[3dem] what is the ideal B factor?

[Alexis Rohou]

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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