Hello,

I guess you are
arguing that by using constraints there are more data available to
refine B-values AND occupancy. You are probably aware that these to
numbers are strongly correlated (>=90%!!) so that it is very tricky to
get get reliable numbers anyhow

sometimes numbers excite me! So this one caught my attention and I decided to entertain myself.

First off, an obvious statement: occupancy defines peak's height and B-factor defines its shape. Therefore one cannot be entirely compensated with the other.

Now let's see if and how occupancy and B-factor are correlated. For this let's take an atom and plot its electron density distribution with occupancy q=1 and some B value; let's call this density rho_ref (reference map). Then let's vary occupancy from 0.1 to 1.0 (with step 0.1) and for each trial occupancy value find such B_opt that corresponding electron density distribution fits rho_ref as good as possible; let's call it rho_opt (map corresponding to optimal B_opt). In the end we will have ten occupancy values and ten corresponding optimal B values so that we can calculate the correlation between two sets of numbers (q, B_opt). In addition let's calculate correlation and R-factor for rho_ref and rho_opt.

We will repeat the numerical experiment defined above with:
a) different starting B values (10, 30, 50, 80),
b) different atoms H, C, S,
c) exact electron density distribution as well as its Fourier image of 2A resolution.

Attached script does it all in one go. Also it illustrates the beauty of CCTBX that allows to do this so easily!

Here are the numbers:

