Nonlinear mathematical models for the origin of asymmetry in biological molecules.

The origin of chemical chirality is probably associated with a difference in the initial concentrations of two separate populations of primeval organic molecules and possible even two types of primeval organisms and amplified by nonlinear kinetic processes leading to the death of one population. This theory, as originally developed by F. C. Frank of the University of Bristol, is reviewed in this paper with additional derivations, discussions and generalizations. The possible effect of asymmetry in the rate constant is compared to the role of statistical fluctuations, and it is shown, that within the simple model investigated here that the role of statistical fluctuations is much more important for the death of one isomer. In the unlikely absence of any fluctuations, the non-linear kinetic processes amplify the asymmetry in the rate constant and lead to the death of one enanthiomorph. The role of spatial diffusion is discussed, and it is shown that in the presence of a local excess of one enanthiomorph this excess would have spread in space and grown, destroying the opposite enanthiomorph. If the total population of both enanthiomorphs was exactly composed of equal parts of both types, but local fluctuation increased one type at one place and decreased the same type at a different location, the diffusion and growth rate would have caused spatial separation in the population of both enanthiomorphs. For general nth order nonlinear symmetric rate processes (incorporating multitudes of reactions and general diffusion), it is shown that if initially two populations of enanthiomorphs were exactly the same at all locations, then for all times both populations would have increased and remained equal to each other. Mathematical model was constructed for stereoscopic autocatalysis suggested by Calvin. It was found that under certain special local conditions in the presence of large fluctuation it is possible indeed to have growth of only one type of isomer. Various approximate methods and numerical solutions are presented in order to facilitate the handling of nonlinear rate equations.