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.
Kurt Gödel's Open World

Today marks Kurt Gödel's one hundred and eleventh birthday. Along with
Aristotle, Gödel is often considered the greatest logician in history. But
I belie...
1 day ago
2 comments:
Yeah, I definitely have a linear algebra fetish. That class would have made me pick up a math minor if several years of calculus hadn't preceded it.
(also i appreciate "she" instead of "he") :)
alright, then it's glassware and linear algebra  I guess that's what gets most people going =)
and sure: with all that political correctness around, I can't make all the people in my texts male
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