James,
The case you’ve chosen is not a good illustration of the relationship between atomic B and resolution. The problem is that during scaling of Fcalc to Fobs also B-factor difference between the two sets of numbers is minimized. In the simplest form with two constants Koverall and Boverall it looks like this:
sum_to_be_minimized = sum (FOBS**2 - Koverall * FCALC**2 * exp(-1/d**2 * Boverall) )
Then one can include bulk solvent correction, anisotripic scaling, … In PHENIX it gets quite complex.
Hence, almost regardless of the average model B you will always get the same map, because the “B" of the map will reflect the B of the FOBS. When all atomic Bs are equal then they are also equal to average B.
best, dusan
> On 7 Mar 2020, at 01:01, CCP4BB automatic digest system <[log in to unmask]> wrote:
>
>> On Thu, 5 Mar 2020 01:11:33 +0100, James Holton <[log in to unmask]> wrote:
>>
>>> The funny thing is, although we generally regard resolution as a primary
>>> indicator of data quality the appearance of a density map at the classic
>>> "1-sigma" contour has very little to do with resolution, and everything
>>> to do with the B factor.
>>>
>>> Seriously, try it. Take any structure you like, set all the B factors to
>>> 30 with PDBSET, calculate a map with SFALL or phenix.fmodel and have a
>>> look at the density of tyrosine (Tyr) side chains. Even if you
>>> calculate structure factors all the way out to 1.0 A the holes in the
>>> Tyr rings look exactly the same: just barely starting to form. This is
>>> because the structure factors from atoms with B=30 are essentially zero
>>> out at 1.0 A, and adding zeroes does not change the map. You can adjust
>>> the contour level, of course, and solvent content will have some effect
>>> on where the "1-sigma" contour lies, but generally B=30 is the point
>>> where Tyr side chains start to form their holes. Traditionally, this is
>>> attributed to 1.8A resolution, but it is really at B=30. The point
>>> where waters first start to poke out above the 1-sigma contour is at
>>> B=60, despite being generally attributed to d=2.7A.
>>>
>>> Now, of course, if you cut off this B=30 data at 3.5A then the Tyr side
>>> chains become blobs, but that is equivalent to collecting data with the
>>> detector way too far away and losing your high-resolution spots off the
>>> edges. I have seen a few people do that, but not usually for a
>>> published structure. Most people fight very hard for those faint,
>>> barely-existing high-angle spots. But why do we do that if the map is
>>> going to look the same anyway? The reason is because resolution and B
>>> factors are linked.
>>>
>>> Resolution is about separation vs width, and the width of the density
>>> peak from any atom is set by its B factor. Yes, atoms have an intrinsic
>>> width, but it is very quickly washed out by even modest B factors (B >
>>> 10). This is true for both x-ray and electron form factors. To a very
>>> good approximation, the FWHM of C, N and O atoms is given by:
>>> FWHM= sqrt(B*log(2))/pi+0.15
>>>
>>> where "B" is the B factor assigned to the atom and the 0.15 fudge factor
>>> accounts for its intrinsic width when B=0. Now that we know the peak
>>> width, we can start to ask if two peaks are "resolved".
>>>
>>> Start with the classical definition of "resolution" (call it after Airy,
>>> Raleigh, Dawes, or whatever famous person you like), but essentially you
>>> are asking the question: "how close can two peaks be before they merge
>>> into one peak?". For Gaussian peaks this is 0.849*FWHM. Simple enough.
>>> However, when you look at the density of two atoms this far apart you
>>> will see the peak is highly oblong. Yes, the density has one maximum,
>>> but there are clearly two atoms in there. It is also pretty obvious the
>>> long axis of the peak is the line between the two atoms, and if you fit
>>> two round atoms into this peak you recover the distance between them
>>> quite accurately. Are they really not "resolved" if it is so clear
>>> where they are?
>>>
>>> In such cases you usually want to sharpen, as that will make the oblong
>>> blob turn into two resolved peaks. Sharpening reduces the B factor and
>>> therefore FWHM of every atom, making the "resolution" (0.849*FWHM) a
>>> shorter distance. So, we have improved resolution with sharpening! Why
>>> don't we always do this? Well, the reason is because of noise.
