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Harmonic restraints

The harmonic biasing method may be used to enforce fixed or moving restraints, including variants of Steered and Targeted MD. Within energy minimization runs, it allows for restrained minimization, e.g. to calculate relaxed potential energy surfaces. In the context of the Colvars module, harmonic potentials are meant according to their textbook definition:

$\displaystyle V(\xi) = \frac{1}{2} k \left(\frac{\xi - \xi_0}{w_{\xi}}\right)^2$ (13.34)

There are two noteworthy aspects of this expression:
  1. Because the standard coefficient of $ 1/2$ of the harmonic potential is included, this expression differs from harmonic bond and angle potentials historically used in common force fields, where the factor was typically omitted resulting in a non-standard definition of the force constant.
  2. The variable $ \xi$ is not only centered at $ \xi_0$ , but is also scaled by its characteristic length scale $ w_{\xi}$ (keyword width). The resulting dimensionless variable $ z = (\xi - \xi_0)/w_{\xi}$ is typically easier to treat numerically: for example, when the forces typically experienced by $ \xi$ are much smaller than $ k/w_{\xi}$ and $ k$ is chosen equal to $ \kappa_{\mathrm{B}}T$ (thermal energy), the resulting probability distribution of $ z$ is approximately a Gaussian with mean equal to 0 and standard deviation equal to 1.

    This property can be used for setting the force constant in umbrella-sampling ensemble runs: if the restraint centers are chosen in increments of $ w_{\xi}$ , the resulting distributions of $ \xi$ are most often optimally overlapped. In regions where the underlying free-energy landscape induces highly skewed distributions of $ \xi$ , additional windows may be added as needed, with spacings finer than $ w_{\xi}$ .

Beyond one dimension, the use of a scaled harmonic potential also allows a standard definition of a multi-dimensional restraint with a unified force constant:

$\displaystyle V(\xi_{1}, \ldots, \xi_{M}) = \frac{1}{2} k \sum_{i=1}^{M} \left(\frac{\xi_{i} - \xi_0}{w_{\xi}}\right)^2$ (13.35)

If one-dimensional or homogeneous multi-dimensional restraints are defined, and there are no other uses for the parameter $ w_{\xi}$ , width can be left at its default value of $ 1$ .

A harmonic restraint is defined by a harmonic {...} block, which may contain the following keywords:

Tip: A complex set of restraints can be applied to a system, by defining several colvars, and applying one or more harmonic restraints to different groups of colvars. In some cases, dozens of colvars can be defined, but their value may not be relevant: to limit the size of the colvars trajectory file, it may be wise to disable outputValue for such ``ancillary'' variables, and leave it enabled only for ``relevant'' ones.

Moving restraints: steered molecular dynamics

The following options allow to change gradually the centers of the harmonic restraints during a simulations. When the centers are changed continuously, a steered MD in a collective variable space is carried out.

Note on restarting moving restraint simulations: Information about the current step and stage of a simulation with moving restraints is stored in the restart file (state file). Thus, such simulations can be run in several chunks, and restarted directly using the same colvars configuration file. In case of a restart, the values of parameters such as targetCenters, targetNumSteps, etc. should not be changed manually.

Moving restraints: umbrella sampling

The centers of the harmonic restraints can also be changed in discrete stages: in this cases a one-dimensional umbrella sampling simulation is performed. The sampling windows in simulation are calculated in sequence. The colvars trajectory file may then be used both to evaluate the correlation times between consecutive windows, and to calculate the frequency distribution of the colvar of interest in each window. Furthermore, frequency distributions on a predefined grid can be automatically obtained by using the histogram bias (see [*]).

To activate an umbrella sampling simulation, the same keywords as in the previous section can be used, with the addition of the following:

Changing force constant

The force constant of the harmonic restraint may also be changed to equilibrate [76].

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Next: Computing the work of Up: Biasing and analysis methods Previous: Metadynamics   Contents   Index