Derivative, heavy atom phases from

Simulated annealing has also proven useful with the reciprocal space equivalent of the real space search problem. More conventionally known as the molecular replacement problem, this reciprocal space search problem occurs when at the outset of the crystallographic experiment a reasonably detailed approximate model of the macromolecule is already available and the intention is to altogether avoid the painful acquisition of heavy-atom phases (Rossman, 1972). The absence of heavy-atom derived phases differentiates this reciprocal space version of the search problem from its real space analog previously discussed. 

Referring to figure Bl.8.5 the radii of the tliree circles are the magnitudes of the observed structure amplitudes of a reflection from the native protein, and of the same reflection from two heavy-atom derivatives, dl and d2- We assume that we have been able to detemiine the heavy-atom positions in the derivatives and hl and h2 are the calculated heavy-atom contributions to the structure amplitudes of the derivatives. The centres of the derivative circles are at points - hl and - h2 in the complex plane, and the three circles intersect at one point, which is therefore the complex value of The phases for as many reflections as possible can then be... 

Multiple isomorphous replacement allows the ab initio determination of the phases for a new protein structure. Diffraction data are collected for crystals soaked with different heavy atoms. The scattering from these atoms dominates the diffraction pattern, and a direct calculation of the relative position of the heavy atoms is possible by a direct method known as the Patterson synthesis. If a number of heavy atom derivatives are available, and... 

Once a suitable crystal is obtained and the X-ray diffraction data are collected, the calculation of the electron density map from the data has to overcome a hurdle inherent to X-ray analysis. The X-rays scattered by the electrons in the protein crystal are defined by their amplitudes and phases, but only the amplitude can be calculated from the intensity of the diffraction spot. Different methods have been developed in order to obtain the phase information. Two approaches, commonly applied in protein crystallography, should be mentioned here. In case the structure of a homologous protein or of a major component in a protein complex is already known, the phases can be obtained by molecular replacement. The other possibility requires further experimentation, since crystals and diffraction data of heavy atom derivatives of the native crystals are also needed. Heavy atoms may be introduced by covalent attachment to cystein residues of the protein prior to crystallization, by soaking of heavy metal salts into the crystal, or by incorporation of heavy atoms in amino acids (e.g., Se-methionine) prior to bacterial synthesis of the recombinant protein. Determination of the phases corresponding to the strongly scattering heavy atoms allows successive determination of all phases. This method is called isomorphous replacement. 

NCP crystals. There were two facets to this approach. First, it was necessary to reconstitute NCPs from a defined sequence DNA that phased precisely on the histone core to circumvent the random sequence disorder. It was obvious that the DNA was important for the quality of the diffraction from NCP crystals but the role of histone heterogeneity was not so clear. Heavy atom derivatives (i.e., electron rich elements bound in specific positions on the proteins) were not readily prepared by standard soaking experiments, due to a paucity of binding sites. Hence, it was necessary to selectively mutate amino acid residues in the histones to create binding sites for heavy atoms. 

The addition of one or more heavy atoms to a macromolecule introduces differences in the diffraction pattern of the derivative relative to that of the native. If this addition is truly isomorphous, these differences will represent the contribution from the heavy atoms only thus the problem of determining atomic positions is initially reduced to locating the position of a few heavy atoms. Once the positions of these atoms are accurately determined, they are used to calculate a set of phases for data measured from the native crystal. Although, theoretically, one needs only two isomorphous derivatives to determine the three-dimensional structure of a biological macromolecule, in practice more than two are needed. This is due to errors in data measurement and scaling and in heavy-atom positions, as well as lack of isomorphism. 

Once the heavy-atom position has been determined, its structure factor amplitude f h and phase an can be calculated. Since the structure factor amplitudes for the native (Fp) and derivative (Fp ) are experimentally measured quantities, it is thus possible to calculate the protein phase angle ap from the following equations ... 

From Eq. 3 and Fig. 6.3a it is clear that with only one heavy-atom derivative (single isomor-phous replacement SIR) the resultant phase will have two values (ap and apb) one of these phases will represent that of one structure and the other of its mirror image. But, since proteins contain only L-amino acids, this phase ambiguity must be eliminated using a second derivative, the anomalous component of the heavy atom or by solvent levelling (Wang, 1985), as shown diagrammatically in Fig. 6.3b. 

As can be seen from Eq. 4, a Fourier synthesis requires phase angles as input, thus it cannot be used to locate heavy-atom positions in a derivative if no phase information exists. However, it can be used to determine such positions in a derivative, if phases are already available from one or more other derivatives. As in the case of a difference Patterson, the Fourier s)mthesis here also employs difference coefficients. They are of the form ... 

Difference Fourier techniques are most useful in locating sites in a multisite derivative, when a Patterson map is too complicated to be interpretable. The phases for such a Fourier must be calculated from the heavy-atom model of other derivatives in which a difference Patterson map was successfully interpreted, and should not be obtained from the derivative being tested, in order not to bias the phases. Also, difference Fourier techniques can be used to test the correctness of an already identified heavy-atom site by removing that site from the phasing model and seeing whether it will appear in... 

