Finite concentration effects on diffusion-controlled reactions

Numerical calculation of protein-ligand binding rates through solution of the Smoluchowski equation using smoothed particle hydrodynamics

Background

The calculation of diffusion-controlled ligand binding rates is important for understanding enzyme mechanisms as well as designing enzyme inhibitors.

Methods

We demonstrate the accuracy and effectiveness of a Lagrangian particle-based method, smoothed particle hydrodynamics (SPH), to study diffusion in biomolecular systems by numerically solving the time-dependent Smoluchowski equation for continuum diffusion. Unlike previous studies, a reactive Robin boundary condition (BC), rather than the absolute absorbing (Dirichlet) BC, is considered on the reactive boundaries. This new BC treatment allows for the analysis of enzymes with “imperfect” reaction rates.

Results

The numerical method is first verified in simple systems and then applied to the calculation of ligand binding to a mouse acetylcholinesterase (mAChE) monomer. Rates for inhibitor binding to mAChE are calculated at various ionic strengths and compared with experiment and other numerical methods. We find that imposition of the Robin BC improves agreement between calculated and experimental reaction rates.

Conclusions

Although this initial application focuses on a single monomer system, our new method provides a framework to explore broader applications of SPH in larger-scale biomolecular complexes by taking advantage of its Lagrangian particle-based nature.

Gating and Intermolecular Interactions in Ligand-Protein Association: Coarse-Grained Modeling of HIV-1 Protease

Most biological processes are initiated or mediated by the association of ligands and proteins. This work studies multistep, ligand-protein association processes by Brownian dynamics simulations with coarse-grained models for HIV-1 protease (HIVp) and its neutral ligands. We report the average association times when the ligand concentration is 100 μM. The influence of crowding on the simulated binding time was also studied. HIVp has flexible loops that serve as a gate during the ligand binding processes. It is believed that the flaps are partially closed most of the time in its free state. To accelerate our simulations, we fixed a part of the HIVp and reparameterized our coarse-grained model, using atomistic molecular dynamics simulations, to reproduce the "gating" motions of HIVp. HIVp-ligand interactions changed the gating behavior of HIVp and helped ligands diffuse on HIVp surface to accelerate binding. The structural adjustment of the ligand toward its final stable state was the limiting step in the binding processes, which is highly system dependent. The intermolecular attraction between the ligands and crowder proteins contributes the most to the crowding effects. The results highlight broader implications in recognition pathways under more complex environment that considers molecular dynamics and conformational changes. This work brings insights into ligand-protein associations and is helpful in the design of targeted ligands.

Crowding effects on association reactions at membranes

The effect of macromolecular crowding on the binding of ligands to a receptor near membranes is studied using Brownian dynamics simulations. The receptor is modeled as a reactive patch on a hard surface and the ligands and crowding agents are modeled as spheres that interact via a steep repulsive interaction potential. When a ligand collides with the patch, it reacts with probability p(rxn). The association rate constant (k(infinity)) can be decomposed into contributions from diffusion-limited (k(D)) and reaction-limited (k(R)) rates, i.e., 1/k(infinity) = 1/k(D) + 1/k(R). The simulations show that k(D) is a nonmonotonic function of the volume fraction of crowding agents for receptors of small sizes. k(R) is always an increasing function of the volume fraction of crowding agents, and the association rate constant k(infinity) determined from both contributions has a qualitatively different dependence on the macromolecular crowding for high and low values of the reaction probability p(rxn). The simulation results are used to predict the velocity of the membrane protrusion driven by actin filament elongation. Based on the simple model where the protrusive force on the membrane is generated by the intercalation of actin monomers between the membrane and actin filament ends, we predict that crowding increases the local concentration of actin monomers near the filament ends and hence accelerates the membrane protrusion.

