SELECTION BETWEEN MULTIPLE PERIODIC REGIMES IN A BIOCHEMICAL SYSTEM - COMPLEX DYNAMIC BEHAVIOR RESOLVED BY USE OF ONE-DIMENSIONAL MAPS

A model biochemical system was analyzed in which 2 autocatalytic enzyme reactions are coupled in series, in conditions where multiple stable periodic regimes coexist for the same set of parameter values. How the periodic regimes are reached from different initial conditions was determined. The structure of the attraction basins is generally simple in the case of 2 coexisting limit cycles (birhythmicity). This structure and the associated behavior may become highly complex. The system exhibits enhanced sensitivity to initial conditions when the boundaries of the attraction basins are fractal. In the latter case, it becomes difficult to predict the evolution towards either 1 of 2 limit cycles, a phenomenon known as final state sensitivity. These complex phenomena can be explained in a unified and simple manner by means of 1 dimensional return maps derived from the time evolution of the model and from 5th degree polynomial equations. Experimental tests of the sensitivity to initial conditions in chemical systems presenting birhythmicity are suggested. The physiological significance of the results is discussed with respect to the sensitivity of regulatory systems admitting multiple stable biological rhythms.