.. _rmsd:
Root mean square deviations in structure
----------------------------------------
| :ref:`gmx rms `, :ref:`gmx rmsdist `
| The *root mean square deviation* (:math:`RMSD`) of certain atoms in a
molecule with respect to a reference structure can be calculated with
the program :ref:`gmx rms ` by least-square fitting the structure to the
reference structure (:math:`t_2 = 0`) and subsequently calculating the
:math:`RMSD` (:eq:`eqn. %s `).
.. math:: 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}}
:label: eqnrmsd
| where :math:`M = \sum_{i=1}^N m_i` and :math:`{\bf r}_i(t)` is the
position of atom :math:`i` at time :math:`t`. **Note** that fitting
does not have to use the same atoms as the calculation of the
:math:`RMSD`; *e.g.* a protein is usually fitted on the backbone atoms
(N,C:math:`_{\alpha}`,C), but the :math:`RMSD` can be computed of the
backbone or of the whole protein.
Instead of comparing the structures to the initial structure at time
:math:`t=0` (so for example a crystal structure), one can also calculate
:eq:`eqn. %s ` with a structure at time :math:`t_2=t_1-\tau`. This
gives some insight in the mobility as a function of :math:`\tau`. A
matrix can also be made with the :math:`RMSD` as a function of
:math:`t_1` and :math:`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 :math:`RMSD` can be computed using a fit-free method
with the program :ref:`gmx rmsdist `:
.. math:: 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}}
:label: eqnrmsdff
where the *distance* **r**\ :math:`_{ij}` between atoms at time
:math:`t` is compared with the distance between the same atoms at time
:math:`0`.