Hello,
1. The reduction in number of parameters is not that significant. On
average, you have some 8 atoms per residue, and thus 32 parameters with
individual B-factors (4 per atom). With grouped B-factors you have 26,
about 20% difference. While it sounds like a lot (hey, I just removed
2000 parameters from my 350 residue model!), it only has an effect
similar to improving resolution by ~6%, e.g. from 2.8A to 2.65A. Worthy
goal, no doubt, but will hardly lead to dramatic improvement that would
justify these crazy jumps in B-factors.
Apart from improving data-to-parameters ratio, another argument for
going from individual isotropic ADP to group isotropic ADP, is that at
low resolution you don't see atoms individually and therefore there is
not much sense in refining parameters for them individually.
2. The total number of parameters is not the same as the effective
number of parameters. Since individual B-factors are restrained, they
don't really contribute one degree of freedom per atom. I don't know of
any reliable estimate of effective number of parameters in the
literature, but my personal guess based on several ways to estimate it
is that it may take about 5 restraints to compensate for one parameter.
B-factors are restrained by bond and by angle, thus giving 2-3
restraints per B-factor. So, every individual B-factor only contributes
half a parameter, thus ~4 parameters per residue. This is very much
comparable with two essentially unrestrained (as confirmed by their wild
variations) grouped B-factors.
That's exactly why ("effective number of parameters in presence of
restraints") individual B-factor refinement works well in phenix.refine
at lower resolutions, where people traditionally tempt to switch to
group isotropic ADP. Going to lower resolution, I only switch to group
ADP refinement if I really have to.
By the way, I always assumed (based on B-factor behavior) that grouped
B-factor refinement is unrestrained.
Group B-factor refinement is a constrained refinement: indeed, you
constrain all B-factors to be identical within the group (similarly, as
TLS refinement is a group constrained anisotropic refinement). It is
"unrestrained" in a sense that there is no restraints used to make the
group B-factors similar between adjacent residues. I will add this
functionality to phenix.refine at some point.
Pavel.
