J. C. KENDREW and M. F. PERUTZ 



Let us now consider as a real example the linear Fourier projection 

 of horse haemoglobin. An x-ray study of this protein showed that 

 the phase angles for the reflexions from a certain set of planes 

 could be found by experiment ; symmetry conditions were such that 

 these phase angles were restricted to or n. The Fourier series con- 

 sisted of only seven terms, each of which forms a cosine wave (Figure 5) ; 

 the first term has a wave-length equal to the whole length of the unit 

 cell, its amplitude (i.e. its coefficient) corresponds to that of the first 

 order diffracted ray and its phase angle is 0, which means that the 

 wave has a maximum at the centre of the unit cell. The second term 



1st Order Term 

 2nd Order Term 



3rd Order Term 

 4th Order Term 



5th Order Term 

 6th Order Term 

 7th Order Term 



Figure 5. Graphical represen- 



\ s\ s\ s\ /n , figure d. urapnicai represen- 



\ / \ / \ ° / \ / \ / tation of Fourier summation 



\J \r . XJ . \J \/ Each sinusoidal wave repre- 



wwwv 



Sum o/ Fourier Terms 



repre- 

 sents one Fourier component. 

 The linear electron density 

 projection of haemoglobin ob- 

 tained by summing all the 

 components is shown at the 

 bottom. The origin is marked 

 by a small circle for each curve. 



10 



20 £ 



has a wave-length corresponding to one half of the length of the unit 

 cell, its amplitude corresponds to that of the second order diffracted 

 ray and its phase angle is n, which means that the wave has a minimum 

 at the centre of the unit cell ; and so on. When all the terms are summed 

 they make up the curve shown at the bottom of the figure which gives 

 the distribution of electron density through the unit cell projected on 

 to a line. It will be noted that this distribution shows four peaks of 

 approximately equal height which are spaced just under 9 A apart. 

 This fact provided valuable information on the structure of the haemo- 

 globin molecule and will be recalled later on when the Patterson 

 equivalent of this curve is discussed. 



In order to calculate the electron density distribution in a unit cell 

 projected on to a plane, the intensities of the reflexions from all those 



168 



