Lattice Models of Polymers: Entanglement Complexity and Confined Geometries

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The lattice models of Statistical Mechanics have proved to be powerful tools for studying phase-change behaviour and equilibrium properties for polymers in solution. For such models, a polymer is considered to be any long chain molecule made of repeated units called monomers and 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. 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 a highly simplified model, it is useful for predicting and understanding  phenomena that result from the fact that polymers are large flexible molecules. 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 by us. At the same time, there is interest in understanding the entanglement complexity of polymers confined in pores or capsids (such as DNA in a viral capsid) and also interest in determining the role of entanglements in polymer crystallization and polymer melts. For example, there are open questions about the relationship between how a polymer is packed in a capsid and its entanglement complexity, and for polymer melts, about the best approach for measuring the extent of entanglement. Lattice models for investigating these questions have also been developed by us. 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. The results 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. More generally, the results will add to our overall understanding of lattice models of polymers and phase transitions in polymer systems.