Structure Similarity and Overlap

Suppose we are dealing with two or more protein structures that contain regions in which the backbone atoms are almost congruent. We wish to superpose the two structures by moving one with respect to the other so that the corresponding atoms in the well-fitting regions are optimally matched. There are two aspects to this problem  

Choosing the corresponding sets of atoms from the two structures. 

Rigid-Body Motions and Least-Squares Fitting 

The general numerical approach employs rigid body motions and least-squares fitting. Given two sets of points xt,i = 1,2. N and yui= 1,2. N (here xt and yt are vectors specifying atomic coordinates), find the best rigid motion of the points y h) such that the sum of the squares of the deviations E xt — Yt 2 is a minimum. 

Two mathematical facts that we apply without proof are  

With protein structures, we can observe similarities in topology or even in structural details. Comparison of protein structures can reveal distant evolutionary relationships that would not be detected by sequence aligmnent alone. Comparing and overlapping two protein stractures quantitatively remain an active area of development in structural biochemistry. Methods for protein comparison generally rely on a fast full search of the protein structure database. 

In devising measures of similarity and difference between two proteins, it is sometimes clearer to note how to proceed in comparing sequences, than to do so in comparing structures. If the amino acid sequences of two proteins can be aligned, then we can either count the number of identical residues, or use a similarity index between amino acids. Such an index would take the form of a 20 x 20 matrix, M, such that each entry corresponds to a pair of amino acids and My gives a measure of the similarity between any pair 

finding the best match of the corresponding atoms/substructures. 

---