# 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 $$0$$.