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Refinement Using Time-Averaged Distance Restraints

NMR-derived structures can be refined with time-averaged NOE distance restraints (Torda, Scheek, and van Gunsteren, 1990,1989; 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 time-average 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

$\begin{displaymath} \bar{R}(t) = \left(\frac{1}{\tau_t} \int_0^t e^{-t'/\tau_t} R(t-t')^{-m}dt'\right)^{-1/m} \end{displaymath}$

(20.22)

where $\tau_t$ is the characteristic time for the exponential decay (in seconds), and the integral is taken over all time steps since the time-averaging 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 $R^{-6}$ ; however, for times much less than the correlation time for molecular tumbling, the NOE signal varies as $R^{-3}$ . Hence

should be set to 3 for molecular dynamics simulations in the picosecond time regime. The exponential decay term ( $e^{-t'/\tau_t}$ ) ensures that the rates of change of $\bar{R}(t)$ and $E_{NOE}$ remain approximately equally responsive to the current structure throughout the trajectory. Time-averaging is not possible with the 3D NOE-NOE potential and the high-dimensional potential.

In practice, a slightly different form of the above equation is used to calculate $\bar{R}(t)$ ; for a discrete number of time points, the equation becomes

$\begin{displaymath} \bar{R}(t) = \left(\frac{\Delta t}{\tau_t} \sum_{t'=0}^t e^{-t'/\tau_t} R(t-t')^{-m}\right)^{-1/m} \end{displaymath}$

(20.23)

where $\Delta t$ is the length of one time step. To avoid having to store all distances and re-evaluate the sum at each time point, we use a recursive form of this equation:

$\begin{displaymath} \bar{R}(t) = \left(\frac{R(t)^{-m}}{\tau} + e^{-1/\tau} \bar{R}(t-\Delta t)^{-m}\right)^{-1/m} \end{displaymath}$

(20.24)

where $\tau=\tau_t/\Delta t$ , and the step size is assumed to be constant. The time-constant for $\tau$ is in units of time-steps of the molecular dynamics simulation.

The initial values for $\bar{R}(t)$ 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,

$\begin{displaymath} \bar{F}(t) = -\nabla E_{NOE} = -\frac{nK_{NOE}}{2} (\bar{R}(t) - d)^{n-1}\nabla \bar{R}(t). \end{displaymath}$

(20.25)

From Eq. 20.22

$\begin{displaymath} \nabla \bar{R}(t) = \frac{1}{\tau} \left[\frac{\bar{R}(t)}{R(t)}\right]^{m+1} \nabla R(t). \end{displaymath}$

(20.26)

The exact form $\nabla R(t)$ 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,

$\begin{displaymath} \nabla R(t) = \frac{{\bf R}(t)}{R(t)} \end{displaymath}$

(20.27)

where ${\bf R}(t)$ denotes the vector joining the positions of the two atoms. Thus in the usual case, where

and

$\begin{displaymath} \bar{F}(t) = - \frac{K_{NOE}}{\tau}(\bar{R}(t) - d) \left[\frac{\bar{R}(t)}{R(t)}\right]^4 \frac{{\bf R}(t)}{R(t)} \end{displaymath}$

(20.28)

Note the fourth-power term with respect to $\bar{R}(t)/R(t)$ ; this may give rise to occasional large forces. To circumvent this problem, Torda, Scheek, and van Gunsteren (1989) proposed an alternative force:

$\begin{displaymath} \bar{F}(t) = -K_{NOE}(\bar{R}(t)-d)\frac{{\bf R}(t)}{R(t)}. \end{displaymath}$

(20.29)

Integrating this force term leads to a time-dependent NOE energy term, hence this force is nonconservative. In X-PLOR the force field can be chosen by setting FORCe to either CONServative (Eq. 20.28) or NONConservative (Eq. 20.29).

X-PLOR can also accumulate running-averages of the distances using the RAVErage statement. The running-average is calculated from

$\begin{displaymath} < R(t) > = \left(\frac{1}{t}\sum_0^t[R(t-t')]^{-3}\delta t'\right)^{-1/3} \end{displaymath}$

(20.30)

Again, we actually use a recursive form:

$\begin{displaymath} < R(t) > = \left( \frac{R(t)^{-3} + (N-1)<R(t-\Delta t)>^{-3}}{N}\right)^{-1/3} \end{displaymath}$

(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 time-steps 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 running-averages.

Xplor-NIH 2025-03-21