Classical and non-classical methods for describing the kinetics of enzymatic reactions.
Back in 1913 Michaelis and Menten [1] proposed a scheme of enzymatic catalysis with an irreversible yield.
E + S ↔ ES → E + P
Scheme I
The logic of Scheme I, the method for its description, provides for the formation of the enzyme-substrate complex ES from the reacting enzyme E and substrate S, followed by its decomposition, resulting in the formation of product P, and the enzyme and part of the substrate, returning to the reactive mixture, continue to participate in the enzymatic reaction. Similar to the scheme of this simplest one-sided reaction with one substrate, many others can be composed, with different stages, enzymes, substrates, products, and united by a common logic, by the classical method of their description. The basis of this method is the idea of stationarity:
when [S] >>[E], [ES] = const.
Square brackets indicate the concentration of reagents.
Scheme I corresponds to the equation of material balance for the enzyme:
[E] 0 = [E] + [ES], (I-1)
([E] 0 is the initial concentration of the enzyme) and the ratio:
obtained by integration in (I-3), taking into account:
d [P] / dt = -d [S] / dt, (I-5)
and the equations of material balance for the substrate:
[S] 0 = [S] + [ES] + [P]. (I-6)
The Michaelis-Menten integral equation is in good agreement with experimental data. At the same time, it, like its differential form, is not free from shortcomings.
Firstly, the range of application of this equation should be limited by the condition of excess substrate.
Secondly, the stationarity condition cannot be satisfied.
Indeed, equations (I-3), (I-4) were derived from the premise of stationarity:
[ES] = const. But then, in accordance with (I-3): [S] = const. And, therefore, from (I-5) and (I-3):
[S] = 0, [ES] = 0, [P] = const,
those. the reaction does not occur at all, and from (I-6) it also follows that this lack of reaction corresponds to [P] = [S] 0, the entire substrate is processed into the product. Thus, the stationarity condition, if it can be satisfied, is only to a certain extent, is fundamentally approximate in nature [ES] ≈ const. The question remains open whether we get this while repeating the above line of reasoning:
[S] ≈ 0, [ES] ≈ 0, [P] ≈ const,
those. previous results with the same degree of approximation as the stationary conditions?
It should be noted that if the Michaelis-Menten equation (I-3) is obtained from any other premises than the stationary condition, then when it is substituted into (I-2), we again come to the stationary condition.
The logical question is whether a logical scheme of the enzymatic reaction is possible, which would:
a) It was a modification of the classical scheme.
b) Allowed to correctly derive the equation for the enzymatic reaction with an irreversible yield of the product without the condition of excess substrate.
c) Under certain conditions, it allowed to arrive at results close to those obtained by the Michaelis-Menten formula in its differential and integral expression.
The material composition of Scheme I is undeniable; a substrate and an enzyme are necessary for the formation of a product. The formation of the enzyme-substrate complex is also undoubted, without it the catalytic ability of the enzyme is inexplicable. In addition, the formation of the complex is confirmed by experiments [3]. It is natural to assume that the complex breaks up, because the enzyme is capable of processing the amount of substrate exceeding its content, which requires the return of the enzyme released from the complex into the reactive mixture. Doubt is caused only by a similar fate, prescribed by Scheme I to the substrate released from the complex. There is no conclusive evidence in favor of the fact that the substrate emerging from the complex in all cases, without fail, again participates in its formation like an enzyme.
Methods of unsteady kinetics determine the reaction rate constants. But the determination of K-1 is not carried out as a result of a direct measurement of the corresponding speed, but is somehow connected with calculations based on the ratio (I-2). On the other hand, for a number of enzymatic reactions, K-1 = 0, K-1 ≈ 0 [4-5].
The concept of reversibility is usually associated with the return to the reactive mixture of all the reactants involved in the direct reaction, which, for example, refers to the substrate in the reaction proceeding according to Scheme I. It involves the return of a part of the substrate to the reactive mixture and the conversion of its other part into a product. The ratio of these parts is not regulated and doubts about the mandatory return of the substrate to the reactive mixture do not dissipate. Nevertheless, the assumption is valid that when one of the reactants has the opportunity to return to the reactive mixture, such a possibility is realized.
