Extraction of well-fitting substructures: Root-mean-square deviation and the difference distance matrix

The extraction of well-fitting substructures of two or more sets of proteins has applications to analysis of mechanisms of conformational change in proteins, including pathways of evolution, to classification of protein folding patterns, and to evaluation of protein structure predictions. Many methods are known for extracting some substantial common substructure with low root-mean-square deviation (r.m.s.d.). A harder problem is addressed here: finding all common substructures with r.m.s.d. less than a prespecified threshold. Our approach is to consider the minimum value of the maximum distance between corresponding points, corresponding to superposition in the Chebyshev norm. Using the properties of Chebyshev superposition, we derive relationships between the r.m.s.d. and the maximum element of the difference matrix, two common measures of structural similarity. The results provide a basis for developing algorithms and software to identify all well-fitting subsets.