.21 Upon integration one obtains
(25)
and thus the perturbation solutions are
These equations all have the same initial condition that x,, = pn = 0 when t = 0. Whether or not the above solutions can be used depends on the nature of their convergence. In principle they describe the course of x and p even for the case that kq + 0. For both the enolase reactionI2 and the fumarase reaction" the Michaelis constants for the substrate and product are within a factor of 10 of each other a t most pH values. Thus the only perturbation terms that will be written out explicitly here will be those terms needed for the case that 0.1 6 k4/kl 10. The first two perturbation terms for x and p derived from equations 25 and 26 are
> eo (or kleo >> k4eo) mere discarded. Other terms went through maxima (or minima) and the values of these terms at their maxima (or minima) were compared with xrl and were discarded if small compared to Xa. A similar procedure was used in discarding terms in $1 and P2. In Fig. 4 x,:eo is plotted v s . t for two values of k1 'k4: in A kJk4 = 10 and in B kl/k4 = 0.1. The value of kl is the same as used in Fig. 1-3 and the values for kz and ka are such as to keep the Michaelis constant of the substrate the same as for Fig. 1-3 while still maintaining the equilibrium constant for the over-all reaction. T o calculate the time course of x/eo as given by the steady-state a p proximation these equations were used
Equation 33 is obtained from equation 16 by setting 1 = 0 and equation 34 is obtained by substituting equation 33 into equation 17 and integrating.22 For curve A the value of x calculated ( k g - kl)xl (k4 - k1)*x2agrees very from MO closely with the steady-state solution while in curve B the agreement is not so good (the third perturbation term is needed in this case). It appears that the steady-state approximation is very good for the fumarase reaction a t least when 0.1 kl'k4 10. Furthermore the perturbation solution appears to converge very rapidly within this region. The equilibrium values of x and $ obtained from the perturbation solution are
+
+
eo is applied after t h e homogeneous equation isintegrated
(22) R . A. Alberty, A d v a n c e s in E n z y m o l o g y , 17, 42 (1958). (23) T h e coefficient of t h e transient term in Q O is always negligible eo and w a s compared to t h e coefficient of t h e other term when SO neglected here.
>>
Oct. 5, 1958
CORRELATION OF
5151
RATECONSTANTS AND EQUILIBRIUM CONSTANTS
calculations, to Dr. Wolfgang Wasow for suggesting the use of a perturbation solution and to Dr. Leonard Peller for suggestions concerning this reSeaXCh.
and
the summations in equations 35 and 36 converge and yield 0.7-
O6 .s.s. Equations 39 and 40 are the correct equilibrium expressions when SO>> eo. If k4 > kl the inequalities 37 and 39 will always be satislied and therefore the summations in equations 35 and 36 always converge t o give the correct equilibrium expressions. However when kd < k1 the inequalities will not always be satisfied and then the perturbation solution is not always applicable. The h t term in equation 35 is (Q) eq and the first term in equation 36 is the second terms are (XI),, and PI)^, etc. Thus by using particular values of the rate constants, so and eo the number of perturbation terms needed to give any desired agreement with the correct value can be determined. In making the numerical calculations it appeared that this was also a good criterion for determining the number of perturbation terms needed to make the time dourse of the over-all reaction agree with the steady state solution. Acknowledgments.-The authors are indebted to Miss Selma Hayman for many of the numerical
[CONTRIBUTION No. 2335
FROM THE
I I
XL0
II
0.3..
I
I
2 3 0
I
100 t (MIN.),
1
200
300
Fig. 4.-Plots of x/eo vs. t: Curve A was calculated using = 7.9 X 108 M-' min.-', ks = 6.58 X lo4 min.-', kr = 1.49 X lo*min.-l, k , = 7.9 X 107 M-l min.-', so = lo-' M and eo = 3 X M. Curve B was calculated using the same values for RI and SO but with ke = 2.07 X lo4 min.-l, ka = 4.67 X lo4 min.-l, k, = 7.9 x 108 M-l rnin.-l and eo = 10-1' M. The first two perturbation terms were used in calculating the solid curve. The dashed curves were calculated using the steady-state assumption.
kl
MADISON, WISCONSIN
GATESAND CRELLIN LABORATORIES OF CHEMISTRY, CALIFORNIA INSTITUTEOF TECHNOLOGY]
Statistical Factors in the Correlation of Rate Constants and Equilibrium Constants BY SIDNEYW. BEN SON'^^ RECEIVED APRIL 26, 1958 It is shown that in comparisons of equilibrium constants with each other or of rate constants among a set of rate constants, it is the intrinsic or "chemical" constants which should be compared and not the observed constants. These chemical
constants, Kaom(or kohsm), are related to the observed constants K (or R ) by the relation: Koham = K/K.; kch?m = k / K * # where KOand K *#are the ratios of symmetry numbers for reactant and product species in equilibrium and chemical reaction, respectively. This leads t o some interesting changes in the "relative" base strengths of amines. The symmetry corrections are derivable from statistical mechanics and are equivalent t o some of the more intuitive methods in current use. In the case of the Br6nsted relation, correlating general acid-base catalytic behavior with acid strength, it leads t o a consistent method of assigning symmetry corrections t o both R and Kion. The need for such corrections in other "linear free-energy relationships" is pointed out.
Introduction S i n e the early of Br(jnsted and pedersena in correlating the catalytic rate constants of adds and bases with their ionization cons tan^, it has (1) Visiting Research Fellow in Chemistry in the Division of Chemistry and Chemicd Enginwring. On aabbatiorl leave from the Chemistry Department, University of Southern California. (2) T h e author wisher & acknowledge his apptedation to the National Science Foundation for a Senior Poet-DoctotaI Fellowship which made possiMe the prcscnt work. (3) J. N BrOnsted and P.Pedersen, 2. fihysik. Chcm., 108, 185 (1924).
become popular to extend this procedure to other reactions. Thus the Hammett4 treatment Of the acidities of substituted benzene derivatives and their correlation with rate constants is a further examde of a correlation which is now come to be as the t i f i n e m free energy" know\ relationship. It was early realized*J that such correlations offer ambiguities when species of (4) L. P. Hammett, Chcm. Revs.,17, 125 (1985); Trans. Faraday SOL.,84, 156 (1688). (5) J. N.Brtlnsted, Chcm. Reus., 6 , 322 (1928).
