  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.

Biological Physics;Statistical Mechanics;Biomolecules