Time-Dependent Michaelis-Menten Theory for Enzyme-Substrate-Inhibitor Systems, Exams of Medical Sciences

Download Time-Dependent Michaelis-Menten Theory for Enzyme-Substrate-Inhibitor Systems and more Exams Medical Sciences in PDF only on Docsity! 3888 Time-Dependent Michaelis-Menten Kinetics for an Enzyme-Substra te-Inhibitor Sy s tern S. I. Rubinow and Joe1 L. Lebowitz Contribution from the Biomathematics Division, Cornell University Graduate School of Medical Sciences, New York, New York Institute, New York, New York. 10021, and the Sloan-Kettering Received November 29, 1969 Abstract: The results of kinetic studies involving an enzyme and two substances competing for the same enzymatic site may be very different when the roles of these substances as substrate and inhibitor are reversed. If the substance used as an inhibitor has a much larger affinity for the enzyme than the substrate, then a plot of inverse reaction velocity of substrate L’S. inverse initial substrate concentration for different values of initial inhibitor concentration will not have, under regular experimental conditions, the usual Michaelis-Menten form. The results to be expecied depend essentially on the time of observation as measured from the start of the reaction. The deviations from the Michaelis-Menten form should be significant in experiments of the type carried out recently by Miller and Balis in their investigation of the enzyme Escherichia coli L-asparagine amidohydrolase reacting with the substrates asparagine and glutamine. Other experiments which are expected to require the time-dependent theory for their understanding are indicated. he results of experiments on simple enzyme acti- T vated reactions are usually described and analyzed in terms of the Michaelis-Menten theory.’ This de- scription predicts a simple linear relationship between the reciprocal reaction velocity and the reciprocal initial substrate concentration so- l. The approximate nature of this description is well known; nevertheless it is widely successful in its applications. The theory was developed initially for a reaction involving an enzyme and a substrate with which it reacts. If there are two substrates present which com- pete for the same enzymatic site, the reaction is said to be “fully c~mpeti t ive.”~ In such a reaction, the substrate that is singled out for measurement of its reaction velocity is referred to as the “substrate.” The second substrate is called the “inhibitor.” Obviously, the roles of inhibitor and substrate may be interchanged in a second study. For such a fully competitive reaction, Michaelis- Menten theory still predicts a linear relation between the reciprocal reaction velocity of the substrate (vS)-l and SO-’. As presented by Briggs and Haldane,4 this relationship takes the form Here V,,,” is the maximum value of the reaction veloc- ity us, KM is the Michaelis constant, io is the initial in- hibitor concentration, and the superscripts s and i denote substrate and inhibitor, respectively. The con- stants KM and V,,, depend on various reaction rate constants and the initial enzyme concentration. When a Lineweaver-Burk plot2 is made of eq 1 for different values of io, the result is a family of straight lines with the common intercept l/VmaXs on the ordinate axis so-1 = 0. Recently, Miller and Balk5 investigated the activ- (1) L. Michaelis and M. L. Menten, Biochem. Z., 49,333 (1913). (2) H. Lineweaver and D. Burk, J . Amer. Chem. SOC., 56,658 (1934). ( 3 ) M. Dixon and E. C. Webb, “Enzymes,” Academic Press, Inc., New York, N. Y., 1964. (4) G. E. Briggs and J. B. S. Haldane, Biochem. J., 19,338 (1925). (5) H. K. Miller and M. E. Balis, Biochem. Pharmacol,, 18, 2225 (1969). ities of the enzyme E. coli L-asparagine amidohydrolase reacting with the substrates asparagine and glutamine, separately and together. When they utilized asparagine as a substrate in the presence of several concentrations of glutamine, they found that the results of this experi- ment were in agreement with Michaelis-Menten theory as expressed by eq 1. However, in a second