The occurrence and the significance of limit cycle behavior in controlled biochemical systems

Abstract

Phase-plane methods indicate that limit cycle behavior will not arise in chemical control systems involving negative feedback and two components if the rate law for the uninhibited step is described by first order kinetics, a rectangular hyperbola, or a sigmoidal relationship. However, in multi-component systems limit cycles arise even if the rate laws for the uninhibited steps are first order. A mapping of the range of values of the constants in the differential equations describing the feedback mechanism indicates that there is a finite range of these constants for which limit cycles arise. The relationships between these constants, the level of the variables expected at the physically important singularity, and the amplitude of the oscillations suggest a possible chemical basis for sustained rhythmic “spike” functions in biological systems.