Lattice Models of Polymers: Entanglement Complexity and Confined Geometries

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A polymer is any large molecule made up of repeated molecular units called monomers. The lattice models of Statistical Mechanics have proved to be simple but powerful for studying phase-change behaviour and equilibrium properties for polymers in solution. For this, a polymer chain is represented by a walk on a grid or lattice so that each monomer in the chain is one step apart from its neighbours. Advantageously, the resulting lattice polymer has flexibility (it can take on a variety of shapes) and the requirement that two monomers cannot be in the same location (the excluded volume effect) is easily incorporated. Although highly simplified, it is useful for predicting and understanding any properties which are a consequence of the fact that a polymer is a large flexible molecule. At the same time, these models are of intrinsic mathematical interest due to the wealth of challenging open mathematical questions, many of which have arisen from polymer physics and more recently from molecular biology. For example, enzymes act on DNA to remove entanglements in order for normal cellular processes to proceed. To study this, a model in which two ends of a lattice polymer chain are joined into a ring can be used to represent the large-scale structure of the DNA molecule, and models of enzyme action on DNA have been developed. At the same time, there is interest in understanding the entanglement complexity of polymers confined in nanochannels or capsids (such as DNA in a viral capsid) or when stretched as in single-molecule atomic force microscopy experiments. There are open questions for all these cases regarding the connection between how the polymer is stretched or packed and its entanglement complexity. Lattice models for investigating these questions have been developed. I plan to continue my research on lattice models of polymers using combinatorial analysis and computer simulations in order to better understand enzyme action on DNA and the entanglement complexity of confined polymers. In doing so, I am committed to promoting equity, diversity, and inclusion in my research program. I am aware that women and minorities are historically underrepresented in the sciences and will therefore encourage applications from women, members of visible minorities, indigenous persons, and persons with disabilities to join my research team. The results of this research program will be of interest to molecular biologists and polymer chemists. In the case of enzyme action on DNA, an improved understanding of their action has the potential to impact Canadians by leading to improved cancer treatments. At the same time, the results will add to our general understanding of lattice models of polymers and phase transitions in polymer systems.  Trainees involved in the research program will benefit from the development of critical thinking and problem-solving abilities, as well as technical, collaborative and communication skills.