Transient Pattern Formation in Biological Systems

Abstract

This dissertation investigates transient pattern formation in biological systems through a combination of theoretical modeling, stochastic analysis, and simulation. Biological processes are often driven by local interactions and feedback mechanisms that give rise to complex, time-dependent patterns. Given the inherent heterogeneity and noise in living systems, traditional deterministic models are often insufficient to capture the full spectrum of behaviors observed in nature. Here, I develop and analyze different models that are robust to microscopic details while capturing essential dynamical features. One focus of this dissertation is the study of reaction-diffusion phenomena in immune cell signaling. I investigate how neutrophils generate self-regulating, transient chemical waves that coordinate a rapid yet contained response to injury or infection. The models show that the interplay between activators and locally produced inhibitors can naturally limit the spatial extent of these signaling waves, providing a mechanistic basis for preventing overreaction in immune responses. Further, I examine the role of mechanical stress in flow-driven pattern formation within porous media. By representing these media as dynamic networks in which individual conduits adapt through erosion and deposition, I identify critical thresholds that lead to distinct phase behaviors, such as channelization and homogenization. This work not only elucidates the feedback between fluid flow and structural evolution but also offers a simple approach to analyze complex networks. By applying a similar strategy to biological networks, I explore the emergence of optimized biological flow networks. By integrating local mechanical sensing into growth dynamics, I derive conditions under which vascular systems naturally converge toward configurations predicted by Murray’s law—a hallmark of energy-efficient design observed in blood vessels, leaf venation, and even in the foraging networks of slime molds. Finally, I extend the classical mutation models, exemplified by the Luria–Delbruck experiment, to regimes where the effective mutation rate is significantly higher through modern gene-editing techniques. By formulating discrete stochastic models, I reveal novel phase transitions in DNA break-and-repair dynamics and demonstrate how randomness in molecular events influences cell fate and population heterogeneity.