LEAPS-MPS: DNA Knot Shadows Explorations

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。数学和生物学交叉领域的最新进展表明，拓扑方法可以帮助解决科学中的一个基本挑战：DNA是如何作为生物体的组成部分发挥作用的。在实验室实验中，生物学家将环状DNA底物与蛋白质反应，并以DNA形状变化的形式捕获蛋白质特征。拓扑学对象的数学（例如，结、链和缠结）然后应用于这些生物观察来推断蛋白质的作用。该项目的研究目标是对实验获得的蛋白质- dna相互作用产物图像进行分类；精确模拟蛋白质- dna相互作用的链动力学；并分享在这项工作中开发的计算工具，最终为研究相关科学问题的数学家和生物学家创建一个数据库。该项目还旨在吸引代表性不足的学生从事STEM研究和职业，特别是在数学和生物学的交叉领域。为了进一步了解DNA拓扑结构的动力学，PI将应用组合工具来解释蛋白质作用的DNA产物的二维投影。由于DNA结是三维物体，当它们被投射到二维时，一些信息不可避免地会丢失。在以往的工作中，PI在代数上扩展了四平面绞结关系之间的关系。本研究的总体目标是通过关注无交叉信息的结的二维投影，扩展结四联体与组合学的效用。拟议的研究将允许创建未被先前技术分类的四胞胎，同时也改进了先前发现的结距离界限。此外，该项目将继续履行PI的服务使命：通过创建和支持项目来支持那些在STEM中代表性不足的人，这些项目可以提高STEM中女性和有色人种的知名度，扩大她们的声音，并在STEM中创建网络和社区，提供共享资源的机会。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 award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Recent advances at the intersection of mathematics and biology have demonstrated that topological methods can help address one of the fundamental challenges in science: how DNA functions as the building blocks of a living organism. In laboratory experiments, biologists react circular DNA substrates with proteins and capture protein signatures in the form of changes in the shape of the DNA. The mathematics of topological objects (for example, knots, links, and tangles) are then applied to these biological observations to deduce the action of proteins. The project's research objectives are to classify experimentally obtained images of products of protein-DNA interactions; to accurately model strand dynamics of protein-DNA interactions; and to share the computational tools developed in the course of this work, ultimately creating a database for mathematicians and biologists working on related scientific questions. This project is also designed to attract underrepresented students to STEM studies and careers, particularly at the intersection of mathematics and biology.To further the understanding of the dynamics of DNA topology, the PI will apply combinatorial tools to interpret 2-dimensional projections of DNA products of protein action. As DNA knots are 3-dimensional objects, some information is inevitably lost when they are projected into 2-dimensions. In previous work, the PI extended the relationship between skein relation knot quadruples of 4-plats algebraically. The overall goal of this proposed research is to extend the utility of knot quadruples with combinatorics by focusing on the 2-dimensional projections of knots with no crossing information. The proposed study will allow for the creation of quadruples that have not been categorized with previous techniques, while also refining previously found bounds on knot distances. In addition, the project will continue to pursue the PI's service mission: to support those underrepresented in STEM by creating and supporting programs that increase visibility of, and amplify the voices of, women and people of color in STEM, and create networks and community in STEM to provide opportunities to share resources.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.