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 degenerate in any linear fixed-charge model. The accompanying correction to the interaction energy has no fixed sign---it softens like-charge repulsion but can deepen or create attraction away from the isoelectric point. We benchmark three ingredients of the theory against published results; 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 measured second virial coefficients reasonably well but also predicts how charge regulation expands the range of pH and screening strength where the protein--protein interaction is attractive.