Collaborative Research: Dynamics and pattern formation of nonlocal collective motion and assembly

合作研究：非局部集体运动和装配的动力学和模式形成

• 批准号: 1312344
• 负责人: James von Brecht
• 金额: $ 10.91万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1312344&HistoricalAwards=false
• 关键词: Collaborative; Research; Dynamics; pattern; formation

The investigators of this project will study the dynamics and pattern formation of particles or agents that evolve under non-local collective motion laws. Specifically, the PI and co-PI will study systems in which collective behavior manifests non-trivial co-dimension. The mathematics of such particle systems pervades many disciplines, ranging from physics, chemistry and biology to control theory and engineering. Modern applications in these areas include protein folding, colloid stability and the self-assembly of nanoparticles into supramolecular structures. In biology, similar mathematical models help explain the complex phenomena observed in viral capsids, locust swarms and colonies of bacteria. The first phase of this project will apply and expand recently developed mathematical tools to identifiy various physical and chemical interactions that will naturally produce co-dimension one structures. This has a direct application to the study of processes, such as the self-assembly of Polyoxometalate (POM) molecular clusters into spherical supramolecular structures, in which experimental evidence is overwhelming but a theoretical understanding of the underlying formation mechanisms is lacking. The PI, co-PI and their collaborators have made numerous important developments in the mathematical theory of such mechanisms when the interaction is isotropic. This part of the project aims to further develop this theory in a manner that will prove useful to a broad set of researchers in other disciplines. The proposed research project is fundamentally interdisciplinary in that it derives mathematical problems from unexplained phenomena in diverse fields such as chemistry, biology and engineering. The PI and co-PI will apply a broad set of mathematical tools drawing from dynamical systems, partial differential equations, mathematical modeling and computational methods to solve several problems that are subject to active research in these fields, including the so-called 'designer potential problem' in nano self-assembly. By clarifying which physical forces drive POM self-assembly and other related phenomena, such as viral capsid formation, we will provide rigorous solutions to the question of how to design sub-units that form into these importantstructures. One of the demonstrated career goals of the PI is to further the inclusion of underrepresented groups in the field of mathematics. The PI has the personal experience and critical perspective necessary to serve as a mentor for the diverse student body at University of San Francisco. Exposing these students from underrepresented groups to cuttingedge mathematical research, coupled with faculty mentoring, will inspire many of them to pursue graduate degrees in the mathematical sciences. The student research supported by this grant will prove indispensible to the project's recruitment efforts given USF's high proportion of low-income students who are compelled to work part-time while they pursue full-time academic degree programs. The PI has a proven track record of recruiting talented students from underrepresented groups into mathematics and this grant would allow these activities to continue and expand.

该项目的研究人员将研究在非局部集体运动定律下演化的粒子或代理的动力学和模式形成。具体来说，PI和co-PI将研究集体行为表现出非平凡余维的系统。这种粒子系统的数学渗透到许多学科，从物理学，化学和生物学到控制理论和工程学。这些领域的现代应用包括蛋白质折叠、胶体稳定性和纳米颗粒自组装成超分子结构。在生物学中，类似的数学模型有助于解释在病毒衣壳、蝗虫群和细菌菌落中观察到的复杂现象。该项目的第一阶段将应用和扩展最近开发的数学工具，以识别自然产生共维结构的各种物理和化学相互作用。这直接应用于过程的研究，例如多金属氧酸盐（POM）分子簇自组装成球形超分子结构，其中实验证据具有压倒性，但缺乏对潜在形成机制的理论理解。PI，co-PI和他们的合作者在相互作用是各向同性时，在这种机制的数学理论方面取得了许多重要的发展。该项目的这一部分旨在进一步发展这一理论的方式，将证明有用的其他学科的广泛的研究人员。拟议的研究项目从根本上讲是跨学科的，因为它从化学，生物学和工程等不同领域的无法解释的现象中推导出数学问题。PI和co-PI将应用一套广泛的数学工具，从动力系统，偏微分方程，数学建模和计算方法来解决这些领域正在积极研究的几个问题，包括纳米自组装中所谓的“设计师潜在问题”。通过阐明哪些物理力驱动POM自组装和其他相关现象，如病毒衣壳的形成，我们将提供严格的解决方案，如何设计形成这些重要结构的亚单位的问题。PI的职业目标之一是进一步将数学领域代表性不足的群体纳入其中。PI具有作为旧金山大学弗朗西斯科多元化学生团体导师所需的个人经验和批判性观点。将这些来自代表性不足群体的学生暴露于前沿的数学研究，再加上教师的指导，将激励他们中的许多人攻读数学科学的研究生学位。这项赠款支持的学生研究将证明是该项目招聘工作不可或缺的，因为USF的低收入学生比例很高，他们在攻读全日制学位课程时被迫兼职工作。PI在从代表性不足的群体中招募有才华的学生进入数学领域方面有着良好的记录，这笔赠款将使这些活动得以继续和扩大。