Phase problem isomorphous replacement methods

In 1954, Perutz introduced the isomorphous replacement method for determining phases. In this procedure a heavy metal, such as mercury or platinum, is introduced at one or more locations in the protein molecule. A favorite procedure is to use mercury derivatives that combine with SH groups. The resulting heavy metal-containing crystals must be isomorphous with the native, i.e., the molecules must be packed the same and the dimensions of the crystal lattice must be the same. However, the presence of the heavy metal alters the intensities of the spots in the diffraction pattern and from these changes in intensity the phases can be determined. Besides the solution to the phase problem, another development that was absolutely essential was the construction of large and fast computers. It would have been impossible for Perutz to determine the structure of hemoglobin in 1937, even if he had already known how to use heavy metals to determine phases. 

However, when the intensities of the X-rays are recorded in this manner all information of the phase is lost. Thus, the fundamental problem in a structure determination is the phase problem. Until recently, the phase problem in protein crystallography has been solved by the heavy atom isomorphous replacement method (sections 2(d) and (e)), but other methods are also available (sections 2(e) and (f)). 

Because we get one true phase and one false phase, we do not know which one is the correct one to choose. Thus, the next step in the isomorphous replacement method is to resolve the phase ambiguity. Although the phase ambiguity is a mathematical problem, most crystallographers routinely solve it by collecting more experimental data. 

For the determination of the electron density map of a molecule the amplitudes and phases of the waves scattered by a single crystal are required for a number of Bragg reflections. Common X-ray techniques yield the product (Ah vh) (Ah 1waves scattered by the electrons of the molecule into the Bragg reflection H. Thus, the scattering amplitude Ah is obtained, but the phase information is lost. The solution of this phase problem for protein structure determination is based on Perutz and Kendrew s isomorphous replacement method (108-111). In this procedure Bragg reflections have to be measured at least three times, first on a crystal of native molecules, and then on two crystals, in which reference scatterers (for example Hg atoms) have been substituted at well-defined positions. From the difference of the measured intensities one can calculate the relative phases without ambiguity. 

In early 1948 I thought that there was an experimental solution of the phase problem of X-ray crystallography. The idea was to use a double reflection hj followed by I12 which diffracts in the direction of I13 = hi + I12. If hi is set on the sphere of reflection so that it diffracts for any orientation of the crystal about a suitably chosen rotation axis, then hi and I12 should show an interference effect. This idea, beautiful in principle, was defeated by the mosaic character of crystals and possibly also crystal boimdary effects. Our experiment in which hi is 040 of a glycine crystal failed, although some reflections which were forbidden as single diffractions were observed. Shortly thereafter (1951) Bijvoet published his experimental solution to the phase problem by multiple isomorphous replacement methods, and I thought then that his discovery opened the way to solve protein structures. However, I did not start work in this direction until about 1958, and pursued it seriously beginning in 1961. 

To solve protein structures by the isomorphous replacement method is quite difficult (Figure 7.9). The growth of the protein crystals is not easy, and it is necessary to search for the condition of isomorphous replacement or maybe more than two isomorphous replacements. It can be imagined how many trials need to be done during such process. Therefore, the anomalous scattering method is proposed to solve the phase problem of protein structure determination. 

In small-molecule crystallography the phase problem was solved by so-called direct methods (recognized by the award of a Nobel Prize in chemistry to Jerome Karle, US Naval Research Laboratory, Washington, DC, and Herbert Hauptman, the Medical Foundation, Buffalo). For larger molecules, protein aystallographers have stayed at the laboratory bench using a method pioneered by Max Perutz and John Kendrew and their co-workers to circumvent the phase problem. This method, called multiple isomorphous replacement... 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

The problem of phase determination is the fundamental one in any crystal structure analysis. Classically protein crystallography has depended on the method of multiple isomorphous replacement (MIR) in structure determination. However lack of strict isomorphism between the native and derivative crystals and the existence of multiple or disordered sites limit the resolution to which useful phases may be calculated. 

The most demanding element of macromolecular crystallography (except, perhaps, for dealing with macromolecules that resist crystallization) is the so-called phase problem, that of determining the phase angle ahkl for each reflection. In the remainder of this chapter, I will discuss some of the common methods for overcoming this obstacle. These include the heavy-atom method (also called isomorphous replacement), anomalous scattering (also called anomalous dispersion), and molecular replacement. Each of these techniques yield only estimates of phases, which must be improved before an interpretable electron-density map can be obtained. In addition, these techniques usually yield estimates for a limited number of the phases, so phase determination must be extended to include as many reflections as possible. In Chapter 7,1 will discuss methods of phase improvement and phase extension, which ultimately result in accurate phases and an interpretable electron-density map. 

In order to resolve the phase ambiguity from the first heavy-atom derivative, the second heavy atom must bind at a different site from the first. If two heavy atoms bind at the same site, the phases of will be the same in both cases, and both phase determinations will provide the same information. This is true because the phase of an atomic structure factor depends only on the location of the atom in the unit cell, and not on its identity (Chapter 5, Section III.A). In practice, it sometimes takes three or more heavy-atom derivatives to produce enough phase estimates to make the needed initial dent in the phase problem. Obtaining phases with two or more derivatives is called the method of multiple isomorphous replacement (MIR). This is the method by which most protein structures have been determined. 

