## Wednesday 22 August 2007

### RMSD

RMSD (root mean square deviation) is the typical measure to compare different structures of a molecule. The nice thing for people with a linear algebra fetish is that the RMSD is nothing but the distance in the appropriate euclidian space. Let's see why, using this neat script for making LATEX formulae I found at A Zephyr in Time.

I don't know which indeces are least confusing but let's do it like this:
In structure A of our molecule with N atoms, the atoms have the following coordinates:

Structure B is the same molecule but bond lengths and angles are changed (or it is moved in space). The coordinates are

With the chosen coordinates the RMSD is defined as follows:

This reminds the attentive reader of the dot product she has been using since high school.

This is the length of the difference vector divided by , in other words the distance.

If you want some more linear algebra you can go on:

With two coordinate vectors a and b

we define the scalar product (using the Dirac notation to remind us that a typical overlap matrix element can be interpreted as a scalar product)

Now the RMSD between two structures A and B can be computed as the length of the difference vector between the two coordinate vectors a and b (containing the coordinates of all the atoms).

RMSD is brought down to something anyone can handle: a distance. And it is brought down to a clean mathematical construct, e.g. triangle inequalities immediately follow. I thought that was pretty cool.