Patterson map

Patterson function The Fourier transform of observed intensities after diffraction (e.g. of X-rays). From the Patterson map it is possible to determine the positions of scattering centres (atoms, electrons). 

How do we find phase differences between diffracted spots from intensity changes following heavy-metal substitution We first use the intensity differences to deduce the positions of the heavy atoms in the crystal unit cell. Fourier summations of these intensity differences give maps of the vectors between the heavy atoms, the so-called Patterson maps (Figure 18.9). From these vector maps it is relatively easy to deduce the atomic arrangement of the heavy atoms, so long as there are not too many of them. From the positions of the heavy metals in the unit cell, one can calculate the amplitudes and phases of their contribution to the diffracted beams of the protein crystals containing heavy metals. 

The three dimensional structure was obtained by means of single crystal X-ray diffraction. CuKa radiation, a graphite monochromator, and a photomultiplier tube were used to collect 1825 total reflections on an automated diffractometer. Of these, 1162 were used for the analysis. Figure 2 shows a computer generated drawing of halcinonide. The position of the chlorine atom was not clear from the Patterson map, but the direct method program "MULTAN" gave its position. 

In contrast to Fourier synthesis, which yields with electron diffraction data high electrostatic potential at the positions of the atoms, the maps obtained from Patterson synthesis show peaks at the tips of vectors. The length of each vector (drawn from the origin of the Patterson map) corresponds always to the distances between pairs of atoms and the direction each vector points... 

Figure 12. In order to check the experimental Patterson map for a known structural fragment it must be converted into Patterson space first. This is shown in the above sketch for the fragment of a body centered cube which is projected along the 4-fold axis (a). The corresponding motif in Patterson space is shown in (b). Note, that the atom in the center of the fragment defines a 2-fold axis what reduces the number of different peaks in the Patterson map.

Case study 2 Structural solution of zeolite from electron diffraction data, with a help of (a) Direct method, and (b) Patterson Map... 

The Patterson function (Patterson, 1934) is a phaseless Fourier summation similar to that in Eq. 4 but employing as coefficients, thus it can be calculated directly from the experimentally measured amplitudes (Fp) without the need to determine the phase angle. In the case of macromolecules, (Fpn —Fp ) are used as coefficients in Eq. 4 to produce a Patterson map (hence the name difference Patterson). Such a map contains peaks of vectors between atoms (interatomic vectors). Thus in the case of a difference Patterson of macromolecules, it is a heavy-atom vector map. For example if a structure has an atom at position (0.25, 0.11, 0.32) and another atom at position (0.10, 0.35, 0.15), there will be a peak in the Paterson map at position (0.25-0.10, 0.11-0.35, 0.32-0.15), meaning a peak at (0.15, —0.24, 0.17). 

The interpretation of Patterson maps requires knowledge of crystallographic symmetry and space groups. Chapter 4 of Blundell and Johnson (1976) offers a concise review of these topics. The ease of interpretation of these maps depends on the quality of the data, the degree of isomorphism, the number of heavy-atom sites per macromolecule and the... 

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... 

Since protein BLl 1 is nearly globular its location may be determined in a Patterson map with coefficients of [F(wild)-F(mutant)] and may serve, by itself, as a giant heavy-atom derivative. At preliminary stages of structure determination this approach may provide phase information and reveal the location of the lacking protein. 

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. 

Paramecium 17, 20 Paramylon 170,174 Parathion 636s Parathyroid hormone 314 Paratose 180s Partition coefficient 410 Parvalbumin 312-316 Parvoviruses 244 Patterson map 135 Pauling, Linus C. 83, 84 Paxillin 406... 

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... 

Because the Patterson function contains no phases, it can be computed from any raw set of crystallographic data, but what does it tell us A contour map of p(x,y,z) displays areas of high density (peaks) at the locations of atoms. In contrast, a Patterson map, which is a contour map of P(u,v,w), displays peaks at locations corresponding to vectors between atoms. (This is a strange idea at first, but the following example will make it clearer.) Of course, there are more vectors between atoms than there are atoms, so a Patterson map is more complicated than an electron-density map. But if the structure is simple, like that of one or a few heavy atoms in the unit cell, the Patterson map may be simple enough to allow us to locate the atom(s). You can see now that the... 

