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. Research highlights with selected publications:Multiscale methods Data-driven simulation Mathematical modeling in biology Multiscale numerical methods  Intracellular cargo transport is an inherently noisy process, in which antagonistic molecular motors push and pull on cytoskeletal anchors in a microscopic version of tug-of-war. In order to capture the effect of stochastic switching between different steady-states in the nonlinear model, we combined stochastic simulations of individual motor attachment with a population-level PDE of the motor length distribution. This resulted in a model that preserved the stochasticity of the agent based model while improving the efficiency and tractability of simulations. •Y. Park, P. Singh, and T.G. Fai, Coarse-grained stochastic model of myosin-driven vesicles into dendritic spines, SIAM J Appl Math, 82(3), 793-820, 2022 (link, arXiv) Data-driven simulation
 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). We simulate 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 modeling in biology  The
proportional size scaling relationship observed between the cell and nucleus is one of the fundamental rules of life. Although it has been
appreciated for over a hundred years that nuclear size scales with cell size, the mechanisms remain poorly understood. In this work, we propose and analyze a mathematical model based on osmotic force balance that predicts the relationship between cell and nuclear volumes. Introducing cell growth leads to the striking prediction that the nuclear-to-cell size ratio is genetically determined by the fraction of biomolecules imported into the nucleus.• J. Lemière, P. Real-Calderon, L.J. Holt, T.G. Fai, F. Chang, Control of nuclear size by osmotic forces in S. pombe, eLife, 10, e69597, 2022 (link, bioRxiv). Featured in Condensed Matter Physics Journal Club. (highlight)
  Our cells contain extensive machinery to control division, but is it possible that cellular precursors replicated by a very simple mechanism? We 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 |
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