Minimal distance transformations between links and polymers: Principles and examples

TL;DR

This paper generalizes the concept of Euclidean distance to polymers, deriving conditions for minimal transformations involving rotations and translations, with applications to protein folding and structural alignment.

Contribution

It introduces a framework for calculating minimal transformations between polymer configurations, extending classical point distance measures to one-dimensional objects.

Findings

Derived necessary and sufficient conditions for minimal transformations.

Showed convergence of the distance to MRSD for many links.

Applied the framework to protein folding and structural alignment.

Abstract

The calculation of Euclidean distance between points is generalized to one-dimensional objects such as strings or polymers. Necessary and sufficient conditions for the minimal transformation between two polymer configurations are derived. Transformations consist of piecewise rotations and translations subject to Weierstrass-Erdmann corner conditions. Numerous examples are given for the special cases of one and two links. The transition to a large number of links is investigated, where the distance converges to the polymer length times the mean root square distance (MRSD) between polymer configurations, assuming curvature and non-crossing constraints can be neglected. Applications of this metric to protein folding are investigated. Potential applications are also discussed for structural alignment problems such as pharmacophore identification, and inverse kinematic problems in motor learning and control.