


Generalization of Distance to HighDimensional ObjectsWe have recently worked out the equations governing the distance between two space curves, such as those shown below. This is a nonlinear PDE with Lagrange functional constraints. The usefulness of such a metric is to answer such questions as "what is the distance from a random walk configuration to this (smoothed) protein structure: Another example, "what do the X and Y axes mean in this diagram of the protein folding funnel?"
Do you know how to move a piece of string from configuration r_{A}(s) to configuration r_{B}(s) while traveling the minimal amount of distance? See
...and to a beta hairpin: In it's simplest form, the equation for the transformation using up the minimal amount of motion looks like this: and an approximation to it, correlates remarkably well with how long it takes a protein to fold up: When knotting or chain noncrossing becomes important, things get tricky: We can actually look at the taxonomy of folding transformations: Two or more atoms cannot occupy the same space when such selfavoidance is accounted for, the trajectories that atoms take when a protein folds up can be highly nontrivial. Here is one such trajectory for a socalled C_alpha atom (blue bead) in a protein called apomyoglobin, the minimal distance trajectory is curved and even doubles back on itself! See Also: Plotkin SS, "Generalization of Distance to Higher Dimensional Objects" Proc. Natl. Acad. Sci. 104, 14899 (2007) Mohazab, AR, Plotkin, SS, “Minimal distance transformations between links and polymers: Principles and applications” J. Phys Condens Matter 20 244133 (2008) Mohazab AR, Plotkin SS, "Structural alignment using the generalized Euclidean distance between conformations" Int. J. Quant. Chem. 109, 3217–3228 (2009) Das A, Sin BK, Mohazab AR, and Plotkin SS "Unfolded protein ensembles, folding trajectories, and refolding rate prediction" J. Chem. Phys. 139, 121925 (2013) 

