UC Berkeley

In this thesis, we apply a common set of tools to two problems that compose two separate parts. Shared between them is the use of importance sampling, in which techniques such as Monte Carlo and molecular dynamics evolve a system in a way that respects its physical conditions under the effect of thermal fluctuations, and the use of techniques from the finite element method, in which a combination of integration by parts and discretization on function spaces allows one to turn strong differential conditions into weaker integral ones.

In the first part, we consider the modeling of biological membranes, complex materials formed of lipids and proteins that serve as interfacial barriers in cells. At length scales beyond the thickness of a membrane, membrane behavior can be understood using a phenomenological model accounting for fluid in-plane and elastic out-of-plane behavior. A discretization of this model is developed using techniques from the finite element method that is evolved using Metropolis Monte Carlo.Compositional energetics are added to model phase separation into liquid-disordered and liquid-ordered domains that are observed experimentally in multicomponent biological membranes. By including the effect of proteins that can induce domains of the thermodynamically disfavored phase it is found that competition between line tension and curvature prevents macrophase separation and leads to stable microphases, providing a possible explanation for nanoscopic domains hypothesized to exist in cells.

In the second part, we consider numerical methods to find representations of the committor function in rare event processes. The committor function, the probability a configuration will commit to the product state instead of the reactant state, encodes the complete mechanistic information of a process but is costly to compute. Instead, the transition pathway is homogeneously sampled with importance sampling methods in order to solve a variational form of a partial differential equation the committor function satisfies. A neural network is used as a basis function from which optimization informs the neural network parameters. Coupling this process with fitting to empirical estimates of the committor function, the procedure is found to yield accurate estimates of the committor function and reaction rates.