From: Jacob Keller
Date: 6 January 2012 16:25
Dear Crystallographers,
has anyone come across a figure showing a normal diffraction image,
and then next to it the equivalent molecular transform, perhaps with
one image as phases and one as amplitudes? Seems like it would be a
very instructional slide to have to explain how crystallography works
(I know about Kevin Cowtan's ducks and cats--I was looking for
approximately the same but from protein or NA molecules.) I don't
think I have ever seen an actual molecular transform of a protein or
NA molecule.
All the best,
Jacob
----------
From: Jacob Keller
Actually, as a way to make this type of figure, I think there are
programs which output simulated diffraction images, so perhaps I could
just input a .pdb file with some really huge (fake) cell parameters
(10,000 Ang?), and then the resulting spots would be really close
together and approximate the continuous molecular transform. I think
this would amount to the same thing as the molecular transform of the
model itself--am I right?
Does anyone know which software outputs simulated diffraction images?
Jacob
----------
From: Dale Tronrud
James Holton has software for calculating molecule transform images.
Check out http://bl831.als.lbl.gov/~jamesh/nearBragg/. The program
doesn't read PDB format coordinates, just lists of three numbers.
Dale Tronrud
----------
From: Bernhard Rupp (Hofkristallrat a.D.)
This may give some idea:
Illustration of a molecule and its cosine transform:
http://www.ruppweb.org/garland/gallery/Ch6/pages/Biomolecular_Crystallograph
y_Fig_6-16.htm
and sampled by lattice points
http://www.ruppweb.org/garland/gallery/Ch6/pages/Biomolecular_Crystallograph
y_Fig_6-01_PART3.htm
BR
----------
From: James M Holton
You mean something like the animation at the top of this web page?
http://bl831.als.lbl.gov/~jamesh/fastBragg/
This program is a relative of nearBragg, which Dale already mentioned.
-James Holton
MAD Scientist
----------
From: Jacob Keller
I like that animation a lot, as it shows the gradual nature of the
lattice effect, but it is not exactly what I am looking for. I am
actually just curious what the pattern behind the spots looks like for
various molecules, and would like to see an image of that in various
orientations. I guess one way to put it is that I would like to see
what the 1.5-2 Ang diffraction pattern would be for a single,
radiation-damage-impervious protein or RNA/DNA molecule given enough
x-rays and time.
Would the intensities-based transform image be much less complicated
than the phases-based one?
Would larger molecules have more complex patterns, corresponding to
the amount of information in their structures?
JPK
----------
From: Tim Gruene
Dear Jacob,
what do you mean by 'molecular transform'? Would you like to visualise
the summed structure factors from the atoms inside the unit cell?
- - What pattern are you talking about/ what pattern do you expect?
- - What benefit do you expect from seeing the phases? What you can
physically observe on the detector are the intensities - the concept of
'phases' is just a mathematical notion to explain the intensities in
terms of interference from single waves and an atomistic model inside
the crystal.
- - What makes you think the pattern from a larger molecule would have a
more complex pattern?
Cheers,
Tim
- --
- --
Dr Tim Gruene
Date: 6 January 2012 16:25
Dear Crystallographers,
has anyone come across a figure showing a normal diffraction image,
and then next to it the equivalent molecular transform, perhaps with
one image as phases and one as amplitudes? Seems like it would be a
very instructional slide to have to explain how crystallography works
(I know about Kevin Cowtan's ducks and cats--I was looking for
approximately the same but from protein or NA molecules.) I don't
think I have ever seen an actual molecular transform of a protein or
NA molecule.
All the best,
Jacob
----------
From: Jacob Keller
Actually, as a way to make this type of figure, I think there are
programs which output simulated diffraction images, so perhaps I could
just input a .pdb file with some really huge (fake) cell parameters
(10,000 Ang?), and then the resulting spots would be really close
together and approximate the continuous molecular transform. I think
this would amount to the same thing as the molecular transform of the
model itself--am I right?
