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 > <lists...@jiscmail.ac.uk> wrote: > >> On Thu, 5 Mar 2020 01:11:33 +0100, James Holton <jmhol...@lbl.gov> 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 >>> ######################################################################## To unsubscribe from the CCP4BB list, click the following link: https://www.jiscmail.ac.uk/cgi-bin/webadmin?SUBED1=CCP4BB&A=1
