Geometric path collective variables

The geometric path collective variables define the progress along a path, $s$ , and the distance from the path, $z$ . These CVs are proposed by Leines and Ensing[54] , which differ from that[55] proposed by Branduardi et al., and utilize a set of geometric algorithms. The path is defined as a series of frames in the atomic Cartesian coordinate space or the CV space. $s$ and $z$ are computed as

$$s = \frac{m}{M} \pm \frac{1}{2M} \left( \frac{\sqrt{(\mathbf{v}_1... ...ert^2)}-(\mathbf{v}_1 \cdot \mathbf{v}_3)}{\vert\mathbf{v}_3\vert^2} -1 \right)$$ (13.13)

$$z = \sqrt{\left(\mathbf{v}_1 + \frac{1}{2}\left(\frac{\sqrt{(\mat... ...cdot \mathbf{v}_3)}{\vert\mathbf{v}_3\vert^2} -1 \right)\mathbf{v}_4 \right)^2}$$ (13.14)

where $\mathbf{v}_1 = \mathbf{s}_{m} - \mathbf{z}$ is the vector connecting the current position to the closest frame, $\mathbf{v}_2 = \mathbf{z} - \mathbf{s}_{m-1}$ is the vector connecting the second closest frame to the current position, $\mathbf{v}_3 = \mathbf{s}_{m+1} - \mathbf{s}_{m}$ is the vector connecting the closest frame to the third closest frame, and $\mathbf{v}_4 = \mathbf{s}_m - \mathbf{s}_{m-1}$ is the vector connecting the second closest frame to the closest frame. $m$ and $M$ are the current index of the closest frame and the total number of frames, respectively. If the current position is on the left of the closest reference frame, the $\pm$ in $s$ turns to the positive sign. Otherwise it turns to the negative sign.

The equations above assume: (i) the frames are equidistant and (ii) the second and the third closest frames are neighbouring to the closest frame. When these assumptions are not satisfied, this set of path CV should be used carefully.

gspath: progress along a path defined in atomic Cartesian coordinate space.

In the gspath {...} and the gzpath {...} block all vectors, namely $\mathbf{z}$ and $\mathbf{s}_{k}$ are defined in atomic Cartesian coordinate space. More specifically, $\mathbf{z} = \left[\mathbf{r}_{1}, \cdots, \mathbf{r}_{n}\right]$ , where $\mathbf{r}_{i}$ is the $i$ -th atom specified in the atoms block. $\mathbf{s}_{k} = \left[\mathbf{r}_{k,1}, \cdots, \mathbf{r}_{k,n}\right]$ , where $\mathbf{r}_{k,i}$ means the $i$ -th atom of the $k$ -th reference frame.

List of keywords (see also for additional options):

• atoms $\langle\,$ Group of atoms$\,\rangle$
  Context: gspath and gzpath
  Acceptable values: Block atoms {...}
  Description: Defines the atoms whose coordinates make up the value of the component.

• refPositionsCol $\langle\,$ PDB column containing atom flags$\,\rangle$
  Context: gspath and gzpath
  Acceptable values: O, B, X, Y, or Z
  Description: If refPositionsFileN is a PDB file that contains all the atoms in the topology, this option may be provided to set which PDB field is used to flag the reference coordinates for atoms.

• refPositionsFileN $\langle\,$ File containing the reference positions for fitting$\,\rangle$
  Context: gspath and gzpath
  Acceptable values: UNIX filename
  Description: The path is defined by multiple refPositionsFiles which are similiar to refPositionsFile in the rmsd CV. If your path consists of $10$ nodes, you can list the coordinate file (in PDB or XYZ format) from refPositionsFile1 to refPositionsFile10.

• useSecondClosestFrame $\langle\,$ Define $\mathbf{s}_{m-1}$ as the second closest frame?$\,\rangle$
  Context: gspath and gzpath
  Acceptable values: boolean
  Default value: on
  Description: The definition assumes the second closest frame is neighbouring to the closest frame. This is not always true especially when the path is crooked. If this option is set to on (default), $\mathbf{s}_{m-1}$ is defined as the second closest frame. If this option is set to off, $\mathbf{s}_{m-1}$ is defined as the left or right neighbouring frame of the closest frame.

• useThirdClosestFrame $\langle\,$ Define $\mathbf{s}_{m+1}$ as the third closest frame?$\,\rangle$
  Context: gspath and gzpath
  Acceptable values: boolean
  Default value: off
  Description: The definition assumes the third closest frame is neighbouring to the closest frame. This is not always true especially when the path is crooked. If this option is set to on, $\mathbf{s}_{m+1}$ is defined as the third closest frame. If this option is set to off (default), $\mathbf{s}_{m+1}$ is defined as the left or right neighbouring frame of the closest frame.

