Self-assembly of a dimer system

TL;DR

This paper analytically studies the statistical physics of dimer self-assembly, revealing how biophysical systems favor search-limited regimes where monomer encounter rates, rather than combinatorial complexity, govern correct dimer formation.

Contribution

It provides an exact partition function for dimer formation and classifies systems into search-limited and combinatorics-limited types, with implications for biological self-assembly.

Findings

Most biological systems are search-limited in dimerization.

Correct dimerization depends on the product of particle diversity and volume.

Analytical framework applies to DNA, transcription factors, and proteins.

Abstract

In the self-assembly process which drives the formation of cellular membranes, micelles, and capsids, a collection of separated subunits spontaneously binds together to form functional and more ordered structures. In this work, we study the statistical physics of self-assembly in a simpler scenario: the formation of dimers from a system of monomers. The properties of the model allow us to frame the microstate counting as a combinatorial problem whose solution leads to an exact partition function. From the associated equilibrium conditions, we find that such dimer systems come in two types: "search-limited" and "combinatorics-limited", only the former of which has states where partial assembly can be dominated by correct contacts. Using estimates of biophysical quantities in systems of single-stranded DNA dimerization, transcription factor and DNA interactions, and protein-protein interactions, we find that all of these systems appear to be of the search-limited type, i.e., their fully correct dimerization regimes are more limited by the ability of monomers to find one another in the constituent volume than by the combinatorial disadvantage of correct dimers. We derive the parameter requirements for fully correct dimerization and find that rather than the ratio of particle number and volume (number density) being the relevant quantity, it is the product of particle diversity and volume that is constrained. Ultimately, this work contributes to an understanding of self-assembly by using the simple case of a system of dimers to analytically study the combinatorics of assembly.