Hydrodynamic effects on the relative rotational velocity of associating proteins

On the possibility of the existence of orienting hydrodynamic steering effects in the kinetics of receptor-ligand association

In the vast majority of biologically relevant cases of receptor-ligand complex formation, the binding site of the receptor is a small part of its surface, and moreover, formation of a biologically active complex often requires a specific orientation of the ligand relative to the binding site. Before the formation of the initial form of the complex, only long-range, electrostatic and hydrodynamic interactions can act between the ligand approaching the binding site and the receptor. In this context, the question arises whether as a result of these interactions, there is a pre-orientation of the ligand towards the binding site, which to some extent would accelerate the formation of the complex. The role of electrostatic interactions in the orientation of the ligand relative to the binding site of the receptor is well documented. The analogous role of hydrodynamic interactions, although assessed as very significant by Brune and Kim (PNAS 91, 2930-2934, (1994)), is still debatable. In this article, I present the current state of knowledge on this subject and consider the possibilities of demonstrating the orienting effect of hydrodynamic interactions in the processes of receptor-ligand association, in an experimental way supported by computer simulations.

Searching for Hydrodynamic Orienting Effects in the Association of Tri-N-acetylglucosamine with Hen Egg-White Lysozyme

Using stopped-flow fluorometry, we determined rate constants for the formation of diffusional encounter complexes of tri-N-acetylglucosamine (NAG₃) with hen egg-white lysozyme (k_a^WT) and its double mutant Asp48Asn/Lys116Gln (k_a^MT). We defined binding anisotropy, κ ≡ (k_a^WT – k_a^MT)/(k_a^WT + k_a^MT), and determined its ionic strength dependence. Our goal was to check if this ionic strength dependence provides information about the orienting hydrodynamic effects in the ligand-binding process. We also computed ionic strength dependence of the binding anisotropy from Brownian dynamics simulations using simple models of the lysozyme–NAG₃ system. The results of our experiments indicate that in the case of lysozyme and NAG₃ such hydrodynamic orienting effects are rather negligible. On the other hand, the results of our Brownian dynamics simulations prove that there exist molecular systems for which such orienting effects are substantial. However, the ionic strength dependence of the rate constants for the wild-type and modified systems do not exhibit any qualitative features that would allow us to conclude the presence of hydrodynamic orienting effects from stopped-flow experiments alone. Nevertheless, the results of our simulations suggest the presence of hydrodynamic orienting effects in the receptor–ligand association when the anisotropy of binding depends on the solvent viscosity.

Hydrodynamic Steering in Protein Association Revisited: Surprisingly Minuscule Effects of Considerable Torques

We investigate the previously postulated hydrodynamic steering phenomenon, resulting from complication of molecular shapes, its magnitude and possible relevance for protein-ligand and protein-protein diffusional encounters, and the kinetics of diffusion-controlled association. We consider effects of hydrodynamic interactions in a prototypical model system consisting of a cleft enzyme and an elongated substrate, and real protein-protein complexes, that of barnase and barstar, and human growth hormone and its binding protein. The kinetics of diffusional encounters is evaluated on the basis of rigid-body Brownian dynamics simulations in which hydrodynamic interactions between molecules are modeled using the bead-shell method for detailed description of molecular surfaces, and the first-passage-time approach. We show that magnitudes of steering torques resulting from the hydrodynamic coupling of associating molecules, evaluated for the studied systems on the basis of the Stokes-Einstein type relations for arbitrarily shaped rigid bodies, are comparable with magnitudes of torques resulting from electrostatic interactions of binding partners. Surprisingly, however, unlike in the case of electrostatic torques that strongly affect the diffusional encounter, overall effects of hydrodynamic steering torques on the association kinetics, while clearly discernible in Brownian dynamics simulations, are rather minute. We explain this result as a consequence of the thermal agitation of the binding partners. Our finding is relevant for the general understanding of a wide spectrum of molecular processes in solution but there is also a more practical aspect to it if one considers the low level of shape detail of models that are usually employed to evaluate hydrodynamic interactions in particle-based Stokesian and Brownian dynamics simulations of multicomponent biomolecular systems. Results described in the current work justify, in part at least, such a low-resolution description.

