An Analytical Electrostatic Model for Salt Screened Interactions between Multiple Proteins Itay Lotan* and Teresa Head-Gordon* ,² Department of Bioengineering, UniVersity of California, Berkeley, Berkeley, California 94720 Received October 24, 2005 Abstract: We present a new general analytical solution for computing the screened electrostatic interaction between multiple macromolecules of arbitrarily complex charge distributions, assuming they are well described by spherical low dielectric cavities in a higher dielectric medium in the presence of a Debye-Hu¨ ckel treatment of salt. The benefits to this approach are 3-fold. First, by exploiting multipole expansion theory for the screened Coulomb potential, we can describe direct charge-charge interactions and all significant higher-order cavity polarization effects be- tween low dielectric spherical cavities containing their charges, while treating these higher order terms correctly at all separation distances. Second, our analytical solution is general to arbitrary numbers of macromolecules, is efficient to compute, and can therefore simultaneously provide on-the-fly updates to changes in charge distributions due to protein conformational changes. Third, we can change spatial resolutions of charge description as a function of separation distance without compromising the desired accuracy. While the current formulation describes solutions based on simple spherical geometries, it appears possible to reformulate these electrostatic expressions to smoothly increase spatial resolution back to greater molecular detail of the dielectric boundaries. 1. Introduction Atomistic molecular dynamics simulations routinely and ben- eficially address materials problems in the condensed phase. However, there is another set of problems on the supramolec- ular scale where the limitations of size and time scales are reached, examples being the recognition events and subse- quent complexation of multiple proteins in explicit solvent environments, or the study of phase behavior and interfacial properties of colloid systems. Atomistic modeling is too com- putationally demanding to evaluate for times long enough to measure the macromolecules’ traversal over large spatial do- mains and to do this with enough statistical confidence to analyze complex phase behavior, association kinetics, or mechanism of assembly. Fortunately, coarse-grained models may actually be the more sensible approach when large-scale spatial organization or dynamic events occurring over long time scales are operative. Spatial coarse-graining would involve, for ex- ample, removing explicit solvent molecules and ions as well as ignoring an individual macromolecule’s internal motion for some period of time. At large spatial separations between charged macromol- ecules in solution, electrostatic interactions will dominate, so that an appropriate coarse-grained model could describe them as complex charge distributions imbedded in a low dielectric medium surrounded by a high dielectric solvent continuum with salt screening defined by explicit microions or implicitly through a Debye-Hu¨ckel treatment. The elec- trostatic potential and forces and torques are found by solving the full Poisson-Boltzmann equation (PBE) or when expand- ing the exponential and keeping only linear terms, by solving the corresponding linearized PBE. Both linearized and the full PBE are typically solved numerically using either a finite-difference (FD) method, boundary-element (BE) meth- * Corresponding authors e-mail: itayl@berkeley.edu (I.L.) and TLHead-Gordon@lbl.gov (T.H.-G.). ² Schlumberger Fellow, Chemistry Department, Cambridge Uni- versity, Lensfield Road, Cambridge CB2 1EW, United King- dom. 541 J. Chem. Theory Comput. 2006, 2, 541-555 10.1021/ct050263p CCC: $33.50 © 2006 American Chemical Society Published on Web 03/16/2006