# Difference Patterson map

An  application of Patterson methods for solution of crystal structures, typically proteins with heavy-atom derivatives, where the Patterson function is calculated using structure-factor  coefficients  based on the difference between the heavy-atom derivative and the native molecule.

### Discussion

Patterson methods for determining diffraction phases depend on the symmetries of interatomic vectors that show up as peaks in a three-dimensional map of the Patterson function. For small molecules containing a heavy atom, the heavy-atom positions can be determined directly from the Patterson function calculated using measured structure-factor amplitudes. For proteins, there are too few heavy atoms for this approach to be successful. However, if an isomorphous derivative crystal is available (i.e. one whose symmetry and dimensions and contents, with the exception of heavy-atom addition, are minimally changed), a Patterson map of derivative ($$F_{PH}$$) minus native ($$F_{P}$$) structure factors will be dominated by the vectors between the heavy atoms, and thus allow a solution of the coordinates of the heavy atoms.

A true difference Patterson function, representing the difference between the Patterson of the derivative minus the Patterson of the native protein, should be calculated using as coefficients:

$\left | F^2 _{PH}-F^2_P \right |$

In practice, protein crystallographers normally calculate a modulus difference-squared synthesis, also known as an isomorphous difference Patterson, using coefficients $(|F_{PH}-F_P|)^2$.