Kinetics of diffusion-controlled enzymatic reactions with charged substrates

The Debye-Hückel limiting law (DHL) has often been used to estimate rate constants of diffusion-controlled reactions under different ionic strengths. Two main approximations are adopted in DHL: one is that the solution of the linearized Poisson-Boltzmann equation for a spherical cavity is used to estimate the excess electrostatic free energy of a solution; the other is that details of electrostatic interactions of the solutes are neglected. This makes DHL applicable only at low ionic strengths and dilute solutions (very low substrate/solute concentrations). We show in this work that through numerical solution of the Poisson-Nernst-Planck equations, diffusion-reaction processes can be studied at a variety of conditions including realistically concentrated solutions, high ionic strength, and certainly with non-equilibrium charge distributions. Reaction rate coefficients for the acetylcholine-acetylcholinesterase system are predicted to strongly depend on both ionic strength and substrate concentration. In particular, they increase considerably with increase of substrate concentrations at a fixed ionic strength, which is open to experimental testing. This phenomenon is also verified on a simple model, and is expected to be general for electrostatically attracting enzyme-substrate systems.PACS Codes: 82.45.Tv, 87.15.VvMSC Codes: 92C30.

Molecular dynamics study on the influence of quencher concentration on the reaction rate for ionic systems

The influence of concentrations of reagents on the rate of reaction: A+B-->C+B for low density equimolar mixtures of spherically symmetric ions immersed in the Brownian medium has been investigated by performing large scale molecular dynamics simulations. The Coulomb potential of ion-ion interactions is truncated at the cutoff distance large enough to make the kinetics of the reaction independent of its value. The simulations have been performed at conditions close to that for quenching reactions for fluophores. One of the simulation results is that the excess in the rate coefficient Delta k is always positive and converges to a constant value which is two to three orders in magnitude higher than that for the soft spheres immersed in the Brownian medium [Litniewski, J. Chem. Phys. 124, 114502 (2006)]. Delta k is approximately proportional to c however, if the concentration is high, positive deviations [O(c(2))] are noticeable. The simulation results are compared with simple model that bases on the superposition approximation. The model predicts most of the properties of Delta k. The predicted values are about 30%-40% lower than that from the simulations.

Toward realistic modeling of dynamic processes in cell signaling: quantification of macromolecular crowding effects

One of the major factors distinguishing molecular processes in vivo from biochemical experiments in vitro is the effect of the environment produced by macromolecular crowding in the cell. To achieve a realistic modeling of processes in the living cell based on biochemical data, it becomes necessary, therefore, to consider such effects. We describe a protocol based on Brownian dynamics simulation to characterize and quantify the effect of various forms of crowding on diffusion and bimolecular association in a simple model of interacting hard spheres. We show that by combining the elastic collision method for hard spheres and the mean field approach for hydrodynamic interaction (HI), our simulations capture the correct dynamics of a monodisperse system. The contributions from excluded volume effect and HI to the crowding effect are thus quantified. The dependence of the results on size distribution of each component in the system is illustrated, and the approach is applied as well to the crowding effect on electrostatic-driven association in both neutral and charged environments; values for effective diffusion constants and association rates are obtained for the specific conditions. The results from our simulation approach can be used to improve the modeling of cell signaling processes without additional computational burdens.

Continuum simulations of acetylcholine consumption by acetylcholinesterase: a Poisson-Nernst-Planck approach

The Poisson-Nernst-Planck (PNP) equation provides a continuum description of electrostatic-driven diffusion and is used here to model the diffusion and reaction of acetylcholine (ACh) with acetylcholinesterase (AChE) enzymes. This study focuses on the effects of ion and substrate concentrations on the reaction rate and rate coefficient. To this end, the PNP equations are numerically solved with a hybrid finite element and boundary element method at a wide range of ion and substrate concentrations, and the results are compared with the partially coupled Smoluchowski-Poisson-Boltzmann model. The reaction rate is found to depend strongly on the concentrations of both the substrate and ions; this is explained by the competition between the intersubstrate repulsion and the ionic screening effects. The reaction rate coefficient is independent of the substrate concentration only at very high ion concentrations, whereas at low ion concentrations the behavior of the rate depends strongly on the substrate concentration. Moreover, at physiological ion concentrations, variations in substrate concentration significantly affect the transient behavior of the reaction. Our results offer a reliable estimate of reaction rates at various conditions and imply that the concentrations of charged substrates must be coupled with the electrostatic computation to provide a more realistic description of neurotransmission and other electrodiffusion and reaction processes.

