Diffusion-controlled reactions between a spherical target and dumbbell dimer by Brownian dynamics simulation

Modeling the Self-Assembly of Protein Complexes through a Rigid-Body Rotational Reaction-Diffusion Algorithm

The reaction-diffusion equations provide a powerful framework for modeling nonequilibrium, cell-scale dynamics over the long time scales that are inaccessible by traditional molecular modeling approaches. Single-particle reaction-diffusion offers the highest resolution technique for tracking such dynamics, but it has not been applied to the study of protein self-assembly due to its treatment of reactive species as single-point particles. Here, we develop a relatively simple but accurate approach for building rigid structure and rotation into single-particle reaction-diffusion methods, providing a rate-based method for studying protein self-assembly. Our simplifying assumption is that reactive collisions can be evaluated purely on the basis of the separations between the sites, and not their orientations. The challenge of evaluating reaction probabilities can then be performed using well-known equations based on translational diffusion in both 3D and 2D, by employing an effective diffusion constant we derive here. We show how our approach reproduces both the kinetics of association, which is altered by rotational diffusion, and the equilibrium of reversible association, which is not. Importantly, the macroscopic kinetics of association can be predicted on the basis of the microscopic parameters of our structurally resolved model, allowing for critical comparisons with theory and other rate-based simulations. We demonstrate this method for efficient, rate-based simulations of self-assembly of clathrin trimers, highlighting how formation of regular lattices impacts the kinetics of association.

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.

Brownian dynamics simulation of protein-protein diffusional encounter

Protein association events are ubiquitous in biological systems. Some protein associations and subsequent responses are diffusion controlled in vivo. Hence, it is important to be able to compute bimolecular diffusional association rates for proteins. The Brownian dynamics simulation methodology may be used to simulate protein-protein encounter, compute association rates, and examine their dependence on protein mutation and the nature of the physical environment (e.g., as a function of ionic strength or viscosity). Here, the theory for Brownian dynamics simulations is described, and important methodological aspects, particularly pertaining to the correct modeling of electrostatic forces and definition of encounter complex formation, are highlighted. To illustrate application of the method, simulations of the diffusional encounter of the extracellular ribonuclease, barnase, and its intracellular inhibitor, barstar, are described. This shows how experimental rates for a series of mutants and the dependence of rates on ionic strength can be reproduced well by Brownian dynamics simulations. Potential future uses of the Brownian dynamics method for investigating protein-protein association are discussed.

Solvent effects on protein motion and protein effects on solvent motion. Dynamics of the active site region of lysozyme

The stochastic boundary molecular dynamics methodology is applied to the active site of the enzyme lysozyme. A comparison is made of in vacuo dynamics results from the stochastic boundary method and a full conventional molecular dynamics simulation of lysozyme. Excellent agreement between the two approaches is obtained. The influence of solvent on the residues in the active site region is explored and it is shown that both the structure and dynamics are affected. Of particular importance for the structure of the protein is the solvation of polar residues and the stabilization of like-charged ion pairs. The magnitude of the fluctuations is only slightly altered by the solvent; the overall increase in the root-mean-square fluctuations, relative to the vacuum run, is 11%. The solvent effect on dynamical properties is found not to be simply related to the solvent viscosity. Both the solvent exposure and dynamic aspects of protein-solvent interactions, including the relative time scales of the motions, are shown to play a role. The effects of the protein on solvent dynamics and structure are also observed to be significant. The solvent molecules around atoms in charged, polar and apolar side-chains show markedly different diffusion coefficients as well as exhibiting different solvation structures. One key example is the water around apolar groups, which is much less mobile than bulk water, or water solvating polar groups.

Computer-aided molecular design

Theoretical chemistry, as implemented on fast computers, is beginning to yield accurate predictions of the thermodynamic and kinetic properties of large molecular assemblies. In addition to providing detailed insights into the origins of molecular activity, theoretical calculations can be used to design new molecules with specific properties. This article describes two types of calculations that show special promise as design tools, the thermodynamic cycle-perturbation method and the Brownian reactive dynamics method. These methods can be applied to calculate equilibrium and rate constants that describe many aspects of molecular recognition, stability, and reactivity.