The macroscopic dynamics created by complex systems of microscopic cells can be better understood using mathematical modeling. In this dissertation, we study the dynamics of two different biological systems: phototaxis exhibited by cyanobacterium Synechocystis sp. and cancer immune interactions as affected by the protein B7-H1.

Synechocystis sp., a common unicellular freshwater cyanobacterium, has been used as a model organism to study phototaxis, i.e. motion in the direction of a light source. This microorganism displays a number of additional characteristics such as delayed motion, surface dependence, and a quasi-random motion, where cells move in a seemingly disordered fashion instead of in the direction of the light source. These unexplained motions are thought to be modulated by local interactions between cells.

In this work, we formulate a model of local interactions between phototactic cells in order to study the structure of their quasi-random motion. We present a stochastic dynamic particle system modeling interacting phototactic cells. We extend our model of local interactions to include global forcing due to light. We also add an activation process of cells as they become affected by the presence of light. We study the parameter space of our model by deriving a system of ordinary differential equations that describe the dynamics of the system in one dimension. The simulations of our model are consistent with experimentally observed phototactic motion.

The second part of this dissertation focuses on the surface protein B7-H1. This protein, also called PD-L1 and CD274, is found on carcinomas of the lung, ovary, colon and melanomas but not on most normal tissues. B7-H1 has been experimentally determined to be an anti-apoptotic receptor on cancer cells, where B7-H1-positive cancer cells have been shown to be immune resistant, and in vitro experiments and mouse models have shown that B7-H1-negative tumor cells are significantly more susceptible to being repressed by the immune system. We derive a mathematical model for studying the interaction between cytotoxic T cells and tumor cells as affected by B7-H1. By integrating experimental data into the model, we isolate the parameters that control the dynamics and obtain insights on the mechanisms that control apoptosis.