protein. Also, amino acid residues that are not conserved can be replaced
with alanine. Computers are used to place the resulting structural model
in the unit cell of the unknown protein crystal. For any given position
and orientation of the model, the resulting diffraction pattern can be
calculated by using eqn 15.6.
Isomorphous replacement
In this case, the phase problem is overcome by use of special modifications
of the protein using a biochemical approach. Heavy metal compounds are
added to the solution containing the crystals. These metal compounds are
designed to bind to specific amino acid groups such as cysteine. Crystals
of the protein with the heavy metal are measured using diffraction, and
a differenceapproach can be used to determine the phase.
Consider one specific diffraction peak at
a given hklvalue. The native protein will
have a structure factor that can be repre-
sented as a vector with an amplitude FP
rotated by an angle αP(Figure 15.14). The
protein with a bound heavy metal will
have a different amplitude FPHand differ-
ent angle αPH. The difference between
these two vectors is due to the heavy
metal that has an amplitude FHand angle
αH. Since the angles associated with FP
and FPHare unknown, the tips of their vectors can lie anywhere along
two circles. However, since the difference between these two vectors is
the vector FH, if the circle associated with FPHis displaced by the vector
FH, then the correct solution must lie at the points where the two circles
intersect, and the correct angle is determined. Since two independent inter-
section points are found, at least two derivatives are needed to determine
the correct one. Normally, the phases are found using several derivatives
to minimize the error.
Anomalous dispersion
An assumption of the analysis of the X-ray diffraction is that the only
effect of the atom is to scatter the X-ray beam. However, if the energy
of the X-rays is at a value corresponding to a transition energy for a
certain atom, then the phase will include a contribution associated with
the transition of electrons for that specific atom, termed the anomalous
dispersion. For proteins, the anomalous dispersion is large for any bound
transition metals. By comparing the diffraction at different wavelengths,
with energies near and away from the transition energies, the differences
in the diffraction arising from the anomalous dispersion can be identified
CHAPTER 15 X-RAY DIFFRACTION AND EXAFS 329
| FP |FPFHFPH FH| FPH || FPH || FP |Figure 15.14
Diagram used to
calculate phases
from the native
protein, FP, heavy
metal, FH, and
derivative, FPH,
scattering factors.