Resolution: None (exact map)----------------------------------------------------------
atom: H
    B: 10
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 1.00 1.00 2.00 3.00 5.00 6.00 7.00 9.00 10.00
      CC(rho_ref,rho_opt): 0.95 0.95 0.95 0.97 0.98 0.99 0.99 1.00 1.00 1.00
      R(%)               : 34.94 34.94 34.94 28.30 23.09 14.83 11.36 8.19 2.55 0.00
      CC(q,B):   0.97
    B: 30
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 4.00 7.00 11.00 14.00 17.00 21.00 24.00 27.00 30.00
      CC(rho_ref,rho_opt): 0.85 0.91 0.95 0.97 0.98 0.99 1.00 1.00 1.00 1.00
      R(%)               : 54.39 42.43 34.42 26.09 20.84 16.16 10.59 6.81 3.29 0.00
      CC(q,B):   1.00
    B: 50
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 3.00 9.00 15.00 20.00 26.00 31.00 36.00 41.00 45.00 50.00
      CC(rho_ref,rho_opt): 0.84 0.91 0.95 0.97 0.98 0.99 1.00 1.00 1.00 1.00
      R(%)               : 55.33 42.14 32.91 26.59 20.01 15.15 10.72 6.64 3.59 0.00
      CC(q,B):   1.00
    B: 80
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 8.00 18.00 27.00 35.00 43.00 51.00 59.00 66.00 73.00 80.00
      CC(rho_ref,rho_opt): 0.85 0.92 0.95 0.97 0.98 0.99 1.00 1.00 1.00 1.00
      R(%)               : 53.74 41.14 32.57 26.13 20.44 15.31 10.64 6.86 3.32 0.00
      CC(q,B):   1.00
atom: C
    B: 10
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
      CC(rho_ref,rho_opt): 0.79 0.88 0.93 0.96 0.98 0.99 0.99 1.00 1.00 1.00
      R(%)               : 62.77 49.83 39.70 31.37 24.32 18.23 12.89 8.14 3.87 0.00
      CC(q,B):   1.00
    B: 30
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 4.00 7.00 10.00 13.00 16.00 19.00 21.00 24.00 27.00 30.00
      CC(rho_ref,rho_opt): 0.83 0.91 0.95 0.97 0.98 0.99 0.99 1.00 1.00 1.00
      R(%)               : 56.51 43.81 34.49 27.06 20.84 15.45 12.21 7.76 3.72 0.00
      CC(q,B):   1.00
    B: 50
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 6.00 12.00 17.00 22.00 27.00 32.00 36.00 41.00 46.00 50.00
      CC(rho_ref,rho_opt): 0.82 0.91 0.95 0.97 0.98 0.99 0.99 1.00 1.00 1.00
      R(%)               : 57.64 42.58 33.79 26.68 20.63 15.32 11.47 7.06 3.02 0.00
      CC(q,B):   1.00
    B: 80
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 11.00 20.00 28.00 36.00 44.00 52.00 59.00 66.00 73.00 80.00
      CC(rho_ref,rho_opt): 0.84 0.91 0.95 0.97 0.98 0.99 1.00 1.00 1.00 1.00
      R(%)               : 54.69 41.90 33.51 26.57 20.56 15.22 10.98 7.06 3.41 0.00
      CC(q,B):   1.00
atom: S
    B: 10
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
      CC(rho_ref,rho_opt): 0.82 0.89 0.93 0.96 0.98 0.99 0.99 1.00 1.00 1.00
      R(%)               : 59.32 47.29 38.28 30.75 24.21 18.39 13.15 8.39 4.03 0.00
      CC(q,B):   1.00
    B: 30
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 4.00 8.00 11.00 14.00 17.00 19.00 22.00 25.00 27.00 30.00
      CC(rho_ref,rho_opt): 0.82 0.91 0.95 0.97 0.98 0.99 0.99 1.00 1.00 1.00
      R(%)               : 57.49 42.56 33.98 26.75 20.48 16.72 11.57 6.91 4.03 0.00
      CC(q,B):   1.00
    B: 50
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 8.00 13.00 19.00 23.00 28.00 33.00 37.00 42.00 46.00 50.00
      CC(rho_ref,rho_opt): 0.84 0.91 0.95 0.97 0.98 0.99 0.99 1.00 1.00 1.00
      R(%)               : 54.25 43.18 32.90 27.17 20.88 15.31 11.27 6.64 3.21 0.00
      CC(q,B):   1.00
    B: 80
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 13.00 22.00 30.00 38.00 46.00 53.00 60.00 67.00 73.00 80.00
      CC(rho_ref,rho_opt): 0.84 0.91 0.95 0.97 0.98 0.99 1.00 1.00 1.00 1.00
      R(%)               : 53.98 41.82 33.40 26.32 20.16 15.32 10.91 6.83 3.57 0.00
      CC(q,B):   1.00
Resolution: 2.0 ----------------------------------------------------------------------
atom: H
    B: 10
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 6.00 10.00
      CC(rho_ref,rho_opt): 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
      R(%)               : 5.93 5.93 5.93 5.93 5.93 5.93 5.93 5.28 2.66 0.00
      CC(q,B):   0.72
    B: 30
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 1.00 1.00 1.00 4.00 10.00 16.00 21.00 26.00 30.00
      CC(rho_ref,rho_opt): 0.99 0.99 0.99 0.99 0.99 0.99 1.00 1.00 1.00 1.00
      R(%)               : 18.17 18.17 18.17 18.17 16.35 12.62 8.83 5.66 2.50 0.00
      CC(q,B):   0.95
    B: 50
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 1.00 2.00 12.00 20.00 27.00 33.00 39.00 45.00 50.00
      CC(rho_ref,rho_opt): 0.97 0.97 0.97 0.98 0.99 0.99 1.00 1.00 1.00 1.00
      R(%)               : 27.44 27.44 26.91 21.35 16.76 12.70 9.30 5.94 2.66 0.00
      CC(q,B):   0.99
    B: 80
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 6.00 20.00 31.00 41.00 50.00 58.00 66.00 73.00 80.00
      CC(rho_ref,rho_opt): 0.93 0.94 0.96 0.98 0.99 0.99 1.00 1.00 1.00 1.00
      R(%)               : 36.70 34.44 27.69 22.24 17.34 13.03 9.33 5.80 2.84 0.00
      CC(q,B):   1.00
atom: C
    B: 10
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 1.00 1.00 1.00 1.00 1.00 1.00 3.00 6.00 10.00
      CC(rho_ref,rho_opt): 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
      R(%)               : 5.96 5.96 5.96 5.96 5.96 5.96 5.96 4.65 2.67 0.00
      CC(q,B):   0.75
    B: 30
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 1.00 1.00 1.00 5.00 11.00 16.00 21.00 26.00 30.00
      CC(rho_ref,rho_opt): 0.99 0.99 0.99 0.99 0.99 0.99 1.00 1.00 1.00 1.00
      R(%)               : 18.37 18.37 18.37 18.37 15.93 12.16 8.96 5.75 2.54 0.00
      CC(q,B):   0.95
    B: 50
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 1.00 3.00 12.00 20.00 27.00 33.00 39.00 45.00 50.00
      CC(rho_ref,rho_opt): 0.97 0.97 0.97 0.98 0.99 0.99 1.00 1.00 1.00 1.00
      R(%)               : 27.82 27.82 26.75 21.70 17.07 12.95 9.48 6.06 2.71 0.00
      CC(q,B):   0.99
    B: 80
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 7.00 21.00 32.00 41.00 50.00 58.00 66.00 73.00 80.00
      CC(rho_ref,rho_opt): 0.93 0.94 0.96 0.98 0.99 0.99 1.00 1.00 1.00 1.00
      R(%)               : 37.14 34.45 27.63 22.10 17.64 13.25 9.48 5.90 2.88 0.00
      CC(q,B):   1.00
atom: S
    B: 10
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 1.00 1.00 1.00 1.00 1.00 1.00 3.00 7.00 10.00
      CC(rho_ref,rho_opt): 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
      R(%)               : 5.88 5.88 5.88 5.88 5.88 5.88 5.88 4.60 1.99 0.00
      CC(q,B):   0.76
    B: 30
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 1.00 1.00 1.00 6.00 11.00 17.00 21.00 26.00 30.00
      CC(rho_ref,rho_opt): 0.99 0.99 0.99 0.99 0.99 0.99 1.00 1.00 1.00 1.00
      R(%)               : 18.46 18.46 18.46 18.46 15.43 12.30 8.45 5.84 2.59 0.00
      CC(q,B):   0.96
    B: 50
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 1.00 4.00 13.00 21.00 28.00 34.00 40.00 45.00 50.00
      CC(rho_ref,rho_opt): 0.97 0.97 0.97 0.98 0.99 0.99 1.00 1.00 1.00 1.00
      R(%)               : 28.30 28.30 26.67 21.63 16.92 12.75 9.18 5.68 2.81 0.00
      CC(q,B):   0.99
    B: 80
      trial q            : 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
      B_opt              : 1.00 9.00 22.00 33.00 43.00 51.00 59.00 66.00 73.00 80.00
      CC(rho_ref,rho_opt): 0.93 0.94 0.96 0.98 0.99 0.99 1.00 1.00 1.00 1.00
      R(%)               : 37.91 34.26 27.88 22.25 17.20 13.22 9.35 6.10 2.99 0.00
      CC(q,B):   0.99

What we see here is:
- correlation of q and B is indeed approaches 100%;
- map correlation is greater than 90% in most cases except a few corner cases;
- the last column in all tests is an obvious sanity check (CC=1, R=0 if exact B and q are used);
- R-factors are greater than zero except a trivial case. This is the key that makes it possible to deconvolute q and B.

All the best,
Pavel