UC Merced

Scientific discovery depends on our ability to model the physical world accurately --- and for many of the most consequential problems in biology and medicine, the geometry of the domain is not a detail to be approximated away but the very mechanism driving the phenomenon of interest. This dissertation develops and applies computational methods for partial differential equations on complex, irregular, and continuously deforming geometries, demonstrating that spatial resolution reveals biological insights that well-mixed or geometry-free approaches fundamentally cannot. The first contribution is the Least Squares Discretization (LSQD) method, a meshless framework for solving elliptic PDEs on scattered point distributions without requiring a mesh, a weak formulation, or quadrature rules. The method achieves h-P convergence across a broad range of geometries, boundary conditions, and problem configurations, remains robust in degenerate cases where traditional formulations may be ill-posed, and includes a built-in error estimator, all while requiring minimal mathematical background to implement. The second contribution applies this spatially resolved framework to characterize how the electrostatic surface potential of the SARS-CoV-2 spike protein evolves as it opens to expose its receptor binding domain. Solving the nonlinear Poisson-Boltzmann equation across 71 frames of a molecular dynamics trajectory, we find that geometric reorganization of the spike actively reshapes the electric field in a manner that may facilitate selective binding to the ACE2 receptor: a conclusion accessible only through spatially resolved computation on a deforming surface. The third contribution develops the first spatially resolved, multi-generation computational model of prion aggregate inheritance in budding yeast. Solving a reaction-diffusion PDE system on a deforming cell geometry across a parallelized binary lineage tree, we demonstrate that aggregate burden decreases monotonically with generation index across numerous biochemical parameter combinations investigated, that this lineage memory is deterministic and geometric in origin, and that the per-generation dilution follows a geometric decay law whose parameters predict aggregate clearance in deeper simulations with remarkable accuracy. Together these contributions establish that geometry is not merely a computational challenge to be overcome, but an essential feature of the science itself.