Chiral symmetry breaking: Frank model for the evolution of homochirality described by population dynamics

Abstract

The intrinsic asymmetric nature of biological compounds, fundamental for the origin of Life on Earth, and the explanation and description of emergence and evolution of homochirality of predominant species represent an important and intriguing challenge for many scientists (in 2005, the editors of Science magazine included the problem among the 125 most important scientific puzzles). Autocatalytic reactions, where a reaction product catalyzes the production of itself and suppresses its enantiomer, can represent a mechanism for the evolution of homochirality, and the model proposed by Frank (Biochim Biophys Acta 11:459–463, 1953) still represents a milestone in the field. Anyway, with due distinctions, the basic idea underlying the Frank model could be mathematically traced back to the Lotka–Volterra equations (also commonly known as the predator–prey equations), describing biological species competition: then, in principle, this inscribes the problem basically within that branch of Mathematics called population dynamics. In the present work, alternatively to the deterministic or stochastic numerical solutions of Frank’s equations describing the evolution of a system towards the homochirality, I tried to describe the “proliferation” of the chiral species by the conceptual methods and tools often adopted in biomathematics (study of geometric progressions and so on). Interestingly, this approach led to solutions about the time-dependent populations of the two enantiomers of in a simple closed form (of course, our results have been quantitatively compared with results obtained from deterministic and stochastic models). Besides this, the length of the time required for the complete evolution of the system to homochirality (also called, sometimes in literature, relaxation time) as a function of the total amount of molecules and of the initial excess of one enantiomer has been explicitly obtained, and, mathematically, it resulted to be exactly twice the value of the time where is located the point of inflection of the curves describing the dynamics of the evolution of two enantiomers. Finally, an explicit realistic estimation of these times, compared with experimental and theoretical data available in literature, is given in the work.