Root mean square deviations in structure

The root mean square deviation (\(RMSD\)) of certain atoms in a molecule with respect to a reference structure can be calculated with the program gmx rms by least-square fitting the structure to the reference structure (\(t_2 = 0\)) and subsequently calculating the \(RMSD\) ((1)).
(1)\[RMSD(t_1,t_2) ~=~ \left[\frac{1}{M} \sum_{i=1}^N m_i \|{\bf r}_i(t_1)-{\bf r}_i(t_2)\|^2 \right]^{\frac{1}{2}}\]
where \(M = \sum_{i=1}^N m_i\) and \({\bf r}_i(t)\) is the position of atom \(i\) at time \(t\). Note that fitting does not have to use the same atoms as the calculation of the \(RMSD\); e.g. a protein is usually fitted on the backbone atoms (N,C:math:_{alpha},C), but the \(RMSD\) can be computed of the backbone or of the whole protein.

Instead of comparing the structures to the initial structure at time \(t=0\) (so for example a crystal structure), one can also calculate (1) with a structure at time \(t_2=t_1-\tau\). This gives some insight in the mobility as a function of \(\tau\). A matrix can also be made with the \(RMSD\) as a function of \(t_1\) and \(t_2\), which gives a nice graphical interpretation of a trajectory. If there are transitions in a trajectory, they will clearly show up in such a matrix.

Alternatively the \(RMSD\) can be computed using a fit-free method with the program gmx rmsdist:

(2)\[RMSD(t) ~=~ \left[\frac{1}{N^2}\sum_{i=1}^N \sum_{j=1}^N \|{\bf r}_{ij}(t)-{\bf r}_{ij}(0)\|^2\right]^{\frac{1}{2}}\]

where the distance r\(_{ij}\) between atoms at time \(t\) is compared with the distance between the same atoms at time \(0\).