A Practical Solution for Aligning and Simplifying Pairs of Protein Backbones under the Discrete Fréchet Distance

Aligning and comparing two polygonal chains in 3D space is an important problem in many areas of research, like in protein structure alignment. A lot of research has been done in the past on this problem, using RMSD as the distance measure. Recently, the discrete Fréchet distance has been applied to align and simplify protein backbones (geometrically, 3D polygonal chains) by Jiang et al., with insightful new results found. On the other hand, as a protein backbone can have as many as 500∼600 vertices, even if a pair of chains are nicely aligned, as long as they are not identical, it is still difficult for humans to visualize their similarity and difference. In 2008, a problem called CPS-3F was proposed to simplify a pair of 3D chains simultaneously under the discrete Fréchet distance. However, it is still open whether CPS-3F is NP-complete or not. In this paper, we first present a new practical method to align a pair of protein backbones, improving the previous method by Jiang et al. Finally, we present a greedy-and-backtrack method, using the new alignment method as a subroutine, to handle the CPS-3F problem. We also prove two simple lemmas, giving some evidence to why our new method works well. Some preliminary empirical results using some proteins from the Protein Data Bank (PDB), with comparisons to the previous method, are presented.