Phase problem

The phase problem is the name given to the problem of loss of information (the phase) from a physical measurement. The name itself 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 have been developed over the years to solve it.

Overview
Light detectors, such as photographic plates or CCDs, measure only the intensity of the light hitting upon them. This measurement is incomplete (even when neglecting other degrees of freedom 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 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.

In x-ray crystallography, the diffraction data when properly assembled gives the amplitude of the 3D Fourier transform of the molecule's electron density 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 patterns (also called Fraunhofer diffraction) giving rise to a similar type of phase problem.

Solutions in x-ray crystallography
In x-ray crystallography, there are several ways to recover the lost phases. A powerful solution is the multiwavelength anomalous diffraction (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 when using the MAD method.

Other methods of experimental phase determination include multiple isomorphous replacement (MIR), where heavy atoms are inserted into structure (usually by synthesizing proteins with analogs or by soaking), and single wavelength anomalous diffraction(SAD).

Phases can also be inferred by using a process called molecular replacement, 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 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 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.