Robust biased Brownian dynamