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

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

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

$$\bar{R}(t) = \left(\frac{R(t)^{-m}}{\tau} + e^{-1/\tau} \bar{R}(t-\Delta t)^{-m}\right)^{-1/m}$$ (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, $R(t)$, or to the restraint distances, $d$, 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,

$$\bar{F}(t) = -\nabla E_{NOE} = -\frac{nK_{NOE}}{2} (\bar{R}(t) - d)^{n-1}\nabla \bar{R}(t).$$ (20.25)

From Eq. 20.22

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

$$\nabla R(t) = \frac{{\bf R}(t)}{R(t)}$$ (20.27)

where ${\bf R}(t)$ denotes the vector joining the positions of the two atoms. Thus in the usual case, where $n=2$ and $m=3$,

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

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

$$< R(t) > = \left(\frac{1}{t}\sum_0^t[R(t-t')]^{-3}\delta t'\right)^{-1/3}$$ (20.30)

Again, we actually use a recursive form:

$$< R(t) > = \left( \frac{R(t)^{-3} + (N-1)<R(t-\Delta t)>^{-3}}{N}\right)^{-1/3}$$ (20.31)

where $N$ 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.