Function Patterson

It was shown in Section 1.5 that the inverse Fourier transform of intensity l(s) 

In the context of diffraction from crystals where I(s) is nonzero only at the reciprocal lattice points, Equation (1.84) can be written, in analogy to (3.12), as 

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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). 

The integrand in this expression will have a large value at a point r if p(r) and p(r+s) are both large, and P s) will be large if this condition is satisfied systematically over all space. It is therefore a self- or autocorrelation fiinction of p(r). If p(r) is periodic, as m a crystal, F(s) will also be periodic, with a large peak when s is a vector of the lattice and also will have a peak when s is a vector between any two atomic positions. The fiinction F(s) is known as the Patterson function, after A L Patterson [14], who introduced its application to the problem of crystal structure detemiination. 

TO Yeates. The asymmetric regions of rotation functions between Patterson functions of arbitrarily high symmetry. Acta Crystallogr A 49 138-141, 1993. 

Derivation of the Structure.—The observed intensities reported by Ludi et al. for the silver salt have been converted to / -values by dividing by the multiplicity of the form or pair of forms and the Lorentz and polarization factors (Table 1). With these / -values we have calculated the section z = 0 of the Patterson function. Maxima are found at the positions y2 0, 0 1/2, and 1/21/2. These maxima represent the silver-silver vectors, and require that silver atoms lie at or near the positions l/2 0 2,0 y2 z, V2 V2 z. The section z = l/2 of the Patterson function also shows pronounced maxima at l/2 0,0 y2, and y2 x/2, with no maximum in the neighborhood of y6 ys. These maxima are to be attributed to the silver-cobalt vectors, and they require that the cobalt atom lie at the position 0 0 0, if z for the silver atoms is assigned the value /. Thus the Patterson section for z = /2 eliminates the structure proposed by Ludi et al. 

The molecular replacement method assumes similarity of the unknown structure to a known one. This is the most rapid method but requires the availability of a homologous protein s structure. The method relies on the observation that proteins which are similar in their amino acid sequence (homologous) will have very similar folding of their polypeptide chains. This method also relies on the use of Patterson functions. As the number of protein structure determinations increases rapidly, the molecular replacement method becomes extremely useful for determining protein phase angles. 

The model protein is used to search the crystal space until an approximate location is found. This is, in a simplistic way, analogous to the child s game of blocks of differing shapes and matching holes. Classical molecular replacement does this in two steps. The first step is a rotation search. Simplistically, the orientation of a molecule can be described by the vectors between the points in the molecule this is known as a Patterson function or map. The vector lengths and directions will be unique to a given orientation, and will be independent of physical location. The rotation search tries to match the vectors of the search model to the vectors of the unknown protein. Once the proper orientation is determined, the second step, the translational search, can be carried out. The translation search moves the properly oriented model through all the 3-D space until it finds the proper hole to fit in. 

The isomorphous replacement method requires attachment of heavy atoms to protein molecules in the crystal. In this method, atoms of high atomic number are attached to the protein, and the coordinates of these heavy atoms in the unit cell are determined. The X-ray diffraction pattern of both the native protein and its heavy atom derivative(s) are determined. Application of the so-called Patterson function determines the heavy atom coordinates. Following the refinement of heavy atom parameters, the calculation of protein phase angles proceeds. In the final step the electron density of the protein is calculated. 

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). 

Many people have recognized that the rotation function suffers from some drawbacks and have tried to improve the score by using origin-removed Patterson functions, normalized structure factors E-values, etc. (Briinger, 1997). 

Patterson and Pattersons Fifty years of the Patterson function... 

Patterson and Pattersons fifty years of the Patterson function J. P. Glusker, B. K. Patterson, and M. Rossi, editors... 

Interatomic vectors. Although, in the absence of knowledge of the signs of the Fourier terms, it is not possible to deduce directly the actual positions of the atoms in the cell, it is theoretically possible to deduce interatomic vectors, that is, the lengths and directions of lines joining atomic centres. Patterson (1934,1935 a) showed that aFourier synthesis employing values of F2 (which are of course all positive) yields this information. The Patterson function... 

Calculations of the Patterson function may be carried out in exactly the same way as those of electron densities. Bragg s optical method may also be used indeed, in general it may be applied more readily to the formation of vector maps, since (the signs of the jF2 coefficients being all positive) the question of phase adjustment does not arise. The optical method has been shown to give a correct vector map for the 6 projection of haemoglobin. ... 

In a similar way, it can be shown that if a crystal has a plane of symmetry perpendicular to its b axis, the Patterson function has maxima along the 6 axis of the cell (the line 0, y, 0, in Fig. 229) which indicate the distance of atoms from the plane of symmetry. For a glide plane perpendicular to b, with a translation c/2, the distance of atoms from this plane are indicated by maxima along the line 0, y,... 

There are other circumstances in which some of the atomic coordinates in a crystal can be discovered by evaluation of the Patterson function over a particular plane or along a particular line. For instance, it may be known, from a consideration of the space-group and the equivalent positions in the unit cell, that there is one particular atom at the origin of the cell and others somewhere on the plane y = L The... 

Patterson function will show maxima on this plane at positions which give immediately the actual coordinates of these atoms. Similar considerations were used in the determination of the structure of potassium sulphamate NH2SO3K (Brown and Cox, 1940) it was known that the y coordinates of the potassium ions are 0 and while those of the sulphur atoms are and consequently, the Patterson function on the plane y = l shows maxima at positions corresponding to K-S vectors. Atomic positions are not given directly, but can be derived from the positions of Patterson peaks by a consideration of the equivalent positions in the space-group. 

The Patterson function has been the most useful and generally applicable approach to the solution of the phase problem, and over the years a number of ingenious methods of unraveling the Patterson function have been proposed. Many of these methods involve multiple superpositions of ports of the map. or "image-seeking with known vectors. Such processes are ideally suited lo machine compulation. Whereas the great increase in the power of x-ray methods of structure determination in the past few years has come simply front our ability lo compute a three-dimensional Patterson function, it is reasonable lo expect that, as machine methods of unraveling the Patterson function are developed, this power will increase many fold. 

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... 

To obtain the Patterson function solely for the heavy atoms in derivative crystals, we construct a difference Pattersonfunction, in which the amplitudes are (AF)2 = (IFHpl — IFpl)2. The difference between the structure-factor amplitudes with and without the heavy atom reflects the contribution of the heavy atom alone. The difference Patterson function is... 

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. 

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... 

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. 

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 ... 

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