5. Phase determination by multiple isomorphous replacement

A simple statement of the problem is that given |F_{P}|,
|F_{PH}| (measured
experimentally), and |F_{H}|,
f_{H}
(calculated from the site coordinates), the unknown
f_{P}
is to be determined. Consider first the hypothetical case where all quantities are
error-free.

5.1. Representation of the phase problem in the complex plane

5.1.1. Centric case

The structure will always appear centrosymmetric when projected down an evenfold
axis. So for example P2_{1} (unique b) has one centric zone:
h0l; P2_{1}2_{1}2 has three: hk0,
h0l and 0kl; P1 and R3 have none. The centric zone reflections give a ready
advantage in MIR studies, because
f_{P}
can take only 1 of 2 possible values differing
by 180° (e.g. 0/180° or 90/270°).

For each possibility of
f_{P},
|F_{PH}|(calc) is compared with |F_{PH}|(obs).

5.1.2. Acentric case

For a non-centric zone reflection,
f_{P}
can take any value. In general there are two
solutions symmetric about
f_{H}, unless
f_{P} =f_{H} or
f_{H} + 180°, in which case
the acentric case is the same as the centric. This phase ambiguity is the reason that
it takes at least two derivatives (or one with anomalous scattering) to solve the phase
problem.

Traditionally, the determination of the phase in the acentric case has been taught by
"Harker construction" (phase circles). However this is not a practical method for a
computer program, and in any case proper account has to be taken of the
experimental and other errors. Consequently all software uses probability density
functions (see section 5.2).

So far it has been assumed that |F_{P}|,
|F_{PH}|, |F_{H}| and
f_{H} are known precisely; in
practice all suffer from experimental error.

Atoms show anomalous scattering when the incident x-rays have a wavelength near
to the absorption edge of the atoms. Part of the radiation is absorbed by the atom
and re-emitted with a change in phase. The scattering factor for the atom can be
written in terms of real and imaginary parts:

where
Df' is the dispersion component
of the anomalous scattering and
Df" the
absorption component. The imaginary component scatters out of phase with the
primary wave, leading to absorption. Both
Df' and
Df" vary with the wavelength
especially near the absorption edge. The absorption coefficients for C, H, O and N
are small enough to be ignored. However, for the heavy metal atoms of a derivative,
these coefficients cannot be ignored, especially as the absorption edge for these
elements is close to the wavelength of X-rays commonly used. The anomalous
differences are largest when |F_{P}| and
|F_{H}| are approximately perpendicular.

By reflecting the complex structure factors F_{P},
F_{H} and F_{PH} for the reflection (-h-k-l)
through the real axis so that they superimpose on those for the reflection (hkl), it will
be seen that |F_{PH}+| and |F_{PH}-| are
unequal, i.e. Friedel's relation |F(hkl)| =
|F(-h-k-l)| no longer holds. The pair |F_{PH}+| and
|F_{PH}-| are treated as two separate derivatives.

5.2. Probability density function

5.2.1. SIR case

The probability density function (PDF) is defined as:

where K_{j} is a normalisation factor and
s(|D_{j}|)
is the r.m.s. lack of closure weighted
by the phase probability. Note that this formulation ignores zone enhancement,
centricity and non-isomorphism factors (see Read, R. in 1991 CCP4 Study Weekend
Proceedings, pp 69-79).

Most programs use this, but the equation assumes independent information from
each derivative, and therefore over-emphasises the contribution of F_{P} (see R. Read,
1991 for rigorous derivation).

5.2.3. Visualisation of the PDF's

If the PDF's
P(f_{P}) and
P_{j}(f_{P})
are plotted as functions of
f_{P}, either in
Cartesian coordinates or in polar coordinates, P will
have maxima at the most probable phase(s). Also in the polar plots the centroid of
the probability-weighted unit circle is shown. The phase of the centroid is the best
phase (minimum r.m.s. error in electron density). The radius of the centroid is the
figure of meritm» cos(phase error). For electron density maps the Fourier
coefficient m|F_{P}|exp(if_{P}) is used;
for difference maps m|F_{PH}-F_{P}|exp(if_{P}) is used.

Example 1

For better resolution, click on the pictures.

Cartesian

polar

The figures show the individual PDF's (P_{1}, P_{2}, P_{3})
and product PDF's (P_{123}, P_{12}) for
one reflection from 3 simulated derivatives, for |F_{P}| = 10.0.

Here, P_{1} and P_{2} are bimodal, whereas P_{3} is unimodal
because |F_{PH3}| = |F_{P}| +
|F_{H3}|.
The phasing power |F_{H}|/s(|D|) determines the sharpness of the density function
at the most probable phase, whereas the figure of merit determines the precision of
the best phase. Multiplication of probabilities improves both the figure of merit and
the phasing power. Addition of the low phasing power derivative 3 in this case
makes little difference (compare P_{123} and P_{12}).

Example 2

For better resolution, click on the pictures.

Cartesian

polar

The figures show the effect of changing
f_{H} to 340° for derivative 3
(P_{3}), while
keeping all the other parameters the same. Even though derivative 3 has low phasing
power, this considerably reduces the overall figure of merit, so even one poorly
phased derivative can negate the accumulated effect of several good derivatives.

Example 3

For better resolution, click on the pictures.

Cartesian

polar

The figures show the effect of varying the phasing power in the SIR case.
f_{H} has
also been varied in order to separate the centroids in order to make the effect clear.

From this it can be seen that a phasing power below 0.5 does not contribute
significantly to the phasing; often a cutoff of 1.0 is used, but at this level the
contribution to the phasing is clearly significant.