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Phase Problem
In physics, the phase problem is the problem of loss of information concerning the phase that can occur when making a physical measurement. The name comes from the field of X-ray crystallography, where the phase problem has to be solved for the determination of a structure from diffraction data. The phase problem is also met in the fields of imaging and signal processing. Various approaches of phase retrieval have been developed over the years. Overview Light detectors, such as photographic plates or CCDs, measure only the intensity of the light that hits them. This measurement is incomplete (even when neglecting other degrees of freedom such as polarization and angle of incidence) because a light wave has not only an amplitude (related to the intensity), but also a phase (related to the direction), and polarization which are systematically lost in a measurement. In diffraction or microscopy experiments, the phase part of the wave often contains valuable information on the stud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase (waves)
In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it varies by one full turn as the variable t goes through each period (and F(t) goes through each complete cycle). It may be measured in any angular unit such as degrees or radians, thus increasing by 360° or 2\pi as the variable t completes a full period. This convention is especially appropriate for a sinusoidal function, since its value at any argument t then can be expressed as \phi(t), the sine of the phase, multiplied by some factor (the amplitude of the sinusoid). (The cosine may be used instead of sine, depending on where one considers each period to start.) Usually, whole turns are ignored when expressing the phase; so that \phi(t) is also a periodic function, with the same period as F, that repeatedly scans the same range of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fraunhofer Diffraction
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when plane waves are incident on a diffracting object, and the diffraction pattern is viewed at a sufficiently long distance (a distance satisfying Fraunhofer condition) from the object (in the far-field region), and also when it is viewed at the focal plane of an imaging lens. In contrast, the diffraction pattern created near the diffracting object (in the near field region) is given by the Fresnel diffraction equation. The equation was named in honor of Joseph von Fraunhofer although he was not actually involved in the development of the theory. This article explains where the Fraunhofer equation can be applied, and shows Fraunhofer diffraction patterns for various apertures. A detailed mathematical treatment of Fraunhofer diffraction is given in Fraunhofer diffraction equation. Equation When a beam of light is partly blocked by an obstacle, some of the light is scattered around the o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multi-wavelength Anomalous Dispersion
Multi-wavelength anomalous diffraction (sometimes Multi-wavelength anomalous dispersion; abbreviated MAD) is a technique used in X-ray crystallography that facilitates the determination of the three-dimensional structure of biological macromolecules (e.g. DNA, drug receptors) via solution of the phase problem. MAD was developed by Wayne Hendrickson while working as a postdoctoral researcher under Jerome Karle at the United States Naval Research Laboratory. The mathematics upon which MAD (and progenitor Single wavelength anomalous dispersion) were based were developed by Jerome Karle, work for which he was awarded the 1985 Nobel Prize in Chemistry (along with Herbert Hauptman). See also * Single wavelength anomalous dispersion (SAD) * Multiple isomorphous replacement (MIR) *Anomalous scattering *Anomalous X-ray scattering *Patterson map The Patterson function is used to solve the phase problem in X-ray crystallography. It was introduced in 1935 by Arthur Lindo Patterson while he w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Single-wavelength Anomalous Dispersion
Single-wavelength anomalous diffraction (SAD) is a technique used in X-ray crystallography that facilitates the determination of the structure of proteins or other biological macromolecules by allowing the solution of the phase problem. In contrast to multi-wavelength anomalous diffraction, SAD uses a single dataset at a single appropriate wavelength. One advantage of the technique is the minimization of time spent in the beam by the crystal, thus reducing potential radiation damage to the molecule while collecting data. SAD is sometimes called "single-wavelength anomalous dispersion", but no dispersive differences are used in this technique since the data are collected at a single wavelength. Today, selenium-SAD is commonly used for experimental phasing due to the development of methods for selenomethionine incorporation into recombinant proteins. See also * Multi-wavelength anomalous dispersion (MAD) * Multiple isomorphous replacement (MIR) *Anomalous scattering *Anomalous X-ray ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiple Isomorphous Replacement
Multiple isomorphous replacement (MIR) is historically the most common approach to solving the phase problem in X-ray crystallography studies of proteins. For protein crystals this method is conducted by soaking the crystal of a sample to be analyzed with a heavy atom solution or co-crystallization with the heavy atom. The addition of the heavy atom (or ion) to the structure should not affect the crystal formation or unit cell dimensions in comparison to its native form, hence, they should be isomorphic. Data sets from the native and heavy-atom derivative of the sample are first collected. Then the interpretation of the Patterson difference map reveals the heavy atom's location in the unit cell. This allows both the amplitude and the phase of the heavy-atom contribution to be determined. Since the structure factor of the heavy atom derivative (Fph) of the crystal is the vector sum of the lone heavy atom (Fh) and the native crystal (Fp) then the phase of the native Fp and Fph vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphous Replacement
