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A discrete stochastic formulation for reversible bimolecular reactions via diffusion encounter. (English) Zbl 1352.92070

Summary: The classical models for irreversible diffusion-influenced reactions can be derived by introducing absorbing boundary conditions to over-damped continuous Brownian motion (BM) theory. As there is a clear corresponding stochastic process, the mathematical description takes both Kolmogorov forward equation for the evolution of the probability distribution function and the stochastic sample trajectories. This dual description is a fundamental characteristic of stochastic processes and allows simple particle-based simulations to accurately match the expected statistical behavior. However, in the traditional theory using the back-reaction boundary condition to model reversible reactions with geminate recombinations, several subtleties arise: It is unclear what the underlying stochastic process is, which causes complications in producing accurate simulations; and it is non-trivial how to perform an appropriate discretization for numerical computations. In this work, we derive a discrete stochastic model that recovers the classical models and their boundary conditions in the continuous limit. In the case of reversible reactions, we recover the back-reaction boundary condition, unifying the back-reaction approach with those of current simulation packages. Furthermore, all the complications encountered in the continuous models become trivial in the discrete model. Our formulation brings to attention the question: With computations in mind, can we develop a discrete reaction kinetics model that is more fundamental than its continuous counterpart?

MSC:

92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
92C40 Biochemistry, molecular biology
60J50 Boundary theory for Markov processes
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
65C35 Stochastic particle methods
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