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


On 1/9/2012 11:13 PM, Jacob Keller wrote:
> 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
>
>
>
> On Fri, Jan 6, 2012 at 6:23 PM, James M Holton<[log in to unmask]> wrote:
>> 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
>>
>> On Jan 6, 2012, at 5:44 PM, Jacob Keller<[log in to unmask]> wrote:
>>
>>> 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
>>>
>>> On Fri, Jan 6, 2012 at 10:25 AM, Jacob Keller
>>> <[log in to unmask]> wrote:
>>>> 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
>>>>
>>>> --
>>>> *******************************************
>>>> Jacob Pearson Keller
>>>> Northwestern University
>>>> Medical Scientist Training Program
>>>> email: [log in to unmask]
>>>> *******************************************
>>>
>>>
>>> --
>>> *******************************************
>>> Jacob Pearson Keller
>>> Northwestern University
>>> Medical Scientist Training Program
>>> email: [log in to unmask]
>>> *******************************************
>
>