Hybrid stochastic simulation

Hybrid stochastic simulation is sub-class of stochastic simulations, designed to simulate part of Brownian trajectories avoiding to simulate the entire trajectories. This approach is particularly relevant when a Brownian particle evolves in an infinite space. Trajectories are then simulated only in the neighborwod of small targets. Otherwise, explicit analytical expression are used to map the initial point to a distribution located on a imaginary surface around the targets. This algorithm was developed in.[1][2]

This approach allows to simulate gradient cues in an open space, diffusing molecules that have to bind to small receptors in cells and many more cases.

Principle of the algorithm

This algorithm avoids the explicit simulation long trajectories with large excursions and thus it circumvents the need for an arbitrary cutoff distance for our infinite domain. The algorithm consists of mapping the source position ${\displaystyle x_{0}}$ to a half-sphere containing the absorbing windows. Inside the sphere, classical Brownian simulations can be run, until the particle is absorbed or exits through the sphere surface. The detailed algorithm consists of the following steps:

1. The source releases a particle at position ${\displaystyle x(t=0)=x_{0}}$.
2. If ${\displaystyle |x(t)|>R'}$, we map the particle's position to the surface of the sphere S(R), using the distribution of exit point ${\displaystyle P_{map}}$. In three dimensions, there is a finite probability for a Brownian particle to escape to infinity upon which the trajectory is terminated.
3. In the first time step,${\displaystyle t=\Delta t,|x^{t}|>R',}$ we use the mapping ${\displaystyle P_{map}}$ to map the particle's position to the sphere S(R). Thus mapping leads to a sequence of mapped position ${\displaystyle x_{1},..x_{n}}$until the particle is absorbed. Note that for the mapping, there is finite probability that the particle escape to infinity, in that case, we terminate the trajectory.
4.  The Euler-Maruyama scheme can be used to perform a Brownian step: ${\displaystyle x(t)=x(t-\Delta t)+{\sqrt {2D\Delta t}}\mathbf {R} ,}$ where ${\displaystyle \mathbf {R} }$ is a vector of standard normal random variables.
5. When either ${\displaystyle x(t)\cdot \mathbf {e} _{z}\leq 0}$ (in the case of half-space) or${\displaystyle |x(t)|\leq 1}$ (in the case of the sphere), and ${\displaystyle |x(t)-x_{i}|<\epsilon }$ for any i, we consider that the particle is being absorbed by window i and terminate the trajectory.
6.    If the particle crossed any reflective boundary, go back to step 3 to generate a new position. Otherwise return to step 2.

Mapping the source for a ball in 3D

${\displaystyle P_{map}(x|y)={\frac {|y|}{R}}{\frac {1}{4\pi }}{\frac {\beta ^{2}-1}{(1+\beta ^{2}-2\beta \cos \gamma )^{3/2}}}}$, with ${\displaystyle \int _{\partial B(R)}P(x,y)dS_{x}=\beta ^{-1}={\frac {R}{|y|}},}$and ${\displaystyle |x||y|\cos \gamma =x\cdot y.}$ It is the first passage probability for hitting a ball before escaping to infinity. The probability distribution of hitting is obtained by normalizing the integral of the flux

Remarks

The choice of the radius R is arbitrary as long as S(R) encloses all windows with a buffer of at least size ${\displaystyle \epsilon }$. The radius R' should be chosen such that frequent re-crossings are avoided, e.g. ${\displaystyle R'\leq R+10{\sqrt {2D\Delta t}}.}$This algorithm can be used to simulate trajectories of Brownian particles at steady-state close to a region of interest. Note that there is no approximation involved.

Stochastic reaction–diffusion simulations

Others classes of stochastic hybrid simulations concern reaction–diffusion simulations .[3] These algorithm are used to study the conversion of species and allow to couple the Fokker-Planck equation to simulate population and single trajectories using Brownian simulations.[4]

References

1. Dobramysl, U., & Holcman, D. (2018). Mixed analytical-stochastic simulation method for the recovery of a Brownian gradient source from probability fluxes to small windows. Journal of computational physics, 355, 22-36.
2. Dobramysl, U., & Holcman, D. (2019). Reconstructing a point source from diffusion fluxes to narrow windows in three dimensions. arXiv preprint arXiv:2001.01562.
3. M. B. Flegg, S. J. Chapman and R. Erban, The two-regime method for optimizing stochastic reaction–diffusion simulations, J. Royal. Soc. Inter. 9 (2011), 859-868.
4. B. Franz, M. B. Flegg, S. J. Chapman and R. Erban, Multiscale reaction-diffusion algorithms: PDE-assisted Brownian dynamics, SIAM J. Appl. Math. 73 (2013), 1224-1247.