[3dem] what is the ideal B factor?

[Carlos Oscar Sorzano]

Dear Alexis and all,

as a very condensed summary of what Jose Luis Vilas and us showed in the 
paper you have mentioned below is that:

1. the Fourier spectrum of a single atom is expected to decay in 
amplitude with frequency (the only way it can be flat is that it is 
infinitely thin). This is well known and comes from the electron atomic 
scattering factors.

2. the Fourier spectrum of a collection of atoms is mostly determined by 
the shape of that collection, more than on the specific nature of the 
atoms being involved (we performed extremely harsh modifications to the 
atoms and the decay did not change significantly).

3. the reasons normally argued to make the spectrum flat, do not apply 
to macromolecules and the reason why B-factor sharpening produces "nice" 
structures is mostly a visualization reason (higher amplitudes at high 
frequencies result in sharper edges whose isosurfaces are easier to 
track and fit with an atomic model).

Because of 2, the decay of the radial average of the Fourier transform 
of a macromolecule cannot be expected to follow any particular shape 
(for instance, a straight line) a priori, that we can estimate its slope 
(also a priori) and force our 3D reconstruction to follow that slope. In 
that regard the question of what is the expected slope is ill-posed. 
 From my point of view, the amplitude correction is much more meaningful 
when performed in the spirit of LocScale of Jakobi and Sachse. You fit 
an atomic model to the map, then convert it into a map, estimate its 
decay and force the map to follow this decay. In this way, the shape of 
the collection of atoms (and their nature) is explicitly taken into 
account. This process can be performed iteratively (with the corrected 
map, you may refine the atomic model, refine the map amplitudes again, 
...). I also like the idea that this process is performed locally.

If we do not want to wait for the atomic model to make the amplitude 
correction, we have devised an alternative based on the local resolution 
(E. Ramirez-Aportela, J.L.Vilas, A. Glukhova, R. Melero, P. Conesa, M. 
Martinez, D. Maluenda, J. Mota, A. Jimenez, J. Vargas, R. Marabini, P.M. 
Sexton, J.M. Carazo, C.O.S. Sorzano. Automatic local resolution-based 
sharpening of cryo-EM maps. Bioinformatics 36: 765-772 (2020)). There is 
no guarantee that it will follow the correct decay, but in practice we 
have observed that it normally approximates the correct decay quite 
closely (there are some examples of this in the paper).

The procedure above of local correction based on local resolution is 
local and it does not require an the atomic model. If we still want to 
do a global correction without an atomic model, procedures like the one 
of phenix 
(https://urldefense.com/v3/__https://journals.iucr.org/d/issues/2018/06/00/ic5102/ic5102.pdf__;!!Mih3wA!WYgLjuUA8Pp3HGHuvN7UtZho6s-8zxd0EX2-_8VsHujZ24uc8ywVLRwGdT5ioYcv2g$ ) 
provides some clue based on the maximum continuity of the isosurface.

Finally, we found that a combination of DeepRes (E. Ramírez-Aportela, J. 
Mota, P. Conesa, J.M. Carazo, C.O.S. Sorzano. DeepRes: A New Deep 
Learning and aspect-based Local Resolution Method for Electron 
Microscopy Maps . IUCR J 6: 1054-1063 (2019)) and BlocRes 
(https://urldefense.com/v3/__https://www.sciencedirect.com/science/article/pii/S1047847713002086__;!!Mih3wA!WYgLjuUA8Pp3HGHuvN7UtZho6s-8zxd0EX2-_8VsHujZ24uc8ywVLRwGdT5wXQqwEg$ ) 
could help to find a B-factor that does not result in overfitting.

In conclusion, from my point of view, there is not an optimal decay 
valid for all proteins, but it depends on each specific protein. And the 
shape of the decay is not a straight line, but arbitrary depending on 
its shape.

I hope these reflections helped a bit.

Cheers, Carlos Oscar

On 8/26/20 7:05 AM, Alexis Rohou wrote:
> Dear colleagues,
>
> I hope you may be able to help me get my head around something.
>
> When considering the radially-averaged amplitudes of an ideal 3D 
> protein structure, the expectation (as laid out in Fig1 of Rosenthal & 
> Henderson, 2003 (PMID: 14568533), among others) is that in the 
> Wilson-statistics regime (q > 0.1 Å^-1, let’s say), amplitudes will 
> decay in a Gaussian manner, or linearly when plotted on a log scale 
> against q^2, reflecting the decay of structure factors.
>
> This expectation is certainly met when simulating maps from PDB files, 
> as described nicely for example by Carlos Oscar Sorzano and colleagues 
> recently (Vilas et al., 2020, PMID: 31911170). Let’s call the rate of 
> decay of this ideal curve B_ideal, the “ideal” B factor.
>
> Assuming for a moment that noise has a flat spectrum (reasonable so 
> long as shot noise is dominant), one may follow in Rosenthal & 
> Henderson’s footsteps and draw a horizontal line on our plot to 
> represent the noise floor. As more averaging is carried out, the noise 
> floor is lowered relative to our protein’s amplitude profile. As more 
> particles are averaged (without error, let’s say) the intersection 
> between the protein’s ideal radial amplitude profile and the noise 
> floor moves to higher and higher frequencies.
>
> This is the basis for the so-called ResLog plots, where one charts the 
> resolution as a function of the number of averaged particles. The 
> slope of the ResLog plot is related to the slope of the radial 
> amplitude profile of the protein. Assuming no additional sources of 
> errors (i.e. ideal instrument and no processing errors), B_ideal (the 
> slope of the ideal protein amplitude profile) can be computed from the 
> slope of the ResLog plot via B_ideal = 2.0/slope.
>
> Now, to my question. By looking at the slope of a schematic Guinier 
> plot generated using Wilson statistics and atomic scattering factors 
> for electrons, I estimated a B_ideal of approximately 50 Å^2 (decay of 
> ~ 1.37 natural log in amplitude over 0.1 Å^-2). The problem is that 
> recent high-resolution studies have reported ResLog-estimated B 
> factors of 32.5 Å^2 (Nakane et al., 2020) and 36 Å^2 (Yip et al., 
> 2020), leading me to wonder what is wrong in the above model.
>
> I see several possibilities:
>
> (1)   B_ideal is actually significantly less than 50 Å^2. This would 
> be consistent with the empirical observation that “flattening” maps’ 
> amplitude spectrum (i.e. assuming B-ideal = 0 Å^2) gives very nice 
> maps. Either:
>
>     a.     I mis-estimated B_ideal when reading the simulated
>     amplitude spectrum plot. Has anyone done this (i.e. fit a B factor
>     to a simulated map’s amplitude spectrum, or to a simulated
>     spectrum)? What did you find?
>
>     b.     The simulations using atomic scattering factors and Wilson
>     statistics do not correctly capture the actual amplitude profile
>     of proteins, which is actually much flatter than the atomic
>     scattering factors suggest.
>
> (2)   B_ideal actually is ~ 50 Å^2, but the assumption of a flat noise 
> spectrum is wrong. I guess that if the true noise spectrum were also 
> decaying at a function of q^2, this would cause the ResLog plot to 
> report “too small” a B factor
>
> What do you think?
>
> Cheers,
> Alexis
>
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