Harker sections

Unit-cell symmetry can also simplify the search for peaks in a three-dimensional Patterson map. For instance, in a unit cell with a 2X axis (twofold screw) on edge c, recall (equivalent positions, Chapter 4, Section II.H) that each atom at (x,y,z) has an identical counterpart atom at (-x,-y,V2 + z). The vectors connecting such symmetry-related atoms will all lie at (u,v,w) = (2x,2y,V2) in the Patterson map (just subtract one set of coordinates from the other), which means they all lie in the plane that cuts the Patterson unit cell at w = l/2. Such planes, which contain the Patterson vectors for symmetry-related atoms, are called Harker sections or Harker planes. If heavy atoms bind to the protein at... 

A Patterson function yielded the positions of the two Xe and one Au atoms and refined to R = 0.21. The Patterson map confirmed the choice of the centric space group since all the Harker sections had either y = 0 or y = /i. A difference Fourier based on the set of phases generated by the least-squares refmement gave the positions of all of the fluorine atoms. Least-squares refinement incorporating... 

FIGURE 9.2 A section from a difference Patterson map calculated between a heavy atom derivative and native diffraction data (known as a difference Patterson map). This map is for a mercury derivative of a crystal of bacterial xylanase. The plane of Patterson density shown here corresponds to all values of u and w for which v =. Because the space group of this crystal is P2, this section of the Patterson map is a Harker section containing peaks denoting vectors between 2t symmetry related heavy atoms. 

Consider another example, that seen in Figure 9.1b. A 2i axis along z would correspond to equivalent positions in the unit cell for all atoms of x, y, z and -x, -y, z + j. The vector between any two symmetry equivalent points in space will have u,v,w components equal to the difference of their coordinates. Thus vectors between equivalent positions will be u = 2x, v = 2y, w = Here the Harker section containing the corresponding peaks between symmetry related atoms, screw axis related atoms, will be the two-dimensional plane for which w = j. 

Second, algebraic differences between the equivalent positions for the space group are formed. For each pair of equivalent positions, one coordinate difference will turn out to be a constant, namely 0, 5, 3, 5, depending on the symmetry operator. These define the Harker sections for that space group, which are the planes having one coordinate u,v, or w constant, and that will contain peaks corresponding to vectors between symmetry equivalent atoms. In focusing attention only on Harker sections, the Patterson coordinates u,v,w... 

Consider the example in Figure 9.10 of a 2i axis along z in the unit cell of a crystal. The corresponding Harker peaks will be at 2x, 2y, If, in examining the w = section of the Patterson map, we find a propitious peak at U,V,, then we can properly conclude that the real space coordinates of the heavy atom are at x = U/2, y = V/2. Again, we are able to deduce actual, real space positions of the heavy atoms in the unit cell, x and y, from coordinates of peaks on the appropriate Harker section of the Patterson map, U and V. 

Next, the Harker sections of the Patterson map are contoured. An example is shown in Figure 9.11. As for electron density maps, this is done by drawing contours, lines of equal density, at regular intervals around specified values of the Patterson function, as discussed in Chapter 10, to obtain a topographical map of the Patterson density. In this manner the major peaks on the section are defined. Remember, however, that a peak in the Patterson map actually represents the end of a vector from the origin of the Patterson map, and this vector, when it appears on a Harker section, is the vector between some atom at x, y, z and the corresponding atom at a symmetry equivalent position. 

FIGURE 9.11 The w = j plane of the difference Patterson map for the K2HgI4 heavy atom derivative of the hexagonal crystal form of the protein canavalin. The space group is P6, so w = is a Harker section. The derivative crystal contained two major K2HgI4 substitution sites and one minor substitution site per asymmetric unit. The Patterson peaks corresponding to those sites are marked with crosses. Note that the Patterson peak corresponding to the minor site cannot be discriminated from noise peaks in the Patterson map as is often the case. 

For each symmetry relationship (and corresponding difference between equivalent positions), it will be found that one of the three Patterson coordinates is a constant, 0 or j or and so on. These define the Harker sections of the Patterson map. For example, if one of the Patterson relationships is u = 2x, v = and w = 2z, then the Harker section , j, w is first calculated and investigated. On this section, like the real unit cell of the crystal whose periodicity it reflects, u and w will vary from 0 to 1 and v will be constant, j. It is generally useful to separately calculate the individual Harker sections, review them, and only then calculate the full map. The appearance of the Harker sections and their content... 

When the unit cell contains several symmetry elements, then each of these will each give rise to a Harker section. The additional Harker sections will in turn yield information about one of the coordinates already known (but not both), as well as information about the missing coordinate. A second Harker section will not only fill out the x, y, z coordinate set for the heavy atom, but will confirm, or serve as a cross check on one of the coordinates deduced from the first Harker section. A third Harker section will, in its turn, provide a cross check on heavy atom coordinates derived from both of the others. 

The use of the Harker sections is made easier when a few heavy atoms are present in the unit cell their Harker maxima can be recognized and used to locate the heavy atoms. If they have a sufficiently high atomic number, they can be used as a good initial model to which one can apply the so-called Method of Fourier Recycling, to obtain the light atom positions and then to recover the complete structure. 

Fig. 21. Vectors in a Patterson map. (a) A peak in a Patterson map indicates that the vector defined between the origin of the Patterson map and the peak in it must be found between atoms in the crystal structure, (b) Harker sections of the Patterson map for a heavy-atom derivative of D-xylose isomerase (Courtesy of H. L. Carrell).

The three Harker sections are shown. The largest nonorigin peaks yield the coordinates of two independent gold sites, Aul and Au2, at (x,y,z) values of (0.2293, 0.0792, 0.2291) and (0.4969, 0.2228, 0.1428). Vectors between Aul and Au2 occur on general sections (not shown here). The origin peak is 1000 and contours start at 25 and increase in intervals of 25. The equivalent positions in the space group... 

Figure 7.13 (cont.) (fLaueHg- LaueNv) compared with (ii) the equivalent monochromatic Harker sections for the mercury amylase case. The Laue maps are as interpretable as the monochromatic ones. 

Figure 9.13 (a) A Harker section in the Patterson synthesis calculated using optimised zIano terms as coefficients for Fe cytochrome C4. The agreement between the observed peaks and the predicted positions is self-evident. All the expected Harker peaks occur in the map and are significantly above background noise. 

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