Does anyone know which software outputs simulated diffraction images?
Jacob
----------
From: Dale Tronrud
James Holton has software for calculating molecule transform images.
Check out http://bl831.als.lbl.gov/~jamesh/nearBragg/. The program
doesn't read PDB format coordinates, just lists of three numbers.
Dale Tronrud
----------
From: Bernhard Rupp (Hofkristallrat a.D.)
This may give some idea:
Illustration of a molecule and its cosine transform:
http://www.ruppweb.org/garland/gallery/Ch6/pages/Biomolecular_Crystallograph
y_Fig_6-16.htm
and sampled by lattice points
http://www.ruppweb.org/garland/gallery/Ch6/pages/Biomolecular_Crystallograph
y_Fig_6-01_PART3.htm
BR
----------
From: James M Holton
You mean something like the animation at the top of this web page?
http://bl831.als.lbl.gov/~jamesh/fastBragg/
This program is a relative of nearBragg, which Dale already mentioned.
-James Holton
MAD Scientist
----------
From: Jacob Keller
I like that animation a lot, as it shows the gradual nature of the
lattice effect, but it is not exactly what I am looking for. I am
actually just curious what the pattern behind the spots looks like for
various molecules, and would like to see an image of that in various
orientations. I guess one way to put it is that I would like to see
what the 1.5-2 Ang diffraction pattern would be for a single,
radiation-damage-impervious protein or RNA/DNA molecule given enough
x-rays and time.
Would the intensities-based transform image be much less complicated
than the phases-based one?
Would larger molecules have more complex patterns, corresponding to
the amount of information in their structures?
JPK
----------
From: Tim Gruene
Dear Jacob,
what do you mean by 'molecular transform'? Would you like to visualise
the summed structure factors from the atoms inside the unit cell?
- - What pattern are you talking about/ what pattern do you expect?
- - What benefit do you expect from seeing the phases? What you can
physically observe on the detector are the intensities - the concept of
'phases' is just a mathematical notion to explain the intensities in
terms of interference from single waves and an atomistic model inside
the crystal.
- - What makes you think the pattern from a larger molecule would have a
more complex pattern?
Cheers,
Tim
- --
- --
Dr Tim Gruene
----------
From: Jacob Keller
The diffraction pattern we see results from the convolution of the
crystal lattice with the fourier transform of the electron density, as
I understand it. I guess I am interested in seeing the unconvoluted
transform of the electron density, just to get a feeling for what the
characteristics of those transforms are. As I put it in a previous
message, I am curious what the diffraction would look like from a
theoretical radiation-damage-impervious single molecule of either
protein or nucleic acid. I suspect that for nucleic acids one would
see the stacked bases a la Rosalind Franklin's fiber diffraction
images, and perhaps other interesting features. Maybe there would be a
powder-diffraction-like ring for CC and CN bond lengths? Anyway, I
guess the goal would be to see whether one could find any other
relationships like phase triplets etc.
Jacob
----------
From: James Holton
I think what Jacob is looking for is something akin to the transition between slides 20 and 21 in this PowerPoint:
http://bl831.als.lbl.gov/~jamesh/nearBragg/nearBragg.ppt
But, if you're looking for something with a more complex molecule, this movie might be what you want:
http://bl831.als.lbl.gov/~jamesh/nearBragg/lysozyme/real_recip.wmv
or, for those who can't play wmv, the individual frames are listed as test_???.png under:
http://bl831.als.lbl.gov/~jamesh/nearBragg/lysozyme/
This is an animation of a lysozyme molecule in real space (white dots on a black background) and its diffraction pattern (in false color intensity, no phases). You can see that the molecular transform has a big, bright blob near the origin. This is SAXS, and reflects the overall size and shape of the molecule. You can also see that the rest of the pattern is made up of blobs that are about the same "size" as the SAXS blob (reciprocal size of the molecule). These blobs tend to have a constant phase across their extent, but since the intensity variations are so huge, I decided to do false-color intensity instead for the movie.
