Polylogarithmic equilibrium treatment of molecular aggregation and critical concentrations

TL;DR

This paper presents a comprehensive equilibrium model for molecular aggregation in 1D and 3D systems, using polylogarithms to predict critical concentrations and analyze aggregation behavior without complex kinetic parameters.

Contribution

It introduces a novel equilibrium approach that simplifies the prediction of critical concentrations in molecular aggregation, bypassing complex kinetic modeling.

Findings

The residual monomer concentration converges to its asymptotic value more strongly in 3D aggregates.

Critical nucleation concentration sharply increases with the difference in polymerization and nucleation constants.

Proposes ansatz equations linking critical concentrations to thermodynamic parameters.

Abstract

A full equilibrium treatment of molecular aggregation is presented for prototypes of 1D and 3D aggregates, with and without nucleation. By skipping complex kinetic parameters like aggregate size-dependent diffusion, the equilibrium treatment allows to predict directly time-independent quantities such as critical concentrations. The relationships between the macroscopic equilibrium constants for the different paths are first established by statistical corrections and so as to comply with the detailed balance constraints imposed by nucleation, and the composition of the mixture resulting from homogeneous aggregation is then analyzed using the polylogarithm function. Several critical concentrations are distinguished: the residual monomer concentation in equilibrium (RMC) and the critical nucleation concentration (CNC), that is the threshold concentration of total subunits necessary for initiating aggregation. When increasing the concentration of total subunits, the RMC converges more strongly to its asymptotic value, the equilibrium constant of depolymerization, for 3D aggregates and in case of nucleation. The CNC moderately depends on the number of subunits in the nucleus, but sharply increases with the difference between the equilibrium constants of polymerization and nucleation. As the RMC and CNC can be numerically but not analytically determined, ansatz equations connecting them to thermodynamic parameters are proposed.