On regularization of charge singularities in solving the Poisson-Boltzmann equation with a smooth solute-solvent boundary

Siwen Wang; Emil Alexov; Shan Zhao; Siwen Wang; Emil Alexov; Shan Zhao

doi:10.3934/mbe.2021072

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 2: 1370-1405. doi: 10.3934/mbe.2021072

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On regularization of charge singularities in solving the Poisson-Boltzmann equation with a smooth solute-solvent boundary

1.
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA
2.
Department of Physics and Astronomy, Clemson University, Clemson, SC 29634, USA

Received: 25 October 2020 Accepted: 12 January 2021 Published: 21 January 2021

Numerical treatment of singular charges is a grand challenge in solving the Poisson-Boltzmann (PB) equation for analyzing electrostatic interactions between the solute biomolecules and the surrounding solvent with ions. For diffuse interface PB models in which solute and solvent are separated by a smooth boundary, no effective algorithm for singular charges has been developed, because the fundamental solution with a space dependent dielectric function is intractable. In this work, a novel regularization formulation is proposed to capture the singularity analytically, which is the first of its kind for diffuse interface PB models. The success lies in a dual decomposition – besides decomposing the potential into Coulomb and reaction field components, the dielectric function is also split into a constant base plus space changing part. Using the constant dielectric base, the Coulomb potential is represented analytically via Green's functions. After removing the singularity, the reaction field potential satisfies a regularized PB equation with a smooth source. To validate the proposed regularization, a Gaussian convolution surface (GCS) is also introduced, which efficiently generates a diffuse interface for three-dimensional realistic biomolecules. The performance of the proposed regularization is examined by considering both analytical and GCS diffuse interfaces, and compared with the trilinear method. Moreover, the proposed GCS-regularization algorithm is validated by calculating electrostatic free energies for a set of proteins and by estimating salt affinities for seven protein complexes. The results are consistent with experimental data and estimates of sharp interface PB models.
- Poisson-Boltzmann equation,
- singular charge source,
- regularization,
- diffuse interface,
- Gaussian convolution surface,
- salt effect
Citation: Siwen Wang, Emil Alexov, Shan Zhao. On regularization of charge singularities in solving the Poisson-Boltzmann equation with a smooth solute-solvent boundary[J]. Mathematical Biosciences and Engineering, 2021, 18(2): 1370-1405. doi: 10.3934/mbe.2021072

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Abstract

Numerical treatment of singular charges is a grand challenge in solving the Poisson-Boltzmann (PB) equation for analyzing electrostatic interactions between the solute biomolecules and the surrounding solvent with ions. For diffuse interface PB models in which solute and solvent are separated by a smooth boundary, no effective algorithm for singular charges has been developed, because the fundamental solution with a space dependent dielectric function is intractable. In this work, a novel regularization formulation is proposed to capture the singularity analytically, which is the first of its kind for diffuse interface PB models. The success lies in a dual decomposition – besides decomposing the potential into Coulomb and reaction field components, the dielectric function is also split into a constant base plus space changing part. Using the constant dielectric base, the Coulomb potential is represented analytically via Green's functions. After removing the singularity, the reaction field potential satisfies a regularized PB equation with a smooth source. To validate the proposed regularization, a Gaussian convolution surface (GCS) is also introduced, which efficiently generates a diffuse interface for three-dimensional realistic biomolecules. The performance of the proposed regularization is examined by considering both analytical and GCS diffuse interfaces, and compared with the trilinear method. Moreover, the proposed GCS-regularization algorithm is validated by calculating electrostatic free energies for a set of proteins and by estimating salt affinities for seven protein complexes. The results are consistent with experimental data and estimates of sharp interface PB models.

References

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