Biological systems are modular, and this modularity affects the evolution of biological systems over time and in different environments. Studying the structure of biological systems provides insight into human physiology and evolution in the natural world, which helps us to understand a wide variety of biological phenomena. In this thesis, we use theoretical and analytical methods to study the theory of personalized critical care, the rate of evolution in a rugged fitness landscape, and the structure of cancer networks and neural networks. We seek to explain how the structure of biological systems evolves over time among many possible states. Our results support the idea that changes in environmental pressure stimulates the spontaneous emergence of modular structure.

In the study of prediction of the heart rate response to a spontaneous breathing trial, a non-equilibrium fluctuation dissipation theorem is applied to predict how critically ill patients will respond to this intervention. The result shows that the response of a group of similar patients to the spontaneous breathing trail can be predicted by the non-equilibrium fluctuation dissipation approach. This mathematical method may serve as part of the basis for personalized critical care.

We develop a theory for the dynamics of evolution in a rugged, modular fitness landscape and show analytically how horizontal gene transfer couples to the modularity in the system and leads to more rapid rates of evolution at short times. The model analytically demonstrates a selective pressure for the prevalence of modularity in biology. We use this model to show how the evolution of the influenza virus is affected by the modularity of the proteins that are recognized by the human immune system. Comparison to influenza virus evolution data, the result shows that a modular model of the fitness landscape of the virus better fits the observed data.

We then study gene and tissue networks of breast cancer patients. We find that the likelihood of metastasis in the future is correlated with an increased value of network hierarchy for expression networks of cancer-associated genes. Conversely, future metastasis and quick relapse times are negatively correlated with the values of network hierarchy in the expression network of all host genes, due to the dedifferentiation of host gene pathways and circuits. These results suggest that the hierarchy of gene expression may be useful as an additional biomarker for breast cancer prognosis.

Finally, we study how the modularity of the human brain changes as children develop into adults. The value of modularity calculated from fMRI data is observed to increase during childhood development and peak in young adulthood. We present a model to illustrate how modularity can provide greater cognitive performance at short times, from which we extract a fitness function from the model. Quasispecies theory is used to predict how the average modularity evolves with age, illustrating the increase of modularity during development from children to adults that arises from selection for rapid cognitive function in young adults. We show that modularity may be a potential biomarker for injury, rehabilitation, or disease.