Biological systems generate reliable, information-processing behavior from large numbers of interacting components operating far from equilibrium. At the molecular scale, this behavior emerges through self-assembly governed by local interaction rules, while at the cellular scale it manifests as gene regulatory networks that encode decision-making through nonlinear dynamics and multistability. Both are governed by the same fundamental questions: how do microscopic interaction rules give rise to robust, directly-tunable macroscopic states, and how can we capture the dynamics and fundamental physics of these processes? In this thesis, I describe a rule-based operator algebra for representing multiparticle assemblies in stochastic chemical systems. The formalism enables rigorous analysis of complex formation and dissolution in systems where combinatorial complexity is otherwise intractable, and allows for both analytic and numerical explorations of scaling behavior and other statistical properties. The general language of the formalism renders it well-suited for studying self-assembly processes both in and out of equilibrium, and for examining not only the polymer systems discussed in this thesis but also broader applications in gene regulation and signaling protein-protein interactions. I also discuss underlying dynamics of self-organization toward cell fate determination, and specifically the power of effector concentration as a tuning knob for examining the dynamics of gene regulatory motifs such as three-gene toggle switches and other biologically-relevant circuits in thermodynamic model descriptions. In particular, capturing shifts in the dynamical landscapes of these gene circuits through an explicit role for induction offers an actionable connection to experimental design.
Rebecca Jasmina Rousseau (Wed,) studied this question.