Computational and Modeling Strategies for Cell Motility

We review emerging strategies and progress toward modeling and simulation of 3D cell morphology and dynamics of a motile cell. We then describe our approach, based on phase field modeling, that is flexible enough to incorporate distinct substructures within a cell, how they physically interact, mobile and reacting chemical species, and chemical-mechanical transduction. We posit a modeling foundation which addresses the collective dynamics and morphology of a cell in response to physical forcing or boundary conditions at the cell membrane as well as internal signaling pathways. Our modeling approach is conditioned on the requirement that we can numerically simulate the model equations consistent with laboratory experiments and compare model results with laboratory data of whole cell morphology and dynamics or more specific features such as localized contractile waves. The proposed whole cell model is an integration of sub-models: for cell membrane dynamics, external boundary conditions (with other cells, external buffer, solid substrates), multiple substructures with highly contrasting properties (membrane, cortical layer, cytosol, nucleus), complex viscoelastic constitutive laws and evolving free surface domains for each substructure, biochemical reaction and diffusion of key protein species, and activation (localized forcing) through chemicalmechanical transduction. We summarize the literature on substructure modeling and activation mechanisms, and then present an integrated modeling framework. The aim is to model diverse modes of motile cells; our first target is a model explanation of sustained cell oscillations [69, 40, 92, 22, 91]. In order for the model to have predictive value, the cell model has to exhibit sufficient conditions for cell oscillations, while illustrating mechanisms which would lead to cessation of oscillations and transition to a different dynamic modality. To simulate the model, we have to strike a compromise between biological detail and computational feasibility. The modeling framework is a phase field (or diffuse interface) methodology which has already yielded progress in cell modeling and which provides a platform for future development. The success of the phase field approach rests on the ability to formulate sufficiently accurate energy functionals for each of the substructures, their interaction at interfaces, and spatio-temporal chemical activation within evolving substructures. Our contention is that sufficient progress has ∗Department of Mathematics and NanoCenter, University of South Carolina, Columbia, SC 29208 †Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 ‡Department of Pharmacology, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 §Department of Cell and Developmental Biology and Lineberger Comprehensive Cancer Center, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 ¶Department of Cell and Developmental Biology, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 ∥Department of Mathematics and Institute for Advanced Materials, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599