experiment in which glutamine was utilized as a substrate in the presence of several concentrations of asparagine, the results could not be represented by means of eq 1. If asparagine were absent (io = 0), the results did agree with eq 1 . These authors noted that the maximum reaction velocity for asparagine V,,, was greater than the maximum reaction velocity for glutamine by a factor of about 15. They therefore suggested that the qualitative basis for the paradoxical difference in the two experiments is that in the second experiment the inhibitor substance asparagine disappears rapidly dur- ing the course of the experiment. It was the suggestion of Balis that we investigate theoretically the appropriate modification of eq 1 for such circumstances that moti- vated the present work. Time-Dependent Michaelis-Menten Theory Let S and I denote substrate and inhibitor substances, respectively, which react with an enzyme E at the same enzymatic site. I t is assumed that substrate and enzyme react to form an enzyme-substrate complex C1, which can in turn dissociate to form either the enzyme and substrate, or the enzyme and some products PI. Sim- ilarly, the inhibitor and enzyme react to form an in- hibitor-enzyme complex Cz, which dissociates to form either the enzyme and inhibitor or the enzyme and some products Pz. The reverse reactions of products and enzyme to form complexes is assumed to be negligible. These reactions are represented schematically as follows. k + i k+a S + E e C i + E +P i k- 1 k + z k , r I + E e C2 + E + Pz k- 9 As is well known, the differential equations of the system present a mathematical problem in singular perturba- Journal of the American Chemical Society / 92313 / July 1, 1970 3889 as those that would obtain in the steady-state experi- mental arrangement envisaged above. It is interesting to note from eq 8 that a steady state is possible only if the quantity tion theory.6 These equations may be solved in a formal way by means of an asymptotic expansion in the small parameter eo/so, where eo is the enzyme concentra- tion. (Fortunately, this parameter is always made small in practical investigations of enzyme reactions. For example, in the Miller-Balis experiments previously cited, if we assume that the molecular weight of the enzyme is -150,000, then eo/so - The first term of this expansion, whose derivation is found in the Appendix, is the solution to zero order in eo/so. This solution yields the following expressions for the sub- strate concentrations s ( t ) and the inhibitor concentra- tion i(t) as functions of the time t . K M s In [?] (2) and where 6 = V m a 2 K M S VmaxSKMi (3) (4) and VmaXs = k+8eo Vmaxi = k+reo Differentiating eq 2 and 3 yields (7) The assumptions underlying this approximate solu- tion are essentially equivalent to the hypothesis that the system is in a pseudosteady state.4 The pseudo- steady-state hypothesis may be understood physically as follows. Imagine an experiment in which a constant supply of substrate and inhibitor is provided at rates J, and .II, respectively, and the reaction products PI and P2 are continuously removed. Then a steady state will be established with the values S and i for the sub- strate and inhibitor concentrations, respectively, given by the expressions (8) S = B J s [ l - - - - ] K M S JS JI Vmax V m a 2 VrnaxI (9) These expressions are identical with eq 6 and 7 providing D"(t), v'(t), s(t ) , and i(t) there are replaced by J,, JI, 3, and ?, respectively. Therefore, we may characterize the pseudosteady-state hypothesis as the assumption that, at any instant of time, the relations between the concentrations and the reaction velocities are the same (6) F. G. Heineken, H. M. Tsuchiya, and R. Ark, Marh. Biosci., 1, 95 (1967). (- JS + -I) JI < 1 Vmax' v m a x It is easy to show that the time interval necessary for the pseudosteady state to be established is of the order of [k+'(so + K ~ ~ ) ] - ' and [k+z(io + KMi)]-l for substrate and inhibitor, respectively (see Appendix). For most enzyme-substrate reactions this interval is on the order of a fraction