Stochastic Approach to Enantiomeric Excess Amplification and Chiral Symmetry Breaking

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.Comment: 25 pages, 6 figure

Total Chiral Symmetry Breaking during Crystallization: Who needs a "Mother Crystal"?

Processes that can produce states of broken chiral symmetry are of particular interest to physics, chemistry and biology. Chiral symmetry breaking during crystallization of sodium chlorate occurs via the production of secondary crystals of the same handedness from a single "mother crystal" that seeds the solution. Here we report that a large and "symmetric" population of D- and L-crystals moves into complete chiral purity disappearing one of the enantiomers. This result shows: (i) a new symmetry breaking process incompatible with the hypothesis of a single "mother crystal"; (ii) that complete symmetry breaking and chiral purity can be achieved from an initial system with both enantiomers. These findings demand a new explanation to the process of total symmetry breaking in crystallization without the intervention of a "mother crystal" and open the debate on this fascinating phenomenon. We present arguments to show that our experimental data can been explained with a new model of "complete chiral purity induced by nonlinear autocatalysis and recycling".Comment: 5 pages, 4 figures, Added reference

Chirality Selection in Open Flow Systems and in Polymerization

As an attempt to understand the homochirality of organic molecules in life, a chemical reaction model is proposed where the production of chiral monomers from achiral substrate is catalyzed by the polymers of the same enatiomeric type. This system has to be open because in a closed system the enhanced production of chiral monomers by enzymes is compensated by the associated enhancement in back reaction, and the chiral symmetry is conserved. Open flow without cross inhibition is shown to lead to the chirality selection in a general model. In polymerization, the influx of substrate from the ambience and the efflux of chiral products for purposes other than the catalyst production make the system necessarily open. The chiral symmetry is found to be broken if the influx of substrate lies within a finite interval. As the efficiency of the enzyme increases, the maximum value of the enantiomeric excess approaches unity so that the chirality selection becomes complete.Comment: 8 pages, 4 figure

Complete homochirality induced by the nonlinear autocatalysis and recycling

A nonlinear autocatalysis of a chiral substance is shown to achieve homochirality in a closed system, if the back-reaction is included. Asymmetry in the concentration of two enantiomers or the enantiometric excess increases due to the nonlinear autocatalysis. Furthermore, when the back-reaction is taken into account, the reactant supplied by the decomposition of the enantiomers is recycled to produce more and more the dominant one, and eventually the homochirality is established.Comment: 4 pages, 2 figure

Chiral Crystal Growth under Grinding

To study the establishment of homochirality observed in the crystal growth experiment of chiral molecules from a solution under grinding, we extend the lattice gas model of crystal growth as follows. A lattice site can be occupied by a chiral molecule in R or S form, or can be empty. Molecules form homoclusters by nearest neighbor bonds. They change their chirality if they are isolated monomers in the solution. Grinding is incorporated by cutting and shafling the system randomly. It is shown that Ostwald ripening without grinding is extremely slow to select chirality, if possible. Grinding alone also cannot achieve chirality selection. For the accomplishment of homochirality, we need an enhanced chirality change on crystalline surface. With this "autocatalytic effect" and the recycling of monomers due to rinding, an exponential increase of crystal enantiomeric excess to homochiral state is realized.Comment: 10 pages, 5 figure

An Extended Model for the Evolution of Prebiotic Homochirality: A Bottom-Up Approach to the Origin of Life

A generalized autocatalytic model for chiral polymerization is investigated in detail. Apart from enantiomeric cross-inhibition, the model allows for the autogenic (non-catalytic) formation of left and right-handed monomers from a substrate with reaction rates $\epsilon_L$ and $\epsilon_R$, respectively. The spatiotemporal evolution of the net chiral asymmetry is studied for models with several values of the maximum polymer length, N. For N=2, we study the validity of the adiabatic approximation often cited in the literature. We show that the approximation obtains the correct equilibrium values of the net chirality, but fails to reproduce the short time behavior. We show also that the autogenic term in the full N=2 model behaves as a control parameter in a chiral symmetry- breaking phase transition leading to full homochirality from racemic initial conditions. We study the dynamics of the N -> infinity model with symmetric ($\epsilon_L = \epsilon_R$) autogenic formation, showing that it only achieves homochirality for $\epsilon < \epsilon_c$, where $\epsilon_c$ is an N-dependent critical value. For $\epsilon \leq \epsilon_c$ we investigate the behavior of models with several values of N, showing that the net chiral asymmetry grows as tanh(N). We show that for a given symmetric autogenic reaction rate, the net chirality and the concentrations of chirally pure polymers increase with the maximum polymer length in the model. We briefly discuss the consequences of our results for the development of homochirality in prebiotic Earth and possible experimental verification of our findings

Fluctuation Induced Homochirality

We propose a new mechanism for the achievment of homochirality in life without any autocatalytic production process. Our model consists of a spontaneous production together with a recycling cross inhibition in a closed system. It is shown that although the rate equations for this system predict no chiral symmetry breaking, the stochastic master equation predicts complete homochirality. This is because the fluctuation induced by the discreteness of population numbers of participating molecules plays essential roles. This fluctuation conspires with the recyling cross inhibition to realize the homochirality.Comment: 13 pages, 6 figure

Homochirality and the need of energy

The mechanisms for explaining how a stable asymmetric chemical system can be formed from a symmetric chemical system, in the absence of any asymmetric influence other than statistical fluctuations, have been developed during the last decades, focusing on the non-linear kinetic aspects. Besides the absolute necessity of self-amplification processes, the importance of energetic aspects is often underestimated. Going down to the most fundamental aspects, the distinction between a single object -- that can be intrinsically asymmetric -- and a collection of objects -- whose racemic state is the more stable one -- must be emphasized. A system of strongly interacting objects can be described as one single object retaining its individuality and a single asymmetry; weakly or non-interacting objects keep their own individuality, and are prone to racemize towards the equilibrium state. In the presence of energy fluxes, systems can be maintained in an asymmetric non-equilibrium steady-state. Such dynamical systems can retain their asymmetry for times longer than their racemization time.Comment: 8 pages, 7 figures, submitted to Origins of Life and Evolution of Biosphere

Toward homochiral protocells in noncatalytic peptide systems

The activation-polymerization-epimerization-depolymerization (APED) model of Plasson et al. has recently been proposed as a mechanism for the evolution of homochirality on prebiotic Earth. The dynamics of the APED model in two-dimensional spatially-extended systems is investigated for various realistic reaction parameters. It is found that the APED system allows for the formation of isolated homochiral proto-domains surrounded by a racemate. A diffusive slowdown of the APED network such as induced through tidal motion or evaporating pools and lagoons leads to the stabilization of homochiral bounded structures as expected in the first self-assembled protocells.Comment: 10 pages, 5 figure