Calculating the bimolecular rate of protein-protein association with interacting crowders

We have recently introduced a method termed Poisson-Boltzmann semianalytical method (PB-SAM) for solving the linearized Poisson-Boltzmann equation for large numbers of arbitrarily shaped dielectric cavities with controlled precision. In this work we extend the applicability of the PB-SAM approach by deriving force and torque expressions that fully account for mutual polarization in both the zero- and first-order derivatives of the surface charges, that can now be embedded into a Brownian dynamics scheme to look at electrostatic-driven mesoscale assembly and kinetics. We demonstrate the capabilities of the PB-SAM approach by simulating the protein concentration effects on the bimolecular rate of association of barnase and barstar, under periodic boundary conditions and evaluated through mean first passage times. We apply PB-SAM to the pseudo-first-order reaction rate conditions in which either barnase or barstar are in great excess relative to the other protein (124:1). This can be considered a specific case in which the PB-SAM approach can be applied to crowding conditions in which crowders are not inert but can form interactions with other molecules.