Stochastic Approach to Enantiomeric Excess Amplification and Chiral Symmetry Breaking

TL;DR

This paper investigates stochastic models of chemical reactions related to homochirality, revealing how autocatalysis and recycling influence enantiomeric excess amplification and symmetry breaking through analytical and numerical methods.

Contribution

It introduces a stochastic framework combining master equations and random walk models to analyze chiral symmetry breaking and enantiomeric excess amplification.

Findings

Double-peak distribution indicates chiral symmetry breaking.

Recycling processes lead to unique final probability distributions.

Strong autocatalysis with initial achirality results in enantiomeric excess amplification.

Abstract

Stochastic aspects of chemical reaction models related to the Soai reactions as well as to the homochirality in life are studied analytically and numerically by the use of the master equation and random walk model. For systems with a recycling process, a unique final probability distribution is obtained by means of detailed balance conditions. With a nonlinear autocatalysis the distribution has a double-peak structure, indicating the chiral symmetry breaking. This problem is further analyzed by examining eigenvalues and eigenfunctions of the master equation. In the case without recycling process, final probability distributions depend on the initial conditions. In the nonlinear autocatalytic case, time-evolution starting from a complete achiral state leads to a final distribution which differs from that deduced from the nonzero recycling result. This is due to the absence of the detailed balance, and a directed random walk model is shown to give the correct final profile. When the nonlinear autocatalysis is sufficiently strong and the initial state is achiral, the final probability distribution has a double-peak structure, related to the enantiomeric excess amplification. It is argued that with autocatalyses and a very small but nonzero spontaneous production, a single mother scenario could be a main mechanism to produce the homochirality.