# Root mean square deviations in structure¶

(1)¶\[RMSD(t_1,t_2) ~=~ \left[\frac{1}{M} \sum_{i=1}^N m_i \|{\bf r}_i(t_1)-{\bf r}_i(t_2)\|^2 \right]^{\frac{1}{2}}\]

**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\(_{\alpha}\), 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:

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