CAREER: Entanglement of Active Polymers

该奖项支持活性聚合物的理论、数学和计算研究以及教育，活性聚合物可以通过施加力和改变其形状来驱动生物功能。生物细胞含有活性聚合物——长丝状分子——可以在细胞周期中消耗能量并改变连通性和结构。PI旨在开发一种方法，可以提供洞察这些系统的动态重组。PI的目的是研究这些长丝的多链几何形状和拓扑结构与聚合物之间的活性互连是否可以单独描述在许多情况下解释活性物质长丝力学的关键元素。本研究使用一种涉及拓扑学和计算机模拟领域的数学思想的方法来检验这一假设，以获得可以与实验相比较的结果。PI旨在使用严格的数学方法来理解、建模并最终控制聚合物细丝如何在生物应用的活性物理系统中纠缠。该项目将使人们更好地了解生命物质，并将推动新型软质玻璃材料的智能制造。该项目还支持外展活动，包括公开演讲、大学外展项目和挑战者STEM中心，这些活动可以向潜在的广泛受众展示研究的各个方面。通过这个项目开发的软件和模拟技术将与社区广泛共享。研究结果将由PI及其学生在跨学科会议（包括PI组织的会议）上发表。此外，PI坚定地致力于扩大代表性不足的少数民族和妇女在STEM中的参与；将开设新课程，培养掌握21世纪数学工具的跨学科科学家。技术总结：活性物质被用来对一系列物理系统进行分类，这些系统是由于“活性”成分的存在而失去平衡的，这些“活性”成分通过耗散能量来施加力。传统的聚合物物理学观点对这种体系的动态重组提供了有限的理解。该领域的一个挑战是将孤立的细丝的性质与一组细丝的性质联系起来。这涉及到纠缠聚合物领域的一个重大挑战，即如何测量三维空间中开放曲线的纠缠。本项目将利用拓扑学、建模和仿真技术来测量活性物质细丝中的拓扑纠缠，并为其力学提供一个新的模型。本研究通过定义和研究三维空间和采用周期边界条件的系统中开放曲线集合的Jones多项式，推进了知识的进步，打破了现有的技术壁垒(1)在拓拓学方面；(2)通过提供活性物质丝的粘弹性响应的新模型，在理解材料科学和生物学中的纠缠效应方面。这项工作的目的是建立这种系统行为的预测模型，并有可能通过明智地选择它们的化学成分和结构来控制它们的功能，例如，通过改变活性交联的数量或交联基序的类型。该奖项由材料研究部的凝聚态物质和材料理论项目以及数学科学部的拓扑和数学生物学项目共同资助。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。

NONTECHNICAL SUMMARY This award supports theoretical, mathematical, and computational research, and education on active polymers which can drive biological function by exerting forces and changing their shape. Biological cells contain active polymers - long filamentous molecules - that can consume energy and change connectivity and architecture during the cell cycle. The PI aims to develop a method that can provide insight into the dynamic reorganization of such systems.The PI aims to investigate whether the many-chain geometry and topology of these filaments in combination with active interconnections among polymers alone can describe key elements that account for the mechanics of active matter filaments in many contexts. This research examines this hypothesis using an approach that involves mathematical ideas from the field of topology and computer simulation to obtain results that can be compared to experiments. The PI aims to use rigorous methods from mathematics to understand, model, and eventually control how polymer filaments entangle in active physical systems with biological applications. This project will lead to a better understanding of living matter and will advance the smart manufacturing of new soft glassy materials. This project also supports outreach activities, including public talks, university outreach programs, and the Challenger STEM Center, which can present aspects of the research to a potentially wide audience. Software and simulation techniques developed through this project will be shared broadly with the community. Results will be presented by the PI and her students at interdisciplinary conferences, including those organized by the PI. Additionally, the PI is strongly committed to broadening participation of underrepresented minorities and women in STEM; new courses will be developed to train interdisciplinary scientists in 21st-century mathematical tools. TECHNICAL SUMMARY Active matter is used to classify a range of physical systems that are driven out of equilibrium by the presence of ''active'' constituents that exert forces by dissipating energy. Conventional polymer physics arguments provide limited understanding of the dynamic reorganization of such systems. A challenge in the field is to connect properties of isolated filaments to properties of a collection of filaments. This relates to a big challenge in the field of entangled polymers, which is how to measure entanglement of open curves in 3-space. This project will use topology, modeling, and simulation to measure topological entanglement in active matter filaments and provide a new model for its mechanics. This research advances knowledge and breaks existing technical barriers (1) in topology by defining and studying the Jones polynomial of a collection of open curves in 3-space and in systems employing Periodic Boundary Conditions and (2) in understanding entanglement effects in materials science and biology, by providing a new model for the viscoelastic response of active matter filaments. This work is aimed to lead to predictive modeling of the behavior of such systems with the possibility of controlling their functions by judicious selection of their chemical compositions and structures, for example, by changing the number of active cross-links or the type of cross-linking motifs. This award is jointly funded through the Condensed Matter and Materials Theory Program in the Division of Materials Research, and the Topology and Mathematical Biology Programs in the Division of Mathematical Sciences.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.