On 03/12/2014 09:02 PM, Keller, Jacob wrote:
The Fourier transform of electron density is a complex scattering amplitude
that by the axiom of quantum mechanics is not a measurable quantity. What is
measurable is the module squared of it. In crystallography, it is called either
F^2 (formally equal F*Fbar) or somewhat informally diffraction intensity, after
one takes into account scaling factors. F*Fbar is the Fourier transform of an
electron density autocorrelation function regardless if electron density is
periodic or not. For periodic electron density the structure factors are
described by sum of delta Dirac functions placed on the reciprocal lattice.
These delta functions are multiplied by values of structure factors for
corresponding Miller indices.
Okay, I may have been confused--I thought that the Fourier transform was
essentially acting like an autocorrelation function (since generally Fourier
transforms are similar to autocorrelation functions--not clear on the details
right now), and I had thought I had heard stories of days of yore handwritten
Fourier series calculations to make electron density maps. You're telling me
they had to also back-calculate an autocorrelation function? Times were tough!
Maybe someone from that generation can chime in about how they dealt with this?
Even in today’s easy times, the fastest way to calculate autocorrelation
function is to calculate Fourier transform of the data, calculate F*Fbar and
calculate back Fourier transform of it.
This is interesting case of pseudocrystal, however because there is no crystal
lattice, it is not relevant to (1) or (2). In any case, pentagonal
quasilattices are probably not relevant to macromolecular crystallography.
I tried a few simulations to show what I mean but ran out of time--sorry about
that. I think I'll probably just drop this.
NB Linus Pauling said more forcefully the same prediction about aperiodic
crystals in general not existing, pentagonal or otherwise, but was proven dead
wrong by now-Nobel laureate Dan Shechtman. Maybe someone will come across an
aperiodic protein crystal, or already has and missed it, and stupefy us all.
Someone mentioned to me once seeing personally a ten-fold symmetrical
diffraction pattern from a protein crystal, but she dismissed it with exactly
the argument that Pauling made, I think that it was a twinned cubic space group.
Unless you are interested in finding curious objects, what would you do with
protein quasicrystal? The practices of macromolecular crystallography is about
determining 3-dimensional structure of objects being crystallized. Protein
quasicrystal are really unlikely to diffract to high enough resolution, and even
ignoring all other practical aspects, like writing programs to solve such a
structure, chances of building an atomic model are really slim.
This is easy to test by analyzing diffraction patterns of individual crystals.
In practice, the dominant contribution to angular broadening of diffraction
peaks is angular disorder of microdomains, particularly in cryo-cooled crystals.
However, exceptions do happen, but these rare situations need to be handled on
case by case basis.
The interpretation of the data presented in this article is that variation in
unit cell between microcrystals induce their spatial misalignment. The data do
not show variation of unit cell within individual microscrystalline domains.
Tetragonal lysozyme can adopt quite a few variations of the crystal lattice
during cryocooling. Depending on the conditions used, resulting mosaicity can
vary from 0.1 degree (even for 1mm size crystal) to over 1. degree.
Consequently, measured structure factors from a group of tetragonal lysozyme
crystal can be quite reproducible, or not. As a test crystal, it should be
handled with care.
1 degree mosaicity is not an impediment to high quality measurements. However,
high mosaicity tends to correlate with presence of phase transitions during
cryo-cooling. If such transition happen during cryo-cooling, crystals of the
same protein, even from the same drop, may vary quite a lot in terms of
structure factors. Additionally, even similar values of unit cell parameters are
not guarantee of isomorphism between crystals.
Zbyszek
I was thinking of this paper for example (see last line of abstract). Perhaps
other crystals are different from lysozyme, though, as you mention.
All the best,
Jacob Keller
Acta Crystallogr D Biol Crystallogr. 1998 Sep 1;54(Pt 5):848-53.
A description of imperfections in protein crystals.
Nave C.
Author information
Abstract
An analysis is given of the contribution of various crystal imperfections to
the rocking widths of reflections and the divergence of the diffracted beams.
The crystal imperfections are the angular spread of the mosaic blocks in the
crystal, the size of the mosaic blocks and the variation in cell dimensions
between blocks. The analysis has implications for improving crystal perfection,
defining data-collection requirements and for data-processing procedures.
Measurements on crystals of tetragonal lysozyme at room temperature and 100 K
were made in order to illustrate how parameters describing the crystal
imperfections can be obtained. At 100 K, the dominant imperfection appeared to
be a variation in unit-cell dimensions in the crystal.
PMID: 9757100 [PubMed - indexed for MEDLINE]
--
Zbyszek Otwinowski
UT Southwestern Medical Center
5323 Harry Hines Blvd., Dallas, TX 75390-8816
(214) 645 6385 (phone) (214) 645 6353 (fax)
zbys...@work.swmed.edu