2006 International Conference in honor of Jacqueline Fleckinger. Electron. J. Diff. Eqns., Conference 16 (2007), pp. 155-184.

Reduction for Michaelis-Menten-Henri kinetics in the presence of diffusion

Leonid V. Kalachev, Hans G. Kaper, Tasso J. Kaper, Nikola Popovic, Antonios Zagaris

The Michaelis-Menten-Henri (MMH) mechanism is one of the paradigm reaction mechanisms in biology and chemistry. In its simplest form, it involves a substrate that reacts (reversibly) with an enzyme, forming a complex which is transformed (irreversibly) into a product and the enzyme. Given these basic kinetics, a dimension reduction has traditionally been achieved in two steps, by using conservation relations to reduce the number of species and by exploiting the inherent fast-slow structure of the resulting equations. In the present article, we investigate how the dynamics change if the species are additionally allowed to diffuse. We study the two extreme regimes of large diffusivities and of small diffusivities, as well as an intermediate regime in which the time scale of diffusion is comparable to that of the fast reaction kinetics. We show that reduction is possible in each of these regimes, with the nature of the reduction being regime dependent. Our analysis relies on the classical method of matched asymptotic expansions to derive approximations for the solutions that are uniformly valid in space and time.

Published May 15, 2007.
Math Subject Classifications: 35K57, 35B40, 92C45, 41A60.
Key Words: Michaelis-Menten-Henri mechanism; diffusion; dimension reduction; matched asymptotics.

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Leonid V.~Kalachev
Department of Mathematical Sciences, University of Montana
Missoula, MT 59812, USA
email: kalachevl@mso.umt.edu
Hans G. Kaper
Mathematics and Computer Science Division
Argonne National Laboratory, Argonne, IL 60439, USA
Division of Mathematical Sciences
National Science Foundation, Arlington, VA 22230, USA
email: hkaper@nsf.gov
Tasso J. Kaper
Department of Mathematics and Statistics and Center for BioDynamics
Boston University, Boston, MA 02215, USA
email: tasso@math.bu.edu
Nikola Popovic
Department of Mathematics and Statistics and Center for BioDynamics
Boston University, Boston, MA 02215, USA
email: popovic@math.bu.edu
Antonios Zagaris
Korteweg-de Vries Institute, University of Amsterdam
1018 TV Amsterdam, The Netherlands, Modelling, Analysis and Simulation
Centrum voor Wiskunde en Informatica (CWI)
1090 GB Amsterdam, The Netherlands
email: a.zagaris@cwi.nl

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