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 backreaction 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.
PACS numbers:Natural organic molecules associated with living matters usually have two possibilities in their stereostructures, in a right-handed form(R) or in a mirrorimage left-handed form(S).1,2 These two forms are called enantiomers, and the molecules are said to be chiral. Chiral molecules are optically active to rotate the direction of polarization of plane polarized light. From the energetic point of view, these two enantiomers can exist with an equal probability, but the life on earth utilizes only one type: only levorotatory(L)-amino acids (or S) and dextrorotatory(D)-sugards (or R). This symmetry breaking in the chirality is called the homochirality.The origin of this unique chirality has long intrigued many scientists.1 In order to find the physical origin of this homochirality, initial asymmetry in the primordial molecular environment has to be created by chance or engendered deteminately by external or internal factors, such as the parity breaking effect in the weak interaction, 3,4,5,6 the asymmetry in circularly polarized light, 2,7 or adsorption on optically active crystals. 8 Then the induced small initial chiral asymmetry has to be amplified.Frank has shown theoretically that an autocatalytic reaction of a chemical substance with an antagonistic process can lead to an amplification of enantiometric excess (ee) and to homochirality.9 Many theoretical models are proposed afterwards, but they are often criticized as lacking any experimental support.1 Recently, asymmetric autocatalysis 10 of pyrimidyl alkanol has been studied intensively.11,12,13,14 The enhancement of ee was confirmed, 12 and its temporal evolution was explained by the second-order autocatalytic reaction.13,14 But only with the nonlinear autocatalysis, chirality selection is not complete and the value of ee stays less than unity. Here we show that the complete homochirality is achieved by the back-reaction to recycle the reactant. If the rate of back reaction is small, it takes a long time before the homochirality is achieved.We consider a chemical reaction such that substances A and B react to form substance C. Though reactants A and B are achiral, the product C happened to be chiral in two enantiometric forms; R-isomer (R)-C and S-isomer we assume a closed conserve system such that concentrations R and S at time t vary in proportion to the present amount of the reactants A and B asHere we first neglect the back reaction from (S)-C or (R)-C to A and B. Later, the effect of back reaction is shown to be very important in achieving the complete homochirality....