The Effect of Stochastic Bursting on Biological Clock Precision

We study stochasticity in gene transcription oscillations. The oscillations in the protein concentration occur due to stochastic bursting of gene transcription being modulated by a delayed negative feedback. The oscillator dynamics is such that the mean dynamics (which the system converges towards in the large size limit) supports an attracting limit cycle. However, since there are only a finite-number of particles in biologically-realistic systems, this results in stochastic effects which perturb the system away from the limit cycle. We define an isochronal phase for the oscillation, and argue that it allows a much more powerful analysis than the Linear Noise Approximation. We demonstrate that the isochronal phase is accurate for very long periods of time: this results from the attracting dynamics of the limit cycle damping down stochasticity from the finite-size effects. Furthermore, because the system stays close to the oscillator for very long periods of time, we can obtain an estimate for the increase in errors over time due to the stochastic nature of the chemical reactions. We investigate numerically how various parametric regimes affect the resulting limit cycle dynamics.