Quantitative analysis of a transient dynamics of a gene regulatory network

Abstract

In a stochastic process, noise often modifies the picture offered by the mean field dynamics.In particular, when there is an absorbing state, the noise erases a stable fixed point of the mean field equation from the stationary distribution, and turns it into a transient peak. We make a quantitative analysis of this effect for a simple genetic regulatory network with positive feedback, where the proteins become extinct in the presence of stochastic noise, contrary to the prediction of the deterministic rate equation that the protein number converges to a non-zero value. We show that the transient peak appears near the stable fixed point of the rate equation, and the extinction time diverges exponentially as the stochastic noise approaches zero. We also show how the baseline production from the inactive gene ameliorates the effect of the stochastic noise, and interpret the opposite effects of the noise and the baseline production in terms of the position shift of the unstable fixed point. The order of magnitude estimates using biological parameters suggest that for a real gene regulatory network, the stochastic noise is sufficiently small so that not only is the extinction time much larger than biologically relevant time-scales, but also the effect of the baseline production dominates over that of the stochastic noise, leading to the protection from the catastrophic rare event of protein extinction. * jul@ssu.ac.kr 1 This is also called the transcriptional leakage, but we will refrain from using this terminology because we will be using the word leakage in quite the opposite sense. 2 The concentration is defined as x ≡ m/m, where m is the number of proteins andm is a large number chosen to be of the order of average number of proteins [31]. The ratesr,k 1 , andã of the rate equation, and the corresponding quantities r, k, and a in the master equation, are related byr = r/m,k 1 = k 1 /m, andã = a/m. See appendix E for details.

Formation and synaptic control of active transient working memory representations

“…Similar stabilizing effect of noise in networks have been reported in the context of statistical mechanics (Juel et al, 1997;Ushakov et al, 2005;Kiss et al, 2003;Morrone et al, 2011) and gene regulatory networks (Lee and Lee, 2018). In the neuroscience context, noise has been reported to stabilize rehearsal of long-term memories (Wei and Koulakov, 2014) and aperiodic attractors in networks of the olfactory system (Freeman et al, 1997).…”

Revisiting the Reduction of Stochastic Models of Genetic Feedback Loops with Fast Promoter Switching

“…The reduced chemical master equation is then obtained by writing effective propensities analogous to the non-massaction reaction rates obtained from the deterministic analysis. For example, Hill-type effective protein production rates in the deterministic rate equations result if the gene equilibrates on a much faster timescale than mRNA and protein, i.e., the fast promoter switching limit (see for example (28) for experimental evidence of this limit), and hence by analogy, Hill-type propensities for the protein production rates are commonly used in stochastic simulations of gene regulatory networks (29)(30)(31)(32)(33)(34)(35)(36). All of these studies and many others assume that such effective propensities are justified in the limit of fast promoter switching.…”

Revisiting the reduction of stochastic models of genetic feedback loops with fast promoter switching

“…The reduced chemical master equation is then obtained by writing effective propensities analogous to the non-mass action reaction rates obtained from the deterministic analysis. For example, Hill-type effective protein production rates in the deterministic rate equations result if the gene equilibrates on a much faster timescale than mRNA and protein, i.e the fast promoter switching limit (see for example [21] for experimental evidence of this limit), and hence by analogy, Hill-type propensities for the protein production rates are commonly used in stochastic simulations of gene regulatory networks [22][23][24][25][26][27][28][29]. All of these studies and many others assume that such effective propensities are justified in the limit of fast promoter switching.…”