Deterministic versus stochastic models for circadian rhythms

Circadian rhythms which occur with a period close to 24 h in nearly all living organisms originate from the negative autoregulation of gene expression. Deterministic models based on genetic regulatory processes account for the occurrence of circadian rhythms in constant environmental conditions (e.g. constant darkness), for entrainment of these rhythms by light-dark cycles, and for their phase-shifting by light pulses. At low numbers of protein and mRNA molecules, it becomes necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. We address the effect of molecular noise by considering two stochastic versions of a core model for circadian rhythms. The deterministic version of this core model was previously proposed for circadian oscillations of the PER protein in Drosophila and of the FRQ protein in Neurospora. In the first, non-developed version of the stochastic model, we introduce molecular noise without decomposing the deterministic mechanism into detailed reaction steps while in the second, developed version we carry out such a detailed decomposition. Numerical simulations of the two stochastic versions of the model are performed by means of the Gillespie method. We compare the predictions of the deterministic approach with those of the two stochastic models, with respect both to sustained oscillations of the limit cycle type and to the influence of the proximity of a bifurcation point beyond which the system evolves to a stable steady state. The results indicate that robust circadian oscillations can occur even when the numbers of mRNA and nuclear protein involved in the oscillatory mechanism are reduced to a few tens or hundreds, respectively. The non-developed and developed versions of the stochastic model yield largely similar results and provide good agreement with the predictions of the deterministic model for circadian rhythms.