How accurate are the non-linear chemical Fokker-Planck and chemical Langevin equations?

Abstract

The chemical Fokker-Planck equation and the corresponding chemical Langevin equation are commonly used approximations of the chemical master equation. These equations are derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation and hence their accuracy remains to be clarified. We use the system-size expansion to show that chemical Fokker-Planck estimates of the mean concentrations and of the variance of the concentration fluctuations about the mean are accurate to order ${\Omega }^{-3/2}$ for reaction systems which do not obey detailed balance and at least accurate to order ${\Omega }^{-2}$ for systems obeying detailed balance, where $\Omega$ is the characteristic size of the system. Hence the chemical Fokker-Planck equation turns out to be more accurate than the linear-noise approximation of the chemical master equation (the linear Fokker-Planck equation) which leads to mean concentration estimates accurate to order ${\Omega }^{-1/2}$ and variance estimates accurate to order ${\Omega }^{-3/2}$. This higher accuracy is particularly conspicuous for chemical systems realized in small volumes such as biochemical reactions inside cells. A formula is also obtained for the approximate size of the relative errors in the concentration and variance predictions of the chemical Fokker-Planck equation, where the relative error is defined as the difference between the predictions of the chemical Fokker-Planck equation and the master equation divided by the prediction of the master equation. For dimerization and enzyme-catalyzed reactions, the errors are typically less than few percent even when the steady-state is characterized by merely few tens of molecules.

39 pages, 3 figures, accepted for publication in J. Chem. Phys

References in corpus (2)

• Diamondoid Molecules
• Stochastic theory of large-scale enzyme-reaction networks: Finite copy number corrections to rate equation models

Cited by in corpus (32)

• Markovian Dynamics on Complex Reaction Networks
• Mitochondrial network state scales mtDNA genetic dynamics
• Stochastic modeling of auto-regulatory genetic feedback loops: a review and comparative study
• Noise-induced multistability in chemical systems: Discrete vs Continuum modeling
• The complex chemical Langevin equation
• The linear-noise approximation and the chemical master equation exactly agree up to second-order moments for a class of chemical systems
• Hybrid framework for the simulation of stochastic chemical kinetics
• Accurate Chemical Master Equation Solution Using Multi-Finite Buffers
• State space truncation with quantified errors for accurate solutions to discrete Chemical Master Equation
• Approximate probability distributions of the master equation
• Intrinsic noise of microRNA-regulated genes and the ceRNA hypothesis
• Exact solution of a stochastic protein dynamics model with delayed degradation
• Extending the linear-noise approximation to biochemical systems influenced by intrinsic noise and slow lognormally distributed extrinsic noise
• System size expansion using Feynman rules and diagrams
• Dichotomous noise models of gene switches
• Gaussian approximations for stochastic systems with delay: chemical Langevin equation and application to a Brusselator system
• Analysis of stochastic bifurcations with phase portraits
• Role of relaxation time scale in noisy signal transduction
• Dynamical mean-field theory: from ecosystems to reaction networks
• Statistical physics approaches to subnetwork dynamics in biochemical systems
• Computation of biochemical pathway fluctuations beyond the linear noise approximation using iNA
• Stochastic pH oscillations in a model of the urea-urease reaction confined to lipid vesicles
• A new approach to simulating stochastic delayed systems
• Data-driven model reduction of agent-based systems using the Koopman generator
• The chemical Langevin equation: a path integral view of Gillespie's derivation
• Fluctuations and the role of collision duration in reaction-diffusion systems
• Discrete Flux and Velocity Fields of Probability and Their Global Maps in Reaction Systems
• Motif Analysis for Small-Number Effects in Chemical Reaction Dynamics
• Intrinsic ratchets: A Hamiltonian approach
• Predicting properties of the stationary probability currents for two-species reaction systems without solving the Fokker-Planck equation
• Thermodynamics of switching in multistable non-equilibrium systems
• Relation between the convective field and the stationary probability distribution of chemical reaction networks