Shape as program: Guiding self-assembly with geometric and topological potentials

The self-assembly of macromolecules into functional biological structures is a fundamental process governed by a complex interplay of physical forces. Simulating this process presents a challenging trade-off: the need for energy models that are thermodynamically accurate on the one hand, and computationally efficient on the other. Successfully simulating physical systems perpetuates a positive feedback loop of understanding. The real physics inform the models, and the successful models inform our understanding of the relevant physics. This thesis employs the morphometric approach, a geometric model of the nonpolar contributions to the solvation free energy, in a simulation setting. The investigation begins by establishing an overview of energy minimizers of idealized systems of hard spheres, which are shown to assemble into clusters with biologically relevant geometries that are theoretically impossible to obtain by pairwise potentials. The model is then applied to the self-assembly of two and three tobacco mosaic virus protein subunits. Results demonstrate that the experimentally observed native assembly is found as an energy minimum of the morphometric model alone. This further supports the model's thermodynamic accuracy and highlights the paramount role of shape complementarity in biological assembly. Considering the simulation process purely from an efficiency standpoint reveals the difficulties inherent to the short-range nature of the model. To overcome kinetic barriers and guide assemblies over long ranges, we introduce a biasing, topological potential based on persistent homology. This shape-based potential acts as a long-range simulation guide. By combining the short-range physical accuracy of the morphometric model with the long-range guidance of the topological potential, a synergistic energy function is created. Applied to protein assembly, this hybrid approach increases the success rate of finding correctly assembled structures substantially. This thesis thus makes three primary contributions. First, it establishes that non-polar solvation effects, modeled purely through geometry, can be a primary driver of the appearance of biologically relevant structural motifs. Second, it further demonstrates that this model is sufficient to simulate the assembly of tobacco mosaic virus and to energetically distinguish its experimentally determined nucleation state. Finally, it pioneers the use of persistent homology not merely as a tool for post-hoc analysis, but as an active component within an energy function to solve the critical problem of kinetic trapping. This fusion of local geometry and global topology presents a new and powerful strategy for the efficient simulation of molecular systems.