Abstract
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.
Highlights
We study mathematical models of absolute asymmetric synthesis [1,2] that are used to explain the emergence of biological homochirality, which is, according to Frank [3], a natural property of life
If Ω is not pseudochiral, it models a chemical mechanism that works on a chiral environment, and because of this it cannot be regarded as a network model of absolute asymmetric synthesis
We focus on the first task, which is solved if one exhibits racemic steady states which, after being perturbed, trigger homochiral dynamics
Summary
We study mathematical models of absolute asymmetric synthesis [1,2] that are used to explain the emergence of biological homochirality, which is, according to Frank [3], a natural property of life. We found a particular symmetry in the Jacobian matrices of those models [9] that yields semialgebraic definitions of the instability regions where the symmetry-breaking can be observed Most of those semialgebraic expressions are highly nonlinear and hard to sample. We used Clarke’s Stoichiometric Network Analysis (SNA) [10] to reduce the complexity of those expressions. All those ingredients were put together into an algorithmic tool, and software Listanalchem [11], that can be used to test models proposed to explain the origin of homochirality, and which can help us to build new and better models. We use the developed algorithm to analyze three representative models of biological homochirality taken from the available literature
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