Limit theorems for generalized density-dependent Markov chains and bursty stochastic gene regulatory networks

Abstract

Stochastic gene regulatory networks with bursting dynamics can be modeled mesocopically as a generalized density-dependent Markov chain (GDDMC) or macroscopically as a piecewisedeterministic Markov process (PDMP). Here we prove a limit theorem showing that each family of GDDMCs will converge to a PDMP as the system size tends to infinity. Moreover, under a simple dissipative condition, we prove the existence and uniqueness of the stationary distribution and the exponential ergodicity for the PDMP limit via the coupling method. Further extensions and applications to single-cell stochastic gene expression kinetics and bursty stochastic gene regulatory networks are also discussed and the convergence of the stationary distribution of the GDDMC model to that of the PDMP model is also proved.

Heavy-tailed distributions in a stochastic gene autoregulation model

“…The ξ → ∞ asymptotics of (32) agree with the x → x * behaviour of the tail-zone WKB solution (26) with arbitrary choices of the integration constant A and the constant κ in the offset of the boundary layer (31) (Appendix E). In order to determine the two constants, the ξ → −∞ asymptotics of (32) need to be matched to the x → x * behaviour of the Cramer-zone WKB solution (25).…”

“…The discontinuity in the potential derivative (24) and the mismatch of prefactor magnitudes in ( 25) and ( 33) suggest the presence of a boundary layer near…”

“…More fine-grained models of gene expression are based on a purely discrete [11][12][13][14][15] or a hybrid discrete-continuous state space [16][17][18][19][20]. The drift-jump model can be derived from the fine-grained processes using formal limit procedures [21][22][23][24][25][26].…”

A data-driven method to learn a jump diffusion process from aggregate biological gene expression data

“…Thus, our model is a generalization of diffusion processes driven by Brownian motion. Nowadays, SDEs with Lévy noise are widely used in modeling gene expression (Xu et al, 2016; Jia et al, 2019; Cai and othres, 2019; Chen and Jia, 2020) and the interpretation of burst behavior from a mathematical perspective can be referred in Bokes et al (2012) and Jia (2017).…”

Detailed balance, local detailed balance, and global potential for stochastic chemical reaction networks

“…Thus far, stochastic reaction networks have served as a fundamental model for the single-cell stochastic gene expression dynamics of gene regulatory networks [20, 22-25, 30, 39, 42]. Recently, the limit theorem of Kurtz has been generalized to stochastic gene regulatory networks with bursting dynamics [9,26].…”