title: Modelling biochemical reaction systems by stochastic differential equations with reflection.
abstract: In this paper, we gave a new framework for modelling and simulating biochemical reaction systems by stochastic differential equations with reflection not in a heuristic way but in a mathematical way. The model is computationally efficient compared with the discrete-state Markov chain approach, and it ensures that both analytic and numerical solutions remain in a biologically plausible region. Specifically, our model mathematically ensures that species numbers lie in the domain D, which is a physical constraint for biochemical reactions, in contrast to the previous models. The domain D is actually obtained according to the structure of the corresponding chemical Langevin equations, i.e., the boundary is inherent in the biochemical reaction system. A variant of projection method was employed to solve the reflected stochastic differential equation model, and it includes three simple steps, i.e., Euler-Maruyama method was applied to the equations first, and then check whether or not the point lies within the domain D, and if not perform an orthogonal projection. It is found that the projection onto the closure D¯ is the solution to a convex quadratic programming problem. Thus, existing methods for the convex quadratic programming problem can be employed for the orthogonal projection map. Numerical tests on several important problems in biological systems confirmed the efficiency and accuracy of this approach.
authors.affiliation: School of Mathematics and Statistics, Central South University, Changsha 410083, China; Key Laboratory of Systems Biology, Innovation Center for Cell Signaling Network, Institute of Biochemistry and Cell Biology, Shanghai Institutes for Biological Science, Chinese Academy of Sciences, China. Electronic address: yuanlingniu@csu.edu.cn.
authors.forename: Yuanling
authors.initials: Y
authors.lastname: Niu
authors.affiliation: Department of Computer Science, Oxford University, Wolfson Building, Parks Road, Oxford OX1 3QD, UK; School of Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia.
authors.forename: Kevin
authors.initials: K
authors.lastname: Burrage
authors.affiliation: Key Laboratory of Systems Biology, Innovation Center for Cell Signaling Network, Institute of Biochemistry and Cell Biology, Shanghai Institutes for Biological Science, Chinese Academy of Sciences, China. Electronic address: lnchen@sibs.ac.cn.
authors.forename: Luonan
authors.initials: L
authors.lastname: Chen
mesh_terms: D008954:Models, Biological; D013269:Stochastic Processes
keywords: Biochemical reaction system; Reflected stochastic differential equations; Reflection and correction approach; Stochastic modelling; Stochastic simulation