A closed-form theory of charge-regulated electrostatic interactions between anisotropically charged spheres
Abstract
Proteins and many other colloids carry ionizable surface groups that are both spatially inhomogeneous (patchy) and pH-responsive (charge-regulating). In the description of the electrostatic interactions between such particles, these two aspects are often treated separately, especially from a theoretical perspective. We present a unified, closed-form theory that unites charge regulation and patchiness within the linearized Poisson--Boltzmann approximation. For spherical particles with anisotropic distributions of titratable sites, we derive both the leading-order mean-field interaction energy as well as the Kirkwood--Shumaker fluctuation interaction. The theory exposes a qualitatively new effect between two electroneutral anisotropic particles: proximity-induced charge regulation generates a net monopole, and splits orientational branches that are {mirror-symmetric} in any linear fixed-charge model. The accompanying correction to the interaction energy has no fixed sign---it weakens like-charge repulsion but can deepen an existing attraction or create one where fixed charge predicts none. We benchmark three components of the theory against published results, as no existing simulation probes their combination, which would capture the central prediction of the theory. Lastly, we apply the theory to two proteins---lysozyme and α-chymotrypsinogen A---and show that it not only reproduces the general trends of the measured second virial coefficients but also predicts how charge regulation expands the range of pH and screening strength where the protein--protein interaction is attractive.
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BibTeX
@misc{bozic2026closedform,
title = {{A closed-form theory of charge-regulated electrostatic interactions between anisotropically charged spheres}},
author = {Anže Božič},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.30566},
eprint = {2608.30566},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30566},
}
RIS
TY - UNPB
TI - A closed-form theory of charge-regulated electrostatic interactions between anisotropically charged spheres
AU - Anže Božič
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2608.30566
UR - https://arxiv.org/abs/2608.30566
AB - Proteins and many other colloids carry ionizable surface groups that are both spatially inhomogeneous (patchy) and pH-responsive (charge-regulating). In the description of the electrostatic interactions between such particles, these two aspects are often treated separately, especially from a theoretical perspective. We present a unified, closed-form theory that unites charge regulation and patchiness within the linearized Poisson--Boltzmann approximation. For spherical particles with anisotropic distributions of titratable sites, we derive both the leading-order mean-field interaction energy as well as the Kirkwood--Shumaker fluctuation interaction. The theory exposes a qualitatively new effect between two electroneutral anisotropic particles: proximity-induced charge regulation generates a net monopole, and splits orientational branches that are {mirror-symmetric} in any linear fixed-charge model. The accompanying correction to the interaction energy has no fixed sign---it weakens like-charge repulsion but can deepen an existing attraction or create one where fixed charge predicts none. We benchmark three components of the theory against published results, as no existing simulation probes their combination, which would capture the central prediction of the theory. Lastly, we apply the theory to two proteins---lysozyme and α-chymotrypsinogen A---and show that it not only reproduces the general trends of the measured second virial coefficients but also predicts how charge regulation expands the range of pH and screening strength where the protein--protein interaction is attractive.
ER -