For the first time in my life I fully agree with what Steven says here!  Does that mean we are both getting old Steven or is that just me?

Have fun!

Marin

On 02/09/2018 23:38, Ludtke, Steven J wrote:

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".

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.

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.

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...

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.

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Charles C. Bell Jr., Professor of Structural Biology
Dept. of Biochemistry and Molecular Biology                      (www.bcm.edu/biochem)
Co-Director CIBR Center                                    (www.bcm.edu/research/cibr)

Dear Marin,

quoting from the 1997 paper:

"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."

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?

Cheers,
Dimitry

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