Diffusion in Stochastic Environments: Analysis and Biological Applications

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This project will develop and apply the mathematics of random motion to reveal features of (i) how chemicals react inside a cell, (ii) how brain regions communicate, and (iii) how insects breathe. These diverse applications are united because they all involve the motion of small molecules (such as oxygen). Because the molecules are so small, they move by an inherently random process called diffusion. While theoretical work on diffusion dates back to Einstein and others, the biology driving this project requires significant extensions to the classical theory. In particular, this research will construct a mathematical framework to calculate statistics of diffusion inside a randomly changing environment, and the theoretical results and predictions will be compared to empirical measurements. In addition, this project will train new mathematical biologists since graduate students will be closely involved in the research. Additionally this research and its biological applications will enrich pair of new courses that the PI is developing: a graduate course on applied random processes and a course on probability and statistics for secondary math teachers.In addition to being widely applicable in biology, diffusion in a random environment is a rich mathematical topic. It has two complementary descriptions, either a randomly switching partial differential equation (PDE) or a randomly switching stochastic differential equation (SDE). In either case, the process combines two layers of randomness interacting across spatial scales: Brownian motion at the particle level and a random environment. Analyzing these processes will require combining tools from disparate areas of mathematics to develop new mathematical machinery. In particular, the dual PDE/SDE representation will allow for the combination of PDE methods and probabilistic tools and elucidate new connections between these fields. Furthermore, the mathematical results will provide the tools needed to model a broad collection of biological systems. These models will in turn (i) transform the understanding of a ubiquitous class of biochemical reactions, (ii) uncover basic properties of an essential neural communication mechanism, and (iii) answer a longstanding and fundamental question in insect physiology. Furthermore, the combination of mathematical techniques pioneered in this project will serve as a prototype for future investigations in stochastic spatial processes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.