On Friday, 10 November 2017 00:10:22 Keller, Jacob wrote:
> Dear Crystallographers,
> 
> I have been considering a thought-experiment of sorts for a while, and wonder what you will think about it:
> 
> Consider a diffraction data set which contains 62,500 unique reflections from a 50 x 50 x 50 Angstrom unit cell, with each intensity measured perfectly with 16-bit depth. (I am not sure what resolution this corresponds to, but it would be quite high even in p1, I think--probably beyond 1.0 Angstrom?).

Meh. 
62500 is < 40^3, so ±20 indices on each axis.
50Å / 20 = 2.5Å,  so not quite 2.5Å resolution


> Thus, there are 62,500 x 16 bits (125 KB) of information in this alone, and there is an HKL index associated with each intensity, so that I suppose contains information as well. One could throw in phases at 16-bit as well, and get a total of 250 KB for this dataset.
> 
> Now consider an parallel (equivalent?) data set, but this time instead of reflection intensities you have a real space voxel map of the same 50 x 50 x 50 unit cell consisting of 125,000 voxels, each of which has a 16-bit electron density value, and an associated xyz index analogous to the hkl above. That makes a total of 250 KB, with each voxel a 1 Angstrom cube. It seems to me this level of graininess would be really hard to interpret, especially for a static picture of a protein structure. (see attached: top is a ~1 Ang/pixel down-sampled version of the image below).

All that proves is that assigning each 1x1x1 voxel a separate density value is a very
inefficient use of information.  Adjacent voxels are not independent, and no possible
assignment of values will get around the inherent blockiness of the representation.

I know!  Let's instead of assigning a magnitude per voxel, let's assign a magnitude per
something-resolution-sensitive, like a sin wave.   Then for each hkl measurement we get
one sin wave term.   Add up all the sine waves and what do you get?  Ta da.  A nice map.
 
> Or, if we wanted smaller voxels still, let's say by half, we would have to reduce the bit depth to 2 bits. But this would still only yield half-Angstrom voxels, each with only four possible electron density values.
> 
> Is this comparison apt? Off the cuff, I cannot see how a 50 x 50 pixel image corresponds at all to the way our maps look, especially at around 1 Ang resolution. Please, if you can shoot down the analogy, do.

Aren't Fourier series marvelous?

 
> Assuming that it is apt, however: is this a possible way to see the power of all of our Bayesian modelling? Could one use our modelling tools on such a grainy picture and arrive at similar results?
>
> Are our data sets really this poor in information, and we just model the heck out of them, as perhaps evidenced by our scarily low data:parameters ratios?
> 
> My underlying motivation in this thought experiment is to illustrate the richness in information (and poorness of modelling) that one achieves in fluorescence microscopic imaging. If crystallography is any measure of the power of modelling, one could really go to town on some of these terabyte 5D functional data sets we see around here at Janelia (and on YouTube).
> 
> What do you think?
> 
> Jacob Keller
> 
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> Jacob Pearson Keller
> Research Scientist / Looger Lab
> HHMI Janelia Research Campus
> 19700 Helix Dr, Ashburn, VA 20147
> (571)209-4000 x3159
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> 

-- 
Ethan A Merritt, Dept of Biochemistry
Biomolecular Structure Center,  K-428 Health Sciences Bldg
MS 357742,   University of Washington, Seattle 98195-7742