Earlier, together with P.A. Podrabinek, we proposed a non-classical method for describing the kinetics of enzymatic reactions. Its essence is as follows.
Enzyme E binds to substrate S in the enzyme-substrate complex ES. The enzyme molecule is capable of conformational changes associated with a change in its energy level and the transition of the substrate to the activated state of Sa. As a result, the complex decomposes with the irreversible conversion of the activated substrate Sa into a free (unbound) product P. Therefore, the rate of decomposition of the enzyme-substrate complex determines the rate of formation of the product, which corresponds to the relaxation concept of L.A. Blumenfeld [6]. In accordance with this principle of limitation, the activated substrate Sa is converted into a product and is not returned to the reactive mixture as the initial (not activated) substrate S.
The above ideas correspond to the following scheme of the course of the enzymatic reaction:
E + S → ES → ESa → E + P
Scheme II
For convenience, we will omit the activation stage of the substrate in the enzyme-substrate complex, since this does not affect the ratios.
The Michaelis-Menten Integral Equation (I-4) is a special case of the equation
(I-10), if the latter neglected a small linear quantity [ES] taking into account (I-6). Of
(I-7) it follows that at some time point the maximum reaction rate is reached, the dependence of which on the substrate concentration is described by the Michaelis-Menten equation (I-3) at K-1 = 0. Unfortunately, we did not find data in the literature taking into account [ES]), allowing direct verification of equation (I-10). It should be assumed that it is valid at least in the same cases in which the Michaelis-Menten integral equation is applied.
II
A generalization of the non-classical method for describing the kinetics of enzymatic reactions.
The proposed method can be extended to more complex cases of enzymatic catalysis.
1) N-step (multi-stage) enzymatic reaction.
Consider a multi-stage enzymatic reaction. The reaction stage of the reversible formation of the enzyme-substrate complex and the subsequent stage of the irreversible formation of the product will be called the reaction step. Thus, if an enzymatic reaction has n reversible stages of formation of an enzyme-substrate complex with subsequent stages of irreversible formation of a product, such a reaction is n-stepwise. Scheme III corresponds to such an enzymatic reaction.
E + S1 → ES1 → E + S2 → ES2 → ... E + Si → ESi → ... E + Sn → ESn → E + P.
Scheme III
The index i, indicating the number of the step, runs from 1 to n. Index j runs from 1 to i. Enzyme E reversibly reacts with the Si substrate to form the enzyme - substrate complex ESi. V1i is the rate of the direct reaction, V2i is the rate of the reverse reaction, K1i and K2i are the corresponding rate constants. The substrate leaving the enzyme of the substrate complex irreversibly forms a product, which is the substrate of the next reaction step, up to the formation of the final product P.
Equation (II-1-7) shows how the concentrations of intermediates change over time. The participation in the i-th stage of the reaction only part of the intermediate product from
(i-1) -th reaction step, is easily taken into account in (II-1-1), (II-1-7).
2) Enzymatic reactions of n enzymes with m substrates.
Consider the set of enzymatic reactions of n enzymes with m substrates, in which each enzyme can react with each substrate. During the reaction, (nхm) products are formed.
We will number the enzymes with index i running through the values from 1 to n. We will number the substrates by the index j running through the values from 1 to m. Figure IV shows the reaction of an arbitrarily selected enzyme Ei with an arbitrarily selected substrate Sj to form the enzyme-substrate complex EiSj and the subsequent irreversible formation of the product Pij.
Ei + Sj → EiSj → Ei + Pij
Scheme IV
[Ei] 0 is the initial concentration of the enzyme, [Sj] 0 is the initial concentration of the substrate. V1ij is the rate of the direct reaction, V2ij is the rate of the reverse reaction, K1ij and K2ij are the corresponding reaction rate constants. Substrate concentrations are not zero.
Equation (II-2-6) expresses the time dependence of the concentrations of any substrate and formed products for the totality of reactions of n enzymes with m substrates. At
n = 1, we obtain the reaction of one enzyme with m substrates, and for any two substrates S1 and S2, whose concentrations are not equal to zero, the relation holds:
Equation (II-2-7) may be useful for determining the ratio of reaction rate constants.
3) Irreversible inhibition of the enzymatic reaction.
