Electron density maps phase problem

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. 

Application. Anomalous X-ray diffraction (AXRD), anomalous wide-angle X-ray scattering (AWAXS), and anomalous small-angle X-ray scattering (ASAXS) are scattering methods which are selective to chemical elements. The contrast of the selected element with respect to the other atoms in the material is enhanced. The phase problem of normal X-ray scattering can be resolved, and electron density maps can be computed. 

Data processing occurs in two stages. Initially, diffraction images are reduced to a tabulation of reflection indices and intensities or, after truncation, structure factors. The second stage involves conversion of the observed structure factors into an experimental electron density map. The choice of how to execute the latter step depends on the method used to determine the phase for each reflection. The software packages enumerated above generally focus on exploitation of anomalous signals to overcome the phase problem for a protein of unknown structure. 

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. 

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. 

Phase problem The problem of determining the phase angle (relative to a chosen origin) that is to be associated with each diffracted wave that is combined to give an electron-density map. The measured intensities of diffracted beams give only the squares of the amplitudes, but the relative phases cannot normally be determined experimentally (see Chapter 8). The determination of the relative phases of the Bragg reflections is crucial to the calculation of the correct electron density map. 

When a diffraction grating, such as a crystal, interacts with X rays, the electron density that causes this diffraction can be described by a Fourier series, as discussed in Chapter 6. The diffraction experiment effects a Fourier analysis, breaking down the Fourier series (of the electron density) into its components, that is, the diffracted beams with amplitudes, F[hkl). The relative phases a(hkl) are, however, lost in the process in all usual diffraction experiments. This loss of the phase information needed for the computation of an electron-density map is referred to as the phase problem. The aim of X-ray diffraction studies is to reverse this process, that is, to find the true relative phase and hence the true three-dimensional electron density. This is done by a Fourier synthesis of the components, but it is now necessary to know both the actual amplitude F[hkl) and the relative phase, a[hkl), in order to calculate a correct electron-density map (see Figure 8.1). We must be able to reconstruct the electron-density distribution in a systematic way by approximating, as far as possible, a correct [but so far unknown) set of phases In this way the crystallographer, aided by a computer, acts as a lens for X rays. 

How is it possible to derive phase information when only structure amplitudes have been measured An answer can be found in what are called direct methods of structure determination. By these methods the crys-tallographer estimates the relative phase angles directly from the values of F hkl) (the experimental data). An electron-density map is calculated with the phases so derived, and the atomic arrangement is searched for in the map that results. This is why the method is titled direct. Other methods of relative phase determination rely on the computation of phase angles after the atoms in a trial structure have been found, and therefore they may be considered indirect methods. Thus, the argument that phase information is lost in the diffraction process is not totally correct. The phase problem therefore lies in finding methods for extracting the correct phase information from the experimental data. 

To show how the phase problem is solved in order to be able to calculate an electron density map (see Chapter 8). 

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... 

Another application of simulated annealing is in the real-space search problem of crystallography. This problem arises when the initial electron density map obtained by the heavy-atom method is so poor that no obvious tracing of the polypeptide chain is even partly possible. Typically one expects to see connected tubes of electron density corresponding to mainchain atoms strung along the polypeptide backbone. When the heavy-atom phases are poor, much of this connectivity is lost and the remaining bubbles of isolated density are impossible to interpret. Frequently, when a crude approximation of the expected model is already available from other unrelated sources, this otherwise fatal situation can be overcome. 

A Patterson map, different for each space group, is a unique puzzle that must be solved to gain a foothold on the phase problem. It is by finding the absolute atomic coordinates of a heavy atom, for both small molecule and macromolecular crystals, that initial estimates (later to be improved upon) can be obtained for the phases of the structure factors needed to calculate an electron density map. 

The central problem in crystallography lies in obtaining the phase for every observed structure factor amplitude. We judge how correct a given set of phases is by the result does the electron density map make chemical sense For small molecules, very accurate data is usually available to high resolution (1 A or better), which allows the use of direct methods 9 to obtain the phases rapidly and correctly. The approach uses statistical relationships between the phases of certain reflections. Unfortunately, direct methods are not easily... 

While this makes perfect chemical sense, it causes a problem. Whatever the source of the phases used to calculate a new electron density map, some features of the model (or old electron density map) will show up in the new map, because the phases dominate its appearance (Figure 15), as mentioned in Section 9.03.9.3. If our model (and hence the phases calculated from it) is correct, this is not a problem, but since the process of structure refinement that we discuss below is iterative, the correctness of the model must be assessed carefully. In crystallography, what you see is what you put in — also known as model bias. 

FIGURE 30.11 The phase problem. The experimental data obtained in an X-ray experiment are the intensities of the reflections. By using an inverse Fourier transform, it is possible to calculate electron-density maps from these intensities. However, it is essential for this calculation to know the phase associated with each reflection. Approximate initial phases can be obtained from heavy-atom derivatives, anomalous dispersion or molecular replacement (see text). More accurate phases can be derived from the refined model, once it has been obtained. 

When a suitable model of the unknown crystal structure is available, it can be used to solve the phase problem. Examples are the use of the stracture of human thrombin to solve the structure of bovine thrombin, the use of a known antibody fragment to solve the stracture of an unknown antibody, or the use of the stracture of an enzyme to solve the stracture of an inhibitor complex of the same enzyme in a different crystal form. The model is oriented and positioned in the unit cell of the unknown crystal with the use of rotation and translation functions, and the oriented model is subsequently used to calculate phases and an electron-density map. 

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