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I confess that the jet lag remaining from my China trip impacted the readability of my message.
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<div class="">When I said 'All 3 of these should have the same "resolution" which should extend to 3 Å' I was trying to say that the PDB -> density conversion should have a low-pass filter at the lower sampling Nyquist (3 Å) for all 3 cases. My entire point
was exactly what you are saying about sampling, but the problem doesn't only exist AT Nyquist. Partial ambiguity extends to 1/2 Nyquist. My PDB suggestion was targeted at people whose brains aren't deeply embedded in Fourier space, to give them a visual example
to better understand the issue.
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<font face="Courier" class=""><span style="font-size: 14px;" class="">--------------------------------------------------------------------------------------<br class="">
Steven Ludtke, Ph.D. <<a href="mailto:sludtke@bcm.edu" class="">sludtke@bcm.edu</a>> Baylor College of Medicine <br class="">
Charles C. Bell Jr., Professor of Structural Biology<br class="">
Dept. of Biochemistry and Molecular Biology (<a href="http://www.bcm.edu/biochem" class="">www.bcm.edu/biochem</a>)<br class="">
Academic Director, CryoEM Core (<a href="http://cryoem.bcm.edu" class="">cryoem.bcm.edu</a>)<br class="">
Co-Director CIBR Center (<a href="http://www.bcm.edu/research/cibr" class="">www.bcm.edu/research/cibr</a>)<br class="">
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<div class="">On Sep 3, 2018, at 10:06 AM, Penczek, Pawel A <<a href="mailto:Pawel.A.Penczek@uth.tmc.edu" class="">Pawel.A.Penczek@uth.tmc.edu</a>> wrote:</div>
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As far as I can tell your postulated numerical experiment is not applicable to the problem discussed. Sampling theorem states that only band-limited signals can be properly sampled while X-ray model atomic coordinates have “infinite” resolution. Properly, in
such a case one would have to apply low-pass filtration before sampling and then the three structures you suggest would be identical.</div>
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A minor remark, sampling cannot be done at exact Nyquist frequency as one would get ambiguities you describe and this is what sampling theory states. </div>
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Regards,</div>
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Pawel Penczek</div>
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UT Houston</div>
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PS I resisted for hours and then caved. </div>
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<div class="moz-cite-prefix">On 02/09/2018 23:38, Ludtke, Steven J wrote:<br class="">
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<blockquote type="cite" cite="mid:51D4D9E7-4323-49FE-9FD6-5FDBEB755D96@bcm.edu" class="">
Ok, can't resist chiming in any more.
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<div class="">The basis of this general consensus, IS mathematical. The specific threshold (2/3 Nyquist) which pretty much everyone agreed on back in the 90's, is empirical. The reason (at least the one I've always used) is that you do not have a complete representation
of data at Nyquist. Nyquist corresponds to a +1/-1/+1/-1 sequence in adjacent pixels, if you phase shift this pattern by 90 degrees, it becomes 0,0,0,0. At 1/2 Nqyuist, you can still represent arbitrary sinusoidal waveforms with arbitrary phase. Between 1/2
Nyquist and Nyquist, you get patterns which are dependent on position within the "box". </div>
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<div class="">When you do X-ray crystallography, you are measuring the Fourier intensities experimentally, and provided that you can get the correct phase, you can then oversample the real-space representation of the crystal pattern (with specified phases),
such that it is fully sampled. In CryoEM, we image in real-space, and between 1/2 Nyquist and Nyquist the phases and amplitudes are convolved in spatially dependent ways, such that information is actually lost, and over-sampling cannot recover the information
fully.</div>
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<div class="">This is NOT saying you cannot achieve FSC curves that remain close to 1 all the way to Nyquist. You can, of course, but the resulting reconstruction will not be properly sampled and features will be distorted from what you would see if you had
the same structure measured with 2x finer sampling.</div>
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<div class="">Try this little experiment. Take a PDB model and convert to electron density with 1.5 Å/voxel sampling, repeat the same process, but translate the PDB by 0.75 Å in x/y/z before doing the conversion. Finally, generate a PDB with 0.75 Å/pixel sampling.
