Stationary distributions and metastable behaviour for self-regulating proteins with general lifetime distributions

Abstract

Regulatory molecules such as transcription factors are often present at relatively small copy numbers in living cells. The copy number of a particular molecule fluctuates in time due to the random occurrence of production and degradation reactions. Here we consider a stochastic model for a self-regulating transcription factor whose lifespan (or time till degradation) follows a general distribution modelled as per a multidimensional phase-type process. We show that at steady state the protein copy-number distribution is the same as in a one-dimensional model with exponentially distributed lifetimes. This invariance result holds only if molecules are produced one at a time: we provide explicit counterexamples in the bursty production regime. Additionally, we consider the case of a bistable genetic switch constituted by a positively autoregulating transcription factor. The switch alternately resides in states of up-and downregulation and generates bimodal protein distributions. In the context of our invariance result, we investigate how the choice of lifetime distribution affects the rates of metastable transitions between the two modes of the distribution. The phase-type model, being non-linear and multi-dimensional whilst possessing an explicit stationary distribution, provides a valuable test example for exploring dynamics in complex biological systems.

Postponing production exponentially enhances the molecular memory of a stochastic switch

“…The early dynamics is characterised by a steady shift of the distributions to the right until they are located at x − . This initial dynamics is closely approximated by the LNA (40) (Figure 3, first three timepoints). At much greater times, the weight of the mode around x − begins to decrease, and a new mode around the upper stable fixed point x + arises.…”

“…for the joint distribution of the concentration x of the inactive and the copy number s of the active protein. The LNA (40) implies that: s follows the truncated Poisson distribution (23); x follows the Gaussian distribution with mean equal to the deterministic solution ξ(t) to (30) and a small variance equal to εσ 2 (t).…”

“…The key insight of [29], on which we expand in this work, is that a stochastic autoregulatory circuit extended by a large delay exhibits a near-deterministic behaviour of the inactive protein (the production of which has been initiated but not yet completed). A classical means of eliminating noise in a stochastic model is to increase its system size/volume [30]; the near-deterministic behaviour in large-volume systems can be described using the linear noise approximation (LNA) [31][32][33][34][35] and the Wentzel-Kramers-Brillouin (WKB) approximation [36][37][38][39][40]. These two approaches are complementary: the LNA applies on finite temporal domains [41]; the WKB approximation covers slow metastable dynamics such as transitions between deterministically stable steady states or to a fixation/extinction point [42].…”

A modified fluctuation test for elucidating drug resistance in microbial and cancer cells

“…on the characteristic system (22). Solving the logistic equation for η = η(t) in (22) by separating the variables η and t yields…”

“…Solving the inhomogeneous linear equation for ξ = ξ(t) in (22) by variation of constants gives ξ(t) = ξ 0 e a(t−t0) − a…”