ABSTRACT:
Stochastic models for gene expression frequently exhibit dynamics on different time-scales. One potential scale separation is due to significant differences in the lifetimes of mRNA and the protein it synthesises, 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 a standard two-stage model which assumes transcription, translation, and degradation to be first-order reactions. In particular, we develop a systematic expansion procedure for the resulting propagator probabilities that can in principle be taken to any order in the perturbation parameter. 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.
ADDITIONAL INFORMATION:
Nikola is an applied mathematician whose research interests lie in dynamical systems and differential equations. His current research focuses on the geometric analysis of modelling applications from biology, medicine and neuroscience and on the development of techniques for reducing the complexity of these models. His approach relies on a combination of analysis and numerical simulation; specific techniques include (geometric) singular perturbation theory, geometric desingularisation (“blow-up”), invariant manifold theory and asymptotic analysis, as well as low-rank approximation and hierarchical matrix techniques.