What is it about?
Chemical reaction kinetics are usually modelled by ordinary differential equations. In biochemistry, especially in modelling gene regulation, discrete-state Markov processes are also frequently used. The question is whether there is a regime with somewhat larger but not very large chemical species populations where an Ito stochastic differential equation is a suitable model. What is the correct form of this SDE?
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Why is it important?
By a clever choice of the diffusion term in the stochastic differential equation, much computational time can be saved compared to the standard implementation, increasing with the number of reversible reactions.
Perspectives
The chemical Langevin equation turned out to be a flawed model for chemical kinetics. It inevitably leads to negative chemical population sizes. These negative numbers then appear under square roots and everything breaks down. I have explained this issue in detail on pp65-71 of my DPhil thesis. By a generalised martingale problem approach I discussed in what sense one can get existence and uniqueness of strong solutions and equivalence of the alternative forms (pp71-86). Link to the thesis: http://ora.ouls.ox.ac.uk/objects/uuid:d368c04c-b611-41b2-8866-cde16b283b0d
Dr Bence Mélykúti
Albert-Ludwigs-Universitat Freiburg
Read the Original
This page is a summary of: Fast stochastic simulation of biochemical reaction systems by alternative formulations of the chemical Langevin equation, The Journal of Chemical Physics, April 2010, American Institute of Physics,
DOI: 10.1063/1.3380661.
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Resources
The complex chemical Langevin equation
David Schnoerr, Guido Sanguinetti and Ramon Grima propose to continue the simulation with complex numbers when negative numbers appear under square roots. They argue in this paper that this works fine.
Central limit theorems and diffusion approximations for multiscale Markov chain models
A central limit theorem capturing the fluctuations of the original model around the deterministic limit. The central limit theorem provides a method for deriving an appropriate diffusion (Langevin) approximation.
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