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Generalization of Distance to High-Dimensional Objects

We 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
Plotkin, S. S. “Generalization of distance to higher dimensional objects”, Proc Nat Acad Sci USA. 104 14899-14904 (2007).
for the answer!

This shows the minimal transformation to an alpha helix.

...and to a beta hairpin:

See Also:

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, "Minimal folding pathways for coarse-grained biopolymer fragments" Biophysical Journal 95 5496 (2008)

Mohazab AR, Plotkin SS, "Structural alignment using the generalized Euclidean distance between conformations" Int. J. Quant. Chem. 109, 3217–3228 (2009)