A polyhedral structure controls programmable self-assembly

Modern experimental methods in programmable self-assembly make it possible to precisely design 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 substantial 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. We validate our predictions through detailed, quantitative assembly experiments of nanoscale particles synthesized using DNA origami. Our results uncover physical relationships underpinning many-component programmable self-assembly in equilibrium and form the basis for robust inverse design, applicable to various systems from biological protein complexes to synthetic nanomachines.