Reconciling Mathematical Models of Biological Clocks by Averaging on Approximate Manifolds

Although oscillations are typically described in terms of two variables (phase and amplitude), most oscillating systems contain more than two variables. We present a method for approximating the dynamics of the phase and the amplitude of a quasilinear system of three or more variables. This method is used to analyze Goldbeter's five-variable model of the biological circadian (approximately 24-hour period) clock in the fruit fly Drosophila [Proc. Roy. Soc. London B, 261 (1995), pp. 319--324]. Using this method, we show that Forger, Jewett, and Kronauer's mathematical model [J. Biolog. Rhythms, 14 (1999), pp. 532--537] of the human circadian system (based on the van der Pol equation) is almost identical to Goldbeter's model, even though these models were developed independently. This leads to (1) a biochemical analogue of the van der Pol equation, (2) biochemical support for the numerous mathematical models of circadian systems which use the van der Pol equation, and (3) possible evidence of the similarity between the circadian system in Drosophila and the circadian system in man.