Stochastic Oscillations Induced by Intrinsic Fluctuations in a Self-Repressing Gene

Abstract

Biochemical reaction networks are subjected to large fluctuations attributable to small molecule numbers, yet underlie reliable biological functions. Thus, it is important to understand how regularity can emerge from noise. Here, we study the stochastic dynamics of a self-repressing gene with arbitrarily long or short response time. We find that when the mRNA and protein half-lives are approximately equal to the gene response time, fluctuations can induce relatively regular oscillations in the protein concentration. To gain insight into this phenomenon at the crossroads of determinism and stochasticity, we use an intermediate theoretical approach, based on a moment-closure approximation of the master equation, which allows us to take into account the binary character of gene activity. We thereby obtain differential equations that describe how nonlinearity can feed-back fluctuations into the mean-field equations to trigger oscillations. Finally, our results suggest that the self-repressing Hes1 gene circuit exploits this phenomenon to generate robust oscillations, inasmuch as its time constants satisfy precisely the conditions we have identified.

The influence of nuclear compartmentalisation on stochastic dynamics of self-repressing gene expression

“…We show that model 2 does not exhibit subpoissonian protein spiking as indicated by the fact the Fano Factor is greater than 1, though under other parameter regimes it is shown that this model can exhibit stochastic oscillations, see Wang et al (2014) for details. Plots (c) and (d) show the corresponding results for model 4 with T 1 = 1min −1 and T 2 = 1 × 10 −5 min −1 .…”

“…Though we could not find the existence of deterministic oscillations, inspired by Wang et al (2014), we investigated the existence of the stochastic analogue of such behaviour. Wang et al (2014) introduced a technique for assessing temporal regularity in protein spiking behaviour by dividing the state space into two regions I and II and studying the distribution of the times where the system leaves I to enter II. To avoid spurious transitions, events are only recorded where the protein level crosses successively the mean protein level P and P ± 0.25std(P) where "std(P)" denotes the standard deviation of protein levels.…”

“…A spatial stochastic model of the Hes1 GRN in embryonic stem cells was studied in Sturrock et al (2013) -this model was derived using mass action kinetics and its mean behaviour was studied in Sturrock et al (2014b). Wang et al considered a spatially-homogeneous model of a self-repressing gene without nonlinearity in regulation in Wang et al (2014), and were able to identify certain regions of the parameter space which yielded 'stochastic oscillations'.…”

“…Finally, given the history of studying oscillatory dynamics in gene regulatory networks containing negative feedback loops, we also take the opportunity to investigate whether there exists a relationship between noise and oscillatory dynamics. Specifically, we explore the role of transport in producing stochastic oscillations as defined in Wang et al (2014). …”

“…This confirms the notion that some nonlinearity in gene regulation is needed for observation of oscillatory behaviour. However, motivated by Wang et al (2014), we investigated the existence of stochastic oscillations or spike anti-bunching. For our choice of parameter set, in the well mixed negative feedback model (model 2), we failed to find conditions necessary for stochastic oscillations but in certain translocation rate regimes, we found that the negative feedback model with nuclear compartmentalisation (model 4) could produce stochastic oscillations, suggesting that transportation processes can facilitate stochastic oscillatory behaviour.…”