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