Exact and WKB-approximate distributions in a gene expression model with feedback in burst frequency, burst size, and protein stability

Abstract

The expression of individual genes into functional protein molecules is a noisy dynamical process. Here we model the protein concentration as a jump--drift process which combines discrete stochastic production bursts (jumps) with continuous deterministic decay (drift). We allow the drift rate, the jump rate, and the jump size to depend on the current protein level in an arbitrary fashion to implement feedback in protein stability, burst frequency, and burst size. Two versions of feedback in burst size are considered: in the ``infinitesimally delayed'' version, only those molecules of protein that have been present before a burst started can regulate the size of the burst; in the ``undelayed'' version, newly produced molecules also partake in the regulation of the further growth of a burst. Excluding the infinitesimal delay in burst size, the model is explicitly solvable. With the inclusion of the infinitesimal delay, an exact distribution to the model is no longer available, but we are able to construct a WKB approximation that applies in the asymptotic regime of small but frequent bursts. Comparing the asymptotic behaviour of the two model versions, we report that they yield the same WKB quasi-potential but a different exponential prefactor. We illustrate the difference on the case of a bimodal protein distribution sustained by a sigmoid feedback in burst size: we show that the omission of the infinitesimal delay overestimates the weight of the upper mode of the protein distribution. The analytic results are supported by kinetic Monte-Carlo simulations.