My research is focused on the modeling and simulation of biological phenomena, including the fluid dynamics inside of cells and membrane growth and form in cellular precursors. To model these phenomena I use ideas from subjects such as partial differential equations, graph theory, and differential geometry. Computational approaches are needed because the resulting equations are often too complicated to solve by hand; I work on developing efficient numerical methods to simulate fluid-structure interaction and to incorporate experimental data into simulations. Current research topics with selected publications: Multiscale numerical methods
Simulations on uniform fluid grids can break down when objects come close to touching. I have developed a numerical method for fluid-structure interaction that uses a subgrid model based on lubrication theory to accurately resolve near-contact. This method allows for the efficient simulation of phenomena such as the migration of deformable red blood cells away from blood vessel walls.
• T. G. Fai and C. Rycroft, Lubricated Immersed Boundary Method in Two Dimensions. J. Comput. Phys., 356, 319-339, 2018 (link,preprint on arxiv).   Data-driven simulation
The red blood cell cytoskeleton is an elastic network that may be modeled as a graph of actin-based junctional complexes (nodes) connected by spectrin polymers (edges). I have simulated the effect of cytoskeletal remodeling on the cell’s mechanical response to prescribed stress and strain, using images obtained by cryo-electron microscopy to generate a random graph with realistic statistical properties. This model may be useful for studying the fundamental question of how red blood cells age.
• T. G. Fai, A. Leo-Macias, D. L. Stokes, and C. S. Peskin, Image-Based Model of the Spectrin Cytoskeleton for Red Blood Cell Simulation. PLOS Comput. Biol., 11(6), e1005790, 2017 (link, preprint).
Mathematical models of fluid-structure interaction at cellular and sub-cellular scales
One of the cellular processes that regulates neurons is the delivery of membrane receptors to the synapse. I have developed a reduced model of the trafficking of elastic vesicles into dendritic spines in neurons and performed a bifurcation analysis of the emergent multistable dynamics. The model, which was validated by comparison to full three-dimensional (3D) simulations reproduces many of the behaviors observed in the actual system.
• T. G. Fai, R. Kusters, J. Harting, C. Rycroft, and L. Mahadevan, Active elastohydrodynamics of vesicles in narrow, blind constrictions. Phys. Rev. Fluids, 2, 113601, 2017 (link, preprint on arxiv). 
Our cells contain extensive machinery to control division, but is it possible that cellular precursors replicated by a very simple mechanism? I have performed 3D simulations of a permeable, growing membrane to explore a model of prebiotic cells. These simulations show that a simple mechanical model can give rise to many modes of growth, including uniform expansion and the emergence of thin membrane ridges and deep recesses. • T. Ruiz-Herrero, T. G. Fai, and L. Mahadevan. Dynamics of growth and form in prebiotic vesicles. In revision.
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