Bistable switching asymptotics for the self regulating gene

Abstract

A simple stochastic model of a self regulating gene that displays bistable switching is analyzed. While on, a gene transcribes mRNA at a constant rate. Transcription factors can bind to the DNA and affect the gene's transcription rate. Before an mRNA is degraded, it synthesizes protein, which in turn regulates gene activity by influencing the activity of transcription factors. Protein is slowly removed from the system through degradation. Depending on how the protein regulates gene activity, the protein concentration can exhibit noise induced bistable switching. An asymptotic approximation of the mean switching rate is derived that includes the pre exponential factor, which improves upon a previously reported logarithmically accurate approximation. With the improved accuracy, a uniformly accurate approximation of the stationary probability density, describing the gene, mRNA copy number, and protein concentration is also obtained.

Dichotomous noise models of gene switches

“…[36][37][38][39][40][41][42][43] Most of these schemes still, however, rely on there being a sufficient degree of time scale separation between gene and protein degrees of freedom, which means that switching dynamics must usually be assumed to be very fast. While such an assumption could be a sound one for many gene circuits, the intermediate regime where there is no such time scale separation is also of interest especially for understanding the behavior of mammalian gene circuits.…”

Reconstructing an epigenetic landscape using a genetic ‘pulling’ approach

“…S2). Note, that in the absence of external perturbation, the pre-factor of τ can be accurately found as well [65,72].…”

Feynman-Kac formula for stochastic hybrid systems

“…Hybrid models arise when a partial thermodynamic limit of a biochemical master equation is taken. This yields a piecewise deterministic or stochastic differential equation for the concentrations of proteins and mRNA, while the remaining discrete variables represent the activation states of one or more genes [17][18][19][20][21]. One quantity of interest is the amount of time that a protein concentration remains above some threshold, which can be formulated in terms of the occupation time of a stochastic hybrid system on R + .…”

“…On the other-hand, channel fluctuations in the finite case can lead to noiseinduced neuronal spiking. Another important example is a gene regulatory network, where the continuous variable is the concentration of a protein product and the discrete variable represents the activation state of the gene [17][18][19][20][21]. Yet another example arises in a stochastic formulation of synaptically-coupled neural networks that has a mathematical structure analogous to stochastic gene networks [22].…”

“…In addition, a variety of complementary techniques in applied mathematics and mathematical physics have been used to solve first passage time problems in biological applications of stochastic hybrid systems. These include WKB approximations and matched asymptotics [10,14,15,[18][19][20][21], and path-integrals [26,27].…”