RUI: Theory and simulations of knotting in physical and biological systems ranging from proteins to glueballs

在物理世界中，从操纵DNA的微型酶到人类规模的花园软管，再到跨越几光年的相对论喷气式飞机，在各个尺度上都可以看到纠缠。这些纠缠的功能与它们的物理形式有关。在这个拟议的项目中，PI、合作者和本科生学习打结管的物理形式。具体地说，拟议的项目有两个主要目标：1)模拟相互接触的厚管的运动，2)严格研究开链内的打结。紧密状态下的结构型被用来模拟凝胶电泳中打结的DNA环的相对速度，预测DNA双螺旋的斜率，并对亚原子胶球状态的结构进行分类。PI和合作者已经编写了计算机代码来收紧结的配置。这一规范及其相应的理论已经导致了处理管状物体自接触问题的一般模型。PI和协作者将扩展该模型，以便将其应用于其他物理系统。该项目的第二部分涉及研究开链中的打结。打结蛋白的发现激发了最近对开链打结分类的兴趣。PI和合作者将专注于链内打结的纠缠稳定性，即构型的几何阻力来改变其打结性质。蛋白质链将被比作随机链，以了解打结在蛋白质生命中所起的作用。当人们想到一个结，它通常是由绳子制成的。绳索的物理属性，如厚度，限制了它的操纵方式。例如，一个人如果不切断绳子，就不能穿过绳子。当一个人在一根绳子上打一个结并把它拉紧时，绳子的表面就会与自己接触，绳子就会自然地沿着接触点滑动。模拟这些沿接触的运动是困难的，但在许多领域都有应用，例如研究弹性杆和计算机图形学。PI和合作者编写了一种绳结收紧算法，以一种数学上合理的、物理上直观的方式在绳状材料的自接触中偏转运动。在本项目的第一部分中，该算法将被扩展到研究具有自接触的其他物理系统。一些可能的应用包括测试子弹对形成防弹背心的编织材料的影响，以及分析划船、钓鱼和外科手术结的安全性。数学家通常研究的结的类型是没有自由端的闭合环，与我们在日常生活中看到的有自由端的结不同，例如在鞋带和花园软管中。然而，研究具有自由端的物体中打结的重要性正变得越来越明显。例如，一些蛋白质含有这些类型的结，尽管这些结的功能仍在争论中。由于蛋白质基本上参与了细胞中的每一个过程，当蛋白质折叠进入和离开其活动状态时，打结似乎是一个不必要的障碍。从数学的角度来看，在开链上打结并不是很好的理解，但应该与一个人关于什么是和什么不是“打结”的直觉概念相一致。打结的一条线应该是稳定的，这样，例如，一个人的鞋子就不会松开。PI、合作者和本科生将学习在开链中打结的概念以及链的空间结构与其稳定性之间的关系。最终，这将导致对蛋白质内部打结的深入了解。除了科学目标外，这笔赠款还有广泛的教育目标。本科生将直接得到助学金的支持，获得研究过程中的关键经验，并在专业会议上展示他们的结果。国际和平协会将继续通过讲座和组织跨学科会议，与来自不同领域的学生、非专家和专家建立联系。

Entanglement is seen at every scale in the physical world, from microscopic enzymes manipulating DNA to human-scale garden hoses to relativistic jets spanning light years in distance. The function of these entanglements is related to their physical form. In this proposed project, the PI, collaborators, and undergraduate students study the physical form of knotted tubes. Specifically, the proposed project has two main goals: 1) to model motions of thick tubes in contact with each other, and 2) to rigorously study knotting within open strands. Knot configurations in a tight state have been used to model the relative speed of knotted DNA loops in gel electrophoresis, to predict the slope of the DNA double helix, and to classify the structure of the sub-atomic glueball states. The PI and collaborators have written computer code to tighten knot configurations. This code, and its corresponding theory, has led to a general model for handling the problem of self-contact of tube-like objects. The PI and collaborators will extend the model so that it can be applied to other physical systems. The second portion of the project concerns studying knotting in open chains. The discovery of knotted proteins spurred the recent interest in classifying knotting within open chains. The PI and collaborators will focus on the entanglement stability of the knotting within the chains, i.e. the resistance of the geometry of the configuration to change its knotting properties. Protein chains will be compared to random chains to understand the role that knotting plays in the life of the proteins.When one thinks of a knot, it is usually made out of rope. The rope has physical properties, such as thickness, that limits how it can be manipulated. For example, one cannot pass a rope through itself without cutting the rope. When one ties a knot in a piece of rope and pulls it tight, the surface of the rope comes in contact with itself and the rope slides naturally along the contacts. Modeling these motions along contacts is difficult but has applications in many fields, such as the study of elastic rods and computer graphics. The PI and collaborators have coded a knot tightening algorithm that deflects motions across self-contacts for rope-like materials in a mathematically sound, and physically intuitive fashion. In the first portion of this project, this algorithm will be extended to study other physical systems with self-contact. Some possible applications include testing the effect of a bullet's impact on woven materials forming bullet-proof vests and analyzing the security of boating, fishing, and surgical knots. The type of knots typically studied by mathematicians are closed loops with no free ends, in contrast to the knots we see in everyday life, such as in shoelaces and garden hoses, that have free ends. However, the importance of studying knotting in objects with free ends is becoming increasingly clear. For example, some proteins contain these types of knot, although the function of the knots is still being debated. Since proteins are involved in essentially every process in cells, knots would seem to be an unnecessary obstruction as the protein folds in and out of its active state. Knotting in open strands is not well understood from a mathematical perspective, but should coincide with one's intuitive notion of what is and what is not "knotted". A "knotted" strand should be stable so that, for example, a person's shoes do not come untied. The PI, collaborators, and undergraduate students will study notions of knotting in open strands and the relationship between the spatial structure of the strand and its stability. Ultimately, this will lead to insights into knotting within proteins. In addition to the scientific goals, this grant has broad educational objectives. Undergraduate students will be directly supported by the grant, gaining critical experience in the research process and presenting their results at professional conferences. The PI will continue to be involved in connecting with students, non-specialists, and specialists from different fields through talks and organizing interdisciplinary conferences.