J. C. KENDREW and M. F. PERUTZ 



waves travelling through a solid body. Again each term has its 

 characteristic wave-length and orientation in the unit cell, while its 

 phase angle (if restricted to or tt) will determine whether the wave 

 has its density maximum or minimum at the centre of the unit cell. 

 Summation of such a series involves summation of the density of each 

 of perhaps more than a thousand waves for each of tens of thousands 

 of points in the unit cell. The labour, even if done by machines, is 

 great, but so is the amount of detailed and often new information on 

 molecular structure which can be obtained from such summations. 



The results of three-dimensional Fourier summations are normally 

 represented in the form of a series of plane sections through the unit 

 cell, the electron density distribution within each section being drawn 

 as a contour map. Wherever such a section intersects an atom the 

 contour map will show a peak. It has become customary, therefore, 

 to refer to concentrations of density within the three-dimensional 

 electron cloud quite generally as peaks and this expression has been 

 carried over into the jargon associated with the Patterson functions 

 discussed below. 



THE X-RAY ANALYSIS OF BIOLOGICAL MACROMOLECULES : 

 PATTERSON SUMMATIONS 



Where a macromolecule (such as a protein) forms crystals it is possible 

 to record the diffraction pattern and to deduce the dimensions of the 

 unit cell (usually very large), the crystal symmetry, and hence imme- 

 diately the size of the asymmetric unit (if this is found to be smaller 

 than the size of the molecule, it can be assumed that the latter consists 

 of identical sub-units whose number and symmetry relationship are 

 given by the symmetry properties of the cell). The next step, however, 

 is the difficult one, for the patterns produced are generally very complex 

 and comprise thousands of reflexions, and none of the methods 

 devised for guessing phases in simple crystals can be applied. For 

 example even the heaviest of heavy atoms would make a negligible 

 contribution to the reflexions from so large a unit cell, so it would 

 exert no control over the phases. Only in a single case, studied by one 

 of us 2 has it so far been possible to make a Fourier projection of a 

 protein molecule (haemoglobin), and then only in one dimension and 

 in simplified form. 



We now proceed to discuss a method whereby at any rate some 

 information about the crystal structure can be obtained without know- 

 ledge of any of the phases of the reflexions in the diffraction pattern. 

 This method owes its discovery to A. L. Patterson 3 who found by 

 mathematical reasoning that a Fourier series, summed with the in- 

 tensities (instead of the amplitudes) as coefficients of the terms, has 



170 



