Collaborative Research: A Study of the Transition of Knot Space from Confinement to Relaxation

Western Kentucky University Research Foundation, Bowling Green KY

Investigators

Abstract

Circular molecules confined to a small volume are often modeled by random polygons confined in a sphere and extracted (that is relaxed) circular molecules are modeled by relaxed random polygons without confinement. The PIs propose to explore the geometric changes that occur during the transition of the polygonal knotspace from confinement to relaxation and to establish correlations between these geometric changes and the topological complexity of the polygons. The results of this research project will provide benchmark data on the relationships between certain knot complexity measures and some geometric measures, where all quantities are measured as averages over families of random polygons before and after they are relaxed. The results can guide the evaluation of experimental data such as the data available in the case of the bacteriophage P4 virus. To reach the goal of the proposed research, several critical objectives must be achieved: a) The development of a fast, reliable, and unbiased algorithm to generate large sets of long equilateral random polygons within a confining volume; b) The development of relaxation schemes for equilateral random polygons and their corresponding algorithms; c) Quantification of the effect of topology on geometric changes of random polygons when transitioning from confinement to relaxation and d) Identification of inferences about topological properties of the random polygons using the average geometric properties of the polygons before and after relaxation. The proposed research will provide a systematic study between the relationships between various geometric measures and topological properties of knots in the average sense when the knots under consideration undergo a transition change from volume confinement to relaxation. The proposed research will reveal potentially important and interesting relationships among these quantities and the role of confinement in these relationships. It is well known that macromolecular self-assembly processes are key players in the complex network of interactions that take place in every organism. One of these self-assembly processes is the packing of the genetic material in the capsids of viruses. Little is know about the details of the packing processes, because in a confined small volume DNA is usually condensed and folds in ways that are difficult to quantify experimentally. DNA molecules that are forcefully removed from bacteriophage P4 capsids often form complicated knots that are a result of the packing process. Thus, the extracted DNA carries important information about how the DNA is packed inside the capsids. The question of how to decipher such information is a main motivation of the proposed research. Circular molecules confined to a small volume are often modeled by random polygons confined in a sphere. On the other hand, extracted circular molecules are usually modeled by relaxed random polygons without confinement. The proposed research will explore the geometric changes that occur during the transition of the polygonal knot space from confinement to relaxation and to establish correlations between these geometric changes and the topological complexity of the polygons. The results will provide some essential benchmark data on the relationships between certain knot complexity measures and some geometric measures, which are important in order for us to fully understand the mechanism of DNA packing in a tight space. The PIs their students (ranging from exceptionally talented high-school students, to undergraduates, graduates, and Ph. D. students) will develop mathematical tools and computational models that will be made freely available to the scientific community and/or interested educators. The results of the work can be used in areas such as biology and physics to check the validity of models of highly condensed DNA or tightly packed polymers.