In bioinformatics, the **root-mean-square deviation of atomic positions**, or simply **root-mean-square deviation (RMSD)**, is the measure of the average distance between the atoms (usually the backbone atoms) of superimposed proteins. Note that RMSD calculation can be applied to other, non-protein molecules, such as small organic molecules.^{[1]} In the study of globular protein conformations, one customarily measures the similarity in three-dimensional structure by the RMSD of the Cα 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 quaternions, for the optimal solid body transformation (rotation-translation) that minimizes the RMSD between two sets of vectors.^{[2]} They proved that the quaternion method is equivalent to the well-known Kabsch algorithm.^{[3]} The solution given by Kabsch is an instance of the solution of the *d*-dimensional problem, introduced by Hurley and Cattell.^{[4]} The quaternion solution to compute the optimal rotation was published in the appendix of a paper of Petitjean.^{[5]} 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 another paper of Petitjean.^{[6]}

## The equation

where *δ _{i}* is the distance between atom

*i*and either a reference structure or the mean position of the

*N*equivalent atoms. This is often calculated for the backbone heavy atoms

*C*,

*N*,

*O*, and

*C*or sometimes just the

_{α}*C*atoms.

_{α}Normally a rigid superposition which minimizes the RMSD is performed, and this minimum is returned. Given two sets of points and , the RMSD is defined as follows:

A RMSD value is expressed in length units. The most commonly used unit in structural biology is the Ångström (Å) which is equal to 10^{−10} m.

## Uses

Typically RMSD is used as a quantitative measure of similarity between two or more protein structures. For example, the CASP protein structure prediction competition uses RMSD as one of its assessments of how well a submitted structure matches the known, target structure. Thus the lower RMSD, the better the model is in comparison to the target structure.

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

The study of RMSD for small organic molecules (commonly called ligands when they're binding to macromolecules, such as proteins, is studied) is common in the context of docking,^{[1]} as well as in other methods to study the configuration of ligands when bound to macromolecules. Note that, for the case of ligands (contrary to proteins, as described above), their structures are most commonly not superimposed prior to the calculation of the RMSD.

RMSD is also one of several metrics that have been proposed for quantifying evolutionary similarity between proteins, as well as the quality of sequence alignments.
^{[7]} ^{[8]}

## 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
^{[3]} - GDT – a different structure comparison measure
- TM-score – a different structure comparison measure
- Longest continuous segment (LCS) — A different structure comparison measure
- Global distance calculation (GDC_sc, GDC_all) — Structure comparison measures that use full-model information (not just α-carbon) to assess similarity
- Local global alignment (LGA) — Protein structure alignment program and structure comparison measure

## References

- ^
^{a}^{b}"Molecular docking, estimating free energies of binding, and AutoDock's semi-empirical force field".*Sebastian Raschka's Website*. 2014-06-26. Retrieved 2016-06-07. **^**Coutsias EA, Seok C, Dill KA (2004). "Using quaternions to calculate RMSD".*J Comput Chem*.**25**(15): 1849–1857. doi:10.1002/jcc.20110. PMID 15376254. S2CID 18224579.- ^
^{a}^{b}Kabsch W (1976). "A solution for the best rotation to relate two sets of vectors".*Acta Crystallographica*.**32**(5): 922–923. Bibcode:1976AcCrA..32..922K. doi:10.1107/S0567739476001873. **^**Hurley JR, Cattell RB (1962). "The Procrustes Program: Producing direct rotation to test a hypothesized factor structure".*Behavioral Science*.**7**(2): 258–262. doi:10.1002/bs.3830070216.**^**Petitjean M (1999). "On the Root Mean Square quantitative chirality and quantitative symmetry measures" (PDF).*Journal of Mathematical Physics*.**40**(9): 4587–4595. Bibcode:1999JMP....40.4587P. doi:10.1063/1.532988.**^**Petitjean M (2002). "Chiral mixtures" (PDF).*Journal of Mathematical Physics*.**43**(8): 185–192. Bibcode:2002JMP....43.4147P. doi:10.1063/1.1484559.**^**Jewett AI, Huang CC, Ferrin TE (2003). "MINRMS: an efficient algorithm for determining protein structure similarity using root-mean-squared-distance" (PDF).*Bioinformatics*.**19**(5): 625–634. doi:10.1093/bioinformatics/btg035. PMID 12651721.CS1 maint: multiple names: authors list (link)**^**Armougom F, Moretti S, Keduas V, Notredame C (2006). "The iRMSD: a local measure of sequence alignment accuracy using structural information" (PDF).*Bioinformatics*.**22**(14): e35–39. doi:10.1093/bioinformatics/btl218. PMID 16873492.

### 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. - Damm KL, Carlson HA (2006). "Gaussian-Weighted RMSD Superposition of Proteins: A Structural Comparison for Flexible Proteins and Predicted Protein Structures".
*Biophys J*.**90**(12): 4558–4573. Bibcode:2006BpJ....90.4558D. doi:10.1529/biophysj.105.066654. PMC 1471868. PMID 16565070. - Kneller GR (2005). "Comment on 'Using quaternions to calculate RMSD' [
*J. Comp. Chem.*25, 1849 (2004)]".*J Comput Chem*.**26**(15): 1660–1662. doi:10.1002/jcc.20296. PMID 16175580. S2CID 27004373. - Theobald DL (2005). "Rapid calculation of RMSDs using a quaternion-based characteristic polynomial".
*Acta Crystallogr A*.**61**(Pt 4): 478–480. Bibcode:2005AcCrA..61..478T. doi:10.1107/S0108767305015266. PMID 15973002. - Maiorov VN, Crippen GM (1994). "Significance of root-mean-square deviation in comparing three-dimensional structures of globular proteins" (PDF).
*J Mol Biol*.**235**(2): 625–634. doi:10.1006/jmbi.1994.1017. hdl:2027.42/31835. PMID 8289285.

## External links

- Molecular Distance Measures—a tutorial on how to calculate RMSD
- RMSD—another tutorial on how to calculate RMSD with example code
- Secondary Structure Matching (SSM) — a tool for protein structure comparison. Uses RMSD.
- GDT, LCS and LGA — different structure comparison measures. Description and services.
- SuperPose — a protein superposition server. Uses RMSD.
- superpose — structural alignment based on secondary structure matching. By the CCP4 project. Uses RMSD.
- A Python script is available at https://github.com/charnley/rmsd
- An alternate Python script is available at https://github.com/jewettaij/superpose3d