In this dissertation, we investigate complex, non-linear interactions in biological systems.This work is presented as two independent projects. The mathematics and biology in each differ, yet there is a unity in that both frameworks are interested in biological responses that cannot be reduced to linear causal chains, nor can they be expressed as an accumulation of binary interactions.

In the first part of this dissertation, we use mathematical modeling to study tumor-immune dynamics at the cellular scale.Recent work suggests that LSD1 inhibition reduces tumor growth, increases T cell tumor infiltration, and complements PD1/PDL1 checkpoint inhibitor therapy. In order to elucidate the immunogenic effects of LSD1 inhibition, we create a delay differential equation model of tumor growth under the influence of the adaptive immune response in order to investigate the anti-tumor cytotoxicity of LSD1-mediated T cell dynamics. We fit our model to the B16 mouse model data from Sheng et al. [DOI:10.1016/j.cell.2018.05.052] Our results suggest that the immunogenic effect of LSD1 inhibition accelerates anti-tumor cytoxicity. However, cytotoxicity does not seem to account for the slower growth observed in LSD1 inhibited tumors, despite evidence suggesting immune-mediation of this effect.

In the second part, we consider the partial information decomposition (PID) of response information within networks of interacting nodes, inspired by biomolecular networks.We specifically study the potential of PID synergy as a tool for network inference and edge nomination. We conduct both numeric and analytic investigations of the $\Imin$ and $\Ipm$ PIDs, from [arXiv:1004.2515] and [DOI:10.3390/e20040297], respectively. We find that the $I_\text{PM}$ synergy suffers from issues of non-specificity, while $I_{\text{min}}$ synergy is specific but somewhat insensitive. In the course of our work, we extend the $I_\text{PM}$ and $I_{\text{min}}$ PIDs to continuous variables for a general class of noise-free trivariate systems. The $I_\text{PM}$ PID does not respect conditional independence, while$I_{\text{min}}$ does, as demonstrated through asymptotic analysis of linear and non-linear interaction kernels. The technical results of this chapter relate the analytic and information-theoretic properties of our interactions, by expressing the continuous PID of noise-free interactions in terms of the partial derivatives of the interaction kernel.