  A generalized autocatalytic model for chiral polymerization is investigated
in detail. Apart from enantiomeric cross-inhibition, the model allows for the
autogenic (non-catalytic) formation of left and right-handed monomers from a
substrate with reaction rates $\epsilon_L$ and $\epsilon_R$, respectively. The
spatiotemporal evolution of the net chiral asymmetry is studied for models with
several values of the maximum polymer length, N. For N=2, we study the validity
of the adiabatic approximation often cited in the literature. We show that the
approximation obtains the correct equilibrium values of the net chirality, but
fails to reproduce the short time behavior. We show also that the autogenic
term in the full N=2 model behaves as a control parameter in a chiral symmetry-
breaking phase transition leading to full homochirality from racemic initial
conditions. We study the dynamics of the N -> infinity model with symmetric
($\epsilon_L = \epsilon_R$) autogenic formation, showing that it only achieves
homochirality for $\epsilon < \epsilon_c$, where $\epsilon_c$ is an N-dependent
critical value. For $\epsilon \leq \epsilon_c$ we investigate the behavior of
models with several values of N, showing that the net chiral asymmetry grows as
tanh(N). We show that for a given symmetric autogenic reaction rate, the net
chirality and the concentrations of chirally pure polymers increase with the
maximum polymer length in the model. We briefly discuss the consequences of our
results for the development of homochirality in prebiotic Earth and possible
experimental verification of our findings.
