Field theoretic description of charge regulation interaction

Abstract

In order to find the exact form of the electrostatic interaction between two proteins with dissociable charge groups in aqueous solution, we have studied a model system composed of two macroscopic surfaces with charge dissociation sites immersed in a counterion-only ionic solution. Field-theoretic representation of the grand canonical partition function is derived and evaluated within the mean-field approximation, giving the Poisson-Boltzmann theory with the Ninham-Parsegian boundary condition. Gaussian fluctuations around the mean-field are then analyzed in the lowest order correction that we calculate analytically and exactly, using the path integral representation for the partition function of a harmonic oscillator with time-dependent frequency. The first order (one loop) free energy correction gives the interaction free energy that reduces to the zero-frequency van der Waals form in the appropriate limit but in general gives rise to a mono-polar fluctuation term due to charge fluctuation at the dissociation sites. Our formulation opens up the possibility to investigate the Kirkwood-Shumaker interaction in more general contexts where their original derivation fails.