Contributions of Far-Field Hydrodynamic Interactions to the Kinetics of Electrostatically Driven Molecular Association Maciej Dlugosz,* ,† Jan M. Antosiewicz, ‡ Pawel Zieliń ski, §,∥,⊥ and Joanna Trylska ∥ † Centre of New Technologies, ‡ Department of Biophysics, Faculty of Physics, § Interdisciplinary Centre for Mathematical and Computational Modeling, and ∥ Centre of New Technologies, University of Warsaw, Z ̇ wirki i Wigury 93, 02-089 Warsaw, Poland ⊥ Department of Electronics and Information Technology, Institute of Electronic Systems, Warsaw University of Technology, Nowowiejska 15/19, 00-665 Warsaw, Poland ABSTRACT: We simulated the diffusional encounters in periodic systems of model isotropic and anisotropic molecules using Brownian dynamics. We considered the electrostatic, excluded volume, and far-field hydrodynamic forces between diffusing molecules. Our goal was to estimate to what extent the hydrodynamic interactions influence the association kinetics when the associating partners are oppositely charged and their direct electrostatic interactions are screened by small mobile ions of dissolved salt. Overall, including hydrodynamic interactions decreases the association rate constants. The relative magnitude of this decrease does not depend on the ionic strength for the association of isotropic charged objects. This also holds true for nonspecific association (i.e., without restrictions regarding the relative orientation of binding partners in an encounter complex) of anisotropic objects. However, such dependence is visible for orientation-specific association of anisotropic objects. Moreover, we observe that some orientations of anisotropic molecules are hydrodynamically favorable during their mutual approach, and that such molecules can be hydrodynamically steered toward a particular relative orientation. This hydrodynamic orientational steering is impeded in case of strong electrostatic interactions or steric hindrance. ■ INTRODUCTION Diffusional encounter of molecules is a first stage of many biologically relevant processes. Moreover, diffusion of mole- cules toward the encounter is often the rate-limiting step of the reaction with the relative diffusion of binding partners controlling the kinetics of association. The rates of diffusive encounter of molecules that are measured experimentally can be also predicted theoretically using, for example, Brownian dynamics (BD) simulations, 1−3 a computer simulation technique that is commonly used to study the association kinetics of macromolecules, proteins, and ligands. 4−7 On the basis of such BD simulations, the association rate constant, k, is evaluated as 1−3 β β = − − k k b () 1 (1 ) k b k c D () () D D (1) where k D (x) is the steady-state rate constant of diffusional encounter for two particles with the reaction distance x (where x equals either c or b, and c > b) as described in Smoluchowski’s theory: 8,9 π = k x Dxx () 4 () D (2) with D(x) being the relative translational diffusion coefficient. Smoluchowski’s result was further generalized by Kramers 10 and Debye 11 to account for the forces between particles that may be derived from potentials. The value of k D (x) can be computed analytically; 12,13 the factor β is the fraction of encounter trajectories, that is, the trajectories that satisfy the predefined reaction criteria, estimated on the basis of BD simulations in which thousands of independent trajectories of the two binding partners are generated. 1−3 While the direct intermolecular interactions (i.e., electro- statics, polar and nonpolar solvation effects, van der Waals potentials) between diffusing particles in BD simulations are often described with a high level of sophistication, 14,15 the intermolecular hydrodynamic interactions (HIs), resulting from the fact that the moving solute causes movements of the surrounding solvent and that, in turn, the moving solvent displaces other solute molecules, are usually neglected. The reason is either the computational cost of evaluating HIs for coarse-grained bead models of molecular systems 16−18 or the lack of an appropriate (and computationally efficient) theoretical description of HIs for rigid-body BD simulations of arbitrarily shaped molecules. 19 Only a few attempts to examine the effects of HIs on the kinetics of molecular association have been made so far, either theoretically or through BD simulations. 16,20−27 Friedman 20 estimated analytically that the hydrodynamic effect reduces about 15% the computed rate constant both for Received: February 8, 2012 Revised: April 3, 2012 Published: April 18, 2012 Article pubs.acs.org/JPCB © 2012 American Chemical Society 5437 dx.doi.org/10.1021/jp301265y | J. Phys. Chem. B 2012, 116, 5437−5447