of a second. The times at which vs are usually measured are on the order of minutes, so that the requirements of the theory are readily satisfied in the foregoing respect. Under usual experimental conditions, the times of measurement are such that s ( t ) = so. If, in addition, i ( t ) = io, then eq 6 yields essentially the same result as eq 1. However, as can be seen from (3), s ( t ) = SO does not assure that i( t) = io when 6 is large. In such a case, the full time dependence of eq 2 and 3 is necessary for interpreting the experiments. In order to compare eq 2 and 3 with experiment in the general case, we note that in practice velocities are often measured by observing s ( t ) for small times and assuming that a linear expansion of s ( t ) about the origin is valid. In mathematical terms, a common experi- mental definition of the reaction velocity is With this definition and eq 2 and 3 for s ( t ) and i(t), an expression for [ss(t)]-' which generalizes eq 1 is readily found. The result is 1 - io -1 - i(t) (11) Vmax't where Equation 11 is simplified if we assume, as is usually the case, that the fractional disappearance of s ( t ) is small during the course of the experiment, [so - s(t)]/so << 1. If the logarithm term in eq 12 is expanded in powers of [so - s(t ) ] /so, then it follows that s ( t ) is of the order of the fractional disappearance of s(t), and is small com- pared to unity. Therefore it may sensibly be neglected in eq 11 for comparison with the results of typical ex- periments. Equation 11 together with eq 3 and 12 constitute a convenient time-dependent generalization of Michaelis-Menten theory. Equation 11 reduces to eq 1 in the limit t + 0. If measurements of P ( f ) are to be made minutes after the start of the reaction, it is necessary to consider whether the time-dependent formalism is needed or not. The decision hinges on the value of 6 that appears in eq 3. Thus, if 6 5 1, then during the course of an experiment in which s( t ) stays close to so, i ( f ) will not differ very greatly from io , so Rubinow, Lebowitz 1 Michaelis-Menten Kinetics for an Enzyme-Substrate-Inhibitor System 3892 expected to be sufficient to understand the kinetic aspects of the study. If 6 is large compared to unity, then the time-dependent theory presented herein is needed for a proper interpretation of the results. Acknowledgment. We are grateful to M. Earl Balk for suggesting this problem to us and for many helpful discussions of the work while it was in progress. This work was supported in part by NCI Grant No. CA- 08748 at Sloan-Kettering Institute and by USAFOSR Grant No. 68-1416 at Belfer Graduate School of Science, Yeshiva University. Appendix. Zero-Order Solution of the Kinetic Equations Let e, s, i, cl, and c2 represent the concentrations of the quantities enzyme, substrate, inhibitor, substrate- enzyme complex, and inhibitor-enzyme complex, re- spectively, at any time. Let the subscript zero attached to a symbol denote its value at the initial time t = 0. Introduce the following dimensionless variables and parameters. t' = kleot s' = sjso i' = i/io cl' = cljeo cz' = c2jeo a = eo/so P = idso y = k+2/k+l (1) Ki = (k-i + k + ~ ) / k + i ~ o Kz = (k-2 + k+r)/k+zio ui = k + ~ / k + i ~ o UZ = k+r/k+zio If we now drop the primes, the kinetic equations assume the following form. (3) ~- - y [ - i + ( i + K2 - Uz)c2 + icll (4) d i( t ) dt These equations are to be solved subject to the initial conditions s(0) = i(0) = 1 Cl(0) = cz(0) = 0 The enzyme concentration e satisfies the relation c l + c 2 + e = 1 (7) Equations 2-5 are nonlinear and not susceptible to solution in closed form. We shall assume that a is very small compared to unity. This condition is satis- fied in the usual experimental situation. In addition we assume that the parameters 0 and y are O(1). This suggests that a solution to eq 2-5 be sought by means of an expansion in a. However, the small parameter a multiplies the highest derivative term in two of the equations and therefore the problem presented is classi- fied in the theory of singular