The difference Fourier technique in protein crystallography: errors and their treatment

Abstract

An expression is derived for the error which arises from the use of the difference Fourier technique.Errors due to the use of inaccurate values of AF and phase are considered in addition to those due to the difference Fourier technique itself. It is shown that a difference Fourier map is able to detect much smaller features of electron density than those revealed by a normal Fourier map with the same phases. The experimental conditions required to obtain difference maps in which the error level is minimized are outlined.

Difference Fourier refinement of the structure of DIP-trypsin at 1.5 Å with a minicomputer technique

“…), which shows the differences between the current model and an improved one, or a map computed with terms F c] exp(iq~c) which superimposes n difference maps on the current structure, showing the improved structure in its entirety. Because peaks resulting from a difference synthesis are about half their theoretical height (Henderson & Moffat, 1971), n = 2 was used for most of these maps. Empirically, n = 2 usually produced the best compromise between freedom from noise and appearance of new in formation.…”

Nanoparticle Structure by Coherent X-ray Diffraction

“…Without entering the hutch, a second CXD measurement was made of the in situ modified crystal. To visualise the small changes in real space, the difference Fourier (DF) method 49) was used to create a 3D map of the differences before and after formation of the SAM. This method is preferred to subtraction of images because it is insensitive to any errors that arise from the HIO phasing step.…”

“…These represent equal and opposite relative displacements of the crystal in the forward and backwards direction of Q, according to the DF formalism. 49) If the strains are radial on the surface of the crystal, it is expected that the biggest differences lie along Q. The opposite sign means both ends of the crystal are expanded relative to the sides.…”

Crystallization of Membrane Proteins: Merohedral Twinning of Crystals

“…An additional problem for X-ray analysis caused by perfect twinning is the inability to use the experimental difference Fourier map. Basing on the mathematical consideration it was shown about 40 years ago that the difference Fourier electron density maps are most sensitive, accurate and less susceptible to model bias method for observing limited structural changes [40]. The difference map is simply the Fourier transform of the amplitudes ( )  exc gr F F (where gr F and exc F are the structural factors of the ground and excited state of the protein) and phases are taken from the model of the ground state.…”