From the vast field of inhibition of enzymatic reactions, we consider the case of inhibition of the simplest enzymatic reaction due to the irreversible formation of complex EI by inhibitor I and enzyme E. Moreover, during the enzymatic reaction itself, in the presence of substrate S, the product P is irreversibly formed. Such reactions are shown in Scheme V.
E + S → ES → E + P, E + I → EI.
Scheme V
Let Vs be the rate of formation of the enzyme-substrate complex and Ks be the corresponding rate constant. Vi is the binding rate of the enzyme inhibitor and Ki is the corresponding rate constant. Consider the case where the reaction ceases due to complete binding of the enzyme inhibitor. According to this, the initial concentration of the substrate should be large enough so that the reaction does not stop as a result of its consumption, and the initial concentration of the inhibitor should exceed the initial concentration of the enzyme:
By the equation (II-3-7) it is convenient to determine the concentration of the product formed at the end of the reaction. It is noteworthy that equation (II-3-6) is similar to equation (II-2-7) related to the reaction of a single enzyme with several substrates. The latter case can also be considered as an example of inhibition of the enzymatic reaction.
Conclusion
A modification of the classical scheme of enzymatic catalysis with irreversible product formation is possible. In the framework of the proposed non-classical method, the following provisions are observed:
and). When the enzyme-substrate complex decomposes, the substrate does not return to the reactive mixture.
b) The rate of decomposition of the enzyme-substrate complex limits the rate of the enzymatic reaction.
In accordance with these provisions, the kinetics of the enzymatic reaction is described by equations, in particular cases leading to previously known laws: the Michaelis-Menten equation in its integral expression, relations for the maximum reaction rates.
The proposed non-classical method for describing the kinetics of enzymatic reactions can be extended to more complex cases of enzymatic catalysis:
and). Multi-stage enzymatic reaction.
b) Enzymatic reactions of an arbitrary number of enzymes with an arbitrary number of substrates.
in). Irreversible inhibition of the enzymatic reaction.
A number of dependences deduced in this work need experimental verification.
2.T.Keleti, Fundamentals of Enzymatic Kinetics, Mir, M., (1990).
3.K. Jagi, T. Ozana, J. Biochem. (Japan), 53.162 (1963).
4.M. Dixon, E. Webb, Enzymes, Mir, M., (1966).
5. L. Webb, Inhibitors of enzymes and metabolism, "Mir", M., (1966).
6. L.A. Blumenfeld, Problems of Biological Physics, "Science", M., (1977).
Note
This work has a difficult fate. The fundamental equation was found by me in the early 70's. Together with my father, Pinhos Abramovich Podrabinek, we developed the idea and wrote an article. But it was not possible to publish dissidents, even in scientific journals. The article illegally went abroad, where it was lost ... In 1977-1983. I was honored to be a political prisoner. In 1988, again together with P.A. Podrabinekom, we wrote a new article, slightly revising the previous one. They registered her at a notary's office and, it seems, sent them abroad again. We didn"t get any news about it ... In 2000, having thoroughly expanded the article, I sent it to the journal "Molecular Biology". The magazine did not accept the article as "not suitable for the profile", advising it to offer other publications. Then the article went to the magazine " as in Russian courts: the truth is not needed, evidence is not accepted. However, I do not pretend to have the undoubted truth. But it is clear that the Michaelis-Menten equation is theoretically untenable, and the new method deserves consideration, reasoned criticism, and experimental verification. Therefore, the work must be published! I was not upset. I am inclined to believe that I am doing a favor to magazines by sharing my good with them, i.e. work. In the editorial offices, for some reason they believe otherwise ... More than 100 years ago, they proposed the first equation. Almost 50 years ago, a new equation was found, including the original as a special case. I think that in 100 years the truth would become clear. But I don"t have that kind of time left. And a little tormented conscience. Still, there are conscientious researchers, my modest work could be useful to someone. There is neither strength nor desire to get involved in official science - I am no longer young. That is why I publish it in a form adapted to "samizdat", offering it to the attention of a respected reader. Unfortunately, the text, and especially the formulas, do not adapt very well to the format of the site. Subscripts are not written. The inserts are floating. There may be other troubles while reading ... I apologize for them. And I apologize for the quality of the translation - the machine worked.