Refinement Using TimeAveraged Distance Restraints
NMRderived structures can be refined with timeaveraged NOE distance
restraints (Torda, Scheek, and van Gunsteren, 1989,1990; Pearlman and Kollman, 1991)
using
the TAVErage statement. In this method the NOE restraint potential is
changed so that distance restraints derived from NOE are applied to
the timeaverage of each distance, rather than each instantaneous
distance. Thus R in Eqs. 20.7,
20.10, 20.12, 20.13
is replaced by an averaged distance

(20.22) 
where is the characteristic time for the exponential decay (in
seconds), and the integral is taken over all time steps since the
timeaveraging was turned on or reset. The exponential used for
distance averaging, , is set by the EXPOnent statement. The NOE
signal arises from dipolar interaction, and hence is a function of ;
however, for times much less than the correlation time for molecular
tumbling, the NOE signal varies as . Hence should be set to 3 for
molecular dynamics simulations in the picosecond time regime. The
exponential decay term (
) ensures that the rates of
change of and
remain approximately equally responsive to the current structure
throughout the trajectory. Timeaveraging is not possible with the 3D
NOENOE potential and the highdimensional potential.
In practice, a slightly different form of the above equation is used
to calculate ; for a discrete number of time points, the
equation becomes

(20.23) 
where is the length of one time step. To avoid having to store all
distances and reevaluate the sum at each time point, we use a
recursive form of this equation:

(20.24) 
where
, and the step size is assumed to be
constant. The timeconstant
for is in units of timesteps of the molecular dynamics simulation.
The initial values for can be set to either the current
distances, ,
or to the restraint distances, , using the TAVErage RESEt statement
(CURRent or CONStraint).
The force associated with each NOE restraint is normally taken to be
the spatial derivative of the energy term, e.g. for a square well
potential,

(20.25) 
From Eq. 20.22

(20.26) 
The exact form of depends on how many atoms are involved
in the NOE
restraint, and the choice of averaging. In the simplest case where
just two atoms are involved,

(20.27) 
where denotes the vector joining the positions of the two
atoms. Thus in the usual case, where and ,

(20.28) 
Note the fourthpower term with respect to
; this
may give rise to
occasional large forces. To circumvent this problem,
Torda, Scheek, and van Gunsteren (1989)
proposed an alternative force:

(20.29) 
Integrating this force term leads to a timedependent NOE energy term,
hence this force is nonconservative. In XPLOR the force field can be
chosen by setting FORCe to either CONServative
(Eq. 20.28) or NONConservative
(Eq. 20.29).
XPLOR can also accumulate runningaverages of the distances using the
RAVErage statement. The runningaverage is calculated from

(20.30) 
Again, we actually use a recursive form:

(20.31) 
where is the total number of time (or energy) steps evaluated since
RAVEraging was turned on or reset. Note the absence of the the
exponential decay term used in Eq. 20.22; this
results in all
timesteps being weighted equally. This facility is useful to
calculate the true average over the course of an entire trajectory.
See Section 38.10 for an example for time and
runningaverages.
XplorNIH 20231110