In this thesis I will analyze the dynamics of two-populations of nuclei within a filamentous fungus. The models developed here are designed to directly represent recent experiments studying Neurospora crassa chimera created from two genetically different nuclear populations.

The focus will be on analyzing how dynamics within the network are affected by the geometry of the fungus network by building and validating spatial population genetic models. We start from simulating some intuitive models for nuclear population dynamics during fungal growth, then provide asymptotic and analytic insights on the dynamics involved, and the relationship between population dynamics and the branching structure of the network. Comparisons among the simulation, the simplified model, and existing experiment data will also be discussed.

Mathematical modeling has played a pivotal role in advancing the understanding of gene expression regulation, operating at both the molecular and cellular levels. In this dissertation, stochastic models and simulations will be employed to explore the impact of various forms of cellular compartmentalization, such as the formation of functionally distinct RNA condensates or the establishment of nuclear territories in syncytia, on the robust spatiotemporal control of gene expression within cells. The highlighted model system, multinucleated filamentous fungi, serves as a prime example of global coordination and compartmentalization within expansive cytoplasmic environments. To complement these modeling efforts, image analysis on live cell data will be employed to quantify and localize RNAs, providing a means by which key model parameters can be inferred and the model's validity can be assessed within physiologically relevant contexts.

Understanding complex data requires some form of model analysis, whether it be with machine learning, statistical, or physical models. Such analysis is useful for producing predictions, identifying dynamics from noise, and understanding the system in question. Complex or "black box" models can accurately predict dynamics of the world around us, and are useful tools when matching data is the primary goal. However, these models fail at increasing understanding of the system of study.

This thesis is dedicated to the analysis of simple models, and when, how, and why they work to represent data. Herein, we analyze the application of the simple SIR epidemic model to complex epidemiological dynamics present in the COVID-19 pandemic in the U.S. We show that, despite the model's simplicity and apparently violated assumptions, it still has a place in matching and predicting real data, and we can learn key intuition from this fact. We also identify key contraction arrangements in the network-like slime mold Physarum polycephalum using a low Reynolds number flow model. These contraction arrangements, or modes, are key to understanding how, exactly, the organism gets its wide array of different behaviors. Our analysis places these modes in a simple, physically-understandable context that will allow researchers to connect measurable physical features to real complex behavior.

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.

Microfluidic devices are tiny circuits that flow fluids instead of electrons. Because they are inexpensive and portable, microfluidic devices are ideal for use in areas where medical resources are scarce. Inertial microfluidic devices represent a new direction in microfluidic device design in which high flow speeds are used to exert nonlinear inertial effects on the fluid and on fluid-suspended particles. While inertial microfluidic devices are finding applications in fields such as fluid mixing, particle filtration, flow cytometry (the counting, sorting, and analyzing of cells), the devices are built with essentially no theoretical input due to a lack of models for the nonlinear inertial effects.

Why is there so little theory for inertial microfluidic devices? While there are many numerical methods for simulating inertial migration, because most devices have multiple moving boundaries and rely on three-dimensional effects, simulations are computationally intensive. In many cases, the computational time far exceeds the time needed to build and test a device experimentally. In contrast, asymptotic studies of inertial migration are only valid in limited cases, such as vanishingly small particle sizes.

This thesis is concerned with developing a theory for inertial effects in microfluidic devices for a wide range of complicated geometries. This theory is achieved through the combination of both asymptotic and numerical methods. First, a theory is developed for the inertial lift force on a particle in a square channel. Second, this theory for the inertial lift force is validated against experiment. Third, a theory is developed for the formation of particle chains in a rectangular channel. Finally, a theory is developed for the number of focusing positions in a given channel.