There are many different ways to explain why the blobs have constant phase, but perhaps the easiest to understand is that there can only be so much "information" in the diffraction pattern and therefore both the intensity and the phase must vary smoothly with about the same "granularity". The "grain size" in reciprocal space is the reciprocal size of the molecule because there are no contributions from inter-atomic distances greater than that. This is sometimes called "speckle". Another way to think about it is to consider the centrosymmetric case where the phases are either 0 (positive structure factor) or 180 (negative structure factor). There is no way for a "smooth" structure factor function to go from positive to negative without passing through zero (in between the blobs). A similar "smoothness argument" can be made for the acentric case: both the real and imaginary components have about the same "granularity" as the intensity.
Now, if the molecule packs into a crystal, the unit cell will be about the same size as one molecule, and the Bragg peaks will therefore have a spacing roughly equal to the diameter of the "SAXS blob". This means that the Bragg peaks are just far enough apart for the the "molecular transform" to go through pretty much any change imaginable in the space between them. Darn! Hence: the "phase problem": the phase of any given spot is essentially unrelated to the next.
Of course, in reality the unit cell is actually a little bigger than the molecule within it, so there is some correlation between neighboring spot phases, and this is why solvent flattening works. NCS works in a similar way, but crystallographic symmetry doesn't because then the phase correlations fall exactly onto the symmetry-related Bragg peaks. But I'm not going to go into a discussion of all phasing methods here!
I have not done a movie like the above with phase coloring, but that is possible using the "phase_color.c" program linked from the main nearBragg page:
http://bl831.als.lbl.gov/~jamesh/nearBragg/phase_color.c
This is what I did for the PowerPoint slides. The trick, however, is that nearBragg outputs the components of the scattered wave as it arrives at the detector in sinimage.bin and cosimage.bin. This is "realistic", but the phase of this wave is not the same as the "phase" of the structure factor we are used to thinking about. This is because the structure factor is defined as the ratio of the wave scattered by "the object" to the wave that would be scattered from a single point electron at the origin (see slide 13). So, you need to take the complex number represented by the cosimage/sinimage output by nearBragg and divide it by the cosimage/sinimage output from a nearBragg run using only one atom at "0 0 0". The complex_divide.csh script should be helpful for this.
Alternatively, you can use the "-curved_det" option to nearBragg, then all the detector pixels will be the same distance from the origin and produce a smoothly-varying phase. If the distance is an integral multiple of the wavelength, then the phase at the detector is essentially the same as the structure factor phase. At least, this works in the far field. If the detector is close enough for the size of the sample to be comparable to the size of a pixel, then the situation is not so clear-cut, and I wrote nearBragg.c to try and figure this out.
I'm afraid I don't really have a nice "canned" procedure for making phase-colored images yet because it is hard to figure out which intensity scale is most interesting. I suppose I could put the complex_divide procedure inside of nearBragg, but that would make it run at least 2x slower, and it is already slow enough!
-James Holton
MAD Scientist
----------
From: Yuri Pompeu
to echo Tim's question:
If by pattern you mean the position of the spots on the film, I dont think they would change based on the complexity of the macromolecule being studied. As far I know it, the position of the spots are dictated by the reciprocal lattice points
(therefore the real crystal lattice) (no?)
The intensity will, obviously, vary dramatically...
ps. Very interesting (cool) images James!!!
----------
From: Jacob Keller
No, I meant the non-lattice-convoluted pattern--the pattern arising
from the Fourier-transformed electron density map--which would
necessarily become more complicated with larger molecular size, as
there is more information to encode. I think this will manifest in
what James H called a smaller "grain size."
JPK
----------
From: Dale Tronrud
I've been thinking about these matters recently and had a nifty
insight about exactly this matter. (While this idea is new to me
I doubt it is new for others.)
