SOLVING THE PHASE PROBLEM

Ultimately we can solve the phase problem for a number of reasons (1) We have chemical and physical information about the structure of the molecules making up the crystal that we can use to interpret and improve even a poor electron density image. (2) Information actually does reside in the intensity distribution alone, cryptic information about the relative phases 

Introduction to Macromolecular Crystallography, Second Edition By Alexander McPherson Copyright 2009 John Wiley Sons, Inc. 

Based on the ideas above, and with the exception of some unique cases that present unusual opportunities, the kinds of approaches available to us for solving the phase problem are as follows  

Initially, all crystal structures were solved by constructing a suitable model making use of any physical properties that reflect crystal symmetry and chemical intuition regarding formulae and bonding, computing Fhkt values and comparing them with observed Fm values. This trial and error method was used to solve the structures of fairly simple crystals, such as the metals and minerals described in Chapter 1. (Many of the 

One of the first mathematical tools to come to the aid of the crystallographer was the development of the Patterson function, described in 1934 and 1935. The Patterson function is  

Direct methods replaced trial and error and other methods of deducing model structures in the middle of the 20th century. In these techniques, statistical relationships between the amplitudes and phases of the strong reflections were established, and the mathematical methodology between these quantities was worked out, particularly by Hauptmann and Karle, in the 1940 s and 1950 s. (The Nobel Prize was awarded to these scientists in 1985 for these studies.) A number of algorithms, which exploited the growing power of electronic computers, used this mathematical framework to derive structures directly from the experimental data set of position, intensity and hkl index. The use of these programs allows the structures of molecular compounds with up to 100 or so atoms to be 

Although direct methods are being extended to larger and larger molecules, the structures of large proteins, with more than say 600 atoms, are still inaccessible to this technique. Such structures are mainly determined using the following two methods. 

The first protein structures were derived using a technique called isomorphous replacement (IR), developed in the late 1950 s. The materials used are heavy metal derivatives of protein crystals. To obtain a heavy metal derivative of a protein, the protein crystal is soaked in a solution of a heavy metal salt. The metals most used are Pt, Hg, U, lanthanides, Au, Pb, Ag and Ir. The heavy metal or a small molecule containing the heavy metal, depending upon the conditions used, diffuses into the crystal via channels created by the disordered solvent present. The aim is for the heavy metal to interact with some surface atoms on the protein, without altering the protein structure. This is never exactly achieved, but in suitable cases, the changes in structure are slight. 

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Solving the phase problem in protein crystallography is a requirement for any structural study. The three... 

A survey of conventional methods for solving the phase problem... 

If the Patterson method cannot be applied because the structure has no or too many heavy atoms, it is possible to use another approach for phase determination, the so-called direct methods. By the term direct methods is meant that class of methods which exploits relationships among the structure factors in order to go directly from the observed magnitudes E to the needed phases < ) (Herbert A. Hauptman, Nobel lecture, 9. Dec., 1985). The direct method approach for solving the phase problem uses probability... 

The electron density is everywhere positive (a probability can not be negative ). Nevertheless, that the positivity of the density function is not a necessary prerequisite for solving the phase problem via direct methods was recently shown [20]... 

Solving the phase problem using isomorphous replacement... 

The possibility and feasibility of molecular replacement was demonstrated by Rossmann and colleagues in the 1960s, as part of an effort to use non-crystallographic synnmetry to solve the phase problem for macromolecules (Rossmann, 1990). 

However, in view of experimental errors, 2N additional equations are unlikely to be sufficient to solve the phase problem. In practice, we can only expect a statistically meaningful solution if we include many more equations and identify the solution that agrees most with all equations simultaneously. Eurther-more, since Eq. 1 is non-linear in 4)(hp, we cannot expect to find an analytic solution. Hence, we have to make initial guesses for the unknowns and improve from there. 

The phase angle of a reflection is not observable however, several possibilities of solving the phase problem computationally with the experimentally available values of are at hand and are used routinely in the course of a structure determination24. 

In crystallography, heavy atom derivatives are required to solve the phase problem before electron density maps can be obtained from the diffraction patterns. In nmr, paramagnetic probes are required to provide structural parameters from the nmr spectrum. In other forms of spectroscopy a metal atom itself is often studied. Now many proteins contain metal atoms, but even these metal atoms may not be suitable for crystallographic or spectroscopic purposes. Thus isomorphous substitution has become of major importance in the study of proteins. Isomorphous substitution refers to the replacement of a given metal atom by another metal that has more convenient properties for physical study, or to the insertion of a series of metal atoms into a protein that in its natural state does not contain a metal. In each case it is hoped that the substitution is such that the structural and/or chemical properties are not significantly perturbed. 

G. R. Fleming I want to make the comment to Prof. Shapiro that another way to solve the phase problem is by controlling the phase with light, as described in Scherer et al., J. Chem. Phys. 95, 1487 (1991). 

