126 J. C. KENDREW [9 



THE BASIC CRYSTALLOGRAPHIC PROBLEM 



It is well known that a three-dimensional Fourier summation, carried out 

 with the amplitudes and phases of all the reflexions in the X-ray pattern 

 of a crystal as terms, gives a representation of the electron density in the 

 unit cell of the crystal ; and that the degree of resolution obtained depends 

 on the Bragg spacing of the highest-order reflexions included in the summa- 

 tion, the number of reflexions involved being proportional to the cube of 

 the spacing (e.g. in myoglobin about 400 for 6 Â., 25,600 for 1 -5 Â.). Further- 

 more, if particular sets of reflexions (known as 'zones') are selected and their 

 amplitudes and phases used in a ?H'o-dimensional Fourier synthesis, the result 

 is a representation of the electron density projected along a particular crys- 

 stallographic direction onto a plane. The amplitudes of the reflexions are 

 the square roots of the observed intensities and are therefore measurable 

 by photographic or other means ; but the phases are not directly observable, 

 and in this consists the basic difficulty of X-ray analysis — that experiment 

 gives one just half the information required to solve a structure. The history 

 of the subject is largely a history of attempts to discover ways to get round 

 this difficulty. In simple structures straightforward trial-and-error methods, 

 or sometimes what are in effect sophisticated variants of them, have often 

 sufficed; but in structures as complex as the proteins there is little hope 

 that these will succeed, and indeed in the past they provided only too much 

 free play for the exercise of speculative theories and models of protein 

 structure, without the possibiUty of proving whether any of the specula- 

 tions were wrong or right. The method of isomorphous replacement (and 

 its near relative the heavy-atom method) was also developed for simple 

 structures, and, where chemically practicable, it does often lead to an un- 

 equivocal solution of the structure. It was the successful application of this 

 method to proteins which first put the structure analysis of their crystals 

 onto a firm basis and it is now the principal tool in all laboratories studying 

 protein crystal structure. To illustrate its use we shall describe its applica- 

 tion to myoglobin, first of all in two dimensions. 



The reason for limiting the investigation to two dimensions in the first instance, 

 is simply that in most crystals the phases of certain reflexions are restricted by 

 the symmetry of the crystal to values of or tt ; or, in other words, these so-called 

 real reflexions may be regarded simply as having signs, either positive or negative 

 (unlike the generality of reflexions whose phases may assume any values from 

 to Itt). Thus the question what phase is possessed by a reflexion of this kind can 

 have only two answers, ' + ' or ' — ', whereas to reflexions in general no such restric- 

 tion applies. It is consequently much easier to determine the phases of real re- 

 flexions; and the real reflexions from a given crystal fall into a group or groups 

 (the real zone or zones of reflexions) such that each zone, when it is used as terms 

 of a Fourier synthesis, gives a projection along a particular crystal axis onto a 

 plane, i.e. a two-dimensional synthesis. 



