Page 154 - Macromolecular Crystallography

CHAPTER 10

Phase refinement through

density modification

Jan Pieter Abrahams, Jasper R. Plaisier, Steven Ness,
and Navraj S. Pannu

10.1 Introduction Electron density statistics. At high resolution we
know the shape of the electron density of an atom,
It is impossible to directly measure phases of
in which case we only need to know its exact
diffracted X-rays. Since phases determine how the
location to reconstruct the electron density in its
measured diffraction intensities are to be recom-
immediate vicinity. At lower resolution we can
bined into a three-dimensional electron density,
impose an expected shape on the uni- or mul-
phaseinformationisrequiredtocalculateanelectron
tivariate distributions of electron density within
density map of a crystal structure. In this chapter we
the protein region in a procedure that is known as
discuss how prior knowledge of the statistical distri-
histogram matching.
bution of the electron density within a crystal can be
used to extract phase information. The information
can take various forms, for example: The problem is not so much in understand-
ing the restrictions these types of prior knowledge
Solvent flatness. On average, protein crystals con- impose on (suboptimal) electron density, but rather
tain about 50% solvent, which on an atomic scale in using these restraints in reciprocal space. In prac-
usually adopts a random, non-periodic structure tice an iterative procedure is followed. First the
within the crystal and hence is featureless within electron density of an initial model is calculated,
the averaged unit cell. Therefore, if we know the which is then modified to satisfy the expected,
location of the solvent regions within a macro- previously determined restraints. From the mod-
molecular crystal, we already know a consider- ified map, the diffraction data are recalculated. The
able part of the electron density (i.e. the part that resulting phases are combined with the measured
is flat and featureless), and ‘flattening’ the elec- data and their associated phase probability distri-
tron density of the solvent region can improve the butions. On this basis the currently most probable
density of our macromolecule of interest. phase set is calculated. The procedure is repeated
Non-crystallographic symmetry. Many protein crys- until convergence. Below, we briefly describe the
tals contain multiple copies of one or more mathematical background of these procedures and
molecules within the asymmetric unit. Often discuss some of their essential aspects. We pay
the conformations of such chemically indistin- special attention to visual, geometric concepts, as
guishable but crystallographically non-equivalent we believe them important for developing an in-
molecules are sufficiently alike to treat them as tuitive grasp for the process of density modification
identical. In this case, we can improve the signal to in phase refinement. To this end, we illustrate the
noise ratio of the electron density of our molecule concepts on a one-dimensional, centrosymmetric
of interest by averaging the density of the multiple map, as this allows us to depict phases simply by
copies in the asymmetric unit. their sign.

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