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.
Over the last decade the genetic information of many organisms was sequenced and analyzed. The sequences were used to investigate many aspects of the organisms. However, the sequences do not answer all questions, such as the structure or arrangement of the genetic material. 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 known 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 with current technology. DNA molecules have been forcibly removed from bacteriophage P4 capsids and it has been observed that these molecules are circular and in fact often form complicated knots which are a result of the packing process. How the DNA is packed inside the capsid cannot be determined directly with current experimental techniques, however, the extracted DNA carries important information about how the packing. Various models have been proposed and one way to check the validity of a proposed model is to verify whether its computer simulation produces polygons with a knotting behavior which is consistent with what has been observed experimentally. This National Science Foundation project has supported the development of the foundations and various mathematical tools to start working on the above topic. The goal of this project was not to model DNA in a virus capsid – but to create a mathematically model using equilateral polygons in spherical confinement. This model will answer the questions what knotting behavior may be observed due to tight spherical confinement. To answer this question millions of polygons in confinement must be generated randomly and analyzed. However, no approaches existed which allowed such a large scale study, since no methods existed for generating polygons in confinement completely randomly based on mathematically derived probability distributions. The team developed several novel algorithms to generated true random polygons in tight confinement. The proofs that these algorithms actually work required the development of new tools i.e probability density functions that grantee that all confined polygons are equally likely to be generated.Using the algorithms the PIs generated millions of random polygons in confinement to make sure that large enough samples actually reflect the statistical behavior of the polygons. This study allowed the first glimpse at how polygons entangle themselves in confined spaces – and how the length of the polygons and the tightness of the confinement influence the entanglement. In addition to the knotting behavior, the team also mathematically analyzed other characteristics of the polygons (such bending angles between segemtns) which are related to the 3D structure of the polygons packed in confinement. Our methods offer new capabilities for studying and understanding the entanglement of polymer chains and other objects which are in confined spaces. The application of our methods and related information is not restricted to DNA packing or even biology. There are indications that ‘quantum-flux tubes’ which are formed in ‘confinement’ caused by heat and pressure are knotted and entangled. Our methods are of interest to people working in areas where biology, mathematics, physics, and computation interact. Over the lifetime of the project, many students at various levels were involved – from high school students to undergraduate to mater’s students in mathematics and computer science. The students were involved in projects involving math, programming, and problem solving. Many of the students presented their work at local and regional conferences and some are co-authors of papers. We believe that mentoring students – starting with high school students – during projects, research activities, and subsequent communication of the work is important for more students to decide to become involved in STEM research areas.
