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

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.