RMSD and Symmetry

Coutsias, Evangelos A.; Wester, Michael J.

doi:10.1002/jcc.25802

Cited by 51 publications

(27 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This form emphasizes some additional explicit symmetry that we will see is connected to the role of cube roots in the quartic algebraic solutions (see, e.g., Coutsias & Wester, 2019). We can turn it into an equation for k (p) to be solved in terms of the matrix parameters p k (E) as follows: First we eliminate e using (e À 1 )(e À 2 )(e À 3 )(e À 4 ) = 0 to express the matrix data expressions p k directly in terms of totally symmetric polynomials of the eigenvalues in the form (Abramowitz & Stegun, 1970)…”

Section: Approaches To Algebraic Solutionsmentioning

confidence: 76%

“…A very informative treatment of the features of the quaternion eigenvalue solutions was given by Coutsias, Seok and Dill in 2004, and expanded in 2019(Coutsias et al, 2004Coutsias & Wester, 2019). Coutsias et al not only take on a thorough review of the quaternion RMSD method, but also derive the complete relationship between the linear algebra of the SVD method and the quaternion eigenvalue system; furthermore, they exhaustively enumerate the special cases involving mirror geometries and degenerate eigenvalues that may appear rarely, but must be dealt with on occasion.…”

Section: Further Literaturementioning

confidence: 99%

“…Finally, in real problems, structures such as molecules may appear in mirrorimage or enantiomer form, and such issues were introduced early on by Kabsch (1976Kabsch ( , 1978. There can also be particular symmetries, or very close approximations to symmetries, that can make some of our natural assumptions about the good behavior of the profile matrix invalid, and many of these issues, including ways to treat degenerate cases, have been carefully studied [see, e.g., Coutsias et al (2004) and Coutsias & Wester (2019)]. The latter authors also point out that if a particular data set M(E) produces a negative smallest eigenvalue 4 such that j 4 j > opt , this can be a sign of a reflected match, and the negative rotation matrix R opt ¼ ÀRðqð 4 ÞÞ may actually produce the best alignment.…”

Section: Quaternion Transformation Of the 3d Cross-term Formmentioning

confidence: 99%

“…The real, symmetric, traceless profile matrix M(E) in equation (13) appearing in the 3D spatial RMSD optimization problem must necessarily possess only real eigenvalues, and the properties of M(E) permit some particular simplifications in the algebraic solutions that we will discuss. The quaternion RMSD literature varies widely in the details of its treatment of the algebraic solutions, ranging from no discussion at all, to Horn, who mentions the possibility but does not explore it, to the work of Coutsias et al (Coutsias et al, 2004;Coutsias & Wester, 2019), who present an exhaustive treatment, in addition to working out the exact details of the correspondence between the SVD eigensystem and the quaternion eigensystem, both of which in principle embody the algebraic solution to the RMSD optimization problem. In addition to the treatment of Coutsias et al, other approaches similar to the one we will study are due to Euler (Euler, 1733;Bell, 2008), as well as a series of papers on the quartic by Nickalls (1993Nickalls ( , 2009.…”

Section: Algebraic Solution Of the Eigensystem For 3d Spatial Alignmentmentioning

confidence: 99%

“…This issue is significant in diverse application domains. Among these are aligning spacecraft (see, e.g., Wahba, 1965;Davenport, 1968;Markley, 1988;and Markley & Mortari, 2000), obtaining correspondence of registration points in 3D model matching (see, e.g., Faugeras & Hebert, 1983, 1986, matching structures in aerial imagery (see, e.g., Horn, 1987;Horn et al, 1988;Huang et al, 1986;Arun et al, 1987;Umeyama, 1991;and Zhang, 2000), and alignment of matched molecular and biochemical structures (see, e.g., Kabsch, 1976Kabsch, , 1978McLachlan, 1982;Lesk, 1986;Diamond, 1988;Kearsley, 1989Kearsley, , 1990Kneller, 1991;Coutsias et al, 2004;Theobald, 2005;Liu et al, 2010;and Coutsias & Wester, 2019). A closely related task is the alignment of multiple sets of 3D range data, for example in digital-heritage applications (Levoy et al, 2000); the widely used iterative closest point (ICP) algorithm [see, e.g., Chen & Medioni (1991), Besl & McKay (1992) and Bergevin et al (1996), as well as Rusinkiewicz & Levoy (2001) and Nü chter et al (2007)] explicitly incorporates standard alignment methods in individual steps with known correspondences.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The quaternion-based spatial-coordinate and orientation-frame alignment problems

Hanson¹

2020

Acta Cryst Sect A

View full text Add to dashboard Cite

The general problem of finding a global rotation that transforms a given set of spatial coordinates and/or orientation frames (the `test' data) into the best possible alignment with a corresponding set (the `reference' data) is reviewed. For 3D point data, this `orthogonal Procrustes problem' is often phrased in terms of minimizing a root-mean-square deviation (RMSD) corresponding to a Euclidean distance measure relating the two sets of matched coordinates. This article focuses on quaternion eigensystem methods that have been exploited to solve this problem for at least five decades in several different bodies of scientific literature, where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; focusing on these exact solutions exposes the structure of the entire eigensystem for the traditional 3D spatial-alignment problem. The structure of the less-studied orientation-data context is then explored, investigating how quaternion methods can be extended to solve the corresponding 3D quaternion orientation-frame alignment (QFA) problem, noting the interesting equivalence of this problem to the rotation-averaging problem, which also has been the subject of independent literature threads. The article concludes with a brief discussion of the combined 3D translation–orientation data alignment problem. Appendices are devoted to a tutorial on quaternion frames, a related quaternion technique for extracting quaternions from rotation matrices and a review of quaternion rotation-averaging methods relevant to the orientation-frame alignment problem. The supporting information covers novel extensions of quaternion methods to the 4D Euclidean spatial-coordinate alignment and 4D orientation-frame alignment problems, some miscellaneous topics, and additional details of the quartic algebraic eigenvalue problem.

show abstract

Section: Approaches To Algebraic Solutionsmentioning

confidence: 76%

Section: Further Literaturementioning

confidence: 99%

Section: Quaternion Transformation Of the 3d Cross-term Formmentioning

confidence: 99%

Section: Algebraic Solution Of the Eigensystem For 3d Spatial Alignmentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The quaternion-based spatial-coordinate and orientation-frame alignment problems

Hanson¹

2020

Acta Cryst Sect A

View full text Add to dashboard Cite

show abstract

Molecular Dynamics

Wieczorek¹,

Niedziałek²

2020

Encyclopedia of Life Sciences

View full text Add to dashboard Cite

Molecular dynamics (MD) is a handy computer simulation method which unveils principles of a large variety of fundamental processes by mimicking the real life perpetual motion of atoms triggered by interatomic forces. In MD, these forces are expressed through analytical functions and associated parameters, which are commonly referred to as the force field. The forces' effect on the movement of atoms is determined by Newton's second law. Most importantly, MD simulations are able to retrace the exact trajectories of interacting atoms by following their spatial evolution in time from a defined initial configuration. The resulting integrated trajectory of the system can be visualised and analysed with rigorous mathematical expressions of statistical mechanics, which relate averaged microscopic states to macroscopic properties relevant to the problem at hand. Key Concepts Molecular dynamics is a computation tool successfully used to unveil principles of a large variety of fundamental processes down to the atomic resolution. Thanks to recent advances in computer simulations, atomistic MD can reach millisecond timescales, thus emerging as a powerful alternative method that successfully complements experimental measurements in biology. Computational efficiency of molecular dynamics algorithms continues to pose challenges. The collective efforts of many groups of researchers have enabled a great deal of force fields applied to many problems in biology. Biomolecular simulations require the presence of solvent; hence, a reliable representation of water is crucial for achieving realistic results. Several specialised methods stemming from MD allow to investigate various processes of biomolecules: REMD, SMD and TMD.

show abstract

Biomedical‐Optical‐Window Tailored Cyanines for Steerable Inflammatory Bowel Disease Theranostic

Yue,

Ai,

Chi

et al. 2024

Advanced Materials

View full text Add to dashboard Cite

Tailored photophysical properties and chemical activity is the ultimate pursuit of functional dyes for in vivo biomedical theranostics. In this work, the independent regulation of the absorption and fluorescence emission wavelengths of heptamethine cyanines is reported. These dyes retain near‐infrared fluorescence emission (except a nitro‐modified dye) while feature variable absorption wavelengths ranging from 590 to 860 nm. This enables to obtain customized functional dyes that meet the excitation and fluorescence wavelength requirements defined by the optical properties of tissues for in vivo biomedical applications. Typically, a nitro‐modified photothermal active derivative Cy‐Mu‐7‐9 is used, which features strong absorption at 810 nm in PBS, a wavelength that balanced the tissue penetration depth and non‐specific photothermal effect, to realize non‐destructive inflammatory bowel disease (IBD) therapy via photothermal induced up‐regulation of heat shock protein 70 in the intestinal epithelial cells. The corresponding amino‐modified dye Cy‐Mu‐7‐9‐NH2, which can be formed in health enteric cavity by Cy‐Mu‐7‐9 after oral administration, is a fluorescence compound with the emission of 800 nm in PBS. Based on the IBD sensitive transformation of Cy‐Mu‐7‐9 and Cy‐Mu‐7‐9‐NH2, in vivo IBD theranostic and therapeutic effect evaluation is realized via the synergy of fluorescence imaging and photothermal therapy for the first time.

show abstract

RMSD and Symmetry

Cited by 51 publications

References 22 publications

The quaternion-based spatial-coordinate and orientation-frame alignment problems

The quaternion-based spatial-coordinate and orientation-frame alignment problems

Molecular Dynamics

Biomedical‐Optical‐Window Tailored Cyanines for Steerable Inflammatory Bowel Disease Theranostic

Contact Info

Product

Resources

About