Using quaternions to calculate RMSD

@article{Coutsias2004UsingQT,
  title={Using quaternions to calculate RMSD},
  author={Evangelos A. Coutsias and Chaok Seok and Ken A. Dill},
  journal={Journal of Computational Chemistry},
  year={2004},
  volume={25}
}
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 root‐mean‐square deviation (RMSD). We present a simple derivation, based on quaternions, for the optimal solid body transformation (rotation‐translation) that minimizes the RMSD between two sets of vectors. We prove that the quaternion method is equivalent to the well‐known formula due to Kabsch. We analyze the various cases that may… Expand
RMSD and Symmetry
TLDR
A new, robust numerical algorithm is presented that computes the RMSD between two molecules or all the mutual RMSDs of a list of molecules and, if desired, the corresponding rotation matrix in a minimal number of operations as compared to previous algorithms. Expand
Comment on “Using quaternions to calculate RMSD” [J. Comp. Chem. 25, 1849 (2004)]
  • G. Kneller
  • Mathematics, Medicine
  • J. Comput. Chem.
  • 2005
TLDR
The method to find the optimal rotational superposition of two molecular structures, which is based on a representation of rotations by quaternions, is compared to the one by Kabsch, where the elements of the rotation matrix are directly used as variables of the optimization problem. Expand
Rapid determination of RMSDs corresponding to macromolecular rigid body motions
TLDR
This work presents RigidRMSD, a new algorithm that determines a set of RMSDs corresponding to aSet of rigid body motions of a macromolecule in constant time with respect to the number of atoms in the molecule. Expand
RapidRMSD: rapid determination of RMSDs corresponding to motions of flexible molecules
TLDR
This study goes beyond the limits of the rigid-body approximation by taking into account conformational flexibility of the molecule, and model the flexibility with a reduced set of collective motions computed with e.g. normal modes or principal component analysis. Expand
Dual quaternion applications to bioinformatics for kinematical analysis of Protein Backbone
TLDR
The paper presents a method for optimal alignment of a Protein Backbone with respect to another using Kabsch Method for optimal orientation which minimizes the inter-molecular distance measure (Root Mean Square Deviation) RMSD. Expand
Extensions and Exact Solutions to the Quaternion-Based RMSD Problem
We examine the problem of transforming matching collections of data points into optimal correspondence. The classic RMSD (root-mean-square deviation) method calculates a 3D rotation that minimizesExpand
Ellipsoid-Weighted Protein Conformation Alignment
TLDR
This work associates each atom in the protein with a mean-field potential well, whose shape, ellipsoidal in general, is to be inferred from the observed or computed fluctuations of that atom around its mean position, which has an additional advantage of aligning conformations of proteins along their preferred directions of motions. Expand
Quaternion‐based definition of protein secondary structure straightness and its relationship to Ramachandran angles
We describe here definitions of “local helical axis” and “straightness” that are developed using a simple quaternion‐based analysis of protein structure without resort to least‐squares fitting. AsExpand
Geometry optimization made simple with translation and rotation coordinates.
TLDR
A new translation-rotation-internal coordinate (TRIC) system which explicitly includes the collective translations and rotations of molecules, or parts of molecules such as monomers or ligands, as degrees of freedom is introduced. Expand
Determination of the Eckart molecule-fixed frame by use of the apparatus of quaternion algebra
The problem of determining the Eckart molecule-fixed frame for an arbitrary molecule with nuclei displaced from the equilibrium positions is considered. The solution of the problem is formulated byExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 28 REFERENCES
Closed-form solution of absolute orientation using unit quaternions
TLDR
A closed-form solution to the least-squares problem for three or more paints is presented, simplified by use of unit quaternions to represent rotation. Expand
A method for parameter optimization in computational biology.
TLDR
An algorithm called ENPOP (energy function parameter optimization) is described that improves-and sometimes optimizes-the parameters for any given model and for anygiven search strategy that identifies the stable state of that model. Expand
A super-linear minimization scheme for the nudged elastic band method
In this article, we present a superlinear minimization scheme for the nudged elastic band (NEB) method, which determines a minimum-energy path (MEP) of a reaction via connecting intermediateExpand
A note on the rotational superposition problem
A rotation axis vector with magnitude tan (θ/2) for a rotation angle θ and a closely related unit vector of dimension 4 are used to show that : (i) the quadratic residual (weighted sum of squares ofExpand
Quaternion transformation of molecular orientation
For comparing two molecules, since the most general positional relationship is the combination of a translation and a rotation, where the translational component can be removed by referring bothExpand
Exploring the quantum mechanical/molecular mechanical replica path method: a pathway optimization of the chorismate to prephenate Claisen rearrangement catalyzed by chorismate mutase
Abstract. A replica path method has been developed and extended for use in complex systems involving hybrid quantum/classical (quantum mechanical/molecular mechanical) coupled potentials. This methodExpand
A comparison of three recently published methods for superimposing vector sets by pure rotation
Solutions to the rotational superposition problem published recently by Diamond [Acta Cryst. (1988), A44, 211-216] Kearsley [Acta Cryst. (1989), A45, 208-210] and MacKay [Acta Cryst. (1984), A40,Expand
Gene duplications in the structural evolution of chymotrypsin.
  • A. Mclachlan
  • Biology, Medicine
  • Journal of molecular biology
  • 1979
TLDR
It is suggested that the Chymotrypsin barrel originally evolved from a closely-linked dimer of two intertwined half-domains which became united into one by gene duplication, consistent with the gene duplication hypothesis. Expand
CHARMM: A program for macromolecular energy, minimization, and dynamics calculations
CHARMM (Chemistry at HARvard Macromolecular Mechanics) is a highly flexible computer program which uses empirical energy functions to model macromolecular systems. The program can read or model buildExpand
Matrix analysis
TLDR
This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. Expand
...
1
2
3
...