The lower limit to the size of the features in one of these
"scattergrams" is indicated by the scattergram's highest frequency
Fourier component. Its Fourier transform is the Patterson map.
While we usually think of the Patterson map as describing interatomic
vectors, it is also the frequency space for the diffraction pattern.
For a noncrystalline object the highest frequency component corresponds
to the longest Patterson vector or, in other words, the diameter of
the object! The bigger the object, the higher the highest frequency
of the scattergram, and the smaller its features.
Dale Tronrud
----------
From: Jacob Keller
I am trying to think, then, what would the Patterson map of a
Fourier-transformed electron density map look like? Would you get the
shape/outline of the object, then a sharp drop-off, presumably? Is
this used to orient molecules in single-particle FEL diffraction
experiments?
JPK
----------
From: Dale Tronrud
I think you have to be a little more clear as to what you mean
by an "electron density map". If you mean our usual maps that we
calculate all the time the Patterson map is just the usual Patterson
map. It also repeats to infinity, with the infinitely long Patterson
vectors (infinitely high frequency components) being required to
create the Bragg peaks. If you mean an electron density map of a
single object with finite bounds your Patterson map will also have
finite bounds, just with twice the radius.
The Patterson boundary is not a sharp drop-off because there aren't
as many long vectors as short ones, but the distribution depends on
the exact shape of your object. Once you have a Patterson map that
has an isolated edge (no cross-vectors) back calculating the original
object is pretty easy. (Miao, et al, Annu. Rev. Phys. Chem. 2008,
59:387-410)
Dale Tronrud
----------
From: arka chakraborty
Hi all,
I would like to ask some questions regarding this thread..
1) What is exactly meant by "Fourier transformed electron density"?- according to my knowldege performing a fourier transform on the electron density gives you the structure factor back. So, how does it related to what Prof. James H called "non-lattice-convoluted pattern"? It will be really nice if somebody can explain the thing in a " decoded" language?!
And also any articles focusing on the concepts discussed in the entire thread will be very helpful
Regards,
ARKO--
ARKA CHAKRABORTY
CAS in Crystallography and Biophysics
University of Madras
Chennai,India
----------
From: Dale Tronrud
The real question is "The Fourier transform of what?". All of our
usual software takes the electron density of the unit cell and Fourier
transforms it assuming that the unit cell is repeated endlessly throughout
the universe. This results in non-zero values only at integral values
of h, k, and l. If, instead, you assume there is only one unit cell
present in the x-ray beam you will get a continuous function for the
intensity of scattering. You can use the convolution theorem to show
that the value of these two functions are equal at the integral values
(except for a scale factor).
This thread has been about the relationship between the features
one would observe in this continuous function (if it could be measured)
and the electron density. If you could measure the continuous function
you can directly calculate the electron density - there is no phase
problem. The paper I referred to earlier gives the details.
Dale Tronrud
Regards,
----------
From: Colin Nave
Dale
I don't think it is quite that easy. The paper you referred to (Miao, et al, Annu. Rev. Phys. Chem. 2008, 59:387-410) gives a good summary but people have struggled to apply this technique for low contrast biological objects. Early tests were on high contrast objects (low entropy images I guess). These probably met the conditions for which classical direct methods would work. Since then, some reconstructions of cells (probably with a high entropy images) have been carried out. However, the images obtained have not yet shown the features obtainable by electron or x-ray tomography with a lens.
High non-crystallographic symmetry (e.g. as in spherical viruses) in a crystal gives extra information similar to that which would be obtained by sampling the continuous molecular transform of a single subunit. However it is more complex and perhaps richer because the individual subunits are in different orientations and positions - not a simple sampling of the transform a single subunit.
I guess this leads to a question. Does anyone know of a case of a crystal with no non crystallographic symmetry but a very high solvent content for which the structure (at say 2.5A resolution whatever that means) could be solved without resort to additional phasing methods?
Colin
No comments:
Post a Comment