International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F, ch. 12.2, pp. 256-257
Section 12.2.1. The origin of the phase problem^{a}Institut für Pharmazeutische Chemie der Philipps-Universität Marburg, Marbacher Weg 6, D-35032 Marburg, Germany, and ^{b}Max-Planck-Institut für Biochemie, 82152 Martinsried, Germany |
Once a native data set has been collected, the next task is the solution of the structure. There is one major hurdle: the phase problem. To study objects at the atomic level, we must utilize waves with a wavelength in the ångström range, i.e. X-radiation. X-rays interact with electrons and so provide an image of the electron distribution of the sample. Unfortunately, X-rays are refracted by matter only very weakly, and so it is not possible to construct a lens to view molecules at atomic dimensions.^{1}
As shown in Chapter 2.1 , the diffraction obtained from an electron-density distribution is given by where S is perpendicular to the scattered wave and ; θ is the scattering angle and λ is the wavelength. The diffraction pattern is a Fourier transform of the electron density. If we have a crystal with cell parameters a, b and c, then the Laue diffraction conditions require that S lies on a reciprocal lattice such that , where and are the reciprocal-lattice vectors, and h, k and l are the integer indices of the diffracted beam. where V represents the volume of the unit cell, and x, y and z are the fractional coordinates within that cell in the directions of a, b and c.
Since the diffraction pattern is a Fourier transform of the electron density, it follows that the electron density is an inverse Fourier transform of the diffraction pattern:
Thus it should be mathematically straightforward to calculate the electron density from the diffraction pattern. This is, unfortunately, not the case. The function describing the diffracted rays is a complex function with a magnitude and a phase . The diffraction experiment measures the intensities , however; the relationship between and is: where is the complex conjugate of . The measured intensities are related directly to the magnitudes of the diffracted beams; the phase information, however, is lost (Fig. 12.2.1.1): this is the origin of the phase problem.
There are essentially four ways of overcoming the phase problem (Fig. 12.2.1.2):
The method of isomorphous replacement, by which the first macromolecular structures were solved (Green et al., 1954), remains the most widely used technique for ab initio structure determination, although the availability of synchrotrons, with their facility for selecting a desired wavelength, and molecular-biology techniques that allow the direct introduction of anomalous scatterers, such as selenium or tellurium, into the protein of interest (Hendrickson et al., 1990; Budisa et al., 1997) have proven that multiple anomalous dispersion is an exceptionally powerful technique for the solution of novel structures. Patterson search techniques (Rossmann, 1972) are ideal if a similar macromolecular structure is already known, while direct methods are more-or-less confined to very high resolution data (Sheldrick, 1990).
In order to obtain phase information from isomorphous replacement (or from anomalous dispersion), it is necessary to locate the atomic positions of the heavy-atom (or anomalous) scatterers.
References
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