Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic Markov processes

Abstract

<jats:p>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.</jats:p>

Accelerated Bayesian inference of gene expression models from snapshots of single-cell transcripts

“…Piecewise-deterministic Markov processes (PDMP) have become a useful, coarse-grained description of stochastic gene dynamics, where the underlying discrete variable s(t) captures the stochastic dynamics of gene states and the continuous variable λ(t) captures the first moment of downstream gene products [36][37][38][39][40][41][42]. The key FIG.…”

Scaling methods for accelerating kinetic Monte Carlo simulations of chemical reaction networks

“…In these cases, the dynamics in the sub-volume will differ qualitatively from the dynamics in the whole system. For example, scaling to a population size ∼ 1 will introduce bursty behavior, which has unique statistical properties 17,18,21 . Thus, scaling should be limited, but it is not clear how to impose an appropriate lower bound on λ.…”

“…When λ is too small, one or more populations become discrete. Discrete populations have distinct dynamics, which are not captured by a Fokker-Planck equation 17,18,21 . Thus, the analysis of the previous section eventually becomes invalid and errors are introduced as λ decreases and becomes too small.…”

“…Unfortunately, these opposing requirements limit applicability of the method. Finally, hybrid methods have been developed and applied to solve specific problems [12][13][14][15][16][17][18][19][20][21] . A characteristic feature of these methods is the division of state variables into two distinct sets, the discrete variables and the continuous variables.…”

Mixture distributions in a stochastic gene expression model with delayed feedback

“…Generally, however, explicit solutions are unavailable or intractable and one resorts to stochastic simulation or seeks a numerical solution to a finite truncation of the master equation (Munsky and Khammash, 2006;Borri et al, 2016;Gupta et al, 2017). An alternative approach, which often provides useful qualitative insights into the model behaviour, is based on reduction techniques such as quasi-steadystate (Srivastava et al, 2011;Kim et al, 2014) and adiabatic reductions (Bruna et al, 2014;Popovic et al, 2016), piecewise-deterministic framework (Lin and Doering, 2016;Lin and Buchler, 2018), linear-noise approximation (Schnoerr et al, 2017;Modi et al, 2018), or moment closure (Singh and Hespanha, 2007;Andreychenko et al, 2017;Gast et al, 2019).…”