Fig. 6: Dependency graph for a discrete reaction-diffusion system with linear reaction kinetics.
From: Optimal spatial allocation of enzymes as an investment problem
For linear kinetics, the reaction flux \({J}^{{{{{{{{\mathcal{P}}}}}}}}}\) is proportional to the dot product of the enzyme profile e and the substrate profile ρ. The graphs shows that e affects \({J}^{{{{{{{{\mathcal{P}}}}}}}}}\) both directly and indirectly. In fact, for a given e we have a reaction-diffusion operator M, whose inverse can be applied to the substrate source vector A to determine the substrate profile ρ = M−1A. Thus to determine the total derivative \(\frac{d{J}^{{{{{{{{\mathcal{P}}}}}}}}}}{d{{{{{{{\bf{e}}}}}}}}}\), one needs to back-propagate the derivatives through these dependencies.
