Denoting monomers by c, small and large left-handed clusters by x 1, x 2 respectively and right-handed by y 1, y 2, Uwaha (2004) writes down the scheme $$ \frac\rm d c\rm d t = – 2 k_0 c^2 – k_1 c (x_1+y_1) + \lambda_1(x_2+y_2) + \lambda_0(x_1+y_1) , $$ (1.12) $$ \frac\rm d x_1\rm d t = k_0 c^2 – k_u x_1 x_2 – k_c x_1^2 + \lambda_u x_2 + \lambda_0 x_1 , $$ (1.13) $$ \frac\rm d x_2\rm d t = k_1 x_2 c + k_u x_1 x_2 + Epigenetic Reader Domain inhibitor k_c x_1^2 – \lambda_1 x_2 – \lambda_u x_2 , $$ (1.14) $$ \frac\rm d y_1\rm d t = k_0 c^2 – k_u y_1 y_2 – k_c y_1^2 + \lambda_u y_2 + \lambda_0 y_1 , $$ (1.15) $$ \frac\rm d y_2\rm d t
= k_1 y_2 c + k_u y_1 y_2 + k_c y_1^2 – \lambda_1 y_2 – \lambda_u y_2 , $$ (1.16)which models the formation of small chiral clusters (x 1, y 1) from an achiral monomer (c) at rate k 0, small chiral clusters (x 1, y 1) of the same handedness combining to form larger chiral clusters (rate k c ), small and larger clusters combining to form larger clusters (rate k u ), large clusters combining with achiral monomers to form more large clusters at the rate k 1, the break up
of larger clusters into smaller clusters (rate λ u ), GSK2118436 clinical trial the break up of small clusters into achiral monomers (rate λ 0), the break up of larger clusters into achiral monomers (rate λ 1). Such a model can exhibit symmetry-breaking to a solution in which x 1 ≠ x 2 and x 2 ≠ y 2. Uwaha points out that the recycling part of the model (the λ * parameters) are crucial to the formation of a ‘completely’ heptaminol homochiral state.
One problem with such a model is that since the variables are all total masses in the system, the size of clusters is not explicitly included. This can easily be overcome by using a more formal coarse-grained model such as that of Bolton and Wattis (2003). In asymmetric distributions, the typical size of left- and right- handed clusters may differ drastically, hence the rates of reactions will proceed differently in the cases of a few large crystals or many smaller crystals. JPH203 in vivo Sandars has proposed a model of symmetry-breaking in the formation of chiral polymers (2003). His model has an achiral substrate (S) which splits into chiral monomers L 1, R 1 both spontaneously at a slow rate and at a faster rate, when catalysed by the presence of long homochiral chains. This catalytic effect has both autocatalytic and crosscatalytic components, that is, for example, the presence of long right-handed chains R n autocatalyses the production of right-handed monomers R 1 from S, (autocatalysis) as well as the production of left-handed monomers, L 1 (crosscatalysis). Sandars assumes the growth rates of chains are linear and not catalysed; the other mechanism required to produce a symmetry-breaking bifurcation to a chiral state is cross-inhibition, by which chains of opposite handednesses interact and prevent either from further growth.