Effects of Spatially Dependent Mobilities on the Kinetics of the Diffusion-Controlled Association Derived from the First-Passage-Time Approach

Brownian dynamics (BD) simulations and the first-passage-time approach are applied to investigate diffusion-controlled association in a biologically relevant model system consisting of a fixed receptor with an elongated cavity and a capsule-like ligand that fits this cavity precisely. Before the binding at the receptor cavity, the ligand undergoes translational and rotational diffusion, either free or under the influence of electrostatic interactions with the receptor. The spatial dependence of the translational and rotational mobilities of the ligand resulting from its hydrodynamic interactions (HIs) with the receptor is accounted for in BD simulations, and an accurate numerical approach is applied for the evaluation of the spatially dependent mobility tensor of the ligand. Different magnitudes of electrostatic interactions (either attraction or repulsion) between the ligand and receptor are considered. The effective range of receptor-ligand electrostatic interactions is varied to account for their screening under different conditions of ionic strength. The effects of HIs on the kinetics of the diffusion-controlled association, evaluated for different electrostatic properties of binding partners, are thoroughly analyzed and discussed.

Effects of Hydrodynamic Interactions on the Apparent 1D Mobility of a Nonspecifically Bound Protein Following a Helical Path around DNA

We investigated effects of hydrodynamic interactions on diffusivities of proteins that undergo rotation-coupled sliding along DNA. For that, we applied numerical calculations of mobility and friction tensors to systems consisting of detailed bead-shell models of DNA and proteins of different size. Using tensors that result from these calculations along with an expression for the instantaneous energy dissipation rate due to motions of a nonspecifically bound protein that follows a helical track around DNA, we evaluated apparent one-dimensional friction and mobility coefficients for model proteins. The results that we obtained indicate that hydrodynamic interactions between DNA and proteins may substantially (even several-fold) reduce the apparent one-dimensional diffusivity of proteins, when compared with results of other theoretical analyses of the rotation-coupled sliding of proteins along DNA that neglect hydrodynamic effects. Moreover, accounting for hydrodynamic effects decreases the gap between values of diffusion coefficients of proteins on DNA measured experimentally and those estimated based on theoretical calculations and analyses applied to model systems. Altogether, the current study gives insights into the significance of hydrodynamic interactions in determination of the rate of finding target sites by DNA-binding proteins.

Toward an Accurate Modeling of Hydrodynamic Effects on the Translational and Rotational Dynamics of Biomolecules in Many-Body Systems

Proper treatment of hydrodynamic interactions is of importance in evaluation of rigid-body mobility tensors of biomolecules in Stokes flow and in simulations of their folding and solution conformation, as well as in simulations of the translational and rotational dynamics of either flexible or rigid molecules in biological systems at low Reynolds numbers. With macromolecules conveniently modeled in calculations or in dynamic simulations as ensembles of spherical frictional elements, various approximations to hydrodynamic interactions, such as the two-body, far-field Rotne-Prager approach, are commonly used, either without concern or as a compromise between the accuracy and the numerical complexity. Strikingly, even though the analytical Rotne-Prager approach fails to describe (both in the qualitative and quantitative sense) mobilities in the simplest system consisting of two spheres, when the distance between their surfaces is of the order of their size, it is commonly applied to model hydrodynamic effects in macromolecular systems. Here, we closely investigate hydrodynamic effects in two and three-body systems, consisting of bead-shell molecular models, using either the analytical Rotne-Prager approach, or an accurate numerical scheme that correctly accounts for the many-body character of hydrodynamic interactions and their short-range behavior. We analyze mobilities, and translational and rotational velocities of bodies resulting from direct forces acting on them. We show, that with the sufficient number of frictional elements in hydrodynamic models of interacting bodies, the far-field approximation is able to provide a description of hydrodynamic effects that is in a reasonable qualitative as well as quantitative agreement with the description resulting from the application of the virtually exact numerical scheme, even for small separations between bodies.