Alternative methods of solving the phase problem are also used now. When a transition metal such as Fe, Co, or Ni is present in the protein, anomolous scattering of X-rays at several wavelengths (from synchrotron radiation) can be used to obtain phases. Many protein structures have been obtained using this multiple wavelength anomalous diffraction (MAD phasing) method.404 407 408 Selenocysteine is often incorporated into a protein that may be produced in... 

Another vital type of ligand is a heavy-metal atom or ion. Crystals of protein/ heavy-metal complexes, often called heavy-atom derivatives, are usually needed in order to solve the phase problem mentioned in Chapter 2 (Section VI.F). I will show in Chapter 6 that, for the purpose of obtaining phases, it is crucial that heavy-atom derivatives possess the same unit-cell dimensions and symmetry, and the same protein conformation, as crystals of the pure protein, which in discussions of derivatives are called native crystals. So in most structure projects, the crystallographer must produce both native and derivative crystals under the same or very similar circumstances. 

In comparison to the protein structure, this "structure"—a sphere (or very few spheres) in a lattice—is very simple. It is usually easy to "determine" this structure, that is, to find the location of the heavy atom in the unit cell. Before considering how to locate the heavy atom (Section III.C.), I will show how finding it helps us to solve the phase problem. 

DM can be applied to "small" structures (< 1000 atoms in the asymmetric unit). Since a crystal with, say, 10 C atoms requires finding only x, y, and z variables, but typically several thousand intensity data can be collected, then, statistically, this is a vastly overdetermined problem. There are relationships between the contributions to the scattering intensities of two diffraction peaks (with different Miller indices h, k, l, and h, k, / ), due to the same atom at (xm, ym, zm). DM solves the phase problem by a bootstrap algorithm, which guesses the phases of a few reflections and uses statistical tools to find all other phases and, thus, all atom positions xm, ym, zm. How to start ... 

Crystals of the material are grown, and isomorphous derivatives are prepared. (The derivatives differ from the parent structure by the addition of a small number of heavy atoms at fixed positions in each — or at least most — unit cells. The size and shape of the unit cells of the parent crystal and the derivatives must be the same, and the derivatization must not appreciably disturb the structure of the protein.) The relationship between the X-ray diffraction patterns of the native crystal and its derivatives provides information used to solve the phase problem. 

The constmction of synthetic selenocysteine-containing proteins or selenium-containing proteins attracts considerable interest at present, mainly for the reason that it can be used to solve the phase problem in X-ray crystallography. Selenomethionine incorporation has been used mostly uutil now for this purpose. There are also two reports ou uew synthetic selenocysteine-containing proteins. In one case, the active site serine of subtUisin has been converted into a selenocysteine residue by chemical means, with the result that the enzyme gains a predominant esterase instead of protease activity. In the second case, automated peptide synthesis was carried out to produce a peptide in which all seven-cysteine residues of the Neurospora crassa metallothioueiu (Cu) were replaced by selenocysteine. The replacement resulted iu au alteration of both the stoichiometry and the affinity of copper binding. ... 

The conunonly used methods for solving the phase problem required for stmcture solutions are the direct methods and Patterson maps. Direct methods use relationships between phases such as triplets (0 = 4>h + 4>k + -h-k The probabihty of 0 >= 0 increases with the magnitude of the product of the normalized stmcture factors of the three reflections involved. Once these triplets associated with high certainty are identified based on diffraction intensities, they are used to assign new phases based on a set of known phases. Since the number of phase relationships is large the problem is over determined. Another approach is based on the Sayre equation, which is derived based on the relationship between the electron density and its square ... 

X-ray structures are obtained from crystalline samples. Crystallization is a laborious process, and the crystallizability of a protein is unpredictable. Some proteins do not crystallize, and X-ray diffraction is not applicable in these cases. In other cases, heavy atom derivatives needed to solve the phase problem may be difficult to obtain. Furthermore, crystals, although highly hydrated, do not represent the structure of the molecule in a true solution. Movements of protein domains are restricted in the crystaUine state. NMR methods circumvent some of these difficulties because they are applied to proteins in solution. Therefore, in addition to permitting the determination of structures, the NMR technique is useful for revealing dynamic processes such as protein protein interactions. 

The crystal structure of the RNase H that resides at the carboxyl terminus of the 66 kDa RTase subunit was determined by using the 133-amino-acid carboxyl terminal fragment expressed and purified from E. coli. Isomorphous replacement using uranium ions was used to solve the phase problem, and further refinement to 2.4 A resolution was achieved by using the structural information provided by homologous RNase H from E. coli. This structure, in addition to providing insights into the mechanism of RNase H activity, provided useful information for the determination of the RTase structure (Davies et al, 1991). 

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