Models for Mirror Symmetry Breaking via {\beta}-Sheet-Controlled Copolymerization: (i) Mass Balance and (ii) Probabilistic Treatment

TL;DR

This paper develops two theoretical models to explain how chiral amplification occurs in copolymerization processes, one based on mass balance and the other on probabilistic polymerization, providing insights into mirror symmetry breaking.

Contribution

It introduces a coupled nonlinear mass balance model and a probabilistic model for template-induced desymmetrization in copolymerization, advancing understanding of homochiral polymer formation.

Findings

Calculates enantiomeric excesses and chain lengths using the mass balance model.

Identifies conditions for maximizing mirror symmetry breaking through probabilistic analysis.

Provides geometric interpretation of occlusion probabilities in polymerization.

Abstract

Experimental mechanisms that yield the growth of homochiral copolymers over their heterochiral counterparts have been advocated by Lahav and co-workers. These chiral amplification mechanisms proceed through racemic {\beta}-sheet-controlled polymerization operative in both surface crystallites as well as in solution. We develop two complementary theoretical models for these template-induced desymmetrization processes leading to multicomponent homochiral copolymers. First, assuming reversible {\beta}-sheet formation, the equilibrium between the free monomer pool and the polymer strand within the template is assumed. This yields coupled nonlinear mass balance equations whose solutions are used to calculate enantiomeric excesses and average lengths of the homochiral chains formed. The second approach is a probabilistic treatment based on random polymerization. The occlusion probabilities depend on the polymerization activation energies for each monomer species and are proportional to the concentrations of the monomers in solution in the constant pool approximation. The monomer occlusion probabilities are represented geometrically in terms of unit simplexes from which conditions for maximizing or minimizing the likelihood for mirror symmetry breaking can be determined.