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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Holomorphic Regularization Method for Differential Equations System of Enzyme-substrate Reaction

Author(s):

Margarita Ilyinichna Besova

Assistant of the Department of Higher Mathematic, National Research University "Moscow Power Engineering Institute".

besovami@mpei.ru

Vasiliy Ivanovich Kachalov

Doctor of Physical and Mathematical Sciences, Head of the Department of Higher Mathematics, National Research University "Moscow Power Engineering InstituteВ"

vikachalov@rambler.ru

Dmitry Alexandrovich Maslov

Ph.D., Associate Professor of the Department of Higher Mathematics, National Research University "Moscow Power Engineering Institute".

maslovdma@mpei.ru

Abstract:

The mathematical model of classical enzyme-substrate Michaelis-Menten reaction representing the Cauchy problem for two nonlinear differential equations written in dimensionless form is studied. This system of equations belongs to the class of Tikhonov systems since one of the two equations is singularly perturbed. To obtain an approximate solution we use the holomorphic regularization method. The holomorphic regularization method is a logical continuation of the regularization method of S.A. Lomov and unlike other methods that deduce approximations in the form of series that converge asymptotically allows to obtain solutions to nonlinear singularly perturbed problems in the form of series in powers of small parameter that converge in the usual sense. The exposition of holomorphic regularization method for a system of differential equations of Tikhonov type is given. Deduced approximation to the solution of the enzyme-substrate reaction differential equations system is given by uniform formulas both in the boundary layer and outside it. The advantage of using the holomorphic regularization method is obtaining the formulas for approximate solution, which allow to analyze the approximate solution of an enzymatic reaction over the entire time interval under consideration including the boundary layer. Represented plots of the dependence of substrate concentration and enzyme-substrate complex concentration on the time demonstrate the high accuracy of the obtained approximate solutions even for relatively large values of the small parameter. The obtained approximate solution of the system of enzymatic reaction equations is used for deducing the reaction rate of the substrate and the reaction rate of the enzyme-substrate complex that are valid both in the boundary layer and outside it.

Keywords

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