ANALYTICAL SINGULAR VALUE DECOMPOSITION FOR A CLASS OF STOICHIOMETRY MATRICES

We present the analytical singular value decomposition of the stoichiometry matrix for a spatially discrete reaction-diffusion system. The motivation for this work is to develop a matrix decomposition that can reveal hidden spatial flux patterns of chemical reactions. We consider a 1D domain with two subregions sharing a single common boundary. Each of the subregions is further partitioned into a finite number of compartments. Chemical reactions can occur within a compart-ment, whereas diffusion is represented as movement between adjacent compartments. Inspired by biology, we study both (1) the case where the reactions on each side of the boundary are different and only certain species diffuse across the boundary and (2) the case where reactions and diffusion are spatially homogeneous. We write the stoichiometry matrix for these two classes of systems using a Kronecker product formulation. For the first scenario, we apply linear perturbation theory to de -rive an approximate singular value decomposition in the limit as diffusion becomes much faster than reactions. For the second scenario, we derive an exact analytical singular value decomposition for all relative diffusion and reaction time scales. By writing the stoichiometry matrix using Kronecker products, we show that the singular vectors and values can also be written concisely using Kronecker products. Ultimately, we find that the singular value decomposition of the reaction-diffusion stoi-chiometry matrix depends on the singular value decompositions of smaller matrices. These smaller matrices represent modified versions of the reaction-only stoichiometry matrices and the analytically known diffusion-only stoichiometry matrix. Lastly, we present the singular value decomposition of the model for the Calvin cycle in cyanobacteria and demonstrate the accuracy of our formulation. The MATLAB code, available at www.github.com/MathBioCU/ReacDiffStoicSVD, provides routines for efficiently calculating the SVD for a given reaction network on a 1D spatial domain.

Automated summary

This research presents an analytical singular value decomposition method for the stoichiometry matrix of a reaction-diffusion system to reveal spatial flux patterns. Approaches for approximate and exact singular value decompositions are studied for different reaction and diffusion scenarios. The study shows that the singular value decomposition depends on smaller matrix decompositions, providing efficient analysis of the stoichiometry matrix of reaction-diffusion systems.