B-factors are definitely a measure of uncertainty in variance (square) units. The crystal lattice has multiple occurrence of the atoms that are equivalent by crystal symmetry. They will have the same fractional coordinates within the uncertainty of their position relative to the crystal lattice orientation definition. B-factors are the measure of this uncertainty (variance) in somewhat unusual units (Angstrom squared / (8*pi*pi)). The fact that you can directly measure uncertainty by observing the width of the profile of the atomic distribution (shape of the uncertainty function) does not negate that this function represents uncertainty of atomic position relative to the crystal lattice.

As a comment: uncertainty of the centroid of the uncertainty distribution is a second order or recursive uncertainty. As this centroid is deposited as atomic coordinates in pdb files, its uncertainty is a separate subject from the uncertainty (variation) of the atom position in the crystal lattice. Unfortunately theories of uncertainty estimates of uncertainty estimates is more complex and for this reason crystallographers rarely deal productively with uncertainty of the x,y,z coordinates deposited.

A second comment: the B-factor really represents the sum of two uncertainties. One is the uncertainty of atom positions in the crystal lattice. The second is our experimental uncertainty about the knowledge of atom position. The first one has a physical interpretation. The second represents our data and analysis, e.g. phasing.

For these reasons, saying that the B-factor represents uncertainty estimates is very productive because it is all about uncertainty. Independent uncertainties are convolved with each other to produce a final uncertainty function. In terms of the width squared of that function, it represents the sum of the widths squared of the contributors. In fact this observation is behind the Central Limit Theorem.

Zbyszek

On 2021-05-28 19:05, James Holton wrote:
I feel I should point out here that B-factors are NOT a measure of
uncertainty.  They are a width.  This width itself may be uncertain,
as may be the position of the center of the peak, but just because
your peak is broad doesn't mean you don't know where the middle of it
is.

As for why leave the mean variation squared?  I expect it is because
it is supposed to be proportional to temperature. Hence the name
"temperature factor".

-James Holton
MAD Scientist

On 5/27/2021 11:09 AM, Gergely Katona wrote:

Dear Jonathan,

In 1D sd may be intuitive, but in 3D it is not so much. The square
root of a symmetric covariance matrix is not universally defined and
it is not intuitive to me.

Best wishes,

Gergely

Gergely Katona, Professor, Chairman of the Chemistry Program Council


Department of Chemistry and Molecular Biology, University of
Gothenburg

Box 462, 40530 Göteborg, Sweden

Tel: +46-31-786-3959 / M: +46-70-912-3309 / Fax: +46-31-786-3910

Web: http://katonalab.eu, Email: [email protected]

From: CCP4 bulletin board <[email protected]> On Behalf Of
Hughes, Jonathan
Sent: 27 May, 2021 18:53
To: [email protected]
Subject: [ccp4bb] AW: [ccp4bb] (R)MS

hey!

thank y'all for the informative (and swift!) answers! but, if the B
factor (as defined) appears in a mathematical formulation, that
doesn't make it an "appropriate" parameter for mobility/uncertainty.
wouldn't √B be better, in the same way that, for humans, standard
deviation (RMSD) is a more appropriate parameter of variability than
variance? or am i missing something?

cheers

j

Von: Ian Tickle <[email protected]>
Gesendet: Donnerstag, 27. Mai 2021 18:32
An: Hughes, Jonathan <[email protected]>
Cc: [email protected]
Betreff: Re: [ccp4bb] (R)MS

Hi Jonathan

It's historical, it's simply how it appears in the expression for
the Debye-Waller factor, i.e. exp(-B sin^2(theta)/lambda^2).  So it
must have the same units as lambda^2.

Cheers

-- Ian

On Thu, 27 May 2021 at 13:25, Hughes, Jonathan
<[email protected]> wrote:

o yes! but maybe the crystal people could explain to me why the B
factor is the variance (with units of Ų) rather than the standard
deviation (i.e. RMS, with units of Å) when, to my simple mind, the
latter would seem be the more appropriate description of variability
in space?

cheers

jon

Von: CCP4 bulletin board <[email protected]> Im Auftrag von
Pearce, N.M. (Nick)
Gesendet: Donnerstag, 27. Mai 2021 12:38
An: [email protected]
Betreff: Re: [ccp4bb] Analysis of NMR ensembles

If you want something comparable to B-factors don’t forget to put
the MSF in the B-factor column, not the RMSF. Will change the
scaling of the tube radius considerably!

Nick

On 27 May 2021, at 11:16, Harry Powell - CCP4BB
<[email protected]> wrote:

Cool…

Purely for visualisation this does look like the approved CCP4 way -


<Screen Shot 2021-05-27 at 10.13.59.png>

Harry

On 27 May 2021, at 10:01, Stuart McNicholas
<[email protected]> wrote:

Drawing style (right menu in display table) -> Worm scaled by ->
Worm
scaled by NMR variability

in ccp4mg?

This changes the size of the worm but not the colour.

On Thu, 27 May 2021 at 09:56, Harry Powell - CCP4BB
<[email protected]> wrote:

Anyway, thanks to all those who answered my original question -
especially

Tristan: Chimerax (+ his attached script)
Michal, Scott: Theseus
(https://theobald.brandeis.edu/theseus/ [1])
Bernhard: Molmol (https://pubmed.ncbi.nlm.nih.gov/8744573/
[2] )
Rasmus CYRANGE (http://www.bpc.uni-frankfurt.de/cyrange.html
[3]) and https://www.ccpn.ac.uk/ [4] (of course…)
Andrew (uwmn - not sure if this is buildable on a modern box)
Smita: PyMol (not sure if I’m allowed to say that on
ccp4bb…)

or I could script it and use Gesamt or Superpose for the
superposition if I wanted to stay in the ccp4 universe and had the
time to spare ;-)

Harry


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