LSQRMS program. V. Alexandrov, Yale (2000).

The LSQ fitting routine (when the program is called with "lsq" keyword) works 
as follows. Assuming that the set of equivalence pairs for two structures (we 
call them "moving" and "reference" structures, respectively) is determined, 
e.g. found from the corresponding sequence alignment, the structural alignment 
is done in two steps. The first is to find the translation vector (TV) that 
takes the center-of-mass (CM) of one structure to the other (easy: find CM 
coordinates for each structure and subtract the CM vector of the "moving" 
structure from the CM vector of the "fixed" one). The second step is to compute
the matrix transformation (Rotation Matrix) that minimizes RMSD between the two
structures after applying TV transformation to the moving structure from the 
previous step. Note that CM for both structures is calculated based only on 
those C-alphas that are present in the set of equivalence pairs.

After the CM (center-of-mass) of the "moving" structure is first moved to that 
of the "reference" structure (with help of the Translation Vector from the 
previous step) the rotation matrix (RM) is computed using a wide-known method 
suggested by Kabsch (1977). The method is based on finding the solution for the
parameters of the 3D rotation matrix minimizing the RMSD of the CM-transformed 
structures (actually, their equivalence pairs only).

After applying RM to the "moving" structure (preceded by TV translation), the 
RMSD is calculated between the coordinates for all the equivalence pairs. 
If the program is not called with "lsq" argument, it repeats the 
above-mentioned process (skipping the sequence alignment stage at each 
iteration) finding new equivalence pairs and minimizing RMSD in each iteration 
until it converges to some minimum value or until the program reaches the 
maximum number of iterations (currently 100).

If supplied with keyword "structal", the program outputs the resulting 
structures (original "target" and rotated "query" structures + the 
corresponding equivalence pairs) in the original Structal format (Levitt, 
1994). Otherwise, the program outputs only the new (rotated and translated) 
coordinates of the query.


Refs.

1) Kabsch (Kabsch, W (1976) Acta Cryst. A32, 922-923) 
2) Levitt, M., STRUCTAL. A structural alignment program, Stanford U, (1994).