Multiple isomorphous replacement (MIR) is historically the most common approach to solving the phase problem in X-ray crystallography studies of proteins. For protein crystals this method is conducted by soaking the crystal of a sample to be analyzed with a heavy atom solution or co-crystallization with the heavy atom. The addition of the heavy atom (or ion) to the structure should not affect the crystal formation or unit cell dimensions in comparison to its native form, hence, they should be isomorphic. Data sets from the native and heavy-atom derivative of the sample are first collected. Then the interpretation of the Patterson difference map reveals the heavy atom's location in the unit cell. This allows both the amplitude and the phase of the heavy-atom contribution to be determined. Since the structure factor of the heavy atom derivative (Fph) of the crystal is the vector sum of the lone heavy atom (Fh) and the native crystal (Fp) then the phase of the native Fp and Fph vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecular Replacement
Molecular replacement (or MR) is a method of solving the phase problem in X-ray crystallography. MR relies upon the existence of a previously solved protein structure which is similar to our unknown structure from which the diffraction data is derived. This could come from a homologous protein, or from the lower-resolution protein NMR structure of the same protein. The first goal of the crystallographer is to obtain an electron density map, density being related with diffracted wave as follows: \rho(x,y,z)=\frac\sum_h\sum_k\sum_l, F_, \exp(2\pi i(hx+ky+lz)+i\Phi(hkl)). With usual detectors the intensity I=F\cdot F^* is being measured, and all the information about phase (\Phi) is lost. Then, in the absence of phases (Φ), we are unable to complete the shown Fourier transform relating the experimental data from X-ray crystallography (in reciprocal space) to real-space electron density, into which the atomic model is built. MR tries to find the model which fits best experimental i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brute-force Search
In computer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique and algorithmic paradigm that consists of systematically enumerating all possible candidates for the solution and checking whether each candidate satisfies the problem's statement. A brute-force algorithm that finds the divisors of a natural number ''n'' would enumerate all integers from 1 to n, and check whether each of them divides ''n'' without remainder. A brute-force approach for the eight queens puzzle would examine all possible arrangements of 8 pieces on the 64-square chessboard and for each arrangement, check whether each (queen) piece can attack any other. While a brute-force search is simple to implement and will always find a solution if it exists, implementation costs are proportional to the number of candidate solutionswhich in many practical problems tends to grow very quickly as the size of the problem increases ( §Combinator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Patterson Function
The Patterson function is used to solve the phase problem in X-ray crystallography. It was introduced in 1935 by Arthur Lindo Patterson while he was a visiting researcher in the laboratory of Bertram Eugene Warren at MIT. The Patterson function is defined as :P(u,v,w) = \sum\limits_ \left, F_\^2 \;e^. It is essentially the Fourier transform of the intensities rather than the structure factors. The Patterson function is also equivalent to the electron density convolved with its inverse: :P\left(\vec\right) = \rho\left(\vec\right) * \rho\left(-\vec\right). Furthermore, a Patterson map of ''N'' points will have peaks, excluding the central (origin) peak and any overlap. The peaks' positions in the Patterson function are the interatomic distance vectors and the peak heights are proportional to the product of the number of electrons in the atoms concerned. Because for each vector between atoms ''i'' and ''j'' there is an oppositely oriented vector of the same length (between atoms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Methods (crystallography)
In crystallography, direct methods are a family of methods for estimating the phases of the Fourier transform of the scattering density from the corresponding magnitudes. The methods generally exploit constraints or statistical correlations between the phases of different Fourier components that result from the fact that the scattering density must be a positive real number. In two dimensions, it is relatively easy to solve the phase problem directly, but not so in three dimensions. The key step was taken by Hauptman and Karle, who developed a practical method to employ the Sayre equation for which they were awarded the 1985 Nobel prize in Chemistry. The Nobel Prize citation was "for their outstanding achievements in the development of direct methods for the determination of crystal structures." At present, direct methods are the preferred method for phasing crystals of small molecules having up to 1000 atoms in the asymmetric unit. However, they are generally not feasible by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter ''lambda'' (λ). The term ''wavelength'' is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids. Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency of the wave: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths. Wavelength depends on the medium (for example, vacuum, air, or water) that a wav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electron Crystallography
Electron crystallography is a method to determine the arrangement of atoms in solids using a transmission electron microscope (TEM). Comparison with X-ray crystallography It can complement X-ray crystallography for studies of very small crystals ( 1 micrometer) crystals impervious to electrons, which only penetrate short distances. One of the main difficulties in X-ray crystallography is determining phases in the diffraction pattern. Because of the complexity of X-ray lenses, it is difficult to form an image of the crystal being diffracted, and hence phase information is lost. Fortunately, electron microscopes can resolve atomic structure in real space and the crystallographic structure factor phase information can be experimentally determined from an image's Fourier transform. The Fourier transform of an atomic resolution image is similar, but different, to a diffraction pattern—with reciprocal lattice spots reflecting the symmetry and spacing of a crystal. Aaron Klug was the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
