is a nonlinear equation which solves for the electrostatic field, , based on the position dependent dielectric, , the position-dependent accessibility of position to the ions in solution, , the solute charge distribution, , and the bulk charge density, , of ion . While this equation does exactly solve for the electrostic field of a charge distribution in a dielectric, it is very expensive to solve, and therefore not suitable for molecular dynamics.

Forces are contributed by other nearby atoms within a cutoff. The GB force on atom is the derivative of the total GB energy with respect to relative atom distances ,

(8) | |||

(9) | |||

(10) |

where the partial derivatives are included since the Born radius, , is a function of all relative atom distances. The total GB energy of the system is

(11) |

where is the Born radius dependent self energy of atom , and the GB energy between atoms and is given by

(12) |

The dielectric term [73] is

(13) |

and the GB function [74] is

(14) |

As the Born radii of atoms and decrease (increasing screening), the effective distance between the atoms ( ) increases. The implicit solvent implemented in NAMD is the model of Onufriev, Bashford and Case [60,61] which calculates the Born radius as

(15) |

where

(16) |

is the piecewise descreening function [61,35,67]; the seven piecewise regimes are

(17) |

and the values of are

(18) |

Below are defined the derivatives of the above functions which are required for force calculations.

(19) |

(20) |

(21) |

(22) |

(23) | |||

(24) | |||

(25) |

(26) |

(27) |

(28) |

(29) |

Other variables referenced in the above GB equations are - distance between atoms i and j; calculated from atom coordinates.

- - debye screening length; calculated from ion concentration, ; Å for 0.1 M monovalent salt.
- - dielectric constant of solvent.
- - dielectric constant of protein.
- - Born radius of atom .
- - intrinsic radius of atom taken from Bondi [9].
- - intrinsic radius offset; Å by default [61].
- - atom radius scaling factor [35,73].
- - Coulomb's constant, , 332.063711 kcal Å / e .
- [61]