All 3 of these should have the same "resolution" which should extend to 3 Å. Look at the 3-maps. Overlay the PDB. Take a look at the sidechains. Ostensibly, these 3 maps are all "identical", but you will see that they are definitely not...</div>
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<div class="">PS - please note that I am NOT saying that the FFT is a lossy process. It is not, of course. Information is exactly preserved by the FFT. The point, is that an arbitrary real-space periodicity requires 4 pixels, not 2 pixels, to unambiguously
represent. </div>
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<font class="" face="Courier"><span class="" style="font-size: 14px;">--------------------------------------------------------------------------------------<br class="">
Steven Ludtke, Ph.D. <<a href="mailto:sludtke@bcm.edu" class="" moz-do-not-send="true">sludtke@bcm.edu</a>> Baylor College of Medicine <br class="">
Charles C. Bell Jr., Professor of Structural Biology<br class="">
Dept. of Biochemistry and Molecular Biology (<a href="https://urldefense.proofpoint.com/v2/url?u=http-3A__www.bcm.edu_biochem&d=DwMFaQ&c=6vgNTiRn9_pqCD9hKx9JgXN1VapJQ8JVoF8oWH1AgfQ&r=vDDf9rsFxPMXm8JgJa6hc4B9V4qKr7wftnDkLIRdshI&m=MdotH8RfcpHobHuOCcrNUbS_qUtp2JwkYj90wEEZDkA&s=i_cXcKE7a3r7av-4pEfAOfDeAwNCHjzzFtLqvz_X_NY&e=" class="" moz-do-not-send="true">www.bcm.edu/biochem</a>)<br class="">
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Co-Director CIBR Center (<a href="https://urldefense.proofpoint.com/v2/url?u=http-3A__www.bcm.edu_research_cibr&d=DwMFaQ&c=6vgNTiRn9_pqCD9hKx9JgXN1VapJQ8JVoF8oWH1AgfQ&r=vDDf9rsFxPMXm8JgJa6hc4B9V4qKr7wftnDkLIRdshI&m=MdotH8RfcpHobHuOCcrNUbS_qUtp2JwkYj90wEEZDkA&s=7y7cpvtbVPNu68lvGjSyySWTWpWVTs36zQc6BFCzt0o&e=" class="" moz-do-not-send="true">www.bcm.edu/research/cibr</a>)<br class="">
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<div class="">On Sep 2, 2018, at 7:27 PM, Dimitry Tegunov <<a href="mailto:tegunov@GMAIL.COM" class="" moz-do-not-send="true">tegunov@GMAIL.COM</a>> wrote:</div>
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<div class="">Dear Marin,<br class="">
<br class="">
quoting from the 1997 paper:<br class="">
<br class="">
"The crossing point between the two curves here lies at about 78% of the Nyquist frequency. The theoretical limit for all processing lies at the Nyquist frequency; yet, considerably before that limit, practical limitations due to the 2D or 3D interpolation
procedures used limit, or at least interfere with, the information transfer through the processing chain. Moreover, since both 3D maps are processed by the same programs, with the same interpolation routines, the same systematic round-off errors may be introduced
in both reconstructions, which the FSC program may see as common “information”. It is thus good practice not to interpret resolution curves at this high end of the resolution range. The sampling of the data at a sampling interval of 5 Å, in our experience,
effectively limits the attainable resolution to ∼15 Å rather than to the theoretical Nyquist limit of 10 Å."<br class="">
<br class="">
You seem to agree that getting close to Nyquist is possible if sloppy real-space interpolation is avoided. As the latter has been the case for at least the past decade, perhaps it is time to let go of an arbitrary rule derived from personal experience rather
than signal processing fundamentals?<br class="">
<br class="">
Cheers,<br class="">
Dimitry<br class="">
<br class="">
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<pre class="moz-signature" cols="72">--
==============================================================
Prof Dr Ir Marin van Heel
Laboratório Nacional de Nanotecnologia - LNNano
CNPEM/LNNano, Campinas, Brazil
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--------------------------------------------------
Emeritus Professor of Cryo-EM Data Processing
Leiden University
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--------------------------------------------------
I receive many emails per day and, although I try,
there is no guarantee that I will actually read each incoming email. </pre>
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