V is the volume of the unit cell and should not be confused with V(r) = V(xyz). From the fact that only the absolute square of F can be measured, the phase of the complex value of F is lost. This is the phase problem of crystallography, which found its solution by the introduction of the method of isomorphous replacement in the fifties... 

This limited radius of convergence arises from the high dimensionality of the parameter space, but also from what is known as the crystallographic phase problem [2]. With monochromatic diffraction experiments on single crystals one can measure the amplitudes, but not the phases, of the reflections. The phases, however, are required to compute electron density maps by Fourier transformation of the structure factor described by a complex number for each reflection. Phases for new crystal structures are usually obtained from experimental methods such as multiple isomorphous replacement [3]. Electron density maps computed by a combination of native crystal amplitudes and multiple isomorphous... 

The Patterson synthesis (Patterson, 1935), or Patterson map as it is more commonly known, will be discussed in detail in the next chapter. It is important in conjunction with all of the methods above, except perhaps direct methods, but in theory it also offers a means of deducing a molecular structure directly from the intensity data alone. In practice, however, Patterson techniques can be used to solve an entire structure only if the structure contains very few atoms, three or four at most, though sometimes more, up to a dozen or so if the atoms are arranged in a unique motif such as a planar ring structure. Direct deconvolution of the Patterson map to solve even a very small macromolecule is impossible, and it provides no useful approach. Substructures within macromolecular crystals, such as heavy atom constellations (in isomorphous replacement) or constellations of anomalous scattered, however, are amenable to direct Patterson interpretation. These substructures may then be used to solve the phase problem by one of the other techniques described below. 

The classical method for solving the phase problem in macromolecular crystal structures, known as isomorphous replacement, dates back to the earliest days of protein crystallography.10,16 The concept is simple enough we introduce into the protein crystal an atom or atoms heavy enough to affect the diffraction pattern measurably. We aim to figure out first where those atoms are (the heavy atom substructure) by subtracting away the protein component, and then bootstrap — use the phases based on the heavy atom substructure to solve — the structure of the protein. 

The first X-ray photographs of a protein crystal were described 50 years ago by Bernal and Crowfoot [1], These remarkable photographs indicated that a wealth of structural information was available for protein molecules once methods for the solution of the patterns had been developed. At that time the determination of atomic positions even in the crystals of small molecules was a difficult task. In 1954, Perutz and his colleagues [2] showed that the technique of heavy atom isomorphous replacement could be used to solve the phase problem. The method was put on a sound systematic basis by Blow and Crick [3] and extended to include the use of anomalous scattering [4,5]. Until recently, these methods provided the basis for all protein structure determinations. They have been remarkably effective (as illustrated below) and new developments have both increased the size of the problem solvable and provided greater insights. 

The most general method of solving the phase problem for protein crystals is that of multiple isomorphous replacement in which two or more isomorphous heavy-atom derivatives are used.1 The principle of the method is shown in Figure 3. In Figure 3a a circle with radius Fp, the amplitude of a reflection from the native protein, is shown with center at the origin, O. It is assumed that the heavy atoms in at least two derivatives have been located and referred to the same unit cell origin. This can be a difficult problem and mistakes can be made, but... 

To obtain the electron density distribution it is necessary to guess, calculate or indirectly estimate the phases. Various methods have been developed to tackle the phase problem. For proteins the most common strategy is multiple isomorphous replacement in which the protein crystals are soaked in solutions containing salts of heavy metals such as mercury. 

X-ray crystallography is currently the most powerful analytical method by which three-dimensional structure information on biological macromolecules may be obtained at high resolution. Its application is however limited first by the preparation of single crystals suitable for X-ray diffraction and second by the so-called phase problem , that is the calculation of phases of difBaction data. Several approaches are available in order to circumvent this latter problem. The most commonly used methods are the multiple and single isomorphous replacement (MIR, SIR). These methods, as well as multiple anomalous difBaction (MAD), require the preparation of heavy atom derivatives, usually by the introduction of electron-dense atoms at distinct locations of the crystal lattice. This is usually done by crystal soaking experiments. 

The application of molecular replacement relies on similar model proteins. If there are no known proteins that are similar to the proteins we want to study, then in such cases, what we face are the de novo structures, and molecular replacement is not valid any more. Isomorphous replacement or anomalous scattering methods have to apply in order to solve the phase problems. 

Because the two circles give two possible solutions of phases, the phase problem is not completely solved. We can also prepare another isomorphous replacement, and then the third circle would give the unique solution. The direct method provides another way to distinguish which is the correct solution in Figure 7.8. 

There are four important techniques used in surmounting the phase problem trial-and-error, Patterson maps, direct methods, and isomorphous replacement. Most of the simple structures have been determined by trial and error. Structure factors are calculated from an assumed set of coordinates / exp[2jri(Iix -I- ky + IzJ]... 

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