Mechanical and chemical signaling are essential in shaping cellular biophysics. This thesis covers computational models for this field with different emphases.

First, a mechanical model for collective cell motility is discussed. It assumes a contraction-protrusion motile cycle modulated by a molecular clutch which can be stalled by resistant forces. It also includes simplistic pictures of cell-substrate and cell-cell interactions. A number of experimental observations receive a unified explanation under this model, including the tensile stress, edge-confining traction force, and mechanical waves during tissue expansion; the spontaneous revolving of cells along a narrow annulus. After accounting for substrate stiffness gradient, this model also explains collective durotaxis when isolated cells are not durotactic. More specifically, it shows that durotactically biased tissue expansion can be achieved without cell polarity flips, as long as disruptions such as cell divisions keep interior cells non-stalled.

Next is a multi-scale model for the coupling between epithelial-mesenchymal transition (EMT) and extracellular matrix (ECM) stiffness. EMT plays a critical role in cancer progression, and has traditionally been associated with chemical signals, but more evidence has established the importance of biomechanical features of tumor microenvironment such as ECM stiffness. A coupled positive feedback loop is proposed whereby mesenchymal cells secretes more LOXL2 that increases crosslinking of collagen fibers and stiffen the ECM, only to produce mechanosensing signals that further drive EMT. Implications of spatial-temporal heterogeneity are also discussed.

Last, a dynamical model for the ecology of tumor microenvironment is studied. The microenvironment consists of both cancer and immune cells, among others, all competing for limited resource. Cancer is a systems disease that involves failure of immune surveillance where a large fraction of immune cells become pro-tumor. The ordinary differential equations (ODE) model includes the dynamics of resource influx/consumption, and growth of tumor, pro- and anti-tumor immune cells. A reduced variant model with constant resource is compared. The bifurcation diagrams show that aggressively proliferative tumors or overly pro-tumor immune systems would lead to cancer progression. It also sets a mathematical example that Jacobian matrix is not representative of the stability against small but finite perturbations.