Harker planes/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... 

The symmetry of the Patterson function is the same as the Laue symmetry of the crystal. The Patterson function for space groups that have symmetry operations with translational components (screw axes and glide planes) has an added property that is very useful for the determination of the coordinates of heavy atoms. Specific peaks, first described b David Harker, are associated with the vectors between atoms related by these symmetry operators. These peaks are found along lines or sections (Figure 8.17). For example, in the space group P2i2i2i there are atoms at... 

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. 

FIGURE 9.7 Two molecules of tRNA in (a) are related by a twofold symmetry axis along z in the crystal. A point x, y, z, which could be the site of a heavy atom in one molecule, has an identical corresponding site in the dyad-related molecule at — x, —y, z The vector that connects the two sites will be (x, y, z) — (—x, —y, z) = 2x, 2y, 0. This vector, a Harker vector, must appear on the ro = 0 section of the corresponding Patterson map computed from the intensities of the diffraction pattern. In (b) the heavy atom site on the protein molecule at x, y, z appears on the 2i screw axis (along z) related asymmetric unit at —x, —y, Z—. But —, the unit translation, is the same as +, so the difference vector is 2x, 2y,. This Harker vector would appear on the plane of the Patterson map containing points for which w =. ... 

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... 

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. 

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