Matrix Structure
We consider two diffusion situations.
1. The substrate is an exchange flux with the first bin only, but is allowed to freely diffuse to adjacent bins, imposing a directionality on the system.
We denote the "source" substrate to be a point 0. For points in the domain, we have substrate, enzyme, complex, and product at each of these points.
The reactiondiffusion matrix is organized as follows:



Then for the twodimensional case, the matrix appears as follows:
with reversibilities as follows, for their respective fluxes:
and for the threedimensional case:
with reversibilities as follows:
Results
Consider the twobin case:
The elementary flux modes that give rise to a steady state among the internal fluxes for situation 1:








We are of course interested only in fluxes in which an inflow of substrate from leads to a buildup of product. Flux balances 1, 2, 3, and 4 correspond to internal cycles, and therefore are of no interest (the first two are "stationary" cycles, the second two involve diffusion). Flux 5 corresponds simply to the onebin situation, and is of no spatial interest. Flux 6 involves substrate flowing to point 2, such that all of the chemical action occurs in that second bin. Fluxes 7 and 8 are of greatest interest, involving the occurrence of the two forward reactions in different bins with a steadystate maintained by diffusion.
Now, we consider the threebin case:
There are too many flux modes to list independently, so we present the flux mode matrix as computed by efmtool:
Recalling the format of given above, we have the following flux balances: Columns 10, 12, and 19 have the stationary internal flux format:

Observe that there are necessarily of these fluxes.
Meanwhile, Columns 1, 2, 7, 8, 14, and 18 all correspond to internal cycles, but through multiple bins via diffusion. Observe that there is no uptake of substrate and no production of product in these situations. There are necessarily of these fluxes: the first reaction flux may occur at any of these locations, whereas the second reaction flux may occur at any of those locations except the one chosen for the first reaction flux.
It is worth examining the individual structures of the fluxes in which substrate is taken up and product is produced:
 Column 3:
 Column 4:
 Column 5:
 Column 6:
 Column 9:
 Column 11:
 Column 13:
 Column 15:
 Column 16:
 Column 17: