Emergent Lévy behavior in single-cell stochastic gene expression

Abstract

Single-cell gene expression is inherently stochastic; its emergent behavior can be defined in terms of the chemical master equation describing the evolution of the mRNA and protein copy numbers as the latter tends to infinity. We establish two types of "macroscopic limits": the Kurtz limit is consistent with the classical chemical kinetics, while the Lévy limit provides a theoretical foundation for an empirical equation proposed in [Phys. Rev. Lett. 97:168302, 2006]. Furthermore, we clarify the biochemical implications and ranges of applicability for various macroscopic limits and calculate a comprehensive analytic expression for the protein concentration distribution in autoregulatory gene networks. The relationship between our work and modern population genetics is discussed.

10 pages, 2 figures

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• Small protein number effects in stochastic models of autoregulated bursty gene expression
• Stochastic modeling of auto-regulatory genetic feedback loops: a review and comparative study
• Single-cell stochastic gene expression kinetics with coupled positive-plus-negative feedback
• Relaxation rates of gene expression kinetics reveal the feedback signs of autoregulatory gene networks
• Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic Markov processes
• Model reduction methods for classical stochastic systems with fast-switching environments: reduced master equations, stochastic differential equations, and applications
• Martingale structure for general thermodynamic functionals of diffusion processes under second-order averaging
• Stochastic fluctuations can reveal the feedback signs of gene regulatory networks at the single-molecule level
• Kinetic foundation of the zero-inflated negative binomial model for single-cell RNA sequencing data
• Kinetic theories of state- and generation-dependent cell populations
• Limit theorems for generalized density-dependent Markov chains and bursty stochastic gene regulatory networks
• Singularities of invariant densities for random switching between two linear ODEs in 2D
• Detailed balance, local detailed balance, and global potential for stochastic chemical reaction networks