Another much simpler (too simple?) way to approach this is that the
number of parameters is:
Ntorsions + Nbvalues + Nplacement*Nmonomers
Nmonomers is the number of monomers in the asymmetric unit and
Nplacement is the number of parameters needed to place each monomer, so
6 for the general case (or the Nth molecule relative to the first), 5
for polar space groups and 3 for triclinic.
NBvalues depends on the B-value refinement model but for individual
isotropic B-values it would be one "free" B-value parameter for each atom
Ntorsion represents the number of "free" torsion angles
For non-covalently bound atoms, like waters, we need to add another term
of "4 * Nfree_atom" to represent the unrestraint coordinate and B-value
parameters for these atoms.
Of course now we can start arguing about what "free" means ...
Bart
Ian Tickle wrote:
To try and answer Peter & Bernhard's questions in one go: one way of
looking at counting restraints is to think of it in terms of the
reduction in the effective number of parameters that it's equivalent to.
So a bond length restraint is equivalent to reducing the 8 parameters
for 2 bonded atoms by 1 (if the bond is rotated to be along the z axis
then you are effectively removing 1 of the 2 z co-ordinates as a
parameter, the remaining z co-ordinate allows the pair of atoms to move
together freely along the z axis). So a restraint on say a CB-OG bond
reduces the effective parameter count by 1 for each occurrence of the
CB-OG bond in the structure. A restraint is equivalent to an
'observation' of the bond length in *your* structure, not the
observation of the bond length in the amino-acid/peptide structure that
the restraint was derived from.
On the question of restraint independence it's clear that in general all
bond length restraints are independent of each other, and similarly all
angle restraints are independent of each other and of the bond length
restraints. I say 'in general' because for rings it's not quite true:
there must be a geometrical relationship between the bond lengths &
angles in a ring, but fortunately the number of rings in the average
structure is small relative to the total number of restraints and the
error resulting from this assumption is at most 1 or 2 restraints per
ring. As you say torsion restraints are independent except when there's
a VDW contact, but usually the number of VDW contacts is small, except
perhaps at low resolution (so maybe count active VDW restraints as 1/2
each). Planar restraints are more tricky: they are clearly not
independent of the bond angle restraints because if you say restrain a
trigonal atom or an aromatic ring to be planar this imposes restrictions
on the sum of bond angles (360, 540 & 720 for a trigonal atom, 5-ring &
6-ring resp). But the planar restraint obviously has an effect so I
would say count 1 planar restraint per plane (not N-3 as I suggested
earlier).
It starts to get tricky with NCS because although NCS restraints between
say 2 molecules apparently halves the number of effective parameters,
much of this reduction in parameter count will already have been counted
because of the other types of restraint, so maybe the effective number
of NCS restraints is 4 * #atoms per monomer minus the number of other
restraints in 1 monomer (that may not be quite right, I probably need to
think more carefully about this!).
Finally as you say it gets really murky with B-factor difference
restraints because differences between bonded atoms and those related by
a bond angle are clearly not independent, but they are not completely
correlated either. In my view, B-factors are over-restrained anyway
(B-factor differences are larger than people like to believe because of
libration effects): for this reason I now always turn off B-factor
restraints related by a bond angle and keep only those related by a bond
(in Refmac you can now do this by making the corresponding SIGBi
parameter on the BFAC record negative). This has the advantage that
there are then fewer B-factor restraint parameters to be determined by
cross-validation.
HTH!
-- Ian
-----Original Message-----
From: Meyer, Peter [mailto:[EMAIL PROTECTED]
Sent: 14 February 2008 23:04
To: Ian Tickle
Subject: RE: counting constraints?
Ian,
Thanks for the quick answers.
<snip>
different question from the one I answered. Of course you count the
restraints in your structure, not the ones in the dictionary,
so in the
<end snip>
I guess my original question is why is this of course? Or,
why isn't this counting an observation (expected value of a
bond length/angle/etc) multiple times?
And yeah, I did mean restraints when I said constraints.
Thanks again,
Pete
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