Editing Phase problem

{{Short description|Issue in determining wave cycle part}}
In physics, the '''phase problem''' is the problem of loss of information concerning the [[phase (waves)|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.<ref name=":0">{{Cite journal|last=Taylor|first=Garry|date=2003-11-01|title=The phase problem|journal=Acta Crystallographica Section D |volume=59|issue=11|pages=1881–1890|doi=10.1107/S0907444903017815|pmid=14573942|doi-access=free|bibcode=2003AcCrD..59.1881T }}</ref> The phase problem is also met in the fields of [[imaging science|imaging]] and [[signal processing]].<ref name=":1">{{Cite arXiv|last1=Shechtman|first1=Yoav|authorlink1=Yoav Shechtman|last2=Eldar|first2=Yonina C.|authorlink2=Yonina Eldar|last3=Cohen|first3=Oren|last4=Chapman|first4=Henry N.|authorlink4=Henry N. Chapman|last5=Miao|first5=Jianwei|authorlink5=Jianwei Miao|last6=Segev|first6=Mordechai|date=2014-02-28|title=Phase Retrieval with Application to Optical Imaging|class=cs.IT|eprint=1402.7350}}</ref> Various approaches of [[phase retrieval]] have been developed over the years.

==Overview==

Light detectors, such as [[photographic plate]]s or [[charge-coupled device|CCD]]s, measure only the intensity of the light that hits them. This measurement is incomplete (even when neglecting other [[degrees of freedom (physics and chemistry)|degrees of freedom]] such as [[polarization (waves)|polarization]] and [[angle of incidence (optics)|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.<ref name=":1" /> In [[diffraction]] or [[microscopy]] experiments, the phase part of the wave often contains valuable information on the studied specimen. 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]].<ref name=":0" /> 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.

==Phase retrieval ==
{{Main|Phase retrieval}}

There are several ways to [[Phase retrieval|retrieve]] the lost phases. The phase problem must be solved in [[x-ray crystallography]],<ref name=":0" /> [[neutron crystallography]],<ref>{{Cite journal|last1=Hauptman|first1=Herbert A.|last2=Langs|first2=David A.|date=2003-05-01|title=The phase problem in neutron crystallography|journal=Acta Crystallographica Section A|volume=59|issue=3|pages=250–254|doi=10.1107/S010876730300521X|pmid=12714776}}</ref> and [[electron crystallography]].<ref name=":2">{{Cite journal|last=Dorset|first=D. L.|date=1997-03-04|title=Direct phase determination in protein electron crystallography: The pseudo-atom approximation|journal=Proceedings of the National Academy of Sciences|language=en|volume=94|issue=5|pages=1791–1794|doi=10.1073/pnas.94.5.1791|pmc=19995|pmid=9050857|bibcode=1997PNAS...94.1791D|doi-access=free}}</ref><ref name=":3">{{Cite journal|last=Dorset|first=D. L.|date=1996-05-01|title=Direct Phasing in Protein Electron Crystallography – Phase Extension and the Prospects for Ab Initio Determinations|journal=Acta Crystallographica Section A |language=en|volume=52|issue=3|pages=480–489|doi=10.1107/S0108767396001420|pmid=8694993|doi-access=free|bibcode=1996AcCrA..52..480D }}</ref><ref>{{Cite journal|last1=Henderson|first1=R.|last2=Baldwin|first2=J. M.|last3=Downing|first3=K. H.|last4=Lepault|first4=J.|last5=Zemlin|first5=F.|date=1986-01-01|title=Structure of purple membrane from halobacterium halobium: recording, measurement and evaluation of electron micrographs at 3.5 Å resolution|journal=Ultramicroscopy|language=en|volume=19|issue=2|pages=147–178|doi=10.1016/0304-3991(86)90203-2}}</ref>

Not all of the methods of [[phase retrieval]] work with every [[wavelength]] (x-ray, neutron, and electron) used in crystallography.

=== [[Direct methods (crystallography)|Direct (''ab initio)'' methods]] ===
If the crystal diffracts to high resolution (<1.2 Å), the initial phases can be estimated using direct methods.<ref name=":0" />  Direct methods can be used in  [[x-ray crystallography]],<ref name=":0" /> [[neutron crystallography]],<ref>{{Cite journal|last=Hauptman|first=H.|date=1976-09-01|title=Probabilistic theory of the structure invariants: extension to the unequal atom case with application to neutron diffraction|journal=Acta Crystallographica Section A |language=en|volume=32|issue=5|pages=877–882|doi=10.1107/S0567739476001757|bibcode=1976AcCrA..32..877H|doi-access=free}}</ref> and [[electron crystallography]].<ref name=":2" /><ref name=":3" />

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.  

For molecules whose crystals provide reflections in the sub-Ångström range, it is possible to determine phases by [[brute-force search|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-Ångström range.  The technique is limited by processing power and data quality.  For practical purposes, it is limited to "small molecules" and peptides because they consistently provide high-quality diffraction with very few reflections.

=== [[Molecular replacement]] (MR) ===
Phases can also be inferred by using a process called [[molecular replacement]], where a similar molecule's already-known phases are grafted onto the intensities of the molecule at hand, which are observationally 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).

=== [[Isomorphous replacement]] ===

==== ''[[Multiple isomorphous replacement]] (MIR)''  ====
''[[Multiple isomorphous replacement]] (MIR)'', where heavy atoms are inserted into structure (usually by synthesizing proteins with analogs or by soaking)

=== Anomalous scattering ===

==== ''[[Single-wavelength anomalous dispersion]]'' (SAD). ====

==== ''[[multi-wavelength anomalous dispersion|Multi-wavelength anomalous dispersion (MAD)]]'' ====
A powerful solution is the ''[[multi-wavelength anomalous dispersion]]'' (MAD) method. In this technique, atoms' inner electrons{{clarify|reason=Which atoms? Special new atoms that are introduced?|date=July 2016}} 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 (like this one) require excitation at very specific wavelengths, it is necessary to use [[synchrotron radiation]] when using the MAD method.

== Phase improvement ==

=== Refining initial phases ===
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 <math>R_\textrm{free}</math>) 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, so 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.<ref name="Kleywegt2000">{{cite journal|last1=Kleywegt|first1=Gerard J.|title=Validation of protein crystal structures|journal=Acta Crystallographica Section D|volume=56|issue=3|year=2000|pages=249–265|doi=10.1107/S0907444999016364|pmid=10713511|doi-access=free}}</ref>

=== Density modification (phase improvement) ===

==== Solvent flattening ====

==== Histogram matching ====

==== Non-crystallographic symmetry averaging ====

==== Partial structure ====

==== Phase extension ====

==See also==
* [[Coherent diffraction imaging]]
*[[Ptychography]]
*[[Phase retrieval]]

==External links==
* [http://www.ysbl.york.ac.uk/~cowtan/fourier/magic.html An example of phase bias]
* [http://www.ysbl.york.ac.uk/~cowtan/fourier/coeff.html An appropriate use of 'molecular replacement']
* [https://www.xtal.iqf.csic.es/Cristalografia/index-en.html Learning crystallography]

==References==
<references/>

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{{DEFAULTSORT:Phase Problem}}
[[Category:Crystallography]]
[[Category:Inverse problems]]