Hoofdpagina van Kristallografie. Summary
In the case of small molecules, a direct solution of the phase problem is
possible from statistical and algebraic relations among the intensities.
These direct methods alone until now proved to be inconclusive for macromolecules.
The reason for this is clear: the joint probability distribution of three
structure factors depends in first approximation on N"1/2 (N: number of atoms
in the unit cell), so the distribution gets increasingly flattened if N becomes
large. On the other hand, large structures such as proteins have been solved
by the method of isomorphous replacement with heavy atoms. Differences in
diffracted intensities from the derivative and native crystals are used to
locate the heavy atoms. The calculated contributions from these atoms serve
as a reference for evaluation of phases for the native crystal structure.
In general, the phase information from a single isomorphous pair is insufficient.
Thus multiple isomorphous replacements or supplementary information such
as from anomalous scattering, is used to resolve the ambiguity. Anomalous
scattering measurements when made at appropriate multiple wavelengths can
also provide a definitive. experimental solution for the phase problem. This
raises the question why direct methods fail while other techniques succeed.
An efficient way to improve the applicability of direct methods is to reduce
the number of variables (N) involved. This reduction can be achieved for
instance by using data from isomorphous replacement and/or anomalous scattering.
In chapter 2 it is shown that the concept of isomorphous structure factors
can be useful for estimating the doublet and triplet phase-sums present amongst
them. A diffraction ratio is proposed that predicts whether a structure solution
via a pair of isomorphously related data sets may in
principle be feasible. A lower limit to the diffraction ratio turns out to
be required in order to get a triplet phase sum error-level comparable to
that of small structures which are solved routinely by Direct Methods. The
diffraction ratio can be used to maximize the triplet phase sum reliability
before collecting the data, by choosing the optimal wavelength in a single
anomalous scattering experiment, by selecting the most suitable heavy-atom
derivative in a single isomorphous replacement experiment or by selecting
the optimal wavelength-combination in a multiwavelength experiment. Another
remarkable characteristic of the diffraction ratio is its linear relationship
to the average doublet phase sum. It is argued that the doublets are essential
for an accurate estimation of the triplet phase sums.
In chapter 3 several probabilistic and algebraic techniques are discussed
to estimate the doublets. The combination of an algebraic estimation technique
and a new difference Patterson synthesis, the maxima of which are used to
improve iteratively the doublet phase sums, is shown to be successful. In
case of too low diffraction ratios no useful estimates can be obtained from
the joint probability distribution of isomorphous structure factors, even
for small structures. If the differences between isomorphous structure factors
become too small, the normal mathematical procedure turns out to be inadequate
because the very small quantities cannot be expressed in terms of the usual
variables. This led to the introduction of a different type of random variable:
the single difference of isomorphous structure factors.
In chapter 4, based on the use of the single differences as random variables,
an efficient procedure is presented for the derivation of joint probability
distributions of the triplet phase sum present among two isomorphous data
sets. It is shown that the usual probabilistic techniques, applied to these
random variables, finally results in th.e joint probability distribution
of three isomorphous structure factors comprising three doublet and eight
triplet phase sums respectively. A major advantage of the new technique is
that the inherent correlation between the isomorphous data sets is removed
if the mathematical procedure is set up for the small differences themselves.
In chapter 5 the technique of single differences has been applied to the
estimation of quartet phase sums leading to very reliable results even without
the use of the cross terms. The error level reduction for the triplet and
quartet phase sums leads to a phase error small enough for direct-method
applications without knowledge of the heavy atom substructure.
In chapter 6 the doublet phase sums are re-examined in order to solve
their sign ambiguity. Despite the success of the single-difference procedure
in solving the problem of the negative doublets in the case of two isomorphous
data sets including anomalous scattering effects and data sets collected
at two different wavelengths, in single isomorphous replacement case the
problem cannot be solved directly, because of the large number of negative
doublets. However, a method is developed which solves the problem if three
isomorphous data sets are available. Also methods are proposed in the case
of only two isomorphous data sets.
The available evidence suggests that the structure solution, even of large
molecules, is possible by the work described in this thesis. Our next step
will be a new version of the SIMPEL program set which incorporates the described
theoretical development.
Overzicht proefschriften kristallografie Amsterdam.
Hoofdpagina van Kristallografie.