Biological rhythms limit cycles or discontinuous iterative behaviour

Biological rhythms limit cycles or discontinuous iterative behaviour  

The examples of rhythmic behaviour analysed in this book all belong to dynamics of the limit cycle type. In the case of phosphofructokinase, like in that of cAMP synthesis in Dictyostelium or signal-induced Ca oscillations, the analysis of models based on experimental data indeed shows that these systems admit a nonequilibrium steady state that becomes unstable beyond a critical value of some control parameter. It is in these conditions that sustained oscillations occur, in the form of a limit cycle in the phase space. 

The second view did not rely on the existence of an oscillator independent from mitosis and stated that the latter is an integral part of the mechanism underlying the cell division rhythm (Tyson Sachsenmaier, 1978, 1984). The rhythm would thus have a simpler origin, due to the discontinuity of cell division. Tyson Sachsenmaier (1978) compared such a rhythm to the effect of a thin stream of sand constantly falling on a pan on one side of a balance when a critical weight is reached, the pan tips over and, once emptied, recovers its equilibrium position. The sand then accumulates again until the next flip. In this phenomenon, the phase of abrupt decrease that precedes the replenishment phase consists of the discontinuous reversal of the position of the pan. This image could apply to the case of the mitotic cycle the latter could result from the accumulation of a mitotic factor up to a threshold beyond which the discontinuity of cell division would occur. No oscillation should take place in the absence of mitosis. 

Experiments carried out on mitosis in Physarum polycephalum (Kaufmann Wille, 1975 Tyson Sachsenmaier, 1978) did not allow a definitive conclusion in favour of one of the two views. The difficulty came from the fact that, short of the identification of the true biochemical variables driving the cell cycle, it was difficult to demonstrate the existence of a limit cycle in Physarum. Experiments based on the fusion of two plasmodia taken at different phases of the mitotic cycle or on the effect of heat shocks aimed at demonstrating the existence of a singular point from which the limit cycle would be reached with an indefinite phase (Winfree, 1974,1980,1987). These experiments did not allow the distinction to be made between a limit cycle characterized by relaxation oscillations and a discontinuous mechanism of the type discussed above. 

From a theoretical point of view (Andronov, Vitt Khaikin, 1966), the two situations can be approached with comparable methods. 

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