Arithmetic path collective variables

The arithmetic path collective variable in CV space uses the same formula as the one proposed by Branduardi[55] et al., except that it computes $s$ and $z$ in CV space instead of RMSDs in Cartesian space. Moreover, this implementation allows different coefficients for each CV components as described in [56]. Assuming a path is composed of $N$ reference frames and defined in an $M$ -dimensional CV space, then the equations of $s$ and $z$ of the path are

$$s = \frac{\sum_{i=1}^{N} i \exp\left(-\lambda\sum_{j=1}^{M} c_j^2... ...}^{N} \exp\left(-\lambda\sum_{j=1}^{M} c_j^2 \left(x_j-x_{i,j}\right)^2\right)}$$ (13.15)

$$z = -\frac{1}{\lambda} \ln \left(\sum_{i=1}^{N} \exp\left(-\lambda\sum_{j=1}^{M} c_j^2 \left(x_j-x_{i,j}\right) \right)\right)$$ (13.16)

where $c_j$ is the coefficient(weight) of the $j$ -th CV, $x_{i,j}$ is the value of $j$ -th CV of $i$ -th reference frame and $x_{j}$ is the value of $j$ -th CV of current frame. $\lambda$ is a parameter to smooth the variation of $s$ and $z$ .

aspathCV: progress along a path defined in CV space.

This colvar component computes the $s$ variable.

List of keywords (see also for additional options):

• weights $\langle\,$ Coefficients of the collective variables$\,\rangle$
  Context: aspathCV and azpathCV
  Acceptable values: space-separated numbers in a {...} block
  Default value: {1.0 ...}
  Description: Define the coefficients. The $j$ -th value in the {...} block corresponds the $c_{j}$ in the equations.

• lambda $\langle\,$ Smoothness of the variation of $s$ and $z$ $\,\rangle$
  Context: aspathCV and azpathCV
  Acceptable values: decimal
  Default value: inverse of the mean square displacements of successive reference frames
  Description: The value of $\lambda$ in the equations.

• pathFile $\langle\,$ The file name of the path file.$\,\rangle$
  Context: aspathCV and azpathCV
  Acceptable values: UNIX filename
  Description: Defines the nodes or images that constitutes the path in CV space. The CVs of an image are listed in a line of pathFile using space-seperated format. Lines from top to button in pathFile corresponds images from initial to last.

azpathCV: distance from a path defined in CV space.

This colvar component computes the $z$ variable. Options are the same as in .

The usage of azpathCV and aspathCV is illustrated below:

colvar {
  # Progress along the path
  name as
  # Path defined by the CV space of two dihedral angles
  aspathCV {
    pathFile ./path.txt
    weights {1.0 1.0}
    lambda 0.005
    dihedral {
      name 001
      group1 {atomNumbers {5}}
      group2 {atomNumbers {7}}
      group3 {atomNumbers {9}}
      group4 {atomNumbers {15}}
    }
    dihedral {
      name 002
      group1 {atomNumbers {7}}
      group2 {atomNumbers {9}}
      group3 {atomNumbers {15}}
      group4 {atomNumbers {17}}
    }
  }
}
colvar {
  # Distance from the path
  name az
  azpathCV {
    pathFile ./path.txt
    weights {1.0 1.0}
    lambda 0.005
    dihedral {
      name 001
      group1 {atomNumbers {5}}
      group2 {atomNumbers {7}}
      group3 {atomNumbers {9}}
      group4 {atomNumbers {15}}
    }
    dihedral {
      name 002
      group1 {atomNumbers {7}}
      group2 {atomNumbers {9}}
      group3 {atomNumbers {15}}
      group4 {atomNumbers {17}}
    }
  }
}

Path collective variables in Cartesian coordinates

The path collective variables defined by Branduardi et al. [55] are based on RMSDs in Cartesian coordinates. Noting $d_i$ the RMSD between the current set of Cartesian coordinates and those of image number $i$ of the path:

$$s = \frac{1}{N-1} \frac{\sum_{i=1}^{N} (i-1) \exp\left(-\lambda d_i^2\right)} {\sum_{i=1}^{N} \exp\left(-\lambda d_i^2\right)}$$ (13.17)

$$z = -\frac{1}{\lambda} \ln \left(\sum_{i=1}^{N} \exp(-\lambda d_i^2)\right)$$ (13.18)

where $\lambda$ is the smoothing parameter.

These coordinates are implemented as Tcl-scripted combinations of rmsd components. The implementation is available as file colvartools/pathCV.tcl, and an example is provided in file examples/10_pathCV.namd of the Colvars public repository. It implements an optimization procedure, whereby the distance to a given image is only calculated if its contribution to the sum is larger than a user-defined tolerance parameter. All distances are calculated every freq timesteps to update the list of nearby images.