Pseudo-improper dihedral (PID) force field

Pseudo-improper dihedral (PID) force field is the second custom potential. It acts on “bundles” which consist of two central, interacting, residues ( numbered 2 and 5), along with the neighboring residues (1 and 3 for the residue 2 and 4, 6 for the residue 5). The general form of the potential is:

\[ V = \sum V_b(\mathtt{bundle}) \]

where the sum is over all the bundles, i.e. all pairs of non-terminal non-natively-connected residues along with the neighbors; whether the residues separated by 3 others should be included is controlled with the include separated by 3 setting. The bundle term is:

\[ V_b = \sum_i \lambda_i(\psi_{24}) \lambda_i(\psi_{42}) \phi_i(r_{24}) \]

The sum runs over two components of the PID potential: “bb+/-“, and “ss” components. These differ in the choice of the Lennard-Jones potential \(\phi_i\), as well as the parameters of the \(\lambda\) functions.

The \(\lambda\) functions are Gaussian-like, and are parametrized by two parameters: \(\alpha\) and \(\psi_0\). There are two variants of the mode of evaluation of the \(\lambda\) functions:

1. The “cosine” \(\lambda\) function:

   \[\begin{split} \lambda(\psi) = \begin{cases} \frac{1}{2}(1+\cos \alpha(\psi-\psi_0)) & |\alpha (\psi - \psi_0)| < \pi\\ 0 & \textrm{otherwise} \end{cases} \end{split}\]

2. The “algebraic” \(\lambda\) function:

   \[ \lambda(\psi) = \frac{t^2-2|t|+1}{2t^2-2|t|+1}, t = \frac{\alpha (\psi - \psi_0)}{\pi} \]

The terms \(\psi_{42}\) and \(\psi_{24}\) are the improper dihedral angles. For a configuration of three consecutive residues 123 and another residue 4, by \(\psi_{24}\) we shall denote the angle between the planes 123 and 143.

Parameter file entry

# Parameters for the pseudo-improper dihedral (PID) potential.
pseudo-improper dihedral:
  # Whether it's enabled.
  enabled: false

  # Whether to allow for interactions between residues separated by three other
  # residues.
  include (i, i+4): true

  # Parameters for the "lambda" functions.
  lambda:
    # Variant of the lambda function used: Options: "algebraic", "cosine"
    variant: algebraic

    # Function parameters for the "bb plus" term.
    bb+:
      alpha: 6.4 1/rad
      psi_0: 1.05 rad

    # Function parameters for the "bb minus" term.
    bb-:
      alpha: 6.0 1/rad
      psi_0: -1.44 rad

    # Function parameters for the "ss" term.
    ss:
      alpha: 1.2 1/rad
      psi_0: -0.23 rad

  # Parameters for the forces.
  forces:
    # Variant of the forces used; in this case, values other than "lj" and
    # "sink lj" aren't supported. The case for the "sink lj" is somewhat
    # special, as by default it *inherits* the values from the lj params spec
    # below to r_high, and sets r_low to zero. In other words, if sinking lj
    # is enabled, by default the sink extends to zero distance, and the end
    # of the sink is as set in `ss_pairs_csv`.
    variant: lj

    # Parameters for the "bb plus" LJ potential.
    bb+:
      lj params:
        depth: 0.2 eps
        r_min: 5.6 A

    # Parameters for the "bb minus" LJ potential.
    bb-:
      lj params:
        depth: 0.2 eps
        r_min: 6.2 A

    # Parameters for the "ss" LJ potential.
    ss:
      lj params:
        default:
          depth: 1.0 eps
        per pair:
          r_min: *ss_pairs_csv