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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 stu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase (waves)
In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other 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 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 \varphi(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 \varphi(t) is also a periodic function, with the same period as F, that repeatedly ... [...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 and (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 ... [...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 (MAD), SAD uses a single dataset at a single appropriate wavelength. Compared to MAD, SAD has weaker phasing power and requires density modification to resolve phase ambiguity. This downside is not as important as SAD's main advantage: the minimization of time spent in the beam by the crystal, thus reducing potential radiation damage to the molecule while collecting data. SAD also allows a wider choice of heavy atoms and can be conducted without a synchrotron beamline. Today, selenium-SAD is commonly used for experimental phasing due to the development of methods for selenomethionine incorporation into recombinant proteins. SAD is sometimes called "single-wavelength anomalous dispersion", but no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiple Isomorphous Replacement
Isomorphous replacement (IR) 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 vectors ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphous Replacement
Isomorphous replacement (IR) 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 vectors can be s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecular Replacement
Molecular replacement (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_\ell, F_, \exp(2\pi i(hx+ky+\ell z)+i\Phi(hk\ell)). 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 exp ... [...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 Iteration#Computing, systematically checking all possible candidates for whether or not 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 (#Combinatorial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Patterson Function
The Patterson function is used to solve the phase problem in X-ray crystallography X-ray crystallography is the experimental science of determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to Diffraction, diffract in specific directions. By measuring th .... It was introduced in 1935 by Arthur Lindo Patterson while he was a visiting researcher in the laboratory of Bertram Eugene Warren at Massachusetts Institute of Technology, MIT. The Patterson function is defined as P(u,v,w) = \sum_ \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 convolution, 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' positio ... [...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 and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same ''phase (waves), phase'' on the wave, such as two adjacent crests, troughs, or zero crossings. Wavelength is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The multiplicative inverse, inverse of the wavelength is called the ''spatial frequency''. Wavelength is commonly designated by the Greek letter lambda (''λ''). For a modulated wave, ''wavelength'' may refer to the carrier wavelength of the signal. The term ''wavelength'' may also apply to the repeating envelope (mathematics), envelope of modulated waves or waves formed by Interference (wave propagation), interference of several sinusoids. Assuming a sinusoidal wave moving at a fixed phase velocity, wave speed, wavelength is inversely proportion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electron Crystallography
Electron crystallography is a subset of methods in electron diffraction focusing upon detailed determination of the positions of atoms in solids using a transmission electron microscope (TEM). It can involve the use of high-resolution transmission electron microscopy images, electron diffraction patterns including convergent-beam electron diffraction or combinations of these. It has been successful in determining some bulk structures, and also surface structures. Two related methods are low-energy electron diffraction which has solved the structure of many surfaces, and reflection high-energy electron diffraction which is used to monitor surfaces often during growth. The technique date back to soon after the discovery of electron diffraction in 1927-28, and was used in many early works. However, for many years quantitative electron crystallography was not used, instead the diffraction information was combined qualitatively with imaging results. A number of advances from the 195 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
