The Heavy Atom Method

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  

Noncrystallographic symmetry and density modification PATTERSON METHODS 

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 heavy atom method is not applicable to macromolecules, again because of the very large number of atoms involved. It is, however, a very useful approach for the solution of conventional crystal structures having as many as 50 or so atoms in the asymmetric unit. For about the first 40 years of X-ray crystallography it was really the only practical method for deducing crystal structures. 

The heavy atom method for structure determination, though not useful for macromolecular crystals, is an easily understandable approach and it is illustrative of the ideas and devices that are incorporated into the other, more applicable techniques. The heavy atom method only works, it should be said at the outset, when the molecule under study contains at least 

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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 relative stereochemistry of stephadiamine (16) was clarified by X-ray diffraction analysis, using the direct method, and the absolute configuration was solved by the heavy-atom method, using the N-p-bromobenzoyl derivative (6). Stephadiamine (16), a C-norhasubanan alkaloid, is not regarded as a hasubanan congener in the strict sense, but as a new member of oe-amino acid derivatives (6). 

As discussed in section 2.3, the electron diffraction intensities need to be corrected before being employed for structure analysis. An empirical method has been set up to correct simultaneously all kinds of distortions in the diffraction data by referring to the heavy atom method and the Wilson statistic technique in X-ray crystallography. After correction, the intensity of each diffraction beam can approximately lead to the modulus of the corresponding structure factor [26]. 

Various methods have been used to circumvent the phase problem. The earliest method was based on trial-and-error procedure and works well for relatively simple molecules (diatomic and tri-atomic). The most successful method has been the heavy atom method, wherein an electron-dense atom (for example, bromine or... 

Erythromycin A, a widely used, macrolide, antibiotic substance, was crystallized as the hydriodide dihydrate. It would be expected that the iodine atom would be used as the heavy atom in solving the structure, and, indeed, the authors tried the heavy-atom method first.68 Unfortunately, the iodine atom lies on a crystallographic mirror-plane, and so this method failed. Because erythromycin con-... 

The crystal and molecular structure of l-(2-pyridyl)-3-benzoyl-6-bromoindolizine has been obtained by the heavy atom method.129 The larger than normal 0=0 distance (1.32 A compared with 1.22 A for a carboxy group) suggests significant contributions from charged canonical forms. 

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. 

Although the heavy atom method has been successful in establishing many structures, particularly those of alkaloids since alkaloids have a propensity for crystallizing as halide salts, there had been an urgent need to develop a procedure for phase determination that was not dependent on the presence of a heavy atom in a crystal. Such a procedure, now commonly called the direct method of phase determination, has been devised. Karle and Hauptman 13) recognized that the number of unique reflections measured in an X-ray pattern is 25-50 times greater than the number of unknowns in a crystal, the unknown quantities being the three coordinates... 

In a minimum-function map, the origin of the Patterson map is put in turn on each of the known symmetry-related positions of a heavy atom that has already been located from a Patterson map. On each superposition of the origin of the Patterson map onto the various symmetry-related heavy-atom positions, the lowest value at each superimposed grid point in the pairs of maps is recorded. This superposition process is repeated until the structure is revealed. In this way the lighter atoms can be located. The method is an alternative to the heavy-atom method just described and has proved useful in many cases. 

The application of the Patterson technique to locate strongly scattering atoms is often called the heavy atom method (which comes from the fact that heavy atoms scatter x-rays better and the Patterson technique is most often applied to analyze x-ray diffraction data). This allows constructing of a partial structure model ( heavy atoms only), which for the most part define phase angles of all reflections (see Eq. 2.107). The heavy atoms-only model can be relatively easily completed using sequential Fourier syntheses (either or both standard, Eq. 2.133, and difference, Eq. 2.135), sometimes enhanced by a least squares refinement of all found atoms. 

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. 

It may not be obvious how we would locate the x, y, z coordinates of the heavy atom in the unit cell. Indeed it is sometimes not a simple matter to find those coordinates, but as for the heavy atom method described above, it can be achieved using Patterson methods (described in Chapter 9). As we will see later, Patterson maps were used for many years to deduce the positions of heavy atoms in small molecule crystals, and with only some modest modification they can be used to locate heavy atoms substituted into macromolecular crystals as well. Another point. It is not necessary to have only a single heavy atom in the unit cell. In fact, because of symmetry, there will almost always be several. This, however, is not a major concern. Because of the structure factor equation, even if there are many heavy atoms, we can still calculate Juki, the amplitude and phase of the ensemble. This provides just as good a reference wave as a single atom. The only complication may lie in finding the positions of multiple heavy atoms, as this becomes increasingly difficult as their number increases. 

In order to exploit the heavy atom method with crystals of conventional molecules, or to utilize the isomorphous replacement method or anomalous dispersion technique for macro-molecular structure determination, it is necessary to identify the positions, the x, y, z coordinates of the heavy atoms, or anomalously scattering substituents in the crystallographic unit cell. Only in this way can their contribution to the diffraction pattern of the crystal be calculated and employed to generate phase information. Heavy atom coordinates cannot be obtained by biochemical or physical means, but they can be deduced by a rather enigmatic procedure from the observed structure amplitudes, from differences between native and derivative structure amplitudes, or in the case of anomalous scattering, from differences between Friedel mates. 

The "heavy-atom method" (which uses a heavy-metal salt of the alkaloid) was not,in this case, sufficiently informative concerning stereochemistry because formation of the salt involved cleavage of the oxazolidine ring present in the base. This particular X-ray study is of considerable interest because nmr spectral studies reported earlier (Pelletier and Mody, ibid-, 1977, 284) indicated that in... 

Acid scdts are ideally suited for X-ray study. They generally crystallize well. They necessarily contain a metal atom, which may be of atomic number high enough for the heavy-atom method of phase-determination failing this, the metal can often be replaced isomorphously, whereupon... 

The structure was solved by the heavy-atom method, and the absolute configuration determined by examination of Friedel pairs. Rings c and e were shown to exist in envelope conformations, rings A and D were in boat conformations, and rings b and F exist in chair conformations. 

In order to confirm the stereochemical assignments for this compound, an X-ray crystallographic analysis was carried out (275). The structure was solved by the heavy-atom method using Patterson and Fourier syntheses. Least-squares refinement reduced the R value to 13.6%. Two molecules were observed in the asymmetric unit and were found to correspond to the optical antipodes having the same relative stereochemistry. Thus, the stereochemistry is firmly established to be as shown in 686 (275). 

The heavy atom method may be used to determine structures with more than 100 independent atoms. However, it cannot be used to determine structures composed of similar atoms, as is the case for many organic molecules and for many alloys. In order to determine structures of this type, algebraic (direct) methods are available, however, they are beyond the scope of this book. Readers interested in the area of structure determination are referred to the numerous specialized books. 

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