[3dem] what is the ideal B factor?

[Alexis Rohou]

Hi Arjen,

Thanks for chiming in!

I agree with your description of the effect and importance of secondary
structure and other shape information on the amplitude spectrum. However,
your estimate that Wilson stats may only apply beyond 3 Å is way beyond
anything I've heard or read before. Let's take an example. I know not all
proteins are beta-gal, but take a look at Vilas et al (2020), which I cited
earlier, Figure 2. In that simulated example, the radial amplitude spectrum
of randomly-distributed atoms looks (to me) indistinguishable from the
spectrum of the simulated structure from about 0.15 Å^-1 (~6.7 Å), and both
look pretty linear in this log plot. Do you think this is atypical? Or
unrealistic perhaps?

Your point regarding the effect of solvent on the Guinier plot is well
taken, but as you point out this is a significant effect in the low
resolutions only.  In my case, I only measured this "B_ideal" number from
the Wilson stats regime, somewhere around 3 Å resolution (but this was a
Matlab simulation, so I could have measured the slope anywhere really).

To spur you on to more comments, here's a question: you clearly have
thought a lot about map sharpening and the estimation of "total" B factor
from the radial amplitude profile of experimental 3D reconstructions. Now,
what about the B factors estimated from ResLog plots? Would you agree with
me that, reading Rosenthal & Henderson (2003), one might expect the two
types of B factors to be related? Looking at figure 1 from that paper... as
the noise floor drops thanks to averaging, the achieved resolution (the
point where signal amplitudes hit the noise floor) moves to higher
frequencies, and that's what ResLog plots track. Therefore, shouldn't the
amplitude spectrum of the protein itself define the best possible
ResLog-plot-estimated B factor?

Cheers,
Alexis


On Thu, Aug 27, 2020 at 7:49 AM Arjen Jakobi - TNW <A.Jakobi at tudelft.nl>
wrote:

> Hi Alexis,
>
> You bring up an interesting point.
>
> My understanding is that Wilson statistics assumes (and is strictly valid
> only for) independent and uniformly distributed (= random) atoms. This is
> why I think that Wilson statistics as derived in the original paper are
> primarily valid in the high-resolution (better than 3Å) part of the
> Wilson/Guinier plot.
>
> At lower resolution, in particular at those spatial frequencies
> corresponding to repetitive features in real-space, i.e. the regular path
> proteins (secondary structure), and nucleic acids (base stacking) follow in
> 3D space & (ordered) solvent give rise to characteristic features in the
> pair-distribution function. This is what you typically see in a “Guinier
> plot”: this plot is in principle a (rotationally averaged) representation
> of the texture of the macromolecule and will contain characteristic
> deviations from the exponential (or linear in log-plot) decay expected from
> Wilson statistics, because at some resolution/spatial frequencies the
> arrangement of atoms is not random. This is true regardless of whether you
> consider X-ray or EM experiments. This is also a reason why I think
> B-factor estimations if performed including these regions are
> systematically off; the R^2 of linear regression will be poor. Once you are
> moving to higher resolution, let us say 3.0 Å and beyond, a protein
> structure can very well be considered as a collection of randomly
> distributed atoms and here Wilson statistics hold and the slope gives the
> B-factor. If you do a fit in this region of a Guinier plot, e.g. for
> high-resolution ApoF structures, the fit will be very good.
>
> Regarding Carlos Oscar’s statement that the fall-off is “arbitrary
> depending on its shape”, I do not necessarily agree but I guess the point
> he is trying to make is that the radially averaged fall-off will be
> modulated by these effect (e.g. secondary structure) and this could be
> considered a “fingerprint” of the protein in question. In practice, when
> radially averaging over the entire structure, the fall-off, including
> deviations from Wilson statistics, will be very similar for most proteins
> unless they are e.g. all-alpha, all-beta or contain significant amount of
> nucleic acids as e.g. ribosomes.
>
> This aside: Could it be that the difference you observe comes from the
> fact that in your simulated model you do not account for solvent? If you
> make a thought experiment and place your protein in a “vacuum” then this
> would lead to an overestimation of “contrast” at the molecule surface
> compared to the situation where you have solvent. Taking this to your
> simulated structure factor, the calculated structure factor would be
> expected to be systematically larger than the observed structure factor
> amplitude in regions where in the real situation bulk solvent is having
> noticeable effect (e.g. 5Å and below).
>
> If in your case you have derived your B_ideal from a fit in this region
> (e.g. 20 – 4 Å), than the calculated fall-off would probably be steeper
> than the observed amplitude fall-off. If you have fit in the
> high-resolution region then this should have no effect.
>
> Not sure if it helps.
>
> Best,
>
> Arjen
>
>
>
>
>
>
>
> *From: *3dem <3dem-bounces at ncmir.ucsd.edu> on behalf of Alexis Rohou <
> a.rohou at gmail.com>
> *Date: *Thursday, 27 August 2020 at 07:06
> *To: *Carlos Oscar Sorzano <coss at cnb.csic.es>
> *Cc: *3dem <3dem at ncmir.ucsd.edu>
> *Subject: *Re: [3dem] what is the ideal B factor?
>
>
>
> Thank you Carlos Oscar for summarizing your work so succinctly!
>
>
>
> I just would like to pick up on your concluding sentence:
>
>
>
> 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.
>
>
>
> If your point of view is correct, this implies that ResLog plots and the
> resulting B factors should not be compared to each other if they were
> obtained from images of different proteins. This would be quite a departure
> from the field's consensus.
>
>
>
> Cheers,
>
> Alexis
>
>
>
>
>
> On Wed, Aug 26, 2020 at 12:32 AM Carlos Oscar Sorzano <coss at cnb.csic.es>
> wrote:
>
> 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!VErnNytZoocKVvnYY9o4V3zkB9BrMH2GixOlZ1kzKPIfsQtbkybeHTeIx77Vmf5Igw$ 
> <https://urldefense.com/v3/__https:/journals.iucr.org/d/issues/2018/06/00/ic5102/ic5102.pdf__;!!Mih3wA!Rn8ptfdW3i-ZJAsyJRtf84MeTMX0gbKZxAVhSD_WbmT6PxiAn3RA49bpffFudf71Mg$>)
> provides some clue based on the maximum continuity of the isosurface.
>
> Finally, we found that a combination of DeepRes (E. Ramírez-Aportela, J.
> Mota, P. Conesa, J.M. Carazo, C.O.S. Sorzano. DeepRes: A New Deep Learning
> and aspect-based Local Resolution Method for Electron Microscopy Maps .
> IUCR J 6: 1054-1063 (2019)) and BlocRes (
> https://urldefense.com/v3/__https://www.sciencedirect.com/science/article/pii/S1047847713002086__;!!Mih3wA!VErnNytZoocKVvnYY9o4V3zkB9BrMH2GixOlZ1kzKPIfsQtbkybeHTeIx76JCBwqBA$ 
> <https://urldefense.com/v3/__https:/www.sciencedirect.com/science/article/pii/S1047847713002086__;!!Mih3wA!Rn8ptfdW3i-ZJAsyJRtf84MeTMX0gbKZxAVhSD_WbmT6PxiAn3RA49bpffHaOthRoQ$>)
> 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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