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The '''Kabsch algorithm''', named after [[Wolfgang Kabsch]], is a method for calculating the optimal [[rotation matrix]] that minimizes the [[RMSD]] ([[root mean square]]d deviation) between two paired sets of points. It is useful in graphics, [[cheminformatics]] to compare molecular structures, and also [[bioinformatics]] for comparing [[protein]] structures (in particular, see [[root-mean-square deviation (bioinformatics)]]).
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The algorithm only computes the rotation matrix, but it also requires the computation of a translation vector. When both the translation and rotation are actually performed, the algorithm is sometimes called partial [[Procrustes superimposition]] (see also [[orthogonal Procrustes problem]]).
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== Description ==
== and little tails wiggle  ｛key｝ ==


The algorithm starts with two sets of paired points, ''P'' and ''Q''. Each set of points can be represented as an ''N''&times;3 [[matrix (mathematics)|matrix]]. The first row is the coordinates of the first point, the second row is the coordinates of the second point, the ''N''th row is the coordinates of the ''N''th point.
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:<math>\begin{pmatrix}
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x_1 & y_1 & z_1 \\
 
x_2 & y_2 & z_2 \\
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\vdots & \vdots  &  \vdots \\
 
x_N & y_N & z_N \end{pmatrix}</math>
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The algorithm works in three steps: a translation, the computation of a covariance matrix, and the computation of the optimal rotation matrix.
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=== Translation ===
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Both sets of coordinates must be translated first, so that their [[centroid]] coincides with the origin of the [[coordinate system]]. This is done by subtracting from the point coordinates the coordinates of the respective centroid.
 
 
</ul>
=== Computation of the covariance matrix ===
The second step consist of calculating a [[covariance matrix]] ''A''. In matrix notation,
 
:<math> A = P^TQ \, </math>
 
or, using summation notation,
 
:<math> A_{ij} = \sum_{k = 1}^N P_{ki} Q_{kj}, </math>
 
=== Computation of the optimal rotation matrix ===
It is possible to calculate the optimal rotation ''U'' based on the matrix formula <math> U = (A^T A)^{1/2}A^{-1}  </math> but implementing a numerical solution to this formula becomes complicated when all special cases are accounted for (for example, the case of ''A'' not having an inverse).
 
If [[singular value decomposition]] (SVD) routines are available, the optimal rotation, ''U'', can be calculated using the following simple algorithm.
 
First, calculate the SVD of the covariance matrix ''A''.
 
:<math> A = VSW^T \, </math>
 
Next, decide whether we need to correct our rotation matrix to ensure a right-handed coordinate system
 
:<math> d = \operatorname{sign}(\det(W V^T)) \, </math>
 
Finally, calculate our optimal rotation matrix, ''U'', as
 
:<math> U = W \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & d \end{pmatrix} V^T </math>
 
Coutsias, Seok, and Dill<ref name=Coutsias2004>{{cite journal | author = Coutsias EA, Seok C, Dill KA | title = Using quaternions to calculate RMSD | journal = J Comput Chem | volume = 25 | issue = 15 | pages = 1849–1857 | year = 2004 | pmid = 15376254 | doi = 10.1002/jcc.20110}}</ref> have found an equivalent method that uses [[quaternion]]s.
 
=== Generalizations ===
 
The algorithm was described for points in a three-dimensional space. The generalization to ''D'' dimensions is immediate.
 
== External links ==
This SVD algorithm is described in more detail at http://cnx.org/content/m11608/latest/
 
A [[Matlab]] function is available at http://www.mathworks.com/matlabcentral/fileexchange/25746-kabsch-algorithm
 
A free [[PyMol]] plugin easily implementing Kabsch is [http://www.pymolwiki.org/index.php/Cealign Cealign]. [[Visual Molecular Dynamics|VMD]] uses the Kabsch algorithm for its alignment.
 
A [[Python_(programming_language)|Python]] script is available at http://github.com/charnley/rmsd
 
== See also ==
 
[[Wahba's problem|Wahba's Problem]]
 
== References ==
{{reflist}}
 
* Kabsch, Wolfgang, (1976) "A solution for the best rotation to relate two sets of vectors", ''Acta Crystallographica'' '''32''':922. {{doi|10.1107/S0567739476001873}} with a correction in Kabsch, Wolfgang, (1978) "A discussion of the solution for the best rotation to relate two sets of vectors", "Acta Crystallographica", "A34", 827–828 {{doi|10.1107/S0567739478001680}}
 
* Lin Ying-Hung, Chang Hsun-Chang, Lin Yaw-Ling (2004) "A Study on Tools and Algorithms for 3-D Protein Structures Alignment and Comparison", ''International Computer Symposium'', December 15–17, Taipei, Taiwan.
 
[[Category:Bioinformatics algorithms]]

Latest revision as of 20:25, 17 August 2014

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