Phase problem
Encyclopedia
In physics the phase problem is the name given to the problem of loss of information concerning the phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

that can occur when making a physical measurement. The name itself comes from the field of x-ray crystallography
X-ray crystallography
X-ray crystallography is a method of determining the arrangement of atoms within a crystal, in which a beam of X-rays strikes a crystal and causes the beam of light to spread into many specific directions. From the angles and intensities of these diffracted beams, a crystallographer can produce a...

, where the phase problem has to be solved for the determination of a structure from diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

data. The phase problem is also met in the fields of imaging
Imaging science
Imaging science is a multidisciplinary field concerned with the generation, collection, duplication, analysis, modification, and visualization of images . As an evolving field it includes research and researchers from physics, mathematics, electrical engineering, computer vision, computer science,...

and signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

. Various approaches have been developed over the years to solve it.

## Overview

Light detectors, such as photographic plates or CCD
Charge-coupled device
A charge-coupled device is a device for the movement of electrical charge, usually from within the device to an area where the charge can be manipulated, for example conversion into a digital value. This is achieved by "shifting" the signals between stages within the device one at a time...

s, measure only the intensity of the light that hits them. This measurement is incomplete (even when neglecting other degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

such as polarization) because a light wave has not only an amplitude (related to the intensity), but also a phase, which is systematically lost in a measurement. In diffraction or microscopy
Microscopy
Microscopy is the technical field of using microscopes to view samples and objects that cannot be seen with the unaided eye...

experiments, the phase part of the wave often contains valuable information on the studied specimen. Note that the phase problem constitutes a fundamental limitation ultimately related to the nature of measurement in quantum mechanics
Measurement in quantum mechanics
The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus....

.

In x-ray crystallography, the diffraction data when properly assembled gives the amplitude of the 3D Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

of the molecule's electron density
Electron density
Electron density is the measure of the probability of an electron being present at a specific location.In molecules, regions of electron density are usually found around the atom, and its bonds...

in the unit cell. If the phases are known, the electron density can be simply obtained by Fourier synthesis. This Fourier transform relation also holds for two-dimensional far-field diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

patterns (also called Fraunhofer diffraction
Fraunhofer diffraction
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens....

) giving rise to a similar type of phase problem.

## Solutions in x-ray crystallography

In x-ray crystallography
X-ray crystallography
X-ray crystallography is a method of determining the arrangement of atoms within a crystal, in which a beam of X-rays strikes a crystal and causes the beam of light to spread into many specific directions. From the angles and intensities of these diffracted beams, a crystallographer can produce a...

, there are several ways to recover the lost phases. A powerful solution is the multiwavelength anomalous diffraction
Multi-wavelength anomalous dispersion
Multi-wavelength anomalous diffraction is a technique used in X-ray crystallography that facilitates the determination of the three-dimensional structure of biological macromolecules via solution of the phase problem...

(MAD) method. In this technique, atoms' inner electrons absorb x-rays of particular wavelengths, and reemit the x-rays after a delay, inducing a phase shift in all of the reflections, known as the anomalous dispersion effect. Analysis of this phase shift (which may be different for individual reflections) results in a solution for the phases. Since x-ray fluorescence techniques require excitation at very specific wavelengths, it is necessary to use synchrotron radiation
The electromagnetic radiation emitted when charged particles are accelerated radially is called synchrotron radiation. It is produced in synchrotrons using bending magnets, undulators and/or wigglers...

Other methods of experimental phase determination include multiple isomorphous replacement
Multiple isomorphous replacement
Multiple isomorphous replacement or MIR is the most common approach of solving the phase problem in X-ray crystallography. 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...

(MIR), where heavy atoms are inserted into structure (usually by synthesizing proteins with analogs or by soaking), and
Single wavelength anomalous dispersion
Single wavelength anomalous dispersion
Single-wavelength anomalous dispersion 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 dispersion, SAD uses a...

Phases can also be inferred by using a process called molecular replacement
Molecular replacement
Molecular replacement 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 homologous to our unknown structure from which the diffraction data is derived.The first goal of the crystallographer is to...

, where a similar molecule's phases are grafted onto the intensities which are experimentally determined. These phases can be obtained experimentally from a homologous molecule or if the phases are known for the same molecule but in a different crystal, by simulating the molecule's packing in the crystal and obtaining theoretical phases. Generally, these techniques are less desirable since they can severely bias the solution of the structure. They are useful, however, for ligand binding studies, or between molecules with small differences and relatively rigid structures (for example derivatizing a small molecule).

There are two major processes for recovering the phases using the data obtained by regular equipment. One is the direct method, which estimates the initial phases and expanding phases using a triple relation. (A trio of reflections in which the intensity and phase of one reflection can be explained by the other two has a triple relation.) A number of initial phases are tested and selected by this method. The other is the Patterson method, which directly determines the positions of heavy atoms. The Patterson function
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....

gives a large value in a position which corresponds to interatomic vectors. This method can be applied only when the crystal contains heavy atoms or when a significant fraction of the structure is already known. Because of the development of computers, the direct method is now the most useful technique for solving the phase problem.

For molecules whose crystals provide reflections in the sub-angstrom range, it is possible to determine phases by brute force
Brute-force search
In computer science, brute-force search or exhaustive search, also known as generate and test, is a trivial but very general problem-solving technique that consists of systematically enumerating all possible candidates for the solution and checking whether each candidate satisfies the problem's...

methods, testing a series of phase values until spherical structures are observed in the resultant electron density map. This works because atoms have a characteristic structure when viewed in the sub-angstrom range. The technique is limited by processing power, and data quality. For practical purposes, it is limited to "small molecules" since they consistently provide high-quality diffraction with very few reflections.

In many cases, an initial set of phases are determined, and the electron density map for the diffraction pattern is calculated. Then the map is used to determine portions of the structure, which portions are used to simulate a new set of phases. This new set of phases is known as a refinement. These phases are reapplied to the original amplitudes, and an improved electron density map is derived, from which the structure is corrected. This process is repeated until an error term (usually Rfree) has stabilized to a satisfactory value. Because of the phenomenon of phase bias, it is possible for an incorrect initial assignment to propagate through successive refinements, and satisfactory conditions for a structure assignment are still a matter of debate. Indeed, some spectacular incorrect assignments have been reported, including a protein where the entire sequence was threaded backwards.