MURPHYS-HSFS-2014 - 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems, April 7-11, 2014 - Abstract
Popovic, Nikola
Stochastic models for gene expression frequently exhibit dynamics on a number of different time-scales. One potential scale separation is due to significant differences in the lifetimes of mRNA and the protein it synthesises; the ratio of the two degradation rates gives a natural small parameter in the resulting Chemical Master Equation, which allows for the application of perturbation techniques. Here, we develop a dynamical systems framework for the analysis of a family of `fast-slow' models for gene expression that is based on geometric singular perturbation theory. We illustrate our approach by giving a complete characterisation of the resulting dynamics in a standard two-stage model which assumes transcription, translation, and degradation to be birth-and-death processes of first order. In particular, we develop a systematic expansion procedure for the probability-generating function that can in principle be taken to any order in the perturbation parameter, allowing for an approximation of the resulting propagator probabilities to that same order. Finally, we verify our asymptotics by numerical simulation, and we explore its practical applicability, as well as the effects of a variation in the system parameters and the scale separation: while the first-order slow-time correction can improve the steady-state probability distribution in parameter regimes that induce translational bursting, we find that inclusion of the fast asymptotics yields a significantly improved time-dependent approximation, via the resulting `patched' fast-slow propagator probabilities, in regimes where mRNA is frequently transcribed.