Lattice models of polymers: entanglement complexity and phase transitions

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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 alot of 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 a useful model for predicting and understanding any properties which are mainly a consequence of the fact that a polymer is a large flexible molecule made up of repeated units. 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, concerning them. For example, enzymes act on ring-like biopolymers such as circular DNA to remove entanglements from DNA 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 a model of enzyme action on the resulting ring is in development. At the same time, polymer chemists are interested in determining the role of entanglements in polymer crystallization and polymer melts. It is an open question, however, as to the best approach for measuring the extent of entanglement. A model for investigating entanglement measures has also 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 determine good measures of entanglement complexity. The results will be of interest to molecular biologists and polymer chemists. At the same time, the results will add to our general understanding of lattice models of polymers and phase transitions in polymer systems.