Root-mean-square deviation of atomic positions

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 an other paper of Petitjean.^[6]

The equation

\mathrm {RMSD} ={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}\delta _{i}^{2}}}

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_α or sometimes just the C_α atoms.

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

{\begin{aligned}\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{aligned}}

An RMSD value is expressed in length units. The most commonly used unit in structural biology is the Ångström (Å) which is equal to 10⁻¹⁰ 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. Thus the lower RMSD, the better model is in comparison to native structure.

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.

The study of RMSD for small organic molecules (commonly called ligands when their 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) configurations are most commonly not superimposed previous to the calculation of the RMSD.

References

1 2 "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.
1 2 Kabsch W (1976). "A solution for the best rotation to relate two sets of vectors". Acta Crystallographica. 32 (5): 922–923. 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". Journal of Mathematical Physics. 40 (9): 4587–4595. doi:10.1063/1.532988.
↑ Petitjean M (2002). "Chiral mixtures". Journal of Mathematical Physics. 43 (8): 185–192. doi:10.1063/1.1484559.

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.
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

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Root-mean-square deviation of atomic positions

The equation

Uses

See also

References

Further reading

External links