Diffusion in Stochastic Environments: Analysis and Biological Applications

随机环境中的扩散：分析和生物学应用

基本信息

批准号：
1814832
负责人：
Sean Lawley
金额：
$ 25万
依托单位：
University of Utah
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1814832&HistoricalAwards=false
关键词：
Diffusion Stochastic Environments Analysis Biological

项目摘要

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.

该项目将开发并应用随机运动的数学来揭示（i）细胞内化学物质如何反应，（ii）大脑区域如何沟通，以及（iii）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（例如氧气）的运动。由于分子非常小，因此它们通过称为扩散的固有随机过程移动。虽然扩散的理论工作可以追溯到爱因斯坦和其他人，但推动该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内的扩散统计数据，并将理论结果和预测与经验测量进行比较。此外，由于研究生将密切参与研究，该项目还将培养新的数学生物学家。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：一门关于应用随机过程的研究生课程和一门为中学数学教师开设的概率与统计课程。随机环境是一个丰富的数学主题。它有两个互补的描述，随机切换偏微分方程 (PDE) 或随机切换随机微分方程 (SDE)。无论哪种情况，该过程都结合了跨空间尺度相互作用的两层随机性：粒子级别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 PDE/SDE 表示将允许 PDE 方法和概率工具的结合，并阐明这些领域之间的新联系。此外，数学结果将提供对广泛的生物系统进行建模所需的工具。这些模型反过来将（i）改变对普遍存在的生化反应的理解，（ii）揭示重要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，该项目首创的数学技术组合将作为未来随机空间过程研究的原型。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。