System size expansion using Feynman rules and diagrams

Abstract

Few analytical methods exist for quantitative studies of large fluctuations in stochastic systems. In this article, we develop a simple diagrammatic approach to the Chemical Master Equation that allows us to calculate multi-time correlation functions which are accurate to a any desired order in van Kampen's system size expansion. Specifically, we present a set of Feynman rules from which this diagrammatic perturbation expansion can be constructed algorithmically. We then apply the methodology to derive in closed form the leading order corrections to the linear noise approximation of the intrinsic noise power spectrum for general biochemical reaction networks. Finally, we illustrate our results by describing noise-induced oscillations in the Brusselator reaction scheme which are not captured by the common linear noise approximation.

25 pages, 5 figures, under review, Journal of Physics A

References in corpus (8)

• Analytical distributions for stochastic gene expression
• How accurate are the non-linear chemical Fokker-Planck and chemical Langevin equations?
• Amplified biochemical oscillations in cellular systems
• Exact results for noise power spectra in linear biochemical reaction networks
• Limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks
• Population oscillations in spatial stochastic Lotka-Volterra models: A field-theoretic perturbational analysis
• Analytical study of non Gaussian fluctuations in a stochastic scheme of autocatalytic reactions
• Functional-integral based perturbation theory for the Malthus-Verhulst process

Cited by in corpus (8)

• Master equations and the theory of stochastic path integrals
• The linear-noise approximation and the chemical master equation exactly agree up to second-order moments for a class of chemical systems
• Approximate probability distributions of the master equation
• Extending the linear-noise approximation to biochemical systems influenced by intrinsic noise and slow lognormally distributed extrinsic noise
• Statistical physics approaches to subnetwork dynamics in biochemical systems
• The chemical Langevin equation: a path integral view of Gillespie's derivation
• Field theories and quantum methods for stochastic reaction-diffusion systems
• A stochastic field theory for the evolution of quantitative traits in finite populations