Page 158 - Macromolecular Crystallography

PHASE REFINEMENT THROUGH DENSITY MODIFICATION 147

Measurement

Get phase probabilities

Structure factors Map

Recombine phases
No Apply constraints

Convergence? Improved structure factors Improved map

Yes

Done
Reciprocal space Real space

Figure 10.1 A flow chart demonstrating phase refinement in practice with the iterative recycling between real and reciprocal space.

function, then multiply the two functions. Now, do 10.6 Phase recombination in
this for each point in B, moving A to each point in B, Fourier cycling
multiplying the functions, and adding all the prod-
Clearly, for the procedure outlined in Fig. 10.1 to
uct functions. The result is the convolution of Awith
work, we need to take care of several critical steps;
B. See Fig. 10.2 for an example of the convolution
next to reasonable initial phase estimates required
operator.
to formulate the initial restraints, we need a statis-
Mathematically, the convolution operator ⊗ is
tically valid procedure for the combination of the
defined as follows:
phases obtained by back transformation of the real
space restrained map and the initial phase probabil-
∞
C(y) = (A ⊗ B)(y) = A(y − x)B(x) δx (7) itydistribution. Thisrecombinationstepisdiscussed
−∞
below.
A useful property of the Fourier transform and In general, estimates of high resolution phases
convolution is that the Fourier transform of the con- will be less accurate than the ones at low resolution.
volution of two functions is equal to the Fourier The reason is that in the beginning of a structure
transform of the two functions multiplied together. determination, it is easier to establish low resolution
Thus, since the convolution of two functions is contours than high resolution details. This is because
typically a time consuming process, this property, contours are hardly affected by errors at high res-
together with the Fast Fourier Transform, is used to olution. On the other hand, the contrast of high
significantly speed up the process of convolving two resolution features is severely affected by errors at
functions. low resolution. Hence, in phase refinement we gen-
In conclusion, when modifying the density in real erally see the improvement progressing from low to
space is equivalent to a multiplication with another high resolution as we cycle through the procedure.
map, in reciprocal space this results in the con- It makes sense to weight down structure factors
volution of the Fourier transform of both maps (and with erroneous phases. Therefore we need to intro-
vice versa). duce a weighting scheme that typically has a higher