Chiral Symmetry Breaking in Large Peptide Systems

Mathematical Models of Chiral Symmetry-breaking – A Review of General Theories, and Adiabatic Approximations of the APED System

We review the literature surrounding chiral symmetry-breaking in chemical systems, with a focus on understanding the mathematical models underlying these chemical processes. We comment in particular on the toy model of Sandars, Viedma’s crystal grinding systems and the APED model. We include a few new results based on asymptotic analysis of the APED system.

Evolutionary Approach to Biological Homochirality

We study a very simple linear evolutionary model based on distribution of protocells by total enantiomeric excess and without any mutual inhibition and show that such model can produce two species with values of total enantiomeric excess in each of the species approaching $$\pm 1$$ ± 1 when there is a global $$L\leftrightarrow D$$ L ↔ D symmetry. We then consider a scenario when there is a small external global asymmetry factor, like weak interaction, and show that only one of the species remains in such a case, and that is the one, which is more efficient in replication. We perform an estimate of the time necessary to reach homochirality in such a model and show that reasonable assumptions lead to an estimate of around 300 thousand years plus or minus a couple of orders of magnitude. Despite this seemingly large time to reach homochirality, the model is immune to racemization because amino acids in the model follow the lifespan of the protocells rather than the time needed to reach homochirality. We show that not needing mutual inhibition in such evolutionary model is due to the difference in the topology of the spaces in which considered model and many known models of biological homochirality operate. Bifurcation-based models operate in disconnected zero-dimensional space (the space is just two points with enantiomeric excess equal $$-1$$ - 1 and $$1$$ 1 ), whereas considered evolutionary model (in its continuous representation) operates in one-dimensional connected space, that is the whole interval between $$-1$$ - 1 and $$1$$ 1 of total enantiomeric excess. We then proceed with the analysis of the replication process in non-homochiral environment and show that replication errors (the probability to attach an amino acid of wrong chirality) result in a smooth decrease of replication time when total enantiomeric excess of the replicated structure moves away from zero. We show that this decrease in replication time is sufficient for considered model to work.

Spontaneous Emergence of Transient Chirality in Closed, Reversible Frank-like Deterministic Models

To explore abiotic theories related to the origin of biomolecular homochirality, we analyze two entirely reversible kinetic models composed of an enantioselective autocatalysis with limited stereoselectivity that is coupled to an enantiomeric mutual inhibition (Frank-like models). The two models differ in their autocatalytic steps in respect to the formation of monomer species in one model and of dimer species in the other. While fully reversible and running in a closed system, spontaneous mirror symmetry breaking (SMSB) gives rise to transient chiral excursions, even when starting from a strictly achiral situation. Before the SMSB, the two models differ in the main dissipative processes. At the SMSB, the entropy production rate reaches its maximum in both models. Here it is the enantioselective autocatalysis with retention of the winner enantiomer that dominates. During the terminal phase, the enantioselective autocatalysis with inversion prevails, while the entropy production rate vanishes, thus fulfilling the conditions of microscopic reversibility. SMSB does not occur if the autocatalytic rate constant is too strong or too weak. However, when the autocatalysis is relatively weak, the temporary chiral excursions last for long periods of time and could be the starting point of a cascade of asymmetric reactions. The realism of such Frank-like models is discussed from the viewpoint of their relevance to prebiotic chemistry.