Next: The Proposed Method Up: Knowledge-Based B-Factor Restraints for Previous: Introduction
Next: The
Proposed Method Up: Knowledge-Based
B-Factor Restraints for Previous: Introduction
When a model of a macromolecule is refined against low-to-moderate resolution
diffraction data ( 
2Å or lower) the B-factors of atoms fluctuate wildly from one atom
to the next. The lower the resolution the larger the fluctuations. Since
B-factors are usually interpreted as a measure of the amount of motion
that atom experiences, it is not plausible for atoms bonded to one another
to exhibit substantially different B-factors.
The simplest method to remove this problem is to add a restraint to
the refinement function which causes the B-factors of atoms bonded to each
other to be similar. The B-factor restraints in PROLSQ [Hendrickson
& Konnert, 1980] and XPLOR [Brunger
et al., 1987] are based upon Konnert
& Hendrickson (1980). In this approach it is assumed that in each
pair of bonded atoms one of the atoms is ``riding'' on the other. Its B-factor
will contain the motion of the first atom as well as its own relative motion.
Consider three atoms, A, B, and C, with A bonded to B and B in turn bonded
to C. Since A is directly coupled to B the relative motion along the bond
is very restricted. One can derive a restraint on the component of the
anisotropic B-factor along the bond direction. Konnert
& Hendrickson (1980) recommend that this restraint be enforced
within a root mean square value of 2.5Å 
. Konnert & Hendrickson also describe a restraint on the motion, and
therefore the anisotropic B-factor component, along the line connecting
two atoms at the ends of a bond angle, i.e. between A and C. This restraint
was given a target value of 8Å
.
Konnert & Hendrickson briefly discussed the application of restraints on isotropic B-factors. Since an isotropic B-factor has no directional variation, one simply minimized the difference in B-factor for each bonded pair of atoms. No target value was suggested for the isotropic restraint.
In order to illustrate the problem poised by the choice of an isotropic
restraint target value, consider the case of the O 
atom in a serine residue. This atom is bonded to the C 
atom and the anisotropic components of the two atoms along this bond should
be tied together with a precision of 2.5Å
. The C 
-C
-O
bond angle also restricts the motion of the O
atom. To reflect this the anisotropic B-factor of O
along the C
-O
direction can be restrained with a precision of 8Å
. The remaining anisotropic component, perpendicular to the two mentioned,
is essentially unrestricted, because the atom can rotate about the N-C
-C
-O
torsion angle.
Since the isotropic B-factor is the mean of the three principal components of the anisotropic B-factor, it should be restricted less than the most restricted component and more than the least restricted component. The median of the three restraints, 8, is a good estimate of the target value for the isotropic B-factor restraint.
PROLSQ and XPLOR apply restraints on the isotropic B-factors, however
they inappropriately use the anisotropic B-factor target. In these programs
the target value for the root-mean-square difference in B-factor for bonded
atoms is very small, typically in the neighborhood of 2Å
. The models produced by these programs underestimate the variability in
B-factor from atom to atom.
This problem was first noted by Yu, et. al., (1985). The authors compared the fluctuations of the molecular dynamic simulation of a small protein with the B-factors determined crystallographically restrained by this method. The atoms in the simulation showed variations in amplitude of motion two to three times larger than those allowed by the B-factor restraints.
A more significant problem with these restraints is the lack of provision
for the expectation that certain atoms move more than the atoms to which
they are bonded. In the example above, it is anticipated that an O
atom will have a B-factor which is larger than the C
atom. The restraint, however, attempts to equalize the B-factors. This
will cause a bias in the model such that B-factors of serine O
atoms will be systematically underestimated.
In addition one would expect that certain pairs of atoms would exhibit
greater consistency in their B-factors than others. One would like an individual
standard deviation for each class of pairs. For instance, one would expect
that the B-factors of the backbone C
and N atoms would show roughly the same correspondence throughout in the
structure. In contrast some lysine side chains are well ordered while others
are disordered. In the former case the atoms will typically display only
modest increased in B-factors as one moves away from the main chain. In
the disordered cases, however, the B-factors will increase rapidly. Restraints
derived from averaging the lysine side chains must clearly be given a smaller
weight than those applied to the main chain atoms.
While the method of Konnert & Hendrickson (1980) does decrease the fluctuations in B-factor, it does not allow the variations in B-factors that are to be expected based on current knowledge of macromolecular structures.