The influence of interactions between reagents on the excess in the rate of quenching reaction: Molecular dynamics study

The influence of the interactions between reagents on the excess in the rate coefficient, Deltak, for the instantaneous reaction A+B-->C+B have been investigated by performing large scale molecular dynamics simulations for simple soft spheres. The simulation method has enabled us to determine the contributions to Deltak coming from A-B as well as B-B interactions. The simulations have shown that positive values of Deltak that appear both for the liquid and for the Brownian system [M. Litniewski, J. Chem. Phys. 123, 124506 (2005); 124, 114501 (2006)] result from B-B interactions. If B-B interactions were absent, Deltak was always negative. The influence of B-B interactions was about three times higher for the Brownian system than for the liquid. A qualitative explanation for the effect has been proposed basing on a simple model and analyzing the influence of B-B interactions on fluctuations in concentrations of reagents. The influence of A-B interactions was completely negligible except for the liquid at short times, for which the cancellation of A-B interaction noticeably decreased Deltak.

Substrate concentration dependence of the diffusion-controlled steady-state rate constant

The Smoluchowski approach to diffusion-controlled reactions is generalized to interacting substrate particles by including the osmotic pressure and hydrodynamic interactions of the nonideal particles in the Smoluchoswki equation within a local-density approximation. By solving the strictly linearized equation for the time-independent case with absorbing boundary conditions, we present an analytic expression for the diffusion-limited steady-state rate constant for small substrate concentrations in terms of an effective second virial coefficient B2*. Comparisons to Brownian dynamics simulations excluding hydrodynamic interactions show excellent agreement up to bulk number densities of B2*rho0 < approximately = 0.4 for hard sphere and repulsive Yukawa-like interactions between the substrates. Our study provides an alternative way to determine the second virial coefficient of interacting macromolecules experimentally by measuring their steady-state rate constant in diffusion-controlled reactions at low densities.

Gated binding of ligands to HIV-1 protease: Brownian dynamics simulations in a coarse-grained model

The internal motions of proteins may serve as a "gate" in some systems, which controls ligand-protein association. This study applies Brownian dynamics simulations in a coarse-grained model to study the gated association rate constants of HIV-1 proteases and drugs. The computed gated association rate constants of three protease mutants, G48V/V82A/I84V/L90M, G48V, and L90M with three drugs, amprenavir, indinavir, and saquinavir, yield good agreements with experiments. The work shows that the flap dynamics leads to "slow gating". The simulations suggest that the flap flexibility and the opening frequency of the wild-type, the G48V and L90M mutants are similar, but the flaps of the variant G48V/V82A/I84V/L90M open less frequently, resulting in a lower gated rate constant. The developed methodology is fast and provides an efficient way to predict the gated association rate constants for various protease mutants and ligands.

The influence of the quencher concentration on the rate of simple bimolecular reaction: Molecular dynamics study

The paper presents the results of large-scale molecular dynamics simulations of the irreversible bimolecular reaction A+B --> C+B for the simple liquid composed of mechanically identical soft spheres. The systems with the total number of molecules corresponding to 10(7)-10(9) are considered. The influence of the concentration of a quencher (B) on the surviving probability of A and the reaction rate is analyzed for a wide range of the concentrations and for two significantly different reduced densities. It is shown that the quencher concentration dependence effect (QCDE) is, in fact, a composition of two QCDE effects: the short-time QCDE that increases the reaction rate and the long-time QCDE that decreases it. The paper also analyzes the influence of the concentration on the steady-state rate constant, k(ss), obtained by integrating the surviving probability. The excess in k(ss) due to finite quencher concentration changes the sign from negative to positive while going from low to high concentrations. Generally, the excess is extremely weak. It attains a 1% level only if the concentration is very high.