Extraction of geometrically similar substructures: Least-squares and Chebyshev fitting and the difference distance matrix

Abstract

In analysis, comparison and classification of conformations of proteins, a common computational task involves extractions of similar substructures. Structural comparisons are usually based on either of two measures of similarity: the root-mean-square (r.m.s.) deviation upon optimal superposition, or the maximal element of the difference distance matrix. The analysis presented here clarifies the relationships between different measures of structural similarity, and can provide a basis for developing algorithms and software to extract all maximal common well-fitting substructures from proteins. Given atomic coordinates of two proteins, many methods have been described for extracting some substantial (if not provably maximal) common substructure with low r.m.s. deviation. This is a relatively easy task compared with the problem addressed here, i.e., that of finding all common substructures with r.m.s. deviation less than a prespecified threshold. The combinatorial problems associated with similar subset extraction are more tractable if expressed in terms of the maximal element of the difference distance matrix than in terms of the r.m.s. deviation. However, it has been difficult to correlate these alternative measures of structural similarity. The purpose of this article is to make this connection. We first introduce a third measure of structural similarity: the maximum distance between corresponding pairs of points after superposition to minimize this value. This corresponds to fitting in the Chebyshev norm. Properties of Chebyshev superposition are derived. We describe relationships between the r.m.s. and minimax (Chebyshev) deviations upon optimal superposition, and between the Chebyshev deviation and the maximal element of the difference distance matrix. Combining these produces a relationship between the r.m.s. deviation upon optimal superposition and the maximal element of the difference distance matrix. Based on these results, we can apply algorithms and software for finding subsets of the difference distance matrix for which all elements are less than a specified bound, either to select only subsets for which the r.m.s.deviation is less than or equal to a specified threshold, or to select subsets that include all subsets for which the r.m.s. deviation is less than or equal to a threshold.

The Effects of Various Alcohols on the Stability of Melittin: A Molecular Dynamics Study

“…Although some complicated algorithms have been proposed to estimate the structural similarity between proteins, 45,46 the RMSD remains the simplest one for closely related proteins 47 or for the structural change during protein folding/unfolding processes. Thus, the effects of various alcohols on the structural integrity of melittin can be evaluated by plotting the Ca RMSDs of melittin as a function of running time.…”

Analysis of protein structures reveals regions of rare backbone conformation at functional sites

“…The selection criterion then becomes a relational comparison where the discrepancy in each interatomic distance is less than some tolerance. 11 The query on distance matrices can be expressed as a single relational query in SQL. In addition, distance matrices describe protein conformation more robustly than local measures like backbone torsion angles.…”

Stability and Unfolding Mechanism of the N‐terminal β‐Hairpin from [2Fe‐2S] Ferredoxin I by Molecular Dynamics Simulations

“…Although some complicated algorithms have been proposed to measure the structural similarity between proteins, 79,80 the root-mean-square deviation (RMSD) remains the most commonly used. For each MD simulation, the RMSDs of the trajectories recorded every 100 ps interval were calculated for the backbone Ca atom of the b-hairpin during the course of 10 ns MD simulations at 300 K with reference to the starting structure, which has been previously energy minimized.…”