Hi Ardan. The requirement that the particles be identical is also a requirement for having a valid 3-D reconstruction in the first place. That is, the actual equivalence would be that the average radial power spectrum of randomly oriented raw particles, with proper background subtraction and CTF correction, should match the solution scattering curve very closely. If the particles are homogeneous and represent a single 3-D structure, then the radial power spectrum of the structure should also match the solution scattering curve. We did a comparison like this as part of our first GroEL publication over a decade ago, and Joachim Frank published something at a similar time with Ribosomes. EMAN2's standard approach includes making an estimate of the radial structure factor from the raw data (only an estimate), which is used as part of the CTF correction scheme. In short, I think this is all of the issues involved:

1) scattering differences between electrons/xrays (probably a minor effect at the resolutions SAXS usually covers)
2) particle density low enough in both cases that N-body effects don't come into play (ie - particle-particle distance isn't a factor)
3) if averaging raw 2-D particles, background subtraction and CTF correction
4) preferred orientation can play a role when averaging 2-D particles rather than the 3-D reconstruction
5) heterogeneity - if the same in both experiments, isn't a problem if averaging 2-D particles, but is a problem if looking at the 3-D map

However, I'm a bit confused about the overall point of this work. Yes, a decade ago a few people used SAXS and compared to cryo-EM as a self-consistency check of the CTF correction at the time. However, using SAXS routinely with cryo-EM for some sort of validation, in my opinion, is a massive waste of time. SAXS is extremely useful when looking at changes over time, since the specimen isn't frozen, and isn't being damaged as rapidly as in cryo-EM. However, as a tool for determining structures, it is trivially shown that any SAXS curve can be matched by an infinite number of different density distributions. While I agree that with some additional assumptions one can often come up with a reasonable protein envelope, it is far from foolproof. In other words I don't really see it adding anything useful when cryo-EM has been done successfully, unless dynamics is involved.

On Oct 31, 2013, at 5:59 AM, Ardan Patwardhan <[log in to unmask]> wrote:

> Dear Sjors and Richard 
> 
> Many thanks for your replies.
> 
> There is still one thing that I am unclear about: 
> To calculate the radially averaged power spectrum
> 1) Take the 3D density map and Fourier transform (FT) it
> 2) Convert 3D FT to 3D power spectrum (PS)
> 3) Radially average 3D PS to obtain 1D PS
> 
> According to what I understand of SAXS
> (http://en.wikipedia.org/wiki/Biological_small-angle_scattering but also
> other sources): 4) the SAX scattering 1D curve is the FT of the "distance
> distribution function". I understand that the FT of the "distance
> distribution function"
> (http://en.wikipedia.org/wiki/Radial_distribution_function#The_structure_fac
> tor) under the assumption that "all particles are identical" is a structure
> factor function. 
> 
> So for biological macromolecules, if we account for  a) CTF, MTF effects etc
> and b) the fact that the scattering cross sections will be different for EM
> and X-rays, should we expect (3) and (4) to be the same? Is the "all
> particles are identical" assumption valid even for ribosomes and viruses
> where protein + DNA/RNA is present?
> 
> Many thanks best wishes
> 
> -----Original Message-----
> From: RICHARD HENDERSON [mailto:[log in to unmask]] 
> Sent: 30 October 2013 19:20
> To: Ardan Patwardhan; [log in to unmask]
> Subject: Re: [ccpem] Correspondance between rotationally averaged power
> spectrum and SAXS curve
> 
> Ardan,
> 
> Not identical, but definitely similar.  The scattering cross-sections are
> different for X-rays and electrons.  Electron scattering is also very
> sensitive to bonding (electron cloud radius) whereas X-ray scattering is
> proportional to Z with a simpler form-factor.
> 
> However, to the extent that C = N = O and hydrogen scatters less for
> electrons and X-rays, they are similar.
> 
> Richard
> 
> On 30/10/2013 18:09, Ardan Patwardhan wrote:
>> Dear all
>> 
>> I was wondering if anyone could point me to a reference on the 
>> following
>> question:
>> 
>> Is the rotationally averaged power spectrum of a 3DEM reconstruction 
>> (in the absence of CTF effects etc) expected to be identical to a 
>> small angle X-ray scattering (SAXS) curve?
>> 
>> Many thanks and best wishes
>> 
>> Ardan Patwardhan
>> 
>> European Bioinformatics Institute (EMBL-EBI) European Molecular 
>> Biology Laboratory
>> 
>> Wellcome Trust Genome Campus
>> 
>> Hinxton, Cambridge CB10 1SD, UK
>> Tel: +44 1223 492649
>> 
> 
> --
> Richard Henderson
> MRC Laboratory of Molecular Biology
> Francis Crick Avenue
> Cambridge Biomedical Campus
> Cambridge CB2 0QH, U.K.
> tel: +44-(0)1223-267065
> fax: +44-(0)1223-268300
> web: http://www2.mrc-lmb.cam.ac.uk/groups/rh15/
> 
> 
> -----
> 
> From Sjors:
> Hi Ardan,
> 
> No. Differences may for example arise from different MTFs of the detectors
> used in the EM and X-ray experiments (or other envelope-affecting factors).
> 
> Sjors
> 
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----------------------------------------------------------------------------
Steven Ludtke, Ph.D.
Professor, Dept of Biochemistry and Mol. Biol.         (www.bcm.edu/biochem)
Co-Director National Center For Macromolecular Imaging        (ncmi.bcm.edu)
Co-Director CIBR Center                          (www.bcm.edu/research/cibr)
Baylor College of Medicine                             
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