X-Ray Crystallography of Biological Macromolecules 



a physical meaning. Since the intensities are proportional to the 

 squares of the amplitudes and the sign of the (amplitude) 2 will always 

 be positive, the unknown signs of the amplitudes do not enter into 

 the series. But even if the phase angles are not restricted to values of 

 or 7T, it can be shown mathematically that they do not enter into the 

 Fourier series which therefore does not contain any quantities that 

 cannot be measured by direct experiment. The physical meaning of 

 this so-called Patterson synthesis is one of the most difficult conceptions 

 in crystallography. It would hardly seem justified in a survey like the 

 present one to dwell on a method as abstract as Patterson's, were it 

 not for the supreme importance which this method has now assumed 

 in the analysis of macromolecular structures. Nearly all the information 

 which has been derived from x-ray studies of crystalline proteins is 

 based on the results of Patterson syntheses. Moreover, the phase 

 angles which have to be known before an ordinary Fourier summation 

 can be performed, can only be found either if the structure is already 

 approximately known, or if the positions at least of certain heavy 

 atoms are known. In most present-day structure analysis this type of 

 information is sought from Patterson syntheses. The meaning of this 

 synthesis will best be explained by beginning with a number of simpli- 

 fied and sometimes hypothetical cases. For the sake of completeness 

 two mathematical formulae are included in the exposition given below, 

 but the treatment is intended to be intelligible also without these. 



Consider once more the electron density distribution of haemoglobin 

 projected on to a line {Figure 5). It was shown how this was obtained 

 by summing a series of cosine waves whose coefficients were the 

 amplitudes of the different orders of diffracted rays ; the waves were 

 placed either with a maximum or a minimum at the centre of the unit 

 cell, depending on whether the signs of the amplitudes were positive 

 or negative, i.e. the phase angles or it. Suppose we now wish to 

 calculate the corresponding Patterson function. The coefficients of the 

 cosine waves in this case will correspond to the intensities of the different 

 order diffracted rays ; as their signs are all positive, all the cosine 

 waves have to be placed with their maxima at the centre of the unit 

 cell (Figure 7). We shall choose this as the origin. 



The resulting curve has its highest maximum at the origin (obviously, 

 since the maxima of all the constituent waves coincide there) and a 

 number of subsidiary maxima on either side. The first of these (peak 1) 

 is 9 A away from the origin, corresponding to the distance between 

 any pair of neighbouring peaks in the electron density projection. 

 The next maximum (peak 2) is 18 A from the origin, corresponding to 

 the distance between next but nearest peaks in the electron density 

 projection, and so on. In fact the distances of the maxima from the 



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