D. Spatial inhomogenity

By dividing the system into subvolumes, we allow the system to be inhomogeneous, which makes the stochastic simulation approach applicable to many more problems, in particular to biological systems in which some molecules diffuse slowly and reactions are often compartmentalized. As stated at the beginning of this section, the state vector X(t) stores N ⋅ Z numbers, for N species and Z subvolumes. Reactions within a single subvolume are described in the same way as previously, but we also allow molecules to diffuse between subvolumes. The propensity of diffusion between subvolumes p and q is given by

where δz is the distance between the subvolumes, and D_i is the diffusion coefficient for species i.²³ Reactions in separate subvolumes become effectively separate, and in addition we have to consider diffusion. The number of reaction channels becomes approximately

where d is the dimensionality of the system.

In exact methods and existing τ-leaping methods computational complexity is proportional to the number of reaction channels. For any exact algorithm it is also linear in the number of firings Y. If we assume that the number of firings scales proportionally to the size of the system, the scaling is approximately 𝒪(M^′2) for the direct method, 𝒪(M^′logM^′) for the next reaction method (a more efficient SSA implementation, Ref. ¹⁰) and 𝒪(M^′) for all existing τ-leaping methods.

τ-leaping scales only linearly with the number of subvolumes. By design, the number of steps in τ-leaping is largely independent of the number of molecules, at least if the number of molecules is large enough. This means that τ-leaping should scale well for large spatial problems. The caveat is that the number of molecules must be large, across all species, and across all subvolumes. If some species have low numbers, the variance of the change of propensity becomes significant even for small leaps, requiring small time steps to observe the leap condition.