A crystallographic comparison of protein molecules with related conformations

The multiple isomorphous replacement method of protein crystallography has revealed the structures of several proteins, but the method is limited by the need for many isomorphous crystals. This difficulty could be eased to a large extent by using a molecular fragment as a model for the unknown structure, when only one set of X-ray intensities need be measured. This thesis discusses ways to determine the orientation of such a partial structure, and recommends a new method to determine the location of a properly oriented partial structure in the crystal unit cell. These methods were applied to the proteins human lysozyme, hen lysozyme crystallised at a high temperature, and goat α-lactalbumin, for all of which part of the published structure of hen lysozyme was used as a model. Only the application to human lysozyme was successful, and the failure in the case of the high-temperature hen lysozyme is surprising. An application of these methods to the antibiotic thiostrepton, using an almost complete structure from another crystal form as a model was successful. Having found a partial structure which explains the X-ray data moderately well, it is necessary to complete and refine the model. Several of the theories discussed in this thesis assume Wilson statistics, which are shown to apply to human lysozyme surprisingly well. Three types of completion / refinement technique were applied, a least squares refinement of atomic occupancies, direct methods, and the use of maps with Fourier coefficients with phases derived from the partial structure. The first two methods failed, but the third had partial success. The relationship between "difference maps", Sim weighting, and the α-synthesis is discussed; an extension of Luzzati's treatment of maps with Fourier coefficients F exp(ia c ) shows that of these three Fourier syntheses, the α-synthesis has the best (unknown peak height)/(known peak height) ratio, whereas Sim weighting has the best signal/noise ratio – which, incidentally, never exceeds unity for acentric data – where "noise" is defined in a more realistic way than that adopted by Sim. A corrected and extended version of a previous semi-quantitative account of difference maps is given. The application of various Fourier syntheses to human lysozyme structure factor amplitudes proceeded as a test of methods in parallel with the isomorphous replacement study by other workers; it is recommended that both F obs , and difference maps be calculated, using Sim weighting. Small differences in the main chain conformations of hen and human lysozymes can be determined, missing side chain atoms can often be placed, but rearrangements of the structure as determined by the isomorphous replacement study are not always seen on difference maps. The value of the residual for a randomly wrong structure when a mixture of acentric and centric data is used is shown to be within 0.007 of the value which corresponds to a linear interpolation between the all centric and all acentric limiting cases.