An Equilibrium Model for Chiral Amplification in Supramolecular Polymers

We describe a model that rationalizes amplification of
chirality
in cooperative supramolecular copolymerization. The model extends
nucleation-elongation based equilibrium models for growth of supramolecular
homopolymers to the case of two monomer and aggregate types. Using
the principle of mass-balance for the two monomer types, we derive
a set of two nonlinear equations, describing the thermodynamic equilibrium
state of the system. These equations can be solved by numerical methods,
but also analytical approximations are derived. The equilibrium model
allows two-sided growth of the aggregates and can be applied to symmetric
supramolecular copolymerizations, corresponding to the situation in
which the monomers are enantiomerically related, as well as to the
more general case of nonsymmetric supramolecular copolymerizations.
In detail, so-called majority-rules phenomena in supramolecular systems
with isodesmic as well as cooperative growth are analyzed. Comparison
of model predictions with experimental data shows that the model gives
a very good description of both titration and melting curves. When
the system shows cooperative growth, the model leads to a phase diagram
in which the presence of the various aggregate types is given as a
function of composition and temperature.