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MICHAELIS-MENTEN EQUATION

The simplest enzymatic reaction is considered. One substrate (S) and one enzyme (E) participate in it, and one product (P) appears as the result of desintegration of the enzyme-substrate complex (ES):

k₁            k₂          
E + S (ES) E + P, where
k_-1                          

k₁, k_-1 - are the constants of the rates of the direct and counter reactions of the formation of the enzyme-substrate complex; k₂ - is the constant of the rate of the product formation.

This system is described by the following differential equations:

       = - k₁[S] · [E] + k_-1[ES],
       = - k₁[S] · [E] + k_-1[ES] + k₂[ES],
       = - k₁[S] · [E] - k_-1[ES] - k₂[ES],        (1)
       = k₂[ES] = ν_P, where

ν_P - is the rate of the product P formation P,
[E], [S], [ES], [P] - are the concentrations of the enzyme, substrate, enzyme-substrate complex and the final product correspondingly.

As the system preserves constant general concentration of enzyme E₀, so at any time the sum of concentrations of the free enzyme and bonded enzyme is equal to [E] + [ES] = E₀.

Characteristic lifetime τ_Å of the variables [E] and [ES] is defined by the quickest stage of its desintegration with formation of the product PP, that is, the constant rate ν_P of the product formation: τ_Å = 1/k₂. The constant k₂ corresponds to the number of catalytic actions, that is, to the number of the actions of desintegration of ES and formation of P in a unit of time. The time τ_Å is the time of the enzyme circulation, and the constant k₂ is called the number of enzyme circulations.

Characteristic time τ_S of decrease of S in the system and corresponding appearance of P depends on the rate of product formation: τ_S = S/ν_p. Maximal rate of the product formation will be achieved, when the whole amount of enzyme E₀ is in bonded state: ν_p^max = k₂·E₀. So, τ_S^max = S/(k₂·E₀). Under usual conditions the concentration of the substrate S exceeds manifold the concentration of the product, that is, E/S ~ 10^-4 << 1. That is why

τ_S >> τ_E,        (2)

It means that [E] and [ES] are fast varying variables. Their changes are so fast, that they stay all the time around their stationary values. Consequently, they can be described by algebraic equations, which result from equalization to zero the right parts of the second and third equations in the model (1):

= = 0        (3).

If the conditions (2) and (3) are met, the Michaelis-Menten equation, which is well-known in biochemistry, comes out. The Michaelis-Menten equation is the equation of relationship of the stationary rate of an enzymatic reaction on the concentration of the substrate

ν_p = , where

K_M = is the Michaelis constant.