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Florian contacted me directly but I think the question may be relevant
for this bulletin board, especially now that many enter crystallography
without the indepth training that used to be common.
Florian Schmitzberger wrote:
Dear Bart,
I hope you don't mind me asking this question, and I apologize if I
have not given this much prior thought. But so far I never quite
understood why truncation at any <I>/sig<I> (threshold), while
otherwise having complete data in the accepted resolution range,
would cause "Fourier ripples" in the e-density map (and what the
interpretation for these features are). Is this due to an inadequate
scaling of the structure factor amplitudes from an incorrect slope
approximation in the Wilson plot? I suppose I would get the same
effect with datasets of significant incompleteness (in one/several
resolution shells)?
Thank you very much!
Florian
To understand this, and many other concepts in crystallography, you need
to know about the "convolution theorem". I don't mean you need to
understand the details of the math (I don't) but you need to know how
the properties of the math relate to the actual physics of our experiments.
In crystallography we deal with the real space (the electron density in
your crystal) and reciprocal space (the diffraction pattern of the
crystal). The convolution theorem states that a multiplication in one
space leads to a convolution in the other space. In this particular case
you can think of a truncated diffraction data set as the multiplication
of the true, untruncated, diffraction data set and a mask that has
values of 1 for all reflections that you have measured and 0 for all
reflections beyond your resolution cutoff. The Fourier transform of this
spherically symmetrical mask will be a spherically symmetrical function
with a peak at the origin and surrounded by ripples (in two dimensions
this looks like the ripples in a pond after you throw in a stone).
Because the mask multiplication happens in reciprocal space, the
convolution happens in real space. This means that at every point in the
map the value due to your measured diffraction is convoluted with the
ripple pattern. What this does is basically placing a copy of the ripple
pattern at each point in your map with the magnitude of the ripple
pattern scaled by the value of the density due to the measured
diffraction pattern. I'm sure if you Google convolution and
crystallography you will get some figures to help make this clear. The
ripple pattern is easiest to recognize for grid points with a high value
surrounded by grid points with low values, this basically is a heavy
atom like a metal in the structure. The distance between the maxima and
minima in the ripple are related to the resolution at which you
truncate. If you truncate at high resolution the ripples will be close
together, if you truncate at lower resolution they will be further apart.
If you get ripple patterns centered at a few atoms at the surface of the
protein then they can interfere to give localized peaks which, depending
on your truncation resolution, could be at a distance that is normal for
bound water molecules. You could be tempted to build a water in such a
peak in the map but it is just a pure artifact.
If you like to experiment, create Fcalc values to 1.5A for a
metallo-protein structure, let's say myoglobin. Calculate a map from
these Fcalcs. Now truncate the data to 2.5A and calculate another map.
If you look at the metal you will probably see the concentric ripples
around it. It was very clear for a dinuclear copper protein that I
worked on long ago for my thesis.
Bart
