UCLA

Biological networks, such as vascular networks and neural circuits, are ubiquitous in nature. An understanding of these networks can help us understand their response to damages, which could lead to novel treatments. They can also inspire the design of man-made networks, as evolution has millions of years to figure out optimal designs. The advancement in imaging techniques has created high-dimensional data streams, which is difficult to analyze by conventional approaches. On the other hand, quantitative tools are naturally suited for processing large data sets, and they become more and more important in improving our knowledge on biological networks. Among existing tools ranging from network science to stochastic analysis, here we focus on optimization and dynamical system approach. Optimization links biological functions to corresponding network structures, which can give predictions to be compared with the data. The dynamical system approach is suited for analyzing time series data and complex interaction between the vertices, which is often exploited in biological systems for intricate signalings and regulations.

This thesis is devoted to the study of biological networks with optimization and dynamical system, focused on two specific biological systems: microvascular network and bipolar disorder. For microvascular networks, we first study a specific example of embryonic zebrafish trunk network, and reveal the significance of flow uniformity in this network. Then we derive analytical structures of networks with optimal transport efficiency, which is widely regarded as the organizing principle of vascular networks, especially for large vessels such as aorta. To compare the morphologies of transport efficient and uniform flow networks, we develop algorithm that is capable of finding optimal networks with general target functions and constraints, and show that the principle of uniform flow creates more realistic microvascular networks under many different topologies. Finally, we propose an vessel adaptation mechanism based on stress sensing dynamic to explain how microvascular networks stay resilient to noise, and how they grow into uniform flow networks. For bipolar disorder, we mathematically analyze a dynamical model based on the interaction of mood and expectation. We show that bipolar disorder can be viewed as a bifurcation in the direction from normal to cyclic personality. We also consider the case where positive and negative events are sensed differently, and describe the bifurcation in this case. Finally we apply commonly used medicine on the model, and recover clinically observed phenomena on bipolar disorder patients.