Probability distribution-restraints

The histogramRestraint bias implements a continuous potential of many variables (or of a single high-dimensional variable) aimed at reproducing a one-dimensional statistical distribution that is provided by the user. The $M$ variables $(\xi_{1}, \ldots, \xi_{M})$ are interpreted as multiple observations of a random variable $\xi$ with unknown probability distribution. The potential is minimized when the histogram $h(\xi)$ , estimated as a sum of Gaussian functions centered at $(\xi_{1}, \ldots, \xi_{M})$ , is equal to the reference histogram $h_{0}(\xi)$ :

$$V(\xi_{1}, \ldots, \xi_{M}) = \frac{1}{2} k \int\left(h(\xi)-h_{0}(\xi)\right)^2 \mathrm{d}\xi$$ (13.26)

$$h(\xi) = \frac{1}{M\sqrt{2\pi\sigma^2}} \sum_{i=1}^{M} \exp\left(-\frac{(\xi-\xi_{i})^2}{2\sigma^2}\right)$$ (13.27)

When used in combination with a distancePairs multi-dimensional variable, this bias implements the refinement algorithm against ESR/DEER experiments published by Shen et al [68].

This bias behaves similarly to the histogram bias with the gatherVectorColvars option, with the important difference that all variables are gathered, resulting in a one-dimensional histogram. Future versions will include support for multi-dimensional histograms.

The list of options is as follows:

• name: see definition of name (biasing and analysis methods)
• colvars: see definition of colvars (biasing and analysis methods)
• outputEnergy: see definition of outputEnergy (biasing and analysis methods)

• lowerBoundary $\langle\,$ Lower boundary of the colvar grid$\,\rangle$
  Context: histogramRestraint
  Acceptable values: decimal
  Description: Defines the lowest end of the interval where the reference distribution $h_{0}(\xi)$ is defined. Exactly one value must be provided, because only one-dimensional histograms are supported by the current version.

• upperBoundary: analogous to lowerBoundary

• width $\langle\,$ Width of the colvar grid$\,\rangle$
  Context: histogramRestraint
  Acceptable values: positive decimal
  Description: Defines the spacing of the grid where the reference distribution $h_{0}(\xi)$ is defined.

• gaussianSigma $\langle\,$ Standard deviation of the approximating Gaussian$\,\rangle$
  Context: histogramRestraint
  Acceptable values: positive decimal
  Default value: 2 $\times$ width
  Description: Defines the parameter $\sigma$ in eq. 13.28.

• forceConstant $\langle\,$ Force constant (kcal/mol)$\,\rangle$
  Context: histogramRestraint
  Acceptable values: positive decimal
  Default value: 1.0
  Description: Defines the parameter $k$ in eq. 13.27.

• refHistogram $\langle\,$ Reference histogram $h_{0}(\xi)$ $\,\rangle$
  Context: histogramRestraint
  Acceptable values: space-separated list of $M$ positive decimals
  Description: Provides the values of $h_{0}(\xi)$ consecutively. The mid-point convention is used, i.e. the first point that should be included is for $\xi$ = lowerBoundary+width/2. If the integral of $h_{0}(\xi)$ is not normalized to 1, $h_{0}(\xi)$ is rescaled automatically before use.

• refHistogramFile $\langle\,$ Reference histogram $h_{0}(\xi)$ $\,\rangle$
  Context: histogramRestraint
  Acceptable values: UNIX file name
  Description: Provides the values of $h_{0}(\xi)$ as contents of the corresponding file (mutually exclusive with refHistogram). The format is that of a text file, with each line containing the space-separated values of $\xi$ and $h_{0}(\xi)$ . The same numerical conventions as refHistogram are used.

• writeHistogram $\langle\,$ Periodically write the instantaneous histogram $h(\xi)$ $\,\rangle$
  Context: metadynamics
  Acceptable values: boolean
  Default value: off
  Description: If on, the histogram $h(\xi)$ is written every colvarsRestartFrequency steps to a file with the name outputName.$<$ name$>$ .hist.datThis is useful to diagnose the convergence of $h(\xi)$ against $h_{0}(\xi)$ .