A geometrical framework for designer self-assembly

Modern programmable self-assembly platforms provide unprecedented control over the concentrations, shapes, and interactions of the assembling particles. But because it is challenging to predict the effects of these control knobs on the final assembly outcome, we are unable to fully leverage this control for the design of complex and functional nanostructures — especially in economical or reconfigurable assemblies where particles are reused and binding is nondeterministic. Here we show that a significant part of the design space in programmable assembly can be understood instantly, by identifying a class of thermodynamic constraints and their geometrical structure. Using this geometrical structure enables us to predict which structures can, and cannot, be assembled by tuning the design attributes of the assembling particles, and reveals that equivalent assembly outcomes are often achievable through many different parameter choices. Exploiting this degeneracy, we optimize the assembly outcome and assembly kinetics simultaneously, allowing us to drastically reduce assembly times, often by many orders of magnitude. Combining our theoretical results with efficient algorithms from graph theory and computational geometry leads to a versatile framework for rational design of self-assembling structures, with applications ranging from DNA nanotechnology to the assembly of biological protein complexes.