Geometry and physics of knots

KNOTS are usually categorized in terms of topological properties that are invariant under changes in a knot's spatial configuration(1-4). Here we approach knot identification from a different angle, by considering the properties of particular geometrical forms which we define as 'ideal'. For a knot with a given topology and assembled from a tube of uniform diameter, the ideal form is the geometrical configuration having the highest ratio of volume to surface area. Practically, this is equivalent to determining the shortest piece of tube that can be closed to form the knot. Because the notion of an ideal form is independent of absolute spatial scale, the length-to-diameter ratio of a tube providing an ideal representation is constant, irrespective of the tube's actual dimensions. We report the results of computer simulations which show that these ideal representations of knots have surprisingly simple geometrical properties. In particular, there is a simple linear relationship between the length-to-diameter ratio and the crossing number-the number of intersections in a two-dimensional projection of the knot averaged over all directions. We have also found that the average shape of knotted polymeric chains in thermal equilibrium is closely related to the ideal representation of the corresponding knot type. Our observations provide a link between ideal geometrical objects and the behaviour of seemingly disordered systems, and allow the prediction of properties of knotted polymers such as their electrophoretic mobility(5)

Urea and Guanidinium Induced Denaturation of a Trp-Cage Miniprotein

Using a combination of experimental techniques (circular dichroism, differential scanning calorimetry, and NMR) and molecular dynamics simulations, we performed an extensive study of denaturation of the Trp-cage miniprotein by urea and guanidinium. The experiments, despite their different sensitivities to various aspects of the denaturation process, consistently point to simple, two-state unfolding process. Microsecond molecular dynamics simulations with a femtosecond time resolution allow us to unravel the detailed molecular mechanism of Trp-cage unfolding. The process starts with a destabilizing proline shift in the hydrophobic core of the miniprotein, followed by a gradual destruction of the hydrophobic loop and the ?-helix. Despite differences in interactions of urea vs guanidinium with various peptide moieties, the overall destabilizing action of these two denaturants on Trp-cage is very similar