RMS Fit and Alignment

When one has two similar structures, one often wants to compare them. What's the difference between two X-ray structures? How much did the structure change during a simulation? To answer these questions, you must first figure out how to compare two structures, which usually means that you must find the root mean square deviation (RMSD).

Formally, given $N$ atom positions from structure $x$ and the corresponding $N$ atoms from structure $y$ with a weighting factor $w\left(i\right)$ , the RMSD is defined as:

$RMSD\left(N; x,y\right) = {\left[\frac {\sum_{i=1}^N w_i {\parallel x_i - y_i \parallel}^2} {N \sum_{i=1}^N w_i}\right]}^{\frac {1}{2}}$

Using this equation by itself probably won't give you the answer you are looking for. Imagine two identical structures offset by some distance. The RMSD should be 0, but the offset prevents that from happening. What you really want is the minimum RMSD between two given structures; the best fit. There are many ways to do this, but for VMD we have implemented the method of Kabsch (Acta Cryst. (1978) A34, 827-828 or see file Measure.C in the VMD source code). This algorithm computes the transformation, needed to move one structure onto another in order to minimize the RMSD.

With the mathematical prerequisites behind us, we still need to be able to specify how to choose the atoms to compare. If you want to compare all the atoms in both structures, and they both have the same number of atoms, then the problem is easy - $N$ is everything. This occurs most often in MD simulations when the only thing different between two structures are the coordinates.

But what about homologous sequences? In this case, the number of atoms differ because while the number of residues is the same, the sidechains have different numbers of atoms. The usual solution is to determine the RMSD based solely on the backbone atoms or, in some X-ray structures where only the $C_{\alpha}$ atoms have been determined, based on the $C_{\alpha}$ atoms. VMD allows you to fit and align based on any valid atom selection, as long as the atom selection specifies the same number of atoms in each molecule being compared.