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Abstract An integral equation method is presented for the 1D steady-state Poisson-Nernst-Planck equations modeling ion transport through membrane channels. The differential equations are recast as integral equations using Green’s 3rd identity yielding a fixed-point problem for the electric potential gradient and ion concentrations. The integrals are discretized by a combination of midpoint and trapezoid rules, and the resulting algebraic equations are solved by Gummel iteration. Numerical tests for electroneutral and non-electroneutral systems demonstrate the method’s 2nd order accuracy and ability to resolve sharp boundary layers. The method is applied to a 1D model of the K$$^+$$ ${}^{+}$ ion channel with a fixed charge density that ensures cation selectivity. In these tests, the proposed integral equation method yields potential and concentration profiles in good agreement with published results. 

Abstract The 3D reference interaction site model (3D‐RISM) of molecular solvation is a powerful tool for computing the equilibrium thermodynamics and density distributions of solvents, such as water and co‐ions, around solute molecules. However, 3D‐RISM solutions can be expensive to calculate, especially for proteins and other large molecules where calculating the potential energy between solute and solvent requires more than half the computation time. To address this problem, we have developed and implemented treecode summation for long‐range interactions and analytically corrected cut‐offs for short‐range interactions to accelerate the potential energy and long‐range asymptotics calculations in non‐periodic 3D‐RISM in the AmberTools molecular modeling suite. For the largest single protein considered in this work, tubulin, the total computation time was reduced by a factor of 4. In addition, parallel calculations with these new methods scale almost linearly and the iterative solver remains the largest impediment to parallel scaling. To demonstrate the utility of our approach for large systems, we used 3D‐RISM to calculate the solvation thermodynamics and density distribution of 7‐ring microtubule, consisting of 910 tubulin dimers, over 1.2 million atoms. 

Abstract The Poisson–Boltzmann (PB) implicit solvent model is a popular framework for studying the electrostatics of solvated biomolecules. In this model the dielectric interface between the biomolecule and solvent is often taken to be the molecular surface or solvent‐excluded surface (SES), and the quality of the SES triangulation is critical in boundary element simulations of the model. This work compares the performance of the MSMS and NanoShaper surface triangulation codes for a set of 38 biomolecules. While MSMS produces triangles of exceedingly small area and large aspect ratio, the two codes yield comparable values for the SES surface area and electrostatic solvation energy, where the latter calculations were performed using the treecode‐accelerated boundary integral (TABI) PB solver. However we found that NanoShaper is computationally more efficient and reliable than MSMS, especially when parameters are set to produce highly resolved triangulations. 

• To compute protein pKas, a continuum dielectric Poisson-Boltzmann model defined on a molecular domain and a solvent domain is used for computing the related electrostatic free energies (top left). • The PB model in its boundary integral form is accurately solved on the triangulated molecular surface (e.g. BPTI) accelerated by a fast Treecode algorithm (top right). • The method obtains the intrinsic pKa and then computes the protonation probability for a given pH including site-site interactions by going through an energy driven titrating procedure. Comparison with experimental results are provided (bottom left and right).