The Patterson function [92] is defined as the autocorrelation function of the electron density distribution or

where and are respectively position vectors in Patterson space and real space and denotes convolution. By substituting in for and using Eq. we find that

The resulting map calculated using this equation is essentially a map of all the cross vectors between each atom pair in the unit cell. For a structure which contains atoms in the unit cell or equivalently has peaks in its electron density distribution, the Patterson function will contain peaks. of the peaks represent self vectors which are superimposed at the origin, , while peaks represent cross vectors and appear elsewhere in the unit cell. Deconvolution of such Patterson maps is a relatively simple procedure for small but becomes complicated as increases.

Location of the heavy atoms may be achieved by use of suitable coefficients in Eq. which correspond to the Fourier transform of the heavy atom electron density. Ideally one would use the set of 's but these are not measurable quantities. The alternative procedure is to calculate using coefficients which are largely dependent on the heavy atom structure. If the heavy atoms display significant anomalous scattering the coefficients may be used [97]. The values of are measurable and, as can be seen from Eq. , they are strongly correlated with the imaginary part of the heavy atom structure factor . Patterson maps calculated using coefficients are called anomalous difference Patterson maps.

If native protein data are available the coefficients may also be used to calculate isomorphous difference Patterson maps; this is in fact the standard procedure in isomorphous replacement.

Fri Oct 7 15:42:16 MET 1994