X-Ray Crystallography of Biological Macromolecules 



sets of planes which are normal to the plane of projection have to be 

 measured, and their amplitudes form the coefficients of the terms. Each 

 of these terms will now be a two-dimensional wave, corresponding to a 

 wave on a water surface, with a characteristic wave-length and orienta- 

 tion relative to the edges of the projected unit cell. An example of a 

 two-dimensional Fourier projection is shown in Figure 6, which also 

 indicates for comparison the layout of the molecule as viewed in pro- 

 jection. The projection is plotted as a contour map of projected 

 electron density : a peak corresponds to the presence of an atom, and 

 the higher the peak the higher the atomic number of the atom. Generally 

 a projection is made on to each of the three faces of the unit cell in 

 turn ; from the three projections the arrangement of the atoms in 

 space in the unit cell can be deduced. Of course such projections are 

 liable to be complicated by the fact that certain atoms in the unit cell 

 may overlap when viewed in projection on to any given plane, but in 

 simple cases it is generally possible, by comparison between the three 

 projections, to see where such overlapping has occurred. Another 

 difficulty is that hydrogen, since it contains but one electron, makes 

 slight contributions to the reflexions and does not as a rule show up 

 in the final result ; it is then necessary to guess the positions of all 

 hydrogen atoms after the main outlines of the structure have been 

 determined. 



Figure 6. Planar electron 

 density projection of a 

 pyrimidine compound. 

 (Reproduced from Clews 

 and Cochran*). 



Because of overlapping of the atoms in planar projection, or because 

 of the extreme weakness or even absence of the higher order diffracted 

 rays, the resolving power of planar projections may be insufficient to 

 find the exact positions of all the atoms. This happened, for instance, 

 in the x-ray analyses of cholesterol iodide and penicillin ; in such 

 cases it is necessary to calculate the electron density in space by means 

 of a three-dimensional Fourier summation. This requires measurement 

 of the intensities of all the diffracted rays from the crystal in question. 

 As a rule the phase angles of these can be found only when the structure 

 is already approximately known. The terms of this Fourier series will 

 be three-dimensional waves, forming a sinusoidal variation of electron 

 density in space. Such waves may be pictured rather like the density 

 variation produced at a given instant by ' monochromatic ' sound 



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