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.
Knotting and untangling happen continuously in nature. For example, when DNA creates a copy of itself, enzymes called topoisomerases untangle the two copies from each other. If the two copies cannot untangle, the cell dies. For this reason, topoisomerase is a target for drugs. Some chemotherapy drugs (especially for children) and the antibiotic ciproflaxin (a treatment for anthrax exposure) work by keeping topoisomerase from untangling the copies of DNA. The lessons learned from understanding how nature manages knotting and untangling can help in the development new drugs and materials that improve the human condition. Of course, we see knots every day in life: in shoelaces, extension cords, and headphone cables. These knots are called "open knots" because they have two free ends. Mathematical knots are "closed" in the sense that they form a loop, like an extension cord plugged into itself. Traditionally, mathematicians have studied these closed knots because once a knot exists in a closed extension cord, there is no way to manipulate the cord to look like a circle without unplugging the cord. In other words, in the closed extension cord, the knotting is trapped and this makes the knotting much easier to study mathematically. The PI's project involved studying open and closed knotting seen in proteins and subatomic glueballs. Proteins are long chains (or groups of long chains) of molecules which control functions in cells that keep living organisms living. For a protein to function, it must fold into a form called its "native state" quickly and reproducibly. When proteins misfold, terrible things can happen (for example, Mad Cow disease comes from misfolded proteins). Around 1% of proteins have native states that form open knots. The knotting seems like an unnecessary complication to the complicated folding process. One main question is: what proteins are knotted and why are they knotted? The PI and an international, multi-disciplinary group of collaborators discovered new knotted proteins and showed that proteins that perform the same function in different organisms have preserved their knotting despite hundreds of millions of years of evolutionary separation. This suggests that knotting contributes to the function of these proteins and thus is an attribute that could be exploited in designing, for example, new drugs. The group also developed a knotted protein database/website where the public can study knotted proteins and also submit proteins for analysis. Glueballs are subatomic particles, observable only for small fractions of a second as residues of particles colliding in particle accelerator experiments. It has been hypothesized that glueballs form as a roughly uniform closed tube and, due to physical forces, the tube shrinks in length while keeping its diameter fixed. Upon creation, a glueball quickly forms this tightened state where it spends the great majority of its lifetime. While experimental data supports the hypothesis, the data is obtained indirectly and many individual observations are averaged (using a complicated procedure) to obtain one master data point for a given glueball state. If there are two glueballs with similar energy values, different glueballs can be grouped inadvertently into the same master data point and lead to false data. As a part of this grant, the PI and collaborators computed the tightened states for several hundred different types of closed knots and compared the data with known glueball information. The data suggests that there has been some inadvertent merging of data and also provides some "goal energies" where researchers could search for new glueballs. While the several hundred types of knots might seem excessive (and it is when thinking only about glueballs), these tightened knots also were used to model knotted DNA moving through a gel and could prove to be a critical component in designing more robust practical knots, e.g. suture knots. These tightened knots were also made available to the public via a web page. As a part of this project, the PI involved 13 undergraduate students (with majors in mathematics, accounting, computer science, actuarial science, physics, statistics, chemistry, and electrical engineering) in cutting-edge research. Many of these students traveled to professional conferences and the students presented 18 posters and gave nine talks over the course of the grant. The PI traveled in the USA and abroad to communicate the work and establish new collaborations. He gave talks to undergraduates as well as multi-disciplinary audiences of top researchers in the natural sciences. In addition, the PI organized four multi-disciplinary conferences which brought together researchers from disparate research areas who would not normally have opportunities to interact.