perturbations. According to this theory, the solution to eq 2-5 in terms of an ex- pansion in a is asymptotic; Le., it tends to the true solu- tion as a -+ 0, although the series solution probably diverges. The solution we shall present parallels the asymptotic solution presented in ref 6 for an enzyme- substrate system. The reader is referred there for a fuller discussion of the solution and the method for obtaining it. We proceed formally by seeking a solution to eq 2-5 as a power series in a, e.g. m (s,i,cl,cz) = (s(n), i(n), c~(~), ~ ~ ( ~ ) ) a ~ (8) n = O If we substitute (8) into (2)-(5) and equate to zero the co- efficients of like powers in Q, we find that the terms which are zero order in a satisfy the equations ds(0) dt - - - - s ( O ) + ( s ( ' ) + K1 - U1)Ci(O) + s(")c~(") (9) (1 1) (12) 0 = ~ ( 0 ) - ('(0) + Kl)clCO) - s(0)c2(O) 0 = j ( 0 ) - ( i ( 0 ) + K2)cz(0) - '(0) (0) I c1 Equations 11 and 12 are algebraic equations which are readily solved for cl(0) and c2(0) . The resulting expres- sions may be substituted into eq 8 and 9. The latter are then also readily solvable. The result is s(o)(t) + - Ul [ S ( O ) ( ~ ) I ~ U ~ K ~ / U ~ K ~ + K~ In s(o)( t ) + uz C = -Uit (15) j ( O)( t ) = [s( O)( ~ ) ] Y U ~ K ~ / U ~ K Z (16) where C is a constant to be determined. We do not impose on the solution the requirement that it satisfy the initial conditions because the assumed expansion is not valid in the neighborhood o f t = 0. In fact, it can be seen that cl(0)(t) and c2(0)(t) cannot satisfy the initial conditions. Rather, we recognize that the ex- pansion (8) constitutes an "outer" expansion, valid only for t "sufficiently large." We must find another expansion, the "inner" expansion, which is valid for t small. To this end we introduce the new time scale r and new variables defined by r = tja S(T) = s(ar,a) Z(r) = i(ar,a) (17) Cl(7) = c1(ar,a) CZ(7) = c2(ar,a) In terms of these variables, eq 2-5 become _ - a [ - S + ( S + Ki - Ui)G + SC21 (18) dS d r a y [ - I + (I + K2 - UZ)CZ + IC11 (19) - = dl d r dC1 - - - S - (S + K1)CI - SC, d r (21) - - dCz - Py[I - ( I + K2)C2 - ICl] d r Journal of the American Chemical Society 1 92:13 1 July I , 1970 3893 = '/z{1 + Ki + Pr(1 + Kz) * [{ 1 + Ki - Pr(1 + Kz))' + 4Prl'"J (33) It may be observed parenthetically that if Pr << 1, A+ = -(1 + K I ) and A- = -Pr ( l + Kz). The inner expansion and the outer expansion must be asymp- totically equal in their common domain of validity. We assume that there is such a common domain for some intermediate values of t . Therefore, if the inner expan- sion is expanded for large values of T and the outer ex- pansion is expanded for small values of t , the resulting expansions must be asymptotically equal. Upon doing this, we find that the constant C in eq 15 is determined as c = -1 - u1/ruz (34) The complete asymptotic solution to a given order in a may be constructed from the inner and outer solutions (see ref 6 for details). We denote this solution to zero order in a by ?(O), ~ ( o ) , & ( O ) . The result is $o) ( t ) = s(O)(t) = - U1t + 1 + u1/yu2 - (u~/~uz)[s~o~(t)]~~~K1'~lK~ - i(o)(t) = i(o)(t) = [ S ( O ) ( ~ ) I Y U ~ K I / U ~ K ~ Kl In s(O)(t) (35) (36) As before, we assume a solution to these equations of the form m (S,Z,C~,CZ) = C(S(n),l(nj,Cl(n),Cz(n))an (22) n = O Substituting eq 22 into (18)-(21) and equating coeffi- cients of ao, we obtain (1 + Kl + A-)Be'-' (30) where i.,(O)(t) = Kzs(O)( t ) - K1i(O)(t) + KZdo)( t ) + KIKz (1 + Kl + A+)AeX+'/" + (1 + Kl + A-)Be"t/Q (38) where A , B, and X, are given by eq 31-33. If the expressions for ? ( O ) ( t ) and i(O)(t) above are expressed in terms of dimensional variables by the use of (l) , they become eq 2 and 3, respectively, of the text. By comparing (37) and (38) with (13) and (14), re- spectively, we see that the time needed for the pseudo- steady state to be established is just the time necessary for the exponential terms in (37) and (38) to be neg- ligible compared with the nonexponential terms. Rubinow, Lebowitz 1 Michaelis-Menten Kinetics for an Enzyme-Substrate-Inhibitor System