The polyhedral structure underlying programmable self-assembly

Abstract: Experiments have reached a monumental capacity for designing and synthesizing microscopic particles for self-assembly, making it possible to precisely control particle concentrations, shapes, and interactions. However, more physical insight is needed before we can take full advantage of this vast design space to assemble nanostructures with complex form and function. Here we show how a significant part of this design space can be quickly and comprehensively understood by identifying a class of thermodynamic constraints that act on it. These thermodynamic constraints form a high-dimensional convex polyhedron that determines which nanostructures can be assembled at high equilibrium yield and reveals limitations that govern the coexistence of structures, which we verify through detailed, quantitative assembly experiments of nanoscale particles synthesized using DNA origami. Strong experimental agreement confirms the importance of the polyhedral structure and motivates its use as a predictive tool for the rational design of self-assembly. These results uncover fundamental physical relationships underpinning many-component programmable self-assembly in equilibrium and form the basis for robust inverse-design, applicable to a wide array of systems from biological protein complexes to synthetic nanomachines.