Researchers from California develop an efficient, ergodic generation method for off-lattice walks with excluded volume and analyze the effects of thickness on shape and knotting. They spoke to JPhys+ about their recent publication in Journal of Physics A: Mathematical and Theoretical, and about how a love of handcrafts and art can help with an understanding of knot theory.
We are Laura Plunkett, an Assistant Professor of Mathematics at Holy Names University in Oakland, California and Kyle L. Chapman, a lecturer at University of California, Santa Barbara, where we both attended graduate school.
What prompted you to pursue this field of research?
Laura: Low dimensional topology, knot theory in particular, was immediately interesting because I love handcrafts and art. My creative projects gave me intuition and insight about the problems. Also, being able to approach the work from both theoretical and experimental angles diversified and enriched the work for me. I really enjoyed growing as a programmer over the course of researching my thesis and this paper.
While in graduate school, I had already developed many of the tools used in the proof, and I knew the result could be extended to polygons. After a talk about the progress I had made, Kyle began working on an analogous proof for rings. I had been working on proving that the method was transitive, that any configuration could be sampled with my method. Kyle showed that both the method for polygons and mine for walks were not just transitive, but ergodic, which really strengthened the work.
Working with Kyle and our advisor, Ken Millett, has been wonderful. We drew complicated diagrams and talk about them for hours, thinking about edge cases and counter examples. They are wonderful colleagues!
Kyle: I find this area of research very interesting because it is so accessible to visualization. A person can picture the movements of the various objects. I am often drawn to problems where there can be such visualization, such as planar algebra and diagrammatic calculus type problems. In the talk Laura gave on her doctoral research progress, she said “Someone should solve this problem for closed chains” and the problem got stuck in my head. It is one of those questions that seemed like it should have been much easier than it was. We would discuss moves by taking our fingers and moving them around in space to try to build spatial intuition. In the end, as with many problems that seem easy, the challenge was in providing the details.
It is also a nice draw that unlike many areas of mathematics, it is easy to explain the implications. The topological interactions of DNA are one of the tools in certain chemotherapy drugs and antibiotics which means there are direct medical implications, even for those of us who focus fully on the abstracted picture
What is this latest paper all about?
We have created an efficient generation method for random walks with a specified excluded volume, (or thickness,) and have proved that our generation method is ergodic, meaning that it samples the space of all such walks uniformly. These random walks are a good model for a wide variety of polymers, from DNA to plastics. The generated data shows that increasing the excluded volume, even by very small amounts, decreases the formation of knotted conformations dramatically. We have also analyzed how the introduction of excluded volume changes the scaling behavior of the squared radius of gyration of the walks, and have found that the increased growth exponent, seen in other excluded volume models and in walks on the simple cubic lattice, is observed immediately with the introduction of nonzero thickness.
What do you plan to do next?
We are working on the extension of the results in this first paper, and we are studying how the knot type distribution changes as a function of excluded volume. We also plan to include more extreme data in this study, very thin and very thick walks, to see how the trends we observe persist to these boundary cases.
Similar questions can be asked with closed rings, and as with the open chains, the questions we ask next are about the observed data, and how much of an effect extremely small thicknesses can have on the spatial structure of the observed conformations. It would also be useful to be able to consider other structures, such as the question of whether we can extend these methods to hairy chains (chains with small chains attached) or tadpoles (closed chains with an open chain attached). These questions are naturally harder because some of the homogeneity is lost.
- Asymptotics of knotted lattice polygons
- Characteristics of shape and knotting in ideal rings
- Off-lattice random walks with excluded volume: a new method of generation, proof of ergodicity and numerical results
This work is licensed under a Creative Commons Attribution 3.0 Unported License.
Author images owned by Laura Plunkett and Kyle L. Chapman respectively, used with permission. Featured Image taken from Laura Plunkett (nee Zirbel) and Kyle Chapman 2016 J. Phys. A: Math. Theor. 49 135203 © IOP Publishing Ltd, all rights reserved.