An Equilibrium Model for Chiral Amplification in Supramolecular Polymers

Abstract

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.

Binding polynomial in molecular self-assembly

“…Taking into account that, as shown above, the law of mass action is tightly linked to the way the grand partition function Z is defined, the application of the binding polynomial Z b to a particular system with cooperativity must be treated with caution. The cooperative self-and heteroassemblies have attracted a great deal of attention in supramolecular chemistry (see, e.g., [10,18,19]); this issue will not be discussed in the present paper. We must note, however, a specific partial case of cooperativity, reviewed in [20], which is fully compliant with the oligomer partition function (6) and hence with the binding polynomial Z b .…”

General statistical-thermodynamical treatment of one-dimensional multicomponent molecular hetero-assembly in solution

“…If cooperativity of one-dimensional hetero-assembly is involved, such models as the N-STOCH approach or the models from Refs. [8,9] are still the alternative. Nevertheless a potential possibility to introduce cooperativity of hetero-assembly into the equations of the model developed in the present work seems real and is a matter of further publication.…”

“…Non-covalent hetero-assembly of small molecules in solution is currently attracting great attention due to extensive use of this phenomenon in molecular electronics [1][2][3], laser physics [4,5], supramolecular chemistry [2,[6][7][8][9], advanced materials [10][11][12][13], numerous biochemical applications [2,14,15] etc. The key feature of this process is the formation of nanoscale aggregates, which modify the macroscopic physical properties of solute resulting in outcomes of practical importance ranging from discovering of new perspective materials to treatment of human diseases [16].…”

“…[9,17] for review) and two-component hetero-assembly (see Refs. [8,18] for review) leading to formation of aggregates, containing one type or two different types of molecules, respectively, ordered along a certain axis.…”

“…[18,24]). An exception is in special cases of intermolecular assembly [9,22] or in account for cooperative effects [9].…”