Realistic boundary conditions for stochastic simulations of reaction-diffusion processes

Many cellular and subcellular biological processes can be described in terms of diffusing and chemically reacting species (e.g. enzymes). Such reaction-diffusion processes can be mathematically modelled using either deterministic partial-differential equations or stochastic simulation algorithms. The latter provide a more detailed and precise picture, and several stochastic simulation algorithms have been proposed in recent years. Such models typically give the same description of the reaction-diffusion processes far from the boundary of the simulated domain, but the behaviour close to a reactive boundary (e.g. a membrane with receptors) is unfortunately model-dependent. In this paper, we study four different approaches to stochastic modelling of reaction-diffusion problems and show the correct choice of the boundary condition for each model. The reactive boundary is treated as partially reflective, which means that some molecules hitting the boundary are adsorbed (e.g. bound to the receptor) and some molecules are reflected. The probability that the molecule is adsorbed rather than reflected depends on the reactivity of the boundary (e.g. on the rate constant of the adsorbing chemical reaction and on the number of available receptors), and on the stochastic model used. This dependence is derived for each model.Comment: 24 pages, submitted to Physical Biolog

Small Corrections to the Tunneling Phase Time Formulation

After reexamining the above barrier diffusion problem where we notice that the wave packet collision implies the existence of {\em multiple} reflected and transmitted wave packets, we analyze the way of obtaining phase times for tunneling/reflecting particles in a particular colliding configuration where the idea of multiple peak decomposition is recovered. To partially overcome the analytical incongruities which frequently rise up when the stationary phase method is adopted for computing the (tunneling) phase time expressions, we present a theoretical exercise involving a symmetrical collision between two identical wave packets and a unidimensional squared potential barrier where the scattered wave packets can be recomposed by summing the amplitudes of simultaneously reflected and transmitted wave components so that the conditions for applying the stationary phase principle are totally recovered. Lessons concerning the use of the stationary phase method are drawn.Comment: 14 pages, 3 figure

Recovering the stationary phase condition for accurately obtaining scattering and tunneling times

The stationary phase method is often employed for computing tunneling {\em phase} times of analytically-continuous {\em gaussian} or infinite-bandwidth step pulses which collide with a potential barrier. The indiscriminate utilization of this method without considering the barrier boundary effects leads to some misconceptions in the interpretation of the phase times. After reexamining the above barrier diffusion problem where we notice the wave packet collision necessarily leads to the possibility of multiple reflected and transmitted wave packets, we study the phase times for tunneling/reflecting particles in a framework where an idea of multiple wave packet decomposition is recovered. To partially overcome the analytical incongruities which rise up when tunneling phase time expressions are obtained, we present a theoretical exercise involving a symmetrical collision between two identical wave packets and a one dimensional squared potential barrier where the scattered wave packets can be recomposed by summing the amplitudes of simultaneously reflected and transmitted waves.Comment: 32 pages, 5 figures, 1 tabl

A probabilistic model of diffusion through a semi-permeable barrier

Diffusion through semipermeable structures arises in a wide range of processes in the physical and life sciences. Examples at the microscopic level range from artificial membranes for reverse osmosis to lipid bilayers regulating molecular transport in biological cells to chemical and electrical gap junctions. There are also macroscopic analogs such as animal migration in heterogeneous landscapes. It has recently been shown that one-dimensional diffusion through a barrier with constant permeability $\kappa_0$ is equivalent to snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflecting BMs that are restricted to either the left or right of the barrier. Each round is killed when its Brownian local time exceeds an exponential random variable parameterized by $\kappa_0$. A new round is then immediately started in either direction with equal probability. In this paper we use a combination of renewal theory, Laplace transforms and Green's function methods to show how an extended version of snapping out BM provides a general probabilistic framework for modeling diffusion through a semipermeable barrier. This includes modifications of the diffusion process away from the barrier (eg. stochastic resetting) and non-Markovian models of membrane absorption that kill each round of partially reflected BM. The latter leads to time-dependent permeabilities.Comment: 29 pages, 7 figure

Fractional Reaction-Diffusion Equation

A fractional reaction-diffusion equation is derived from a continuous time random walk model when the transport is dispersive. The exit from the encounter distance, which is described by the algebraic waiting time distribution of jump motion, interferes with the reaction at the encounter distance. Therefore, the reaction term has a memory effect. The derived equation is applied to the geminate recombination problem. The recombination is shown to depend on the intrinsic reaction rate, in contrast with the results of Sung et al. [J. Chem. Phys. {\bf 116}, 2338 (2002)], which were obtained from the fractional reaction-diffusion equation where the diffusion term has a memory effect but the reaction term does not. The reactivity dependence of the recombination probability is confirmed by numerical simulations.Comment: to appear in Journal of Chemical Physic

Gene flow across geographical barriers - scaling limits of random walks with obstacles

In this paper, we study the scaling limit of a class of random walks which behave like simple random walks outside of a bounded region around the origin and which are subject to a partial reflection near the origin. If the probability of crossing the barrier scales as $1/\sqrt{n}$ as we rescale space by $\sqrt{n}$ and time by $n$, we obtain a non trivial scaling limit which behaves like reflected Brownian motion until its local time at the origin reaches an independent exponential variable. It then follows reflected Brownian motion on the other side of the origin until its local time at the origin reaches another exponential level, and so on. We give a martingale problem characterisation of this process as well as another construction and an explicit formula for its transition density. This result has applications in the field of population genetics where such a random walk is used to trace the position of one's ancestor in the past in the presence of a barrier to gene flow.Comment: Stochastic Processes and Their Applications, in pres

The snapping out Brownian motion

We give a probabilistic representation of a one-dimensional diffusion equation where the solution is discontinuous at $0$ with a jump proportional to its flux. This kind of interface condition is usually seen as a semi-permeable barrier. For this, we use a process called here the snapping out Brownian motion, whose properties are studied. As this construction is motivated by applications, for example, in brain imaging or in chemistry, a simulation scheme is also provided.Comment: Published at http://dx.doi.org/10.1214/15-AAP1131 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org