Nonlinear mathematical models for the origin of asymmetry in biological molecules

Abstract

The paper deals with a mathematical model of two separate populations of primeval organisms (or polymers), one based on L- and one on D-amino acids. Possible interactions between those two separate, selfreproducing, and mutually antagonistic populations are discussed. Below we list some of the conclusions. For the process of reproduction of L and D enanthiomorphs with the same rate constant k1, i.e. for L → L + L and D → D + D, the L and D enanthiomorphs will grow exponentially in time. The presence of an initial disproportion, due to fluctuation, (eg. LO > DO) will cause also an exponential growth of the difference L-D. In case when K1 for both processes are not equal, (a case postulated by some but not yet proven), both initial disproportion and the difference in rate constant will contribute to increased difference of L-D. In this simple system there is no instability and no sudden death of one type (eg. D). By adding additional process of lethal interaction (Frank's model, Frank, 1953) whenever L encounters D (death rate with rate constant k2) the kinetic equations become nonlinear, i.e. the rate of production k1L and K1D is counteracted by the rate of destruction K2LD. The system now is only stable if initially L = D, exactly. If at any time L will exceed D in a given volume V by a small number △n, due to a fluctuation, L will increase exponentially, but D after some time will die extremely fast, faster than exponentially (as e−Aexp(k1t), where A = K2△n/V). If at any time D will exceed L by a small number, due to a fluctuation, D will grow exponentially and L will be destroyed. The probability that L will grow and that L = D at all times is extremely small and situation is very unstable. It was shown that if the time between fluctuations is greater than the time of reproduction, initial disproportion (fluctuation) predominates over consecutive fluctuations, leading to growth of one and death of the other isomer. The possible effect of asymmetry in the rate constants is compared to the role of statistical fluctuations, and it is shown, that within the simple model investigated that therole of statistical fluctuations is much more important for the death of one isomer. In the unlikely absence of any fluctuations, the nonlinear 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 n-th order nonlinear symmetric rate processes (incorporating multitudes of reactions and 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. This is very unstable situation and any small fluctuation would have quickly selected a winner. The paper consists of detailed mathematical analysis with many intermediate steps for the pedagogical purpose of making the mathematics more accessible to larger audience who may not yet be familiar with handling of nonlinear rate equations. It is hoped by the author that these mathematical models will help to develop an intuition for better understanding of the possible mechanisms in the origin of asymmetry in biological molecules.