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The '''root-mean-square deviation''' ('''RMSD''') is the measure of the average distance between the atoms (usually the backbone atoms) of [[protein structural alignment|superimposed]] [[proteins]]. In the study of globular protein conformations, one customarily measures the similarity in three-dimensional structure by the RMSD of the C&alpha; atomic coordinates after optimal rigid body superposition.

When a dynamical system fluctuates about some well-defined average position, the RMSD from the average over time can be referred to as the ''RMSF'' or [[root mean square fluctuation]]. The size of this fluctuation can be measured, for example using [[Mössbauer spectroscopy]] or [[nuclear magnetic resonance]], and can provide important physical information. The [[Lindemann index]] is a method of placing the RMSF in the context of the parameters of the system.

A widely used way to compare the structures of biomolecules or solid bodies is to translate and rotate one structure with respect to the other to minimize the RMSD. Coutsias, ''et al.'' presented a simple derivation, based on [[quaternion]]s, for the optimal solid body transformation (rotation-translation) that minimizes the RMSD between two sets of vectors.<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> They proved that the quaternion method is equivalent to the well-known [[Kabsch algorithm]].<ref name=Kabsch1976>{{cite journal | author = Kabsch W | title = A solution for the best rotation to relate two sets of vectors | journal = Acta Crystallographica | volume = 32 | pages = 922–923 | year = 1976 | doi = 10.1107/S0567739476001873 | issue = 5}}</ref> The solution given by Kabsch is an instance of the solution of the d-dimensional problem, introduced by Hurley and Cattell.<ref name=HurleyCattell2002>{{cite journal | author = Hurley JR and Cattell RB | title = The Procrustes Program: Producing direct rotation to test a hypothesized factor structure | journal = Behavioral Science | volume = 7 | issue = 2 | pages = 258–262 | year = 1962 | doi = 10.1002/bs.3830070216}}</ref> The [[quaternion]] solution to compute the optimal rotation was published in the appendix of a paper of Petitjean.<ref name=Petitjean1999>{{cite journal | author = Petitjean M | title = On the Root Mean Square quantitative chirality and quantitative symmetry measures | journal = Journal of Mathematical Physics | volume = 40 | issue = 9 | pages = 4587–4595 | year = 1999 | doi = 10.1063/1.532988}}</ref> This [[quaternion]] solution and the calculation of the optimal isometry in the d-dimensional case were both extended to infinite sets and to the continuous case in the appendix A of an other paper of Petitjean.<ref name=Petitjean2002>{{cite journal | author = Petitjean M | title = Chiral mixtures | journal = Journal of Mathematical Physics | volume = 43 | issue = 8 | pages = 185–192| year = 2002 | doi = 10.1063/1.1484559}}</ref>

==The equation==

: <math>\mathrm{RMSD}=\sqrt{\frac{1}{N}\sum_{i=1}^N\delta_i^2}</math>

where ''δ'' is the distance between ''N'' pairs of equivalent atoms (usually ''Cα'' and sometimes ''C'',''N'',''O'',''Cβ'').

Normally a rigid superposition which minimizes the RMSD is performed, and this minimum is returned. Given two sets of <math>n</math> points <math>\mathbf{v}</math> and <math>\mathbf{w}</math>, the RMSD is defined as follows:

: <math>
\begin{align}
\mathrm{RMSD}(\mathbf{v}, \mathbf{w}) & = \sqrt{\frac{1}{n}\sum_{i=1}^{n} \|v_i - w_i\|^2} \\
& = \sqrt{\frac{1}{n}\sum_{i=1}^{n} 
      (({v_i}_x - {w_i}_x)^2 + ({v_i}_y - {w_i}_y)^2 + ({v_i}_z - {w_i}_z)^2)}
\end{align}
</math>

An RMSD value is expressed in length units. The most commonly used unit in [[structural biology]] is the [[Ångström]] (Å) which is equal to 10<sup>–10</sup>m.

==Uses==
Typically RMSD is used to make a quantitative comparison between the structure of a partially folded protein and the structure of the native state. For example, the [[CASP]] [[protein structure prediction]] competition uses RMSD as one of its assessments of how well a submitted structure matches the native state.

Also some scientists who study [[protein folding]] simulations use RMSD as a [[reaction coordinate]] to quantify where the protein is between the folded state and the unfolded state.

==See also==
*[[Root mean square deviation]]
*[[Root mean square fluctuation]]
*[[Quaternion]] – used to optimise RMSD calculations
*[[Kabsch algorithm]] – an algorithm used to minimize the RMSD by first finding the best rotation<ref name="Kabsch1976" />
*[[Global distance test|GDT]] – a different structure comparison measure
*[[Template modeling score|TM-score]] – a different structure comparison measure

==References==
{{Reflist}}
===Further reading===
* Shibuya T (2009). "Searching Protein 3-D Structures in Linear Time." Proc. 13th Annual International Conference on Research in Computational Molecular Biology (RECOMB 2009), ''LNCS'' 5541:1–15.
* {{cite journal |author=Armougom F, Moretti S, Keduas V, Notredame C |year=2006 |title=The iRMSD: a local measure of sequence alignment accuracy using structural information |journal=Bioinformatics |volume=22 |issue=14 |pages=e35–39 |doi=10.1093/bioinformatics/btl218}}
* {{cite journal |author=Damm KL, Carlson HA |year=2006 |title=Gaussian-Weighted RMSD Superposition of Proteins: A Structural Comparison for Flexible Proteins and Predicted Protein Structures |journal=Biophys J |volume=90 |issue=12 |pages=4558–4573 |doi=10.1529/biophysj.105.066654 |pmid=16565070 |pmc=1471868}}
* {{cite journal |author=Kneller GR |year=2005 |title=Comment on 'Using quaternions to calculate RMSD' [''J. Comp. Chem.'' 25, 1849 (2004)] |journal=J Comput Chem |volume=26 |issue=15 |pages=1660–1662 |doi=10.1002/jcc.20296 |pmid=16175580}}
* {{cite journal |author=Theobald DL |year=2005 |title=Rapid calculation of RMSDs using a quaternion-based characteristic polynomial |journal=Acta Crystallogr A |volume=61 |issue=Pt 4 |pages=478–480 |doi=10.1107/S0108767305015266 |pmid=15973002}}
* {{cite journal |author=Maiorov VN, Crippen GM |year=1994 |title=Significance of root-mean-square deviation in comparing three-dimensional structures of globular proteins |journal=J Mol Biol |volume=235 |issue=2 |pages=625–634 |doi=10.1006/jmbi.1994.1017 |pmid=8289285}}

==External links==
* [http://cnx.org/content/m11608/latest/ Molecular Distance Measures]&mdash;a tutorial on how to calculate RMSD
* [http://boscoh.com/protein/rmsd-root-mean-square-deviation RMSD]&mdash;another tutorial on how to calculate RMSD with example code
*[http://www.ebi.ac.uk/msd-srv/ssm/ Secondary Structure Matching (SSM)] &mdash; a tool for protein structure comparison. Uses RMSD.
*[http://wishart.biology.ualberta.ca/SuperPose/ SuperPose] &mdash; a protein superposition server. Uses RMSD.
*[http://www.ccp4.ac.uk/html/superpose.html superpose] &mdash; structural alignment based on secondary structure matching. By the CCP4 project. Uses RMSD.

[[Category:Statistical deviation and dispersion]]