The method of Multiple wavelength Anomalous Diffraction (MAD) is most applicable to problems where there are no available separate native protein diffraction data, e.g. for metallo-proteins where a heavy atom is already bound in the native structure, or to cases where derivative crystals are non-isomorphous and are therefore unsuitable for phasing via isomorphous replacement.
Changes in the heavy atom structure factor are induced by tuning the X-ray energy (wavelength) to
different points in the absorption spectrum of the heavy atom, thus changing the values of the anomalous scattering
factors 
and 
which in turn effect the magnitudes 
and 
. As with the MIR and SIRAS phasing methods a minimum of three different measurements
of the overall structure factor are necessary where the values of 
and 
are different for each. In MAD this may be achieved by the measurement of diffraction
data at at least two X-ray energies providing the minimum requirement of a Bijvoet pair measurement at one of the energies
and a single Bijvoet mate at the other.
Given that we are now dealing only with `derivative' structure factors we shall alter our notation and denote
the Bijvoet pair structure factors for the protein plus heavy atom structure measured at the 
th X-ray energy
by 
and 
.
For a structure containing 
protein atoms
and 
heavy atoms all of the same type we can write
The only X-ray energy dependent terms in Eq. 
are 
and 
.
Fig. illustrates how changes in the anomalous scattering factors of the heavy atoms can
in theory lead to a solution to the phase problem.
Figure: Harker construction for the MAD case. Only structure factors for one X-ray energy are labelled
but phase circles have been drawn for four observations indicated by the vectors 
, 
, 
and 
.
The observed structure factor amplitudes arise from the measurement of Bijvoet pairs at each of two X-ray
energies. The correct solution is given by the point 
. The changes in the anomalous scattering factors have the
effect of offsetting the centres of the phase circles. The vector OC would represent the phase of the native structure
without the heavy atoms present.
The magnitudes of the and components
in Fig. are greatly exaggerated with respect to 
and in reality
have magnitudes which are only a few percent of .
Hendrickson et al. [54] have shown that estimated anomalous diffraction ratios (the ratio
between the rms anomalous difference and the rms structure factor magnitude)
may be calculated for a particular type and number of anomalous scatterer present in a protein
containing non-hydrogen atoms. The Bijvoet difference ratio is given by
and the dispersive ratio by
where an estimate of 
may be obtained using
There are two main approaches to phasing via multiple wavelength methods. The first is an extension of the MIR approach discussed earlier and the second is an algebraic approach which has been developed more recently by Karle [64] and Hendrickson [48] [54].
