9] 



CRYSTALLOGRAPHY OF MYOGLOBIN 



133 



a heavy atom contribution |f^| (with phase ^^) combine to give a reflexion 

 of amphtude |Fp^| and phase ^p^ in the compound crystal. In practice 

 we are presented with the inverse problem : we are given | Fp ] and | Fp^ | 

 (the square roots of the measured intensities) and | f^ | and ^^ (obtained by 

 straightforward computation from the known position of the heavy atoms 

 in the unit cell). We proceed as shown in Fig. 5 (b), drawing two concen- 

 tric circles whose radii are [Fpl and |Fp^l respectively, and marking off" 

 intercepts equal in length and direction to the heavy-atom contribution f^. 



(a) 



(b) 



(c) 



Fig. 5. The vector combination of protein and heavy-atom contributions to a reflexion; 

 for explanation see text. 



It is evident that there are in general two solutions which attribute diff'erent 

 values <f)p and ^p' to the phase of the protein reflexion. This ambiguity 

 cannot be resolved without further information, but as was shown by 

 Bokhoven, Schoone and Bijvoet^ in a different connexion, a decision can 

 be reached if two isomorphous replacements can be made available in the 

 same unit cell. A pair of vector diagrams can now be drawn for every re- 

 flexion, one for each of the isomorphous replacements, and each gives two 

 alternative values of the protein phase angle. If all is well one value of the 

 angle from the first isomorphous derivative should be identical with one 

 value from the second derivative, and this must be the correct phase angle 

 (see Figs. 5 (b) and (c)). Proceeding in this way one can, theoretically, deter- 

 mine the phase angles of all the reflexions in the diff'raction pattern and then 

 compute a three-dimensional Fourier synthesis which should be a correct 

 representation of the electron density throughout the unit cell. 



It is for these reasons that we have thought it worth while to develop a 

 number of different methods of attaching heavy groups to various sites on 

 the myoglobin molecule, even though for two-dimensional work a single 

 isomorphous derivative is sufficient to establish nearly all the signs of the 

 real reflexions. Among the complexes illustrated in Fig. 4, we have found 

 that those with Hgl42-, with AUCI4-, with PCMBS, and with Hg(NH3)2+, 

 are satisfactory for general phase determination. In addition we have been 

 able to prepare crystals containing two or even three different heavy groups 

 simultaneously, e.g. Au and PCMBS, Au and Hg(NH3)2+, PCMBS and 



