Page 156 - Macromolecular Crystallography

PHASE REFINEMENT THROUGH DENSITY MODIFICATION 145

the following type: Here, x solvent is a real space coordinate within the
solvent region and ρ is the mean electron
φ(h j ) =−φ(−h j ) (2) solvent
density of the solvent.
Straight substitution of these equalities into Eq. 1
The number of additional terms based on Eq. 4
reduces the magnitude of the problem; rather than
is determined by the solvent fraction of the crys-
4N unknowns, we are left with 3N unknowns.
tal. If the solvent content is 50% (which is the
average for protein crystals), N independent, addi-
10.3.2 Differences between corresponding tional equations of type 4 are introduced. As these
|F(h j )|’s
equations can be substituted into the Fourier sum-
In order to obtain initial phase estimates, crystallo- mations of Eq. 1, they effectively reduce the number
graphers typically use either experimental phasing of unknowns by 2N times the solvent fraction –
techniques or molecular replacement. Obtaining ini- provided they accurately distinguish disordered sol-
tial phase estimates and their associated phase prob- vent from protein.
ability distributions are treated in other chapters of
this book, and for the remainder of this chapter 10.4.2 Non-crystallographic symmetry (NCS)
we assume that the initial phases and the associ-
ated phase probabilities have been calculated. These If the electron density of one area of the asymmetric
phase probability distributions are conveniently unit is sufficiently similar to that of another (after a
described by Hendrickson–Lattman coefficients. translation and/or a rotation), additional equations
of the following type can be inferred:
P(φ(h j ))
ρ(x unique ) = ρ(Tx unique ) (5)
A j cos(φ(h j ))+B j sin(φ(h j ))+C j cos(2φ(h j ))+D j sin(2φ(h j ))
= K j e
(3) Here, x unique is a real space coordinate within a
Here, P(φ(h j )) is the probability function of a region of density that is repeated elsewhere in the
phase φ(h j ), whilst A j , B j , C j , and D j are its asymmetric unit after a rotation and/or transla-
Hendrickson–Lattman coefficients and K j is a nor- tion defined by the transformation T. Also these
malizing constant. Clearly, P(φ(h j )) cannot be inser- equations can be substituted in the Fourier sum-
ted straight into Eq. 1. However, it does provide mation of Eq. 1, effectively further reducing the
additional equations, one for each phase for which number of unknowns in real space down to the num-
A j , B j , C j ,or D j are non-zero. ber of independent grid points within the fraction of
unique density.
10.4 Real space restraints
10.4.1 Solvent flatness 10.4.3 Electron density statistics
The overall distribution of randomly phased elec-
In disordered solvent regions of the unit cell, the
tron density is Gaussian, whereas a correctly phased
density is featureless and flat. In practice, the loca-
map is expected to have a non-Gaussian distribu-
tion and size of the solvent region are inferred from
tion at resolutions beyond about 2.5 Å. An electron
an initial electron density map and the molecular
density distribution is described by a histogram, in
weight of the molecule. Since the average electron
which for each density value the likelihood is plotted
density of the solvent usually is very similar to
offindingsuchavaluewithintheunitcell. Theshape
that of the protein, automated procedures identify
of this histogram is determined by the resolution of
the solvent mask by determining the regions which
the map (at low resolution, extreme values are less
have the smallest variation in electron density. If we
likely) and the chemical composition (heavy atoms
know which regions of the unit cell are featureless,
will cause more extreme histograms). Proteins share
additional equations of the following type can be
characteristic electron density histograms, provided
inferred:
they do not contain many heavy atoms or large,
ρ(x solvent ) = ρ solvent (4) disordered volumes. This implies that for a given