Mathematical Modeling of Cell-Substrate Interaction
Cells respond to their environment mechanically and biochemically – the stiffness of a substrate or extracellular matrix with which a cell interacts affects the cell’s biochemical response. The goal of this project for 2016 is to continue to develop mathematical models and computational tools that allow us to better understand how substrate properties affects cellular response. Over the past several years, this group has developed a mathematical model of cell-substrate interaction. In this model, which is based on principles of continuum mechanics, both the cell and substrate are treated as two-dimensional deformable continua. The cell is attached to the substrate via linear springs that can break and reform in a stress-dependent manner. The nonlinear governing equations are solved using the finite element method. This results in a large system of nonlinear algebraic equations which require significant computational resources to solve. The solution method has been implemented in both MATLAB and Comsol.
Thus far, the model and simulations have provided some insight into how cells respond to varying substrate stiffness. For example, they have shown that one reason why a cell spreads more on stiffer substrates is because on a stiffer substrate a fixed contractile force leads to a smaller contraction magnitude. The researchers are now investigating the effects of ligand patterning of the substrate on cellular shape and intracellular stresses. To do this, they have coupled the existing model of cell-substrate mechanics to a reaction-diffusion model for the concentrations of active and inactive attachment complexes as well as substrate surface ligand concentrations. The addition of these components results in yet a larger mathematical model.
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