Fig. 9. Approximate Bragg resolution for the first exposure of each of about 200 crystals from H. marismortui SOS subunits that were investigated at XI1, EMBL/DESY, Hamburg (ERG), in August 1986 at —4° to 19 °C. Shading indicates heavy-atom derivative test crystals (undecagold-cluster and tetrakis(acetoxymercuri)methane (TAMM) see paragraph 4 Phase Determination)...

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. 

An entire data set must be collected for each of these derivatives. The evaluation of the phases from these data is a complex mathematical process which usually involves the calculation first of a "difference Patterson projection."406 This is derived by Fourier transformation of the differences between the scattering intensities from the native and heavy atom-containing crystals. The Patterson map is used to locate the coordinates of the heavy metal atoms which are then refined and used to compute the phases for the native protein. 

Despite the fact that the phases for the high resolution map were obtained from only two heavy-atom derivatives (55), the election density map was quite interpretable with the aid of the sequence along most of the peptide chain. This is illustrated in Fig. 5a which shows the high resolution map of the bottom right helix, one of the best resolved sections. Figure 5b shows the same section phased only with the high resolution iodine data Fig. 5c, the Type II crystals phased with iodine and barium at 4 A and Fig. 5d, the Type I crystals phased with the three heavy-atom derivatives at 6 A resolution. 

Fig. 5. (a) The helical section at the lower right of the nuclease as seen from directly above. This section is from the high resolution map phased with both Ba and I in the Type II crystal. The heaviest lines are meant to indicate the groups closest to the viewer, (b) The same helical section at high resolution but phased only with I in Type II crystals, (e) The same helical section at 4 A resolution phased with both Ba and I in Type II crystals, (d) The same helical section at 6 A resolution in Type I crystals phased with three heavy-atom derivatives. 

The second criterion for useful heavy-atom derivatives is that there must be measurable changes in at least a modest number of reflection intensities. These changes are the handle by which phase estimates are pulled from the data, so they must be clearly detectable, and large enough to measure accurately. 

Our heavy-atom derivative allows us to determine, for each reflection hkl, that uhkl has one of two values. How do we decide which of the two phases is correct In some cases, if the two intersections lie near each other, the average of the two phase angles will serve as a reasonable estimate. I will show in Chapter 7 that certain phase improvement methods can sometimes succeed with such phases from only one derivative, in which case the structure is said to be solved by the method of single isornorphous replacement (SIR). More commonly, however, a second heavy-atom derivative must be found and the vector problem outlined previously must be solved again. Of the two possible phase angles found by using the second derivative, one should agree better with one of the two solutions from the first derivative, as shown in Fig. 6.8. 

Figure 6.8 a shows the phase determination using a second heavy-atom derivative F h is the structure factor for the second heavy atom. The radius of the smaller circle is IF Hpl, the amplitude of F Hp for the second heavy-atom derivative. For this derivative, Fp = F Hp — F H. Construction as before shows that the phase angles of F and Fj are possible phases for this reflection. In Fig. 6.8 b, the circles from Figs. 6.7b and 6.8a are superimposed, showing that Fp is identical to F. This common solution to the two vector equations is Fp, the desired structure factor. The phase of this reflection is therefore the angle labeled a in the figure, the only phase compatible with data from both derivatives. 

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. 

As I described earlier, this entails extracting the relatively simple diffraction signature of the heavy atom from the far more complicated diffraction pattern of the heavy-atom derivative, and then solving a simpler "structure," that of one heavy atom (or a few) in the unit cell of the protein. The most powerful tool in determining the heavy-atom coordinates is a Fourier series called the Pattersonfunction P(u,v,w), a variation on the Fourier series used to compute p(x,y,z) from structure factors. The coordinates (u,v,w) locate a point in a Patterson map, in the same way that coordinates (x,y,z) locate a point in an electron-density map. The Patterson function or Patterson synthesis is a Fourier series without phases. The amplitude of each term is the square of one structure factor, which is proportional to the measured reflection intensity. Thus we can construct this series from intensity measurements, even though we have no phase information. Here is the Patterson function in general form... 

In words, the difference Patterson function is a Fourier series of simple sine and cosine terms. (Remember that the exponential term is shorthand for these trigonometric functions.) Each term in the series is derived from one reflection hkl in both the native and derivative data sets, and the amplitude of each term is (IFHpI — IFpl)2, which is the amplitude contribution of the heavy atom to structure factor FHp. Each term has three frequencies h in the u-direction, k in the v-direction, and l in the w-direction. Phases of the structure factors are not included at this point, they are unknown. 

A second means of obtaining phases from heavy-atom derivatives takes advantage of the heavy atom s capacity to absorb X rays of specified wavelength. As a result of this absorption, Friedel s law (Chapter 4, Section III.G) does not hold, and the reflections hkl and —h — k—l are not equal in intensity. This inequality of symmetry-related reflections is called anomalous scattering or anomalous dispersion. 

As I discussed in Section III.C, Patterson methods do not allow us to distinguish between enantiomeric arrangements of heavy atoms, and phases derived from heavy-atom positions of the wrong hand are incorrect. When high-resolution data are available for the heavy-atom derivative, phases and... 

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