Computing the Parameter Values for the Emergence of Homochirality in Complex Networks

The goal of our research is the development of algorithmic tools for the analysis of chemical reaction networks proposed as models of biological homochirality. We focus on two algorithmic problems: detecting whether or not a chemical mechanism admits mirror symmetry-breaking; and, given one of those networks as input, sampling the set of racemic steady states that can produce mirror symmetry-breaking. Algorithmic solutions to those two problems will allow us to compute the parameter values for the emergence of homochirality. We found a mathematical criterion for the occurrence of mirror symmetry-breaking. This criterion allows us to compute semialgebraic definitions of the sets of racemic steady states that produce homochirality. Although those semialgebraic definitions can be processed algorithmically, the algorithmic analysis of them becomes unfeasible in most cases, given the nonlinear character of those definitions. We use Clarke's system of convex coordinates to linearize, as much as possible, those semialgebraic definitions. As a result of this work, we get an efficient algorithm that solves both algorithmic problems for networks containing only one enantiomeric pair and a heuristic algorithm that can be used in the general case, with two or more enantiomeric pairs.