I will show, in a two-dimensional example, how to construct the Patterson map from a simple crystal structure and then how to use a calculated Patterson map to deduce a structure (Fig. 6.10). The simple molecular structure in Fig. 6.10a contains three atoms (dark circles) in each unit cell. To construct the Patterson map, first draw all possible vectors between atoms in one unit cell, including vectors between the same pair of atoms but in opposite directions. (For example, treat 1 —> 2 and 2 — 1 as distinct vectors.) Two of the six vectors (1 — 3 and 3 —> 2) are shown in the figure. Then draw empty unit cells around an origin (Fig. 6.10b), and redraw all vectors with their tails at the origin. The head of each vector is the location of a peak in the Patterson map, sometimes called a Patterson "atom" (light circles). The coordinates (u,v,w)... 

Figure 6.10 Construction and interpretation of a Patterson map. (a) Structure of unit cell containing three atoms. Two of the six interatomic vectors are shown, (h) Patterson map is constructed by moving all interatomic vectors to the origin. Patterson "atoms" (peaks in the contour map) occur at the head of each vector. (c) Complete Patterson map, containing all peaks from (b) in all unit cells. Peak at origin results from self-vectors. Image of original structure is present (origin and two darkened peaks) amid other peaks, (d) Trial solution of map (c).lf origin and Patterson atoms a and b were the image of the real unit cell, the interatomic vector a - b would produce a peak in the small box. Absence of the peak disproves this trial solution.

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... 

If the model and the new protein are indeed similar, and if they are oriented in the same way in unit cells of the same dimensions and symmetry, they should give very similar Patterson maps. We might imagine a trial-and-error method in which we compute Patterson maps for various model orientations and compare them with the Patterson map of the desired protein. In this manner, we could find the best orientation of the model, and then use that single orientation in our search for the best position of the model, using the structure-factor approach outlined earlier. 

For the orientation search (often called a rotation search), the computer is looking for large values of the model Patterson function Pmodel( ,v,w) at locations corresponding to peaks in the Patterson map of the desired protein. A powerful and sensitive way to evaluate the model Patterson is to compute the minimum value of Pmodel(w,v,w) at all locations of peaks in the Patterson map of the desired protein. A value of zero for this minimum means that the trial orientation has no peak in at least one location where the desired protein exhibits a peak. A high value for this minimum means that the trial orientation has peaks at all locations of peaks in the Patterson map of the desired protein. 

As discussed later in the "Results" section, the rotation function contains many peaks. The strongest 100 peaks are selected, and each orientation is refined by least squares to produce the best fit to the ALBP Patterson map. For each refined orientation, a correlation coefficient is computed. The orientation giving the highest correlation coefficient is chosen as the best orientation for the phasing model. 

In Figure 8.4a, the value of the rotation function, which indicates how well the probe and ALBP Patterson maps agree with each other, is plotted vertically against numbers assigned to the 101 orientations that produced best agreement. Then each of the 101 orientations were individually refined further, by finding the nearby orientation having maximum value of the rotation function. In some cases, different peaks refined to the same final orientation. 

Each refined orientation of the probe received a correlation coefficient that shows how well it fits the Patterson map of ALBP. The orientation receiving the highest correlation coefficient was taken as the best orientation of the probe, and then used to refine the position of the probe in the ALBP unit cell. The orientation and position of the model obtained from the molecular replacement search was so good that refinement of the model as a rigid body produced only slight improvement in R. The authors attribute this to the effectiveness of the Patterson correlation refinement of model orientation, stage two of the search. 

I have shown that, in simple systems, Patterson functions can give us valuable clues about distances, even when we know nothing about phases (see Chapter 6, Section III.C). Diffraction from the randomly oriented molecules in a solution or powder would give a spherically averaged diffraction pattern, from which we can compute a spherically averaged Patterson map. Is this map interpretable ... 

The output is the integrated intensity value for that particular reflection. FHKL - This is the structure factor calculating program. The input is a list of hkl s and intensity values. The output consists of E values and phase angles to be used as input to the electron density program. ELECDEN - Calculates the electron density and contours the E-map on a Tektronix 4662 digital plotter. PATTERSON - Used to calculate three-dimensional Patterson maps. 

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