The meaning of the term "e/A^3" as used in Coot has nothing to do
with the charge of an electron. The intention of its authors is to
indicate that the value being represented by the map is the density of
electrons. It is the number of electrons per cubic Anstrom at that
point in space.
I, generally, find it useless to say that a number is a density
(units of "per volume") without saying density of what. This is not a
density of charge, not a density of mass, not a density of rabbits. It
is a density of electrons.
Now the calculations are pretty simple. When the refinement program
produces a model one of its parameters is a scale factor which relates
the relative values of the observed intensity to those calculated from
the model. This scale factor, which is usually not written out in an
obvious place since its value is not very interesting, is just a simple
number, not a function of resolution or anything like that. It's value
is also sensitive to a number of assumptions (like you are modeling
everything in your unit cell) so it is not particularly reliable. This
means you shouldn't take the density values of your map terribly
seriously. The differences in density from place to place is much more
reliable.
This has led to the practice of converting the density values to
normalized values. Here you calculate the rms value of the points in
the map and divide all the density points by that number. "rms" is
simply what is says "Root of the Mean of the Squares". Again this is
just a number, nothing fancy.
The only trick is deciding what region of space to sum. The crystal
is composed of regions which are occupied by ordered molecules (One
hopes this includs the molecule you are interested in!) and regions of
bulk solvent. You could reasonably conclude that the rms's of these
regions should be considered different properties of the map. I'm not
aware of any software that actually tries to calculate an rms for just
the region of the map that contains ordered structure.
What was done in the past was to simply calculate the sum over the
region of the map presented to the program, and this was usually a
rectangular box inscribed over the molecule. The corners of that box
would cover some amount of bulk solvent (and symmetry related molecules)
which depends on a lot of factors which shouldn't be affecting your
choice of contour levels. This method is inconsistent and causes confusion.
The conventional method today is to calculate the rms by summing
over the entire asymmetric unit. This, at least, creates consistency,
and can be calculated from the Fourier coefficients making it an easy
number to come up with. Many programs now use this method, including
Coot (As I understand). It does have the drawback that a crystal with a
large amount of solvent will have a lower rms than one that is very dry,
even if the variation within the ordered structure is the same. You
need to be aware of this when interpreting maps whose contours are based
on the rms. Showing the contour level both as rms and electrons/A^3 is
an attempt to provide a fuller description.
Ian is certainly right -- the rms of a function is not a "sigma".
This confusion is a problem that is endemic in our field! "sigma" is
shorthand for standard deviation which is a measure of the uncertainty
in our knowledge about a value. The rms is not a measure in any way of
uncertainty -- It is simply a description of how variable the values of
the function are. James Holton has written a very nice paper on this
topic, but I don't have the reference on hand.
The excuse for this confabulation is that people believe, in a Fo-Fc
style map, most of the values should represent factors other than errors
in the structural model and therefore one can estimate the uncertainty
of the map by calculating the rms over the map. This assumption is
highly questionable and unreliable. The major problem then arises when
the same logic is extrapolated to a 2Fo-Fc style map. In these maps the
variability in the ordered region of the crystal is all "signal" so
calculating an rms really has nothing to do with uncertainty.
Describing your contours or peaks in an Fo-Fc style map by rms can
sort-of, kind-of, be justified, but it makes no sense for a 2Fo-Fc style
map. If I want to be really serious about deciding if a peak in a map
may be missing atoms, I will leave some known atoms out of the model and
see how the heights of their difference peaks compare to the heights of
the mysterious peaks. This method is fairly insensitive to the
systematic problems that affect both rms and electrons/A^2.
Dale Tronrud
On 4/19/2018 8:30 AM, Ian Tickle wrote:
>
> Hi Mohamed
>
> The RMSD of the electron density (or difference density) is calculated
> by the FFT program using the standard equation that I referenced. I
> would guess that what you see in Coot is copied either from the map
> header or the FFT log file.
>
> I'm not clear what you mean when you refer to 'the e/A^3'. The RMSD (as
> does the electron density from which it's calculated) consists of a pure
> number and a unit of measure, e.g. '1.234 A^-3' (why it's not '1.234
> eA^-3' we won't go into here: suffice it to say that 'e' is a unit of
> charge and 'electron density' is not the same as 'charge density', while
> an electron, or 'e-', is not a unit of measure at all, it's a
> sub-atomic particle: see Wikipedia/Non-SI_units_mentioned_in_the_SI and
> Wikipedia/electron). The relation between 'RMSD' and 'A^-3' is simply
> that the latter is the unit of the former, exactly analogous to the
> relation between 'distance' and 'metre'.
>
> Yes '3 sigma' in this context is not correct: it's '3 RMSD', and indeed
> COOT itself uses the latter terminology (which you see any time you
> change the contour level), so I'm not clear where you got that from.
> The uncertainty ('sigma') of the density does have a value, though this
> is not estimated by any of the standard programs AFAIK. However one
> thing is certain: it's unlikely to equal the RMSD!
>
> Cheers
>
> -- Ian
>
>
> On 19 April 2018 at 15:25, Mohamed Ibrahim <[log in to unmask]
> <mailto:[log in to unmask]>> wrote:
>
> Hi Ian,
>
> Thanks a lot for your answers. They are very informative. I am
> afraid that I was looking in the wrong direction to figure out what
> I seek. So, I am reforming my question; I am trying to figure out
> whether the relation between the RMSD and the e/A^3 is linear or
> not. Therefore, I was looking for how does COOT calculate the RMSD,
> hoping to find the relation between RMSD and e/A^3. If you could
> refer to me a reference that is related to this, it will be great.
> One more question, you mentioned that " it shouldn't be called sigma
> because it's not an uncertainty ", so when we say, for example, this
> map is contoured at 3 sigma, this is a wrong statement?
>
> Cheers,
>
> On Thu, Apr 19, 2018 at 2:57 PM, Ian Tickle <[log in to unmask]
> <mailto:[log in to unmask]>> wrote:
>
>
> Hi, first maps are produced by Refmac, not Coot, and second it
> shouldn't be called sigma because it's not an uncertainty, it's
> a root-mean-square deviation from the mean. The equation for
> the RMSD can be found in any basic text on statistics, e.g. just
> type 'RMSD' in Wikipedia.
>
> Cheers
>
> -- Ian
>
>
> On 19 April 2018 at 13:20, Mohamed Ibrahim
> <[log in to unmask]
> <mailto:[log in to unmask]>> wrote:
>
> Dear COOT users,
>
> Do you know how to extract the equations that COOT uses for
> generating the maps and calculating the sigma values?
>
> Best regards,
> Mohamed
>
> --
>
> --
> /*
>
> ----------------------------------*/
> /*Mohamed Ibrahim
>
> *//* *//*
>
> */
> /*Humboldt University
>
> */
> /*Berlin, Germany
>
>
> */
> /*Tel: +49 30 209347931
>
> */
>
>
>
>
>
> --
>
> --
> /*
>
> ----------------------------------*/
> /*Mohamed Ibrahim
>
> *//* *//*
>
> */
> /*Humboldt University
>
> */
> /*Berlin, Germany
>
>
> */
> /*Tel: +49 30 209347931
>
> */
>
>
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