Computing all persistent subspaces of a reaction-diffusion system

An algorithm is presented for computing for a reaction-diffusion partial differential equation (PDE) system all possible subspaces that can hold a persistent solution of the equation. For this, all possible sub-networks of the underlying reaction network that are distributed organizations (DOs) are identified, because recently it has been shown that a persistent subspace must be a DO. The algorithm computes the hierarchy of DOs starting from the largest by a linear programming (LP) approach using integer cuts. The underlying constraints use elementary reaction closures (ERCs) as minimal building blocks to guarantee local closedness and global self-maintenance, required for a DO. Additionally, the algorithm delivers for each subspace an affiliated set of organizational reactions (SOR) and a minimal compartmentalization that is necessary for this subspace to persist. It is proved that all SORs of a reaction network, as already DOs, form a lattice and that this lattice contains all potentially persistent sets of reactions of all constrained solutions of reaction-diffusion PDEs. This provides a hierarchical structure of all persistent subspaces with regard to the species and also to the reactions of the reaction-diffusion PDE system. Here, the algorithm is described and corresponding python source code is provided. Furthermore, an analysis of its run time is performed and all models from the BioModels database as well as further examples are examined. Apart from the practical implications of the algorithm the results also give insights into the complexity of solving reaction-diffusion PDEs.