On the Stability Analysis of Chiral Networks and the Emergence of Homochirality

We study a particular set of chemical reaction networks related to the emergence of homochirality. Each element of this set is a chemical reaction mechanism intended to produce homochirality. Those mechanisms contains a pair of enantiomers, the central subject of this study, which are involved in a series of reactions that produce and consume them. The other species concentrations are considered constant. The reactions of each mechanism are arranged into six categories, that we have called synthesis, first order decomposition, autocatalytic, second order decomposition, non-enantioselective and inhibition reactions. The reaction networks must satisfy a symmetry constraint that is related to the kinetic and thermodynamic indiscernibility of the isomers. We investigate the emergence of homochirality phenomena in those networks. To this end, we introduce a mathematical notion of homochiral states that we call Frank states, and which seems to be deeply related to the occurrence of homochiral dynamics. We find sufficient and necessary conditions for the existence of Frank states, and we use those results to develop an algorithmic tool. This tool can be used to recognize networks admitting homochiral states, and in given case, it can also be used to construct Rank states of the input-network. We test the mathematical machinery, and the aforementioned algorithm, analyzing the well-established models of Rank and Kondepudi-Nelson. We were able to show that those two networks admit homochiral dynamics. We use our tools to analyze three further network models derived from the Kondepudi-Nelson model and which were adapted to the Strecker synthesis of amino acids.