| Summary: | The Euclidean distance, D, between two points is generalized to the distance between strings or polymers. The problem is of great mathematical beauty and very rich in structure even for the simplest of cases. The necessary and sufficient conditions for finding minimal distance transformations are presented. Locally minimal solutions for one-link and two-link chains are discussed, and the large N limit of a polymer is studied. Applications of D to protein folding and structural alignment are explored, in particular for finding minimal folding pathways. Non-crossing constraints and the resulting untangling moves in folding pathways are discussed as well. It is observed that, compared to the total distance, these extra untangling moves constitute a small fraction of the total movement. The resulting extra distance from untangling movements (Dnx ) are used to distinguish different protein classes, e.g. knotted proteins from unknotted proteins. By studying the ensembles of untangling moves, dominant folding pathways are constructed for three proteins, in particular a knotted protein. Finally, applications of D, and related metrics to protein folding rate prediction are discussed. It is seen that distance metrics are good at predicting the folding rates of 3-state folders. === Science, Faculty of === Physics and Astronomy, Department of === Graduate
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