• fittingAtoms $\langle\,$ The atoms that are used for alignment$\,\rangle$
  Context: gspath and gspath
  Acceptable values: Group of atoms
  Description: Before calculating $\mathbf{v}_1$ , $\mathbf{v}_2$ , $\mathbf{v}_3$ and $\mathbf{v}_4$ , the current frame need to be aligned to the corresponding reference frames. This option specifies which atoms are used to do alignment.

gzpath: distance from a path defined in atomic Cartesian coordinate space.

List of keywords (see also for additional options):

• useZsquare $\langle\,$ Compute $z^2$ instead of $z$ $\,\rangle$
  Context: gzpath
  Acceptable values: boolean
  Default value: off
  Description: $z$ is not differentiable when it is zero. This implementation workarounds it by setting the derivative of $z$ to zero when $z = 0$ . Another workaround is set this option to on, which computes $z^2$ instead of $z$ , and then $z^2$ is differentiable when it is zero.

The usage of gzpath and gspath is illustrated below:

colvar {
  # Progress along the path
  name gs
  # The path is defined by 5 reference frames (from string-00.pdb to string-04.pdb)
  # Use atomic coordinate from atoms 1, 2 and 3 to compute the path
  gspath {
    atoms {atomnumbers { 1 2 3 }}
    refPositionsFile1 string-00.pdb
    refPositionsFile2 string-01.pdb
    refPositionsFile3 string-02.pdb
    refPositionsFile4 string-03.pdb
    refPositionsFile5 string-04.pdb
  }
}
colvar {
  # Distance from the path
  name gz
  # The path is defined by 5 reference frames (from string-00.pdb to string-04.pdb)
  # Use atomic coordinate from atoms 1, 2 and 3 to compute the path
  gzpath {
    atoms {atomnumbers { 1 2 3 }}
    refPositionsFile1 string-00.pdb
    refPositionsFile2 string-01.pdb
    refPositionsFile3 string-02.pdb
    refPositionsFile4 string-03.pdb
    refPositionsFile5 string-04.pdb
  }
}

linearCombination: Helper CV to define a linear combination of other CVs

This is a helper CV which can be defined as a linear combination of other CVs. It maybe useful when you want to define the gspathCV {...} and the gzpathCV {...} as combinations of other CVs.

gspathCV: progress along a path defined in CV space.

In the gspathCV {...} and the gzpathCV {...} block all vectors, namely $\mathbf{z}$ and $\mathbf{s}_{k}$ are defined in CV space. More specifically, $\mathbf{z} = \left[{\xi}_{1}, \cdots, {\xi}_{n}\right]$ , where ${\xi}_{i}$ is the $i$ -th CV. $\mathbf{s}_{k} = \left[{\xi}_{k,1}, \cdots, {\xi}_{k,n}\right]$ , where ${\xi}_{k,i}$ means the $i$ -th CV of the $k$ -th reference frame. It should be note that these two CVs requires the pathFile option, which specifies a path file. Each line in the path file contains a set of space-seperated CV value of the reference frame. The sequence of reference frames matches the sequence of the lines.

List of keywords (see also for additional options):

• useSecondClosestFrame $\langle\,$ Define $\mathbf{s}_{m-1}$ as the second closest frame?$\,\rangle$
  Context: gspathCV and gzpathCV
  Acceptable values: boolean
  Default value: on
  Description: The definition assumes the second closest frame is neighbouring to the closest frame. This is not always true especially when the path is crooked. If this option is set to on (default), $\mathbf{s}_{m-1}$ is defined as the second closest frame. If this option is set to off, $\mathbf{s}_{m-1}$ is defined as the left or right neighbouring frame of the closest frame.

• useThirdClosestFrame $\langle\,$ Define $\mathbf{s}_{m+1}$ as the third closest frame?$\,\rangle$
  Context: gspathCV and gzpathCV
  Acceptable values: boolean
  Default value: off
  Description: The definition assumes the third closest frame is neighbouring to the closest frame. This is not always true especially when the path is crooked. If this option is set to on, $\mathbf{s}_{m+1}$ is defined as the third closest frame. If this option is set to off (default), $\mathbf{s}_{m+1}$ is defined as the left or right neighbouring frame of the closest frame.

• pathFile $\langle\,$ The file name of the path file.$\,\rangle$
  Context: gspathCV and gzpathCV
  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.

gzpathCV: distance from a path defined in CV space.

List of keywords (see also for additional options):

• useZsquare $\langle\,$ Compute $z^2$ instead of $z$ $\,\rangle$
  Context: gzpathCV
  Acceptable values: boolean
  Default value: off
  Description: $z$ is not differentiable when it is zero. This implementation workarounds it by setting the derivative of $z$ to zero when $z = 0$ . Another workaround is set this option to on, which computes $z^2$ instead of $z$ , and then $z^2$ is differentiable when it is zero.

The usage of gzpathCV and gspathCV is illustrated below:

colvar {
  # Progress along the path
  name gs
  # Path defined by the CV space of two dihedral angles
  gspathCV {
    pathFile ./path.txt
    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 gz
  gzpathCV {
    pathFile ./path.txt
    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}}
    }
  }
}