Abstract
Circadian rhythms are endogenous oscillations that occur with a period
close to 24 h in nearly all living organisms. These rhythms originate
from the negative autoregulation of gene expression. Deterministic
models based on such 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. When the numbers
of protein and mRNA molecules involved in the oscillations are small,
as may occur in cellular conditions, 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 the stochastic version of a deterministic model previously
proposed for circadian oscillations of the PER and TIM proteins and
their mRNAs in Drosophila. The model is based on repression of the
per and tim genes by a complex between the PER and TIM proteins.
Numerical simulations of the stochastic version of the model are
performed by means of the Gillespie method. The predictions of the
stochastic approach compare well with those of the deterministic
model with respect both to sustained oscillations of the limit cycle
type and to the influence of the proximity from a bifurcation point
beyond which the system evolves to stable steady state. Stochastic
simulations indicate that robust circadian oscillations can emerge
at the cellular level even when the maximum numbers of mRNA and protein
molecules involved in the oscillations are of the order of only a
few tens or hundreds. The stochastic model also reproduces the evolution
to a strange attractor in conditions where the deterministic PER-TIM
model admits chaotic behaviour. The difference between periodic oscillations
of the limit cycle type and aperiodic oscillations (i.e. chaos) persists
in the presence of molecular noise, as shown by means of Poincar�
sections. The progressive obliteration of periodicity observed as
the number of molecules decreases can thus be distinguished from
the aperiodicity originating from chaotic dynamics. As long as the
numbers of molecules involved in the oscillations remain sufficiently
large (of the order of a few tens or hundreds, or more), stochastic
models therefore provide good agreement with the predictions of the
deterministic model for circadian rhythms.
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