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## 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 variables are interpreted as multiple observations of a random variable with unknown probability distribution. The potential is minimized when the histogram , estimated as a sum of Gaussian functions centered at , is equal to the reference histogram : (13.26) (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 .

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 Lower boundary of the colvar grid Context: histogramRestraint
Acceptable values: decimal
Description: Defines the lowest end of the interval where the reference distribution is defined. Exactly one value must be provided, because only one-dimensional histograms are supported by the current version.

• upperBoundary: analogous to lowerBoundary

• width Width of the colvar grid Context: histogramRestraint
Acceptable values: positive decimal
Description: Defines the spacing of the grid where the reference distribution is defined.

• gaussianSigma Standard deviation of the approximating Gaussian Context: histogramRestraint
Acceptable values: positive decimal
Default value: 2 width
Description: Defines the parameter in eq. 13.28.

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

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

• refHistogramFile Reference histogram  Context: histogramRestraint
Acceptable values: UNIX file name
Description: Provides the values of 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 and . The same numerical conventions as refHistogram are used.

• writeHistogram Periodically write the instantaneous histogram  Description: If on, the histogram is written every colvarsRestartFrequency steps to a file with the name outputName. name .hist.datThis is useful to diagnose the convergence of against .     