A new analysis of electrostatic free energy minimization and Poisson–Boltzmann equation for protein in ionic solvent

• Give Poisson dielectric model a solution splitting formula and prove it to have a unique solution.
• Present a new modified electrostatic free energy minimization problem for protein in ionic solvent.
• Find the first and second Gâteaux derivatives of the target function of the new minimization problem.
• Prove the new modified electrostatic free energy minimization problem to have a unique solution.
• Prove solution equivalence between the new minimization problem and Poisson–Boltzmann equation.

In this paper, a novel solution decomposition of the Poisson dielectric model is proposed to modify a traditional electrostatic free energy minimization problem into one that is well defined for the case of protein in ionic solvent. The target function of this modified problem is shown to be strictly convex, weak sequentially lower semicontinuous, and twice continuously Fréchet differentiable. Its first and second Gâteaux derivatives are then found. Moreover, it is proved that this modified electrostatic free energy minimization problem has a unique solution, and its solution existence and uniqueness is equivalent to that of the Poisson–Boltzmann equation, a widely-used implicit solvent model for computing the electrostatics of biomolecules.