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:

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 non-crossing 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 self-avoidance is accounted for, the trajectories that atoms take when a protein folds up can be highly non-trivial. Here is one such trajectory for a so-called C_alpha atom (blue bead) in a protein called apo-myoglobin, the minimal distance trajectory is curved and even doubles back on itself!