**The basic concept of MIR
phasing **

Isomorphous replacement is the first phasing technique which
became available to macromolecular crystallographers. It is based
on the fact that the structure
factor *F(hkl)* for a certain reflection *hkl*
is a simple summation of all individual atomic scattering
contributions :

This allows us to do simple arithmetics with *F* such
as* F(p+h) = F(p) + F(h)*. Let us assume that *Fp*
is the structure factor of a reflection for the protein, and *Fph*
for the same protein now including a heavy atom. Heavy atoms such
as transition metals, lanthanides, Uranuim, even noble gasses
under pressure, can quite successfully be soaked into crystals
through slovent channels and frequently bind to well defined
sites in the native protein. Such a protein crystal is then
called a heavy
atom derivative (crystal). Based under the assumption that
the derivative crystal retains its structure (i.e., is in fact
isomorphous) we guess that we should be able to derive some
information on the structure factor amplitudes for the heavy
metal from the differences between the derivative and the native
dataset.

Unfortunately, in the absence of phase information we cannot
apply the simple subtraction *F(h) = F(p+h) - F(p) *and
end up with the structure factors (and thus positions) of the
heavy atom in one step, at least not in the generic
non-centrosymmetric case of interest to macromolecular
crystallographers. Note also that *|Fh| = |Fph| - |Fp| *is
not correct! This is obvious from the fact that a negative
difference in the intensities may well occur - only implying that
the phase of *Fh* is somehow opposite to *Fp*. We
need to separate phase terms from magnitudes in order to do math
correctly with a mixture of structure factors *F* and
structure amplitudes* |F|*.

We need to answer the following questions to attack the problem steps by step :

- What is the formal vector mathematics behind the MIR phasing ?
- How do we actually find the heavy atom positions ?
- How do we get the protein phases and how "good" are they really ?

At this point, you may want to briefly review the introduction
to vector representation of
scattering factors. The following vector diagram (Harker
presentation) illustrates the relationship between native and
derivative scattering factors. The
objective of a phasing experiment is to derive the unknown phase *a
(p) *of each protein reflection * Fp*.

From our the experiment, we know
only the magnitudes* **|Fph|*
(derivative) and
*|Fp|*
(protein) which can be represented in the complex plane as a circle of
radius *|Fph|* and *|Fp| **,* respectively. If we **know both the
magnitude and the phase of *** Fh* we
can draw both circles offset by vector

The phase and magnitude of * Fh* can be
calculated easily if we know the positions of a heavy metal (we
will show in the next chapter how to determine these positions). At this point
is is clear that the best phase we can obtain from the 2 solutions is the mean
in between the 2 possibilities, and the
phase error can be quite large. In real cases,

In order to eliminate the * phase
ambiguity* we can prepare a

We have now, at least in theory,
an exact **solution for the phase angle of ****F**p* .* The theory is based on 2
assumptions : a) ideal isomorphism and b) exact heavy atom
positions, neither of which are perfectly met, for practical and
experimental reasons in the first case and for theoretical
reasons in the second. In our picture it means that the phasing
circles may not intersect in exactly in one spot, and another
derivative may be necessary to improve the quality of the phases.
The method is therefore called

What remains at this point is to
investigate how we can actually determine the positions of the
heavy metal atoms with nothing else but the Structure factor
amplitudes of native protein,* **|Fp|*, and
of an isomorphous heavy metal derivative, *|Fph|*.
Click her to proceed to calculate some isomorphous derivative Patterson maps. -
Working on that....

**Back to X-ray Tutorial Index**

**This World Wide Web site conceived and maintained
by Bernhard Rupp. ****
Last revised
Dezember 27, 2009 01:40**