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We present the analytical singular value decomposition of the stoichiometrymatrix for a spatially discrete reaction-diffusion system on a one dimensionaldomain. The domain has two subregions which share a single common boundary.Each of the subregions is further partitioned into a finite number ofcompartments. Chemical reactions can occur within a compartment, whereasdiffusion is represented as movement between adjacent compartments. Inspired bybiology, we study both 1) the case where the reactions on each side of theboundary are different and only certain species diffuse across the boundary aswell as 2) the case with spatially homogenous reactions and diffusion. We writethe stoichiometry matrix for these two classes of systems using a Kroneckerproduct formulation. For the first scenario, we apply linear perturbationtheory to derive an approximate singular value decomposition in the limit asdiffusion becomes much faster than reactions. For the second scenario, wederive an exact analytical singular value decomposition for all relativediffusion and reaction time scales. By writing the stoichiometry matrix usingKronecker products, we show that the singular vectors and values can also bewritten concisely using Kronecker products.  Ultimately, we find that the singular value decomposition of thereaction-diffusion stoichiometry matrix depends on the singular valuedecompositions of smaller matrices. These smaller matrices represent modifiedversions of the reaction-only stoichiometry matrices and the analytically knowndiffusion-only stoichiometry matrix. Our results provide a mathematicalframework that can be used to study complex biochemical systems with metaboliccompartments. MATLAB code for calculating the SVD equations is available at\url{www.github.com/MathBioCU/ReacDiffStoicSVD}.

Analytical singular value decomposition for a class of stoichiometry matrices