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Large facets: Processes on networks

My research is driven by a desire to understand the roles of stochasticity, structure, and evolution in shaping the dynamics of biological systems. I develop and analyze mathematical models, combining methods from probability, dynamical systems, and random graph theory, to shed light on biological issues while generating new mathematical questions. Stochasticity is fundamental to gene expression and ion channel gating; it also plays an under-appreciated role in physiological regulation at larger scales, such as the regulation of sleep and wake behavior. This behavior is governed by networks of neurons in the brain with underlying structures that likely influence dynamics, suggesting a prominent role for stochastic processes evolving on networks. To what extent are the dynamics that we observe shaped by the stochastic nature of biochemical reactions, by physiological structure such as neuronal networks, and by the constraints of evolution? A particular challenge of this question is the need to bridge multiple spatial and temporal scales: from microscopic to macroscopic, from millisecond to millions of years. Below I describe my work using methods such as stochastic processes on graphs and stochastic models in genetics to understand interlocking large and small facets of biological regulation.