Extreme properties of the initial rate of the four-stage reaction of enzyme catalyzed ATP hydrolysis


We investigated the kinetic model of the enzyme catalyzed reaction of the hydrolysis of adenosine triphosphate (ATP) in which ATP (substrate) and MgATP (true substrate) compete for binding with the catalytic site of the enzyme (ATPase). The goal was to accurately establish all kinds of kinetic dependences of the initial rate on the initial concentrations of the enzyme, substrate, and activator inherent in this model. The analysis was carried out under the natural condition of the experiment – the concentration of the reaction product and the rate of elementary reactions in which it participates can be neglected. No additional simplifying assumptions were used. This is necessary in order to reliably identify the reaction proceeding according to the investigated mechanism among others, which can occur in a system containing three substances, (enzyme, substrate and activator), capable of interacting with each other. For this reaction we have proved the existence of a quasi-steady state at any initial reagent concentrations. This made it possible to analyze the dependences of the initial rate on the initial concentrations of the enzyme, substrate, and activator in the general case. It has been proven that (i) the dependence of the initial rate on the initial substrate concentration (ATP) always has a unique maximum point; (ii) the dependence of the initial rate on the initial enzyme concentration always has unique maximum point; (iii) the initial rate monotonically increases with increasing activator concentration (Mg2+) until it reaches the maximum limiting level.

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Karakhim S.O.—article concept, verification of the obtained equations, discussion of the results, writing the paper. Zhuk P.F.—mathematical analysis of the kinetic model, derivation of equations, proof of theorems, analysis and discussion of the results, writing the paper. All authors read and approved the final manuscript.

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Correspondence to S. O. Karakhim.

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Zhuk, P.F., Karakhim, S.O. Extreme properties of the initial rate of the four-stage reaction of enzyme catalyzed ATP hydrolysis. J Math Chem 59, 1785–1807 (2021). https://doi.org/10.1007/s10910-021-01262-2

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  • Multistep reactions
  • Kinetic model
  • Initial reaction rate
  • Quasi steady state approximation
  • Extremum
  • Polynomial

Mathematics subject classification

  • 92C45
  • 65H10