Phase-plane geometries in enzyme kinetics

Abstract

The transient and steady-state behaviour of the reversible Michaelis–Menten mechanism [R] and Competitive Inhibition (CI) mechanism is studied by analysis in the phase plane. Usually, the kinetics of both mechanisms is simplified to give a modified Michaelis–Menten velocity expression; this applies to the CI mechanism with excess inhibitor and to mechanism [R] in the product inhibition limit. In this paper, [R] is treated exactly as a plane autonomous system of differential equations and its true (dynamical) steady state is described by a line-like slow manifold M. Initial velocity experiments for [R] no longer strictly correspond to the hyperbolic law (as in the irreversible Michaelis–Menten mechanism) and this leads to corrections to the standard integrated rate law. Using a new analysis, the slow dynamics of the CI mechanism is reduced from a three-dimensional system to a planar system. In this mechanism transient decay collapses the trajectory flow onto a two-dimensional "slow" surface Σ; motion on Σ can be treated exactly as projected dynamics in the plane. This projected flow may differ in important ways from that of two-step mechanisms, e.g., it may lack a proper steady state. The relevance of these more accurate dynamical descriptions is discussed in relation to experimental design and metabolic function.