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Research

My research investigates the geometry of thickened curves through mathematical experimentation and geometric analysis. A thickened curve is a self-avoiding space curve whose unit-radius tube does not overlap itself, coming at most into contact. This simple model aims to relate geometry and real tube-like materials, by enabling a geometry-centred study that would be challenging with more physically detailed and thereby technically demanding models. For curves, tangling or intertwining is a natural geometric phenomenon, with important consequences for the properties of many materials.

My recent publication with Myf Evans considers the thickened curve as a model for a short open biopolymer, dissolved in a hard-sphere solvent (liquid). A consequence of the fluid environment is that the tube folds to adopt an energetically favourable configuration—a process known as solvation. Our study utilises the morphometric approach to solvation, a geometry-based method for computing the thermodynamic energy of the fluid. Using this approach as a simulation technique we derived a phase diagram of low energy configurations over a comprehensive range of physically realistic solvents. In essence this demonstrates that the solvent can fold a freely formable tube into complex geometries. The two most prominent examples are a double-helical configuration, in which the tube folds back on itself, and an overhand-knot shape (shown right). Our main result is the identification of these geometries as thermodynamically favourable, in particular when compared with the single helix—a prevalent biopolymer structural motif.

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