# Formulating equation for distances between atoms

I'm trying to formulate notation to describe code that calculates the distances between protein atoms (represented as points in 3D space).

Fragments consist of residues (amino acids) and each residue consists of atoms named N, Ca, Cb, and C. Say there are 3 fragments of 5 residues each with an overlap of 2:

1 2 3 4 5              # fragment 1
1 2 3 4 5        # fragment 2
1 2 3 4 5  # fragment 3


The code calculates the mean square distance between each atom of fragment 1 residue 4 and the equivalent atom of fragment 2 residue 1, and so on for f1r5 and f2r2, f2r4 and f3r1, and f2r5 and f3r2.

Here's what I've tried:

$$MSD = \sum_{i=1}^{N-1} \sum_{j=1}^{M} \sum_{k=1}^{|A|} \frac{||\mathbf{x}_{i,j+l-M,k}-\mathbf{x}_{i+1,j,k}||^2} {M|A|} \\ A = \{ N, C_\alpha, C_\beta, C \} \\ N = \text{number of fragments} \\ M = \text{number of overlapping residues} \\ l = \text{number of residues in fragment } i$$

$\mathbf{x}_{i,j,k}$ is a vector containing the 3D coordinates of the kth atom of the jth residue of the ith fragment. In the above example, $N=3$, $M=2$, and $l=5$.

I find this neither attractive nor easy to understand; however, I have been unable to come up with an alternative.

What is a more clear and concise way to express this?

• If one had the chain, it would be the mean of the distances between all possible atom pairs?
– mvw
Commented Mar 20, 2015 at 20:52
• No, in my case M=3 and l=9. I'll edit to clarify. Commented Mar 20, 2015 at 21:07
• not sure if this is helpful. ncbi.nlm.nih.gov/pmc/articles/PMC1781147 Commented Mar 21, 2015 at 3:07