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

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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.