>>> Sharpening up-weights the noise of high-order Fourier terms and
>>> therefore degrades the overall signal-to-noise (SNR) of the map. This
>>> is what I believe Colin would call reduced "contrast". Of course, since
>>> we view maps with a threshold (aka contour) a map with SNR=5 will look
>>> almost identical to a map with SNR=500. The "noise floor" is generally
>>> well below the 1-sigma threshold, or even the 0-sigma threshold
>>> (https://doi.org/10.1073/pnas.1302823110). As you turn up the
>>> sharpening you will see blobs split apart and also see new peaks rising
>>> above your map contouring threshold. Are these new peaks real? Or are
>>> they noise? That is the difference between SNR=500 and SNR=5,
>>> respectively. The tricky part of sharpening is knowing when you have
>>> reached the point where you are introducing more noise than signal.
>>> There are some good methods out there, but none of them are perfect.
>>>
>>> What about filtering out the noise? An ideal noise suppression filter
>>> has the same shape as the signal (I found that in Numerical Recipes),
>>> and the shape of the signal from a macromolecule is a Gaussian in
>>> reciprocal space (aka straight line on a Wilson plot). This is true, by
>>> the way, for both a molecule packed into a crystal or free in solution.
>>> So, the ideal noise-suppression filter is simply applying a B factor.
>>> Only problem is: sharpening is generally done by applying a negative B
>>> factor, so applying a Gaussian blur is equivalent to just not sharpening
>>> as much. So, we are back to "optimal sharpening" again.
>>>
>>> Why not use a filter that is non-Gaussian? We do this all the time!
>>> Cutting off the data at a given resolution (d) is equivalent to blurring
>>> the map with this function:
>>>
>>> kernel_d(r) = 4/3*pi/d**3*sinc3(2*pi*r/d)
>>> sinc3(x) = (x==0?1:3*(sin(x)/x-cos(x))/(x*x))
>>>
>>> where kernel_d(r) is the normalized weight given to a point "r" Angstrom
>>> away from the center of each blurring operation, and "sinc3" is the
>>> Fourier synthesis of a solid sphere. That is, if you make an HKL file
>>> with all F=1 and PHI=0 out to a resolution d, then effectively all hkls
>>> beyond the resolution limit are zero. If you calculate a map with those
>>> Fs, you will find the kernel_d(r) function at the origin. What that
>>> means is: by applying a resolution cutoff, you are effectively
>>> multiplying your data by this sphere of unit Fs, and since a
>>> multiplication in reciprocal space is a convolution in real space, the
>>> effect is convoluting (blurring) with kernel_d(x).
>>>
>>> For comparison, if you apply a B factor, the real-space blurring kernel
>>> is this:
>>> kernel_B(r) = (4*pi/B)**1.5*exp(-4*pi**2/B*r*r)
>>>
>>> If you graph these two kernels (format is for gnuplot) you will find
>>> that they have the same FWHM whenever B=80*(d/3)**2. This "rule" is the
>>> one I used for my resolution demonstration movie I made back in the late
>>> 20th century:
>>> https://bl831.als.lbl.gov/~jamesh/movies/index.html#resolution
>>>
>>> What I did then was set all atomic B factors to B = 80*(d/3)^2 and then
>>> cut the resolution at "d". Seemed sensible at the time. I suppose I
>>> could have used the PDB-wide average atomic B factor reported for
>>> structures with resolution "d", which roughly follows:
>>> B = 4*d**2+12
>>> https://bl831.als.lbl.gov/~jamesh/pickup/reso_vs_avgB.png
>>>
>>> The reason I didn't use this formula for the movie is because I didn't
>>> figure it out until about 10 years later. These two curves cross at
>>> 1.5A, but diverge significantly at poor resolution. So, which one is
>>> right? It depends on how well you can measure really really faint
>>> spots, and we've been getting better at that in recent decades.
>>>
>>> So, what I'm trying to say here is that just because your data has CC1/2
>>> or FSC dropping off to insignificance at 1.8 A doesn't mean you are
>>> going to see holes in Tyr side chains. However, if you measure your
>>> weak, high-res data really well (high multiplicity), you might be able
>>> to sharpen your way to a much clearer map.
>>>
>>> -James Holton
>>> MAD Scientist
>>>
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