Exact results in the large system size limit for the dynamics of the Chemical Master Equation, a one dimensional chain of equations

Abstract

We apply the Hamilton-Jacobi equation (HJE) formalism to solve the dynamics of the Chemical Master Equation (CME). We found exact analytical expressions (in large system-size limit) for the probability distribution, including explicit expression for the dynamics of variance of distribution. We also give the solution for some simple cases of the model with time-dependent rates. We derived the results of Van Kampen method from HJE approach using a special ansatz. Using the Van Kampen method, we give a system of ODE to define the variance in 2-d case. We performed numerics for the CME with stationary noise. We give analytical criteria for the disappearance of bi-stability in case of stationary noise in 1-d CME.

7 pages, 3 figures

References in corpus (4)

• Extinction of metastable stochastic populations
• Thermodynamic Limit of a Nonequilibrium Steady-State: Maxwell-Type Construction for a Bistable Biochemical System
• Exactly solvable dynamics of the Eigen and the Crow-Kimura models
• Evolution Equation of Phenotype Distribution: General Formulation and Application to Error Catastrophe

Cited by in corpus (2)

• Analytical Mechanics in Stochastic Dynamics: Most Probable Path, Large-Deviation Rate Function and Hamilton-Jacobi Equation
• Dynamics of the Chemical Master Equation, a strip of chains of equations in d-dimensional space