# Stochastic simulation Gillespie algorithm for areas instead of volumes?

I am trying to find resources on the Gillespie stochastic simulation algorithm for my system which happens on a surface. The original algorithm was developed for a reactor of volume $V$, but my system is a flat surface of area $A$. My questions are as follows:

1. Are there any publications that do a stochastic simulation on surfaces?
2. Is extending the SSA from a volume to a surface trivial? For example, consider the following second-order reaction: $$R_1+R_2 \rightarrow P_1$$ and suppose that the rate constant is $k$. The stochastic and deterministic constants ($k^\text{stoc}$ and $k^\text{det}$) for a reactor of volume $V$ are related as follows $$k^\text{det}=\frac{N_aV}{2}k^\text{stoch}$$ where $N_a$ is Avogadro's number. Can I simply replace $V$ with the surface area $A$ to get $$k^\text{det}_\text{area}=\frac{N_aA}{2}k^\text{stoch}_\text{area}$$

Please note that I am not looking to discretize space for diffusion. Just like the SSA does not discretize volumes, I should like to summarize the surface by one number (i.e, the surface area).

• Are you performing simulations of a system with many cells or with a single cell? SSA does simulations of a master equation and the normalization with the volume is commonly used. You need to define your constants in such a way that the limit of large N gives the proper rate equations. – Pierre de Buyl Jul 28 '18 at 11:56
• I am performing simulations with many (bacterial) cells. So each 'unit' is a bacterium and not a chemical entity. – Distopia Nov 8 '18 at 19:43