The measured quantities in any crystal diffraction experiment are the set of intensities 
where
The phases 
of the structure factors 
are not measured so that before
an electron density map can be calculated using Eq. 
the phases must be recovered
For structures
containing only a small number of atoms the phases may be determined using direct methods which will
be described later (see Sec. ). Direct methods cannot in general be applied to protein
structures and other means must be used to obtain phase information.
The most common method of ab initio phase determination for proteins is via Multiple Isomorphous
Replacement (MIR). MIR involves introducing heavy atoms into the protein structure such that the
native protein structure, the crystal form and unit cell dimensions are essentially unchanged, i.e. the native
protein crystal and heavy atom derivative crystals should be isomorphous. The change in the
scattered intensities due to the additional scattering power from the heavy atom `give away' its position thus
producing an estimate of the heavy atom structure factor contribution, 
and 
.
Using this phase and the phase of a
second heavy atom with a different position as a reference it is possible to obtain an estimate of
the protein structure factor phase 
. This is most easily understood with the use of the Harker
construction [44]. Assume that a series of diffraction experiments have been performed on a protein and two
derivative structures where the anomalous effects are small. For each reflection measurements have been made
of 
, from the native protein structure, 
from one derivative, and from the other.
Given that the positions of the heavy atoms 
and 
can be found, the phase may be determined via
the Harker method as shown in Fig. . A circle of radius is drawn at the origin 
to
represent all possible phases from 
. From the origin the two
vectors 
and 
are drawn. Centred at the ends of these vectors, two circles of
radii and 
are drawn. The point of intersection 
of all three circles gives an unambiguous
solution for the phase of 
. Use of only
the 1st derivative, for example, produces two possible solutions for the phase of indicated by
points and 
.
Figure: Harker construction for the case of MIR using two isomorphous derivatives. Measurement of
only the native structure factor modulus and that for the first derivative leads to
two possible orientations for the vector given by the two intersection points and .
Inclusion of a second derivative however resolves the ambiguity and clearly selects for the correct
orientation of .
The mathematical treatment of the Harker construction is as follows. For each derivative we can write down an equation using the cosine rule. For the 1st derivative this will be
and since 
we obtain an equation for the phase in terms
of all other known quantities
Similarly for the 2nd derivative we have
The presence of the cosine terms in Eqs. and confirm that use of only one
of the two derivatives gives two possible solutions for . Use of two or more derivatives however
provide a set of simultaneous equations which may be solved to give absolutely.
The underlying idea of such a method for solving the phase problem is the measurement of at least three structure factor moduli from isomorphous structures where the only difference between the structures arises from the presence of a heavy atom structure with respect to the native protein or from changes in the heavy atom scattering from anomalous effects. Anomalous scattering has successfully been used in conjunction with isomorphous replacement to solve the phase problem in this manner [5].
Single Isomorphous Replacement with Anomalous
Scattering (SIRAS) utilises measurements of the native structure factor moduli along with the measurements
of derivative structure factor moduli when the the imaginary part of the atomic scattering factor for the
heavy atom is significant; for example when the incident X-ray energy is tuned so that it is just higher
than the atoms absorption edge energy where 
is near its maximum. Because of the violation of
Friedel's law, this provides three different observations, , 
and 
where the +
and 
refer to the Bijvoet mates. This situation is demonstrated in Fig.
Figure: Harker construction for the method of Single Isomorphous Replacement with Anomalous Scattering (SIRAS).
Measurement of three structure factor moduli, one for the native protein and the Bijvoet mates from a
derivative containing an anomalously scattering heavy atom, and lead to an unambiguous
solution indicated by the vector 
.
The discussion of phasing methods so far has assumed that the heavy atom positions are known. The determination of the heavy atom structure is however non-trivial and is a pre-requisite for MIR and SIRAS phasing as well as for multiple wavelength methods which will be discussed in detail later. The next section deals with methods of finding the heavy atom structure.
