The structure of haemoglobin VIII. A three-dimensional Fourier synthesis at 5.5 Å resolution: determination of the phase angles

Determination of the phase angles of a crystalline protein requires a series of isomorphous heavy-atom compounds, with heavy atoms attached to different sites on the protein molecule. The asymmetric unit of horse oxyhaem oglobin was found to combine with heavy atoms at two different sites which are now known to be sulphydryl groups. Altogether six different heavy - atom com pounds of haemoglobin were made which proved isomorphous on X -ray analysis. The positions of the heavy atoms were determined first by difference Patterson and Fourier projections on the centrosym metric plane of the monoclinic crystals, and later by three-dimensional correlation functions, ( F H 1 — F H 2 ) 2 being used as coefficients, where F H 1 and F H 2 are the structure factors of the two different heavy-atom compounds. The parameters and anisotropic shape factors of the heavy atoms were refined by a three-dimensional least-squares method. For each of the 1200 reflexions in the limiting sphere of (5.5 Å) -1 the structure amplitudes of all seven compounds were combined in an Argand diagram and the probability of the phase angle having a value a was calculated for oc = 0, 5, 10, ..., 355°. The coefficients for the final Fourier summation were then calculated in two different ways. In one method the vector from the origin to the centroid of the probability distribution, plotted around a circle of radius | F |, was chosen as the ‘best F’. The alternative set of coefficients was calculated, using the full, observed, value of F and the most probable value of the phase angle a. The most probable error in phase angle was found to be 23°, and the standard error in electron density to be expected in the final results 0.12 e/Å 3 .