Root Mean Square Deviation

Calculate the root-mean square deviation using quaternions.

Introduction

A widely used way to compare the structures is to translate and rotate one structure with respect to the other to minimize the root mean square deviation (RMSD). We implemented a simple scheme, based on quaternions, for the optimal transformation (rotation-translation) that minimizes the RMSD between two sets of structures. Quaternions are generally represented in the form

$$q = a+b\,\mathbf{i} +c\,\mathbf {j} +d\,\mathbf {k}$$

with a, b, c, and d as real numbers; and i, j, and k as the fundamental quaternion units. The set of quaternions is made a four-dimensional vector space over the real numbers {1, i, j, k} as a basis, where Hamilton defined the quaternion as consisting of a scalar part and a vector part. The vector part of the quaternion can be interpreted as a coordinate in three dimensional space. Operations such as the vector dot product and cross products can be defined in terms of quaternions, and this makes it possible to apply quaternion techniques wherever spatial vectors arise.

Define the Subcommand

rms

> kallisto rms options arguments

options

--out <string> 
(optional)
description: 
 write output to file

arguments

input file is given as (positional) argument

Application

As an example let's compare two conformers of 1-propanol that we have obtained by applying the conformer rotamer ensemble sampling tool.

Lowest energy conformer of 1-propanol.

> cat 1-propanol_lowest.xyz
  12
  
 C         -1.9554949371        0.1467391618        0.0031595607
 C         -0.5906278346       -0.5279387138       -0.0201649611
 C          0.5440986558        0.4958779663        0.0283462055
 H          0.4812068385        1.1678478833       -0.8308000219
 H          0.4590669813        1.0993020658        0.9450529713
 O          1.8195161785       -0.0957487212       -0.0534239359
 H          1.9103706588       -0.7338049177        0.6631507673
 H         -0.5004127933       -1.2028008461        0.8364936998
 H         -0.4854009629       -1.1250023438       -0.9282499098
 H         -2.7476736372       -0.5972665554       -0.0242488945
 H         -2.0700756998        0.8040326560       -0.8554507953
 H         -2.0722381370        0.7410005769        0.9069567477

Higher energy conformer of 1-propanol.

> cat 1-propanol_higher.xyz
12

C       -1.60306996       0.10333519       0.50792736
C       -0.66904416      -0.46962566      -0.55371646
C        0.67345677       0.26436258      -0.61179298
H        1.26292797      -0.10585085      -1.45392921
H        0.49744830       1.34089332      -0.75955140
O        1.47742183       0.05176805       0.52349829
H        0.98773122       0.34094585       1.30125393
H       -0.48213061      -1.52528483      -0.34815476
H       -1.14165995      -0.39229359      -1.53423716
H       -2.56608070      -0.40007121       0.47312929
H       -1.76619136       1.16652831       0.34003517
H       -1.19366144      -0.03197289       1.50775619

Let's calculate the RMSD error and the rotation matrix

> kallisto rms 1-propanol_lowest.xyz 1-propanol_higher.xyz
RMSD [1.12070194] Angstrom
Rotation Matrix
[[ 0.98139458 -0.04965545 -0.18546973]
 [ 0.06170977  0.9963015   0.05979323]
 [ 0.18181471 -0.07012604  0.98082911]]