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.