The Research Of A Nonuniform Ionic Size-modified Poisson-Boltzmann Model

Poisson-Boltzmann(PB)model,as an "implicit solven" model,is widely used in the study of electrostatic interactions of biomolecules in ionic solution.However,this model does not consider the effect of nonuniform ionic sizes in ionic solvent on the distribution of electrostatic potential of biomolecule models.In order to improve this model,the nonuniform ionic sizes modified Poisson-Boltzmann model is studied in this paper.The model can be derived from the electrostatic free energy functional minimization problem with box constraints and partial differential equation(PDE)constraints.However,it is difficult to analyze and numerically solve a nonuniform ionic size modified Poisson-Boltzmann model,because of the solution singularity caused by the Dirac-delta distributions at the fixed point charge.Therefore,in this paper,? is decomposed into three parts,?,? and G,based on the three-term solution decomposition scheme proposed in our previous work,where G has an analytical solution and contains all the singularities in ?,and ? and ? are the solutions of two interface problems,respectively.Then the weak forms of the two interface problems are derived by using the variational method.By handling the singularities in this model,the optimization problem with box constraints and PDE constraints is simplified,and then its solution existence and uniqueness is proved.Finally,based on the three-term solution decomposition scheme and the weak forms of ? and ?,we introduce the steps to solve the PDEs in FEniCS.And then,an active set-type Newton method is proposed in this paper for solving a nonuniform ionic size modified Poisson-Boltzmann model.This algorithm will effectively preserve the positivity of the concentration functions.