Complex analyticity of the nonlinear Poisson-Boltzmann equation for the interface problem with random domains.

Area of Science:

• Computational biophysics
• Mathematical modeling
• Electrostatics

Background:

• The nonlinear Poisson-Boltzmann equation (NPBE) models electrostatic interactions in ionic solutions, crucial for understanding protein behavior.
• Accurate modeling of protein interactions requires accounting for solvent-induced domain perturbations, leading to complex, high-dimensional problems.
• The 'curse of dimensionality' makes direct computation of statistics intractable for high-dimensional random perturbations.

Purpose of the Study:

• To demonstrate the analyticity of the NPBE solution with respect to analytic domain perturbations.
• To develop methods for quantifying uncertainty in electrostatic models of biomolecules.
• To establish a theoretical foundation for applying advanced computational techniques like sparse grids and neural networks to NPBE.

Main Methods:

• Application of the analytic implicit function theorem to establish analyticity.
• Utilizing the domain mapping method to handle perturbations.
• Derivation of a priori bounds for the region of analyticity.
• Testing the method on the Cucurbita Maxima Trypsin Inhibitor I (CMTI-I) molecule.

Main Results:

• Proved analyticity of the NPBE solution concerning analytic domain perturbations, a novel result for nonlinear problems.
• Established a method to derive bounds on the analyticity region.
• Demonstrated convergence rates consistent with analyticity for the CMTI-I molecule.
• Validated the applicability of the theoretical framework to practical biomolecular systems.

Conclusions:

• The analyticity of NPBE solutions with respect to domain perturbations opens avenues for efficient uncertainty quantification.
• The developed methodology is general and applicable to other nonlinear problems in computational science.
• This work bridges the gap between theoretical mathematical analysis and practical computational challenges in biophysics.