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
> 
> +++++++++++++++++++++++++++++++++++++++++++++++++
> Jacob Pearson Keller
> Research Scientist / Looger Lab
> HHMI Janelia Research Campus
> 19700 Helix Dr, Ashburn, VA 20147
> (571)209-4000 x3159
> +++++++++++++++++++++++++++++++++++++++++++++++++
> 

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

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