Diffusion in Stochastic Environments: Analysis and Biological Applications

这个项目将发展和应用随机运动的数学来揭示(1)化学物质如何在细胞内反应，(2)大脑区域如何交流，以及(3)昆虫如何呼吸的特征。这些不同的应用是统一的，因为它们都涉及小分子（如氧）的运动。因为分子非常小，所以它们的移动遵循一种固有的随机过程，称为扩散。虽然关于扩散的理论工作可以追溯到爱因斯坦和其他人，但推动这个项目的生物学需要对经典理论进行重大扩展。特别是，本研究将构建一个数学框架来计算随机变化环境中的扩散统计，并将理论结果和预测与实证测量进行比较。此外，该项目将培养新的数学生物学家，因为研究生将密切参与研究。此外，这项研究及其生物学应用将丰富PI正在开发的两门新课程：一门应用随机过程的研究生课程和一门面向中学数学教师的概率与统计课程。除了在生物学中广泛应用外，随机环境中的扩散也是一个丰富的数学课题。它有两个互补的描述，即随机切换偏微分方程（PDE）或随机切换随机微分方程（SDE）。在这两种情况下，这个过程结合了两层在空间尺度上相互作用的随机性：粒子水平上的布朗运动和随机环境。分析这些过程需要结合不同数学领域的工具来开发新的数学机制。特别是，双重PDE/SDE表示将允许PDE方法和概率工具的组合，并阐明这些领域之间的新联系。此外，数学结果将提供建立广泛生物系统模型所需的工具。这些模型将依次(i)改变对普遍存在的一类生化反应的理解，（ii）揭示基本神经通讯机制的基本特性，以及（iii）回答昆虫生理学中长期存在的基本问题。此外，在这个项目中开创的数学技术的结合将作为未来随机空间过程研究的原型。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.