The Effect of Slow Allosteric Transitions in a Simple Biochemical Oscillator Model

The quasi-steady state approximation (QSSA) has an important role in models of biological control systems. Often internal dynamics is assumed to be sufficiently fast to allow relative simple polynomial functions, e.g. Hill-type kinetics, to be used in a description of minimal dimensionality. Here we investigate the effect of a finite rate alosteric transition, or finite rates for substrate or product binding. The basic system chosen shows only simple limit cycle behaviour when QSSA is invoked, but we report much more complicated dynamics with finite rate allosteric transitions, such as two coexisting stable limit cycles (birythmicity). Such phenomena are usually assumed to require two or more interacting Hill-type controls, but only one is present here. The basic QSSA-simplified system also underestimates the full systems capability to oscillate, which may suggest a physiological role for control by monitoring the allosteric transition rate. Our findings indicate that biomodels assuming fast internal dynamics with resulting Hill-type kinetics may underestimate the potential for complex dynamics in the real systems.