Root mean square deviations in structure

gmx rms, gmx rmsdist

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$_{\alpha}$, 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 $0$.