Stability of a model for cell metabolism
The paper is concerned with explicit stability conditions for a stationary solution of equations describing the cell metabolism. A cell is modeled as a hereditary dynamic system with a specific time delay. As is common practice, a time delay in functional-differential equations is assumed to be given. The stability problem consists in imposing restrictions on coefficients of these equations which would ensure the Lyapunov stability of a fixed solution. The system behavior obeys functional-differential equations of the standard nature, i.e., either ordinary integro-differential equations, or partial integro-differential equations. Mass transfer through a cell is described by parabolic partial differential equations in a domain occupied by the cell with specific differential relationships on a boundary (membrane) of this domain. By excluding mass transport across the protoplasm from the membrane to the nucleus from our consideration, we reduce the initial problem to an ordinary integro-differential equation on the boundary with an integral operator dependent essentially on the molecular diffusion inside the cell. To develop stability conditions we linearize the problem and employ a method of integral inequalities. It is shown that a large number of coefficients determining the mass transport is reduced to three basic dimensionless parameters, one of which characterizes the cell geometry, another characterizes a stationary process of transport, and the third characterizes feedback control in cell metabolism. We derive a stability condition formulated explicitly as an inequality dependent on the above parameters. The dependence of the ultimate parameter of the feedback control on the two others is studied analytically and numerically.
scientific article; zbMATH DE number 1042766
scientific article