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Kabsch Algorithm
The Kabsch algorithm, named after Wolfgang Kabsch, is a method for calculating the optimal rotation matrix that minimizes the RMSD (root mean squared deviation) between two paired sets of points. It is useful in graphics, cheminformatics to compare molecular structures, and also bioinformatics for comparing protein structures (in particular, see root-mean-square deviation (bioinformatics)). The algorithm only computes the rotation matrix, but it also requires the computation of a translation vector. When both the translation and rotation are actually performed, the algorithm is sometimes called partial Procrustes superimposition (see also orthogonal Procrustes problem). Description The algorithm for the rotation of into starts with two sets of paired points, and . Each set of points can be represented as an matrix. The first row is the coordinates of the first point, the second row is the coordinates of the second point, the th row is the coordinates of the th point. C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfgang Kabsch
Wolfgang is a German male given name traditionally popular in Germany, Austria and Switzerland. The name is a combination of the Old High German words ''wolf'', meaning "wolf", and ''gang'', meaning "path", "journey", "travel". Besides the regular "wolf", the first element also occurs in Old High German as the combining form "-olf". The earliest reference of the name being used was in the 8th century. The name was also attested as "Vulfgang" in the Reichenauer Verbrüderungsbuch in the 9th century. The earliest recorded famous bearer of the name was a tenth-century Saint Wolfgang of Regensburg. Due to the lack of conflict with the pagan reference in the name with Catholicism, it is likely a much more ancient name whose meaning had already been lost by the tenth century. Grimm ('' Teutonic Mythology'' p. 1093) interpreted the name as that of a hero in front of whom walks the "wolf of victory". A Latin gloss by Arnold of St Emmeram interprets the name as ''Lupambulus''.E. Fö ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Design Matrix
In statistics and in particular in regression analysis, a design matrix, also known as model matrix or regressor matrix and often denoted by X, is a matrix of values of explanatory variables of a set of objects. Each row represents an individual object, with the successive columns corresponding to the variables and their specific values for that object. The design matrix is used in certain statistical models, e.g., the general linear model. It can contain indicator variables (ones and zeros) that indicate group membership in an ANOVA, or it can contain values of continuous variables. The design matrix contains data on the independent variables (also called explanatory variables) in statistical models which attempt to explain observed data on a response variable (often called a dependent variable) in terms of the explanatory variables. The theory relating to such models makes substantial use of matrix manipulations involving the design matrix: see for example linear regression. A n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wahba's Problem
In applied mathematics, Wahba's problem, first posed by Grace Wahba in 1965, seeks to find a rotation matrix ( special orthogonal matrix) between two coordinate systems from a set of (weighted) vector observations. Solutions to Wahba's problem are often used in satellite attitude determination utilising sensors such as magnetometers and multi-antenna GPS receivers. The cost function that Wahba's problem seeks to minimise is as follows: : J(\mathbf) = \frac \sum_^ a_k\, \mathbf_k - \mathbf \mathbf_k \, ^2 for N\geq2 where \mathbf_k is the ''k''-th 3-vector measurement in the reference frame, \mathbf_k is the corresponding ''k''-th 3-vector measurement in the body frame and \mathbf is a 3 by 3 rotation matrix between the coordinate frames. a_k is an optional set of weights for each observation. A number of solutions to the problem have appeared in literature, notably Davenport's q-method, QUEST and methods based on the singular value decomposition (SVD). Several methods for so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FoldX
FoldX is a protein design algorithm that uses an empirical force field. It can determine the energetic effect of point mutations as well as the interaction energy of protein complexes (including Protein- DNA). FoldX can mutate protein and DNA side chains using a probability-based rotamer library, while exploring alternative conformations of the surrounding side chains. Applications * Prediction of the effect of point mutations or human SNPs on protein stability or protein complexes * Protein design to improve stability or modify affinity or specificity * Homology modeling The FoldX force field The energy function includes terms that have been found to be important for protein stability, where the energy of unfolding (∆G) of a target protein is calculated using the equation: ∆G = ∆Gvdw + ∆GsolvH + ∆GsolvP + ∆Ghbond + ∆Gwb + ∆Gel + ∆Smc + ∆Ssc Where ∆Gvdw is the sum of the Van der Waals contributions of all atoms with respect to the same interacti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Visual Molecular Dynamics
Visual Molecular Dynamics (VMD) is a molecular modelling and visualization computer program. VMD is developed mainly as a tool to view and analyze the results of molecular dynamics simulations. It also includes tools for working with volumetric data, sequence data, and arbitrary graphics objects. Molecular scenes can be exported to external rendering tools such as POV-Ray, RenderMan, Tachyon, Virtual Reality Modeling Language (VRML), and many others. Users can run their own Tcl and Python scripts within VMD as it includes embedded Tcl and Python interpreters. VMD runs on Unix, Apple Mac macOS, and Microsoft Windows. VMD is available to non-commercial users under a distribution-specific license which permits both use of the program and modification of its source code, at no charge. History VMD has been developed under the aegis of principal investigator Klaus Schulten in the Theoretical and Computational Biophysics group at the Beckman Institute for Advanced Science and Technolo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross-entropy Method
The cross-entropy (CE) method is a Monte Carlo method for importance sampling and optimization. It is applicable to both combinatorial and continuous problems, with either a static or noisy objective. The method approximates the optimal importance sampling estimator by repeating two phases:Rubinstein, R.Y. and Kroese, D.P. (2004), The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning, Springer-Verlag, New York . #Draw a sample from a probability distribution. #Minimize the '' cross-entropy'' between this distribution and a target distribution to produce a better sample in the next iteration. Reuven Rubinstein developed the method in the context of ''rare-event simulation'', where tiny probabilities must be estimated, for example in network reliability analysis, queueing models, or performance analysis of telecommunication systems. The method has also been applied to the traveling salesman, quadratic assignment, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PyMol
PyMOL is an open source but proprietary molecular visualization system created by Warren Lyford DeLano. It was commercialized initially by DeLano Scientific LLC, which was a private software company dedicated to creating useful tools that become universally accessible to scientific and educational communities. It is currently commercialized by Schrödinger, Inc. As the original software license was a permissive licence, they were able to remove it; new versions are no longer released under the Python license, but under a custom license (granting broad use, redistribution, and modification rights, but assigning copyright to any version to Schrodinger, LLC.), and some of the source code is no longer released. PyMOL can produce high-quality 3D images of small molecules and biological macromolecules, such as proteins. According to the original author, by 2009, almost a quarter of all published images of 3D protein structures in the scientific literature were made using PyMOL. PyMO ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Python (programming Language)
Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. Python is dynamically-typed and garbage-collected. It supports multiple programming paradigms, including structured (particularly procedural), object-oriented and functional programming. It is often described as a "batteries included" language due to its comprehensive standard library. Guido van Rossum began working on Python in the late 1980s as a successor to the ABC programming language and first released it in 1991 as Python 0.9.0. Python 2.0 was released in 2000 and introduced new features such as list comprehensions, cycle-detecting garbage collection, reference counting, and Unicode support. Python 3.0, released in 2008, was a major revision that is not completely backward-compatible with earlier versions. Python 2 was discontinued with version 2.7.18 in 2020. Python consistently r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigen (C++ Library)
Eigen is a high-level C++ library of template headers for linear algebra, matrix and vector operations, geometrical transformations, numerical solvers and related algorithms. Eigen is open-source software licensed under the Mozilla Public License 2.0 since version 3.1.1. Earlier versions were licensed under the GNU Lesser General Public License. Version 1.0 was released in Dec 2006. Eigen is implemented using the expression templates metaprogramming technique, meaning it builds expression trees at compile time and generates custom code to evaluate these. Using expression templates and a cost model of floating point operations, the library performs its own loop unrolling and vectorization. Eigen itself can provide BLAS and a subset of LAPACK interfaces. New Release 3.4 includes many improvements. See also * List of numerical libraries * Numerical linear algebra Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matlab
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. As of 2020, MATLAB has more than 4 million users worldwide. They come from various backgrounds of engineering, science, and economics. History Origins MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was based on his 1960s PhD thesis. Moler became a math professor at the University of New Mexico a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecular Dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the system. In the most common version, the trajectories of atoms and molecules are determined by numerically solving Newton's equations of motion for a system of interacting particles, where forces between the particles and their potential energies are often calculated using interatomic potentials or molecular mechanical force fields. The method is applied mostly in chemical physics, materials science, and biophysics. Because molecular systems typically consist of a vast number of particles, it is impossible to determine the properties of such complex systems analytically; MD simulation circumvents this problem by using numerical methods. However, long MD simulations are mathematically ill-conditioned, generating cumulative erro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k where , and are real numbers; and , and are the ''basic quaternions''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
