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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:

(13.24) 
Note that this differs from harmonic bond and angle potentials in common
force fields, where the factor of one half is typically omitted,
resulting in a nonstandard definition of the force constant.
The formula above includes the characteristic length scale
of the colvar
(keyword width, see 13.2.1) to allow the definition of a multidimensional restraint with a unified force constant:

(13.25) 
If onedimensional or homogeneous multidimensional restraints are defined, and there are no other uses for the parameter
, the parameter width can be left at its default value of
.
A harmonic restraint is set up by a harmonic {...}
block, which may contain (in addition to the standard option
colvars) the following keywords:

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)

forceConstant
Scaled force constant (kcal/mol)
Context: harmonic
Acceptable values: positive decimal
Default value: 1.0
Description: This defines a scaled force constant
for the harmonic potential (eq. 13.26).
To ensure consistency for multidimensional restraints, it is
divided internally by the square of the specific width
for each colvar involved (which is 1 by default), so that all colvars
are effectively dimensionless and of commensurate size.
For instance, setting a scaled force constant of 10 kcal/mol acting
on two colvars, an angle with a width of 5 degrees and a distance
with a width of 0.5 Å, will apply actual force constants of
0.4 kcal/mol
degree
for the angle and
40 kcal/mol/Å
for the distance.

centers
Initial harmonic restraint centers
Context: harmonic
Acceptable values: spaceseparated list of colvar values
Description: The centers (equilibrium values) of the restraint,
, are entered here.
The number of values must be the number of requested colvars.
Each value is a decimal number if the corresponding colvar returns
a scalar, a ``(x, y, z)'' triplet if it returns a unit
vector or a vector, and a ``(q0, q1, q2, q3)'' quadruplet
if it returns a rotational quaternion. If a colvar has
periodicities or symmetries, its closest image to the restraint
center is considered when calculating the harmonic potential.
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.

targetCenters
Steer the restraint centers towards these
targets
Context: harmonic
Acceptable values: spaceseparated list of colvar values
Description: When defined, the current centers will be moved towards
these values during the simulation.
By default, the centers are moved over a total of
targetNumSteps steps by a linear interpolation, in the
spirit of Steered MD.
If targetNumStages is set to a nonzero value, the
change is performed in discrete stages, lasting targetNumSteps
steps each. This second mode may be used to sample successive
windows in the context
of an Umbrella Sampling simulation.
When continuing a simulation
run, the centers specified in the configuration file
colvarsConfig
are overridden by those saved in
the restart file
colvarsInput
. To perform Steered
MD in an arbitrary space of colvars, it is
sufficient to use this option and enable
outputAppliedForce within each of the colvars involved.

targetNumSteps
Number of steps for steering
Context: harmonic
Acceptable values: positive integer
Description: In singlestage (continuous) transformations, defines the number of MD
steps required to move the restraint centers (or force constant)
towards the values specified with targetCenters or
targetForceConstant.
After the target values have been reached, the centers (resp. force
constant) are kept fixed. In multistage transformations, this sets the
number of MD steps per stage.

outputCenters
Write the current centers to the trajectory file
Context: harmonic
Acceptable values: boolean
Default value: off
Description: If this option is chosen and colvarsTrajFrequency is not zero, the positions of the restraint centers will be written to the trajectory file during the simulation.
This option allows to conveniently extract the PMF from the colvars trajectory files in a steered MD calculation.

outputAccumulatedWork
Write the accumulated work of the moving restraint to the trajectory file
Context: harmonic
Acceptable values: boolean
Default value: off
Description: If this option is chosen, targetCenters is defined, and colvarsTrajFrequency is not zero, the accumulated work from the beginning of the simulation will be written to the trajectory file.
If the simulation has been continued from a previous state file, the previously accumulated work is included in the integral.
This option allows to conveniently extract the PMF from the colvars trajectory files in a steered MD calculation.
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 onedimensional 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 13.5.7).
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 [65].

targetForceConstant
Change the force constant towards this value
Context: harmonic
Acceptable values: positive decimal
Description: When defined, the current forceConstant will be moved towards
this value during the simulation. Time evolution of the force constant
is dictated by the targetForceExponent parameter (see below).
By default, the force constant is changed smoothly over a total of
targetNumSteps steps. This is useful to introduce or
remove restraints in a progressive manner.
If targetNumStages is set to a nonzero value, the
change is performed in discrete stages, lasting targetNumSteps
steps each. This second mode may be used to compute the
conformational free energy change associated with the restraint, within
the FEP or TI formalisms. For convenience, the code provides an estimate
of the free energy derivative for use in TI. A more complete free energy
calculation (particularly with regard to convergence analysis),
while not handled by the Colvars module, can be performed by postprocessing
the colvars trajectory, if colvarsTrajFrequency is set to a
suitably small value. It should be noted, however, that restraint
free energy calculations may be handled more efficiently by an
indirect route, through the
determination of a PMF for the restrained coordinate.[65]

targetForceExponent
Exponent in the timedependence of the force constant
Context: harmonic
Acceptable values: decimal equal to or greater than 1.0
Default value: 1.0
Description: Sets the exponent,
, in the function used to vary the force
constant as a function of time. The force is varied according to a
coupling parameter
, raised to the power
:
, where
,
, and
are the initial, current, and final values
of the force constant. The parameter
evolves linearly from
0 to 1, either smoothly, or in targetNumStages equally spaced
discrete stages, or according to an arbitrary schedule set with
lambdaSchedule.
When the initial value of the force constant is zero,
an exponent greater than 1.0 distributes the effects of introducing the
restraint more smoothly over time than a linear dependence, and
ensures that there is no singularity in the derivative of the
restraint free energy with respect to lambda. A value of 4 has
been found to give good results in some tests.

targetEquilSteps
Number of steps discarded from TI estimate
Context: harmonic
Acceptable values: positive integer
Description: Defines the number of steps within each stage that are considered
equilibration and discarded from the restraint free energy derivative
estimate reported reported in the output.

lambdaSchedule
Schedule of lambdapoints for changing force constant
Context: harmonic
Acceptable values: list of real numbers between 0 and 1
Description: If specified together with targetForceConstant, sets the sequence of
discrete
values that will be used for different stages.
Next: Linear restraints
Up: Biasing and analysis methods
Previous: Metadynamics
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