Analytic solutions for stochastic hybrid models of gene regulatory networks

Abstract

Discrete-state stochastic models are a popular approach to describe the inherent stochasticity of gene expression in single cells. The analysis of such models is hindered by the fact that the underlying discrete state space is extremely large. Therefore hybrid models, in which protein counts are replaced by average protein concentrations, have become a popular alternative. The evolution of the corresponding probability density functions is given by a coupled system of hyperbolic PDEs. This system has Markovian nature but its hyperbolic structure makes it difficult to apply standard functional analytical methods. We are able to prove convergence towards the stationary solution and determine such equilibrium explicitly by combining abstract methods from the theory of positive operators and elementary ideas from potential analysis.

23 pages, 7 figures

References in corpus (8)

• Approximation and inference methods for stochastic biochemical kinetics - a tutorial review
• Genetic Toggle Switch Without Cooperative Binding
• Exact distributions for stochastic gene expression models with bursting and feedback
• Quantitative ergodicity for some switched dynamical systems
• A stochastic and dynamical view of pluripotency in mouse embryonic stem cells
• Stochastic gene expression with a multistate promoter: breaking down exact distributions
• Stochastic Hybrid Models of Gene Regulatory Networks - A PDE Approach
• Exact solution of a model DNA-inversion genetic switch with orientational control

Cited by in corpus (2)

• Linear Hyperbolic Systems on Networks
• Uniform convergence of solutions to stochastic hybrid models of gene regulatory networks