and the structure factors calculated from the refined model of the ordered,
atomic portion of the structure, 
. To avoid bias toward some assumed model of solvent scattering, the ordered
model should have been refined with no attempt at modeling solvent.
Let us consider the average agreement between
and
as a function of resolution ( 
or s). One can do this by splitting reciprocal space into thin shells,
and determining a linear scale factor for each shell. We can determine
the scale factor K for each shell by minimizing the function:
One can rearrange this to show that
Figure i shows K as a function of resolution for the Thermolysin:inhibitor complex TLN:ZGPLL [Tronrud et al., 1987] for data from 20 to 1.6Å.
One can see in the figure that K changes smoothly with resolution. There are two roughly straight sections with a break at about 4Å resolution.
It is best to consider the two sections of the curve separately. In
the high-resolution portion, K does not vary rapidly with resolution.
The slight variation that is evident can be removed by addition of a temperature
factor correction to
, equivalent to adding some constant to the B factor of each atom
in the model. If the B's are properly refined, no correction will
be required, and the K's for the high-resolution data will form
a horizontal line. The line is not horizontal in Figure
i because the individual B factors were biased in refinement
by the lack of a disordered solvent model. (The bias in the example is
limited because only the data between 10 and 1.6Å were used for this
refinement.)
In the low-resolution portion of the curve, K changes very rapidly.
K at 20Å is much smaller than K at 2.0Å. This
implies (by Equation 2) that
is systematically too large at low resolution.
One can understand the behavior of the low resolution portion of the curve by examining the difference between the usual model of a macromolecule and the electron density that actually occupies the crystal. The electron density of a model without a disordered solvent contribution drops to zero outside the envelope of the molecule. Because the crystal contains significant amounts of electron density in these regions the contrast of the model is much greater than it should be. This high contrast results in the inappropriately large size of the low resolution structure factors calculated from the model. It is important to note that the phases of the low-resolution coefficients from these two models will be very similar in spite of the large difference in amplitude.
