Geometry of the steady-state approximation: Perturbation and accelerated convergence methods

Abstract

The time evolution of two model enzyme reactions is represented in phase space Γ. The phase flow is attracted to a unique trajectory, the slow manifold ℳ, before it reaches the point equilibrium of the system. Locating ℳ describes the slow time evolution precisely, and allows all rate constants to be obtained from steady-state data. The line set ℳ is found by solution of a functional equation derived from the flow differential equations. For planar systems, the steady-state (SSA) and equilibrium (EA) approximations bound a trapping region containing ℳ, and direct iteration and perturbation theory are formally equivalent solutions of the functional equation. The iteration’s convergence is examined by eigenvalue methods. In many dimensions, the nullcline surfaces of the flow in Γ form a prism-shaped region containing ℳ, but this prism is not a simple trap for the flow. Two of its edges are EA and SSA. Perturbation expansion and direct iteration are now no longer equivalent procedures; they are compared in a three-dimensional example. Convergence of the iterative scheme can be accelerated by a generalization of Aitken’s δ2 extrapolation, greatly reducing the global error. These operations can be carried out using an algebraic manipulative language. Formally, all these techniques can be carried out in many dimensions.