Download Time-Dependent Michaelis-Menten Theory for Enzyme-Substrate-Inhibitor Systems and more Exams Medical Sciences in PDF only on Docsity!
Time-Dependent Michaelis-Menten Kinetics for an
Enzyme-Substra te-Inhibitor Sy stern
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- Tvated 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~mpetitive.”~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/VmaXson 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. It 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 C 2 +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
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
and
VmaXs = k+8eo Vmaxi^ =^ k+reo
Differentiating eq 2 and 3 yields
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
S = B J s [ l - - - - ] K M S JS JI Vmax V m a 2 VrnaxI
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)
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 ) ] / s o , 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 i o , so
Rubinow, Lebowitz 1 Michaelis-Menten Kinetics for an Enzyme-Substrate-Inhibitor System
180-
5 min
10 min
20 min
50 100 150 200
so mole
Figure 2. The reciprocal mean reaction velocity [o.(t)]-l is shown as a function of the reciprocal initial substrate concentration s0-l for different values of the time. The solid curves are based on eq 11 for the same experimental values of the parameters as those quoted for Figure 1. The dashed curves are based on the approximate form of eq 11 (eq 13), for f = 10,20 min. This form is valid for t sufficiently large so that i(t) << io, while[so - s ( r ) ] / s ~ << 1.
this reaction are K M = 17.5 X 10W M and V,,, = 0.
pmol hr-’/0.4 unit enzyme. For the catalytic oxida- tion of xanthine to uric acid by xanthine oxidase, the
parametric values of the reaction are KM^ =^ 5.4^ X
M , and V,,, = 9.40 pmol hr-’/0.4 unit of enzyme.
The competitive inhibition of xanthine oxidation by 6-mercaptopurine in the presence of xanthine oxidase was observed to obey ordinary Michaelis-Menten
theory for reactions of a fully competitive type.’ We
readily calculate 6 = 0.0027 for this reaction, so that
the agreement with ordinary theory is expected. At the same time, we note that in the reverse situation in which xanthine inhibits the oxidation of 6-mercapto- purine, 6 = 370. Such an experiment has not been performed to the best of our knowledge. When it is, we predict that ordinary Michaelis-Menten theory will not be applicable and that the time-dependent formalism
(7) H. R. Silberman and J. B. Wyngaarden, Biochint. Biophys. Acm, 47, 178 (1961).
Figure 3. The fractional amount of inhibitor concentration ic/)/io is shown as a function of the fractional disappearance of substrate concentration (1 -^ s(t)/so)^ at a given time, for different values^ of^ 6. The figures are based on eq 3.
will be applicable. With regard to the observations, it is recommended that the time at which observations are made be carefully recorded. Other substrates of xanthine oxidase, such as 2,6-diaminopurine, are “slow” when compared with xanthine,8 so that utilizing xan- thine as an inhibitor with them would also result in a large value for 6.
Another example of a fully competitive reaction i n
which 6 is rather small is the catalysis by the enzyme
adenosine deaminase of the dechlorination of 6-chloro-
purine ribonucleoside to yield inosine and chloride ions,
in the presence of adenosine as inhibitor. The reaction
parameters for adenosine diaminase with adenosine as
substrate are V,,, = 400 pmol min-l/mg of enzyme, KM
= 8.3 X 10-j M ; with 6-chloropurine ribonucleoside as
substrate, V,,, = 100 pmol min-’jmg of enzyme, KM --
6.4 X M. With adenosine as a substrate and
6-chloropurine ribonucleoside as inhibitor, 6 = 0.032,
so that the time-independent theory should apply as is
o b ~ e r v e d. ~The reverse situation i n which 6-chloro- purine ribonucleoside is utilized i n the presence of adenosine diaminase with adenosine as an inhibitor has a value of 6 = 31 associated with it. This value is
not as^ large as^ in^ the previously cited example, although
we would still expect some deviations from the classical
theory if the experiment is performed.
Conclusions
When two substances react with an enzyme in a
fully competitive manner, the theoretical expressions for the temporal disappearance of each of them assume
a symmetric form. I n kinetic studies of such a system,
the substance whose reaction velocity is measured is called the substrate and the other substance is called
the inhibitor. Thus, there is a dual choice as to which
substance plays the role of substrate and which sub- stance plays the role of inhibitor. The parameter^6 is a measure of the relative “fastness” or “slowness” of the two substances. When 6 is less than unity, the “slow” substance is being utilized as inhibitor. When 6 is greater than unity, the “fast” substance is being uti- lized as inhibitor. If 6 is less than or comparable with unity, then ordinary Michaelis-Menten theory may be ( 8 ) J. B. Wyngaarden, J. B i d. Chem., 224, 453 (1957). (9) J. G. Cory and R. J. Suhadolnik, Biochemistry 4, 1733 (1965).
Rubinow, Lebowitz / Michaelis-Menten Kinetics for an EnzPiiie-Suhstrate-In/lihiror System
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.
~- - y [ - i + ( i + K 2 - 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 t o 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 t o eq 2-5 as a power series in a, e.g. m
(s,i,cl,cz) = n = O(s(n),i(n), c~(~), ~ ~ ( ~ ) ) a ~ (8)
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)
0 = ~ ( 0 ) - ('(0) + Kl)clCO) - s(0)c2(O)
0 = j ( 0 ) - ( i ( 0 ) + K2)cz(0) - (^) I '(0) (^) c1 (0)
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. T o 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
- =dl a y [ - I + (I + K2 - UZ)CZ + IC11 (19) d r
dC1- - - S - (S + K1)CI - SC, d r
- -dCzd r - Py[I - ( I + K2)C2 - ICl] (21)
Journal of the American Chemical Society 1 92:13 1 July I , 1970
