Abstract
Single-cell experiments show that gene expression is stochastic and bursty, a feature that can emerge from slow switching between promoter states with different activities. In addition to slow chromatin and/or DNA looping dynamics, one source of long-lived promoter states is the slow binding and unbinding kinetics of transcription factors to promoters, i.e. the non-adiabatic binding regime. Here, we introduce a simple analytical framework, known as a piecewise deterministic Markov process (PDMP), that accurately describes the stochastic dynamics of gene expression in the non-adiabatic regime. We illustrate the utility of the PDMP on a non-trivial dynamical system by analysing the properties of a titration-based oscillator in the non-adiabatic limit. We first show how to transform the underlying chemical master equation into a PDMP where the slow transitions between promoter states are stochastic, but whose rates depend upon the faster deterministic dynamics of the transcription factors regulated by these promoters. We show that the PDMP accurately describes the observed periods of stochastic cycles in activator and repressor-based titration oscillators. We then generalize our PDMP analysis to more complicated versions of titration-based oscillators to explain how multiple binding sites lengthen the period and improve coherence. Last, we show how noise-induced oscillation previously observed in a titration-based oscillator arises from non-adiabatic and discrete binding events at the promoter site.
Highlights
Gene expression is fundamentally a stochastic biochemical process that arises from thermal fluctuations
We address the following questions: what are the dynamical consequences of non-adiabatic binding? What kind of modelling framework accurately describes the non-stationary dynamics of gene regulatory networks in the non-adiabatic regime? To answer these questions, we use a model of titration-based clocks to illustrate the effects of nonadiabatic binding on dynamics and to show how an analytical framework, known as a piecewise deterministic Markov process (PDMP), accurately describes the stochastic dynamics of the full model in the non-adiabatic regime
While the PDMP can be numerically simulated for any given state, the evolution of the transcription factors (TFs) concentrations is described by a set of nonlinear ordinary differential equations (ODEs) that do not allow for analytic solutions
Summary
Gene expression is fundamentally a stochastic biochemical process that arises from thermal fluctuations. The adiabatic approximation is commonly used to reduce the number of dynamical variables (e.g. promoter states) in gene regulatory networks It is a bold assumption because experiments [4,5,6,7] show that promoter dynamics (e.g. the binding and unbinding events of TFs) can take place at a comparable, or even slower, timescale than the downstream processes of gene expression. This observation has motivated theoretical studies into the effects of slow or non-adiabatic binding on gene regulatory networks.
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