Dielectric Boundary Force In Molecular Solvation With The Poisson–boltzmann Equation: A Shape Derivative Approach

The classical Poisson-Boltzmann(PB)model,as an implicit solvent continuum model,is widely used in the study of electrostatic interactions of biomolecules in inoic solution.The electrostatic free energy is given through the electrostatic potential which is the solution to the boundary-value problem of the Poisson–Boltzmann equation.The shape derivative of electrostatic free energy with respect to solute-solvent interface is derived.Thus,the dielectric boundary force in the normal direction is derived.Firstly,the model of the classical PB and the functional of electrostatic free energy are introduced.The explicit formula of dielectric boundary force based on classical PB model is derived.In this model,however,the influence of ion size in ionic solvent on the electrostatic potential energy distribution of the biomolecular model is neglected.To consider the influence of ion size in the electrostatic force,the uniform size-modefied Poisson-Boltzmann equation is considered in this paper.The dielectric boundary forces of the two kinds of electrostatic free energies are analyzed with the uniform ionic size PB equation.In order to deal with the nonhomogeneous boundary conditions and two phase dielectric coe cient ",two auxiliary functions are introduced which satisfy the homogeneous and nonhomogeneous Poisson equations,respectively.The continuity and di?erentiability of electrostatic potential with respect to boundary variations are analyzed.The lemmas of boundary variations are obtained.The explicit formula of the dielectric boundary force based on two functionals of electrostatic free energy are derived.It is could be concluded that the dielectric boundary forces of the two functionals of electrostatic free energy are the generalized formulas compared to the classical PB model.