Asymptotic Designability of Self-Assembled Structures

Self assembly is heavily constrained by kinetic and thermodynamic factors. In some situations, thermodynamic constraints can be avoided with simple recipes, but it is not obvious how to apply these recipes to more complicated assembly scenarios, especially when the number of component species is limited, or when components bind promiscuously. In such cases, it is often unclear whether a target structure (or group of structures) can be made into the thermodynamic ground state, making it hard to predict what structures can be assembled at high yield. Here, we introduce the concept of Asymptotic Designability, which allows us to quickly identify which structures are thermodynamically stable at high yield, and to determine the degree of control over the relative yields within a group of designable structures. We show that checking for Asymptotic Designability can be formulated as a Linear Program, which can be solved efficiently, and that optimizing relative yields is equivalent to solving a system of linear equations. Our asymptotic theory is exact in the limit of infinite binding energies, but it is also predictive for finite energies and concentrations, which allows us to take kinetic constraints into account. These results enable an exact quantification of the fundamental thermodynamic constraints on self-assembly, and they provide a starting point for the description of more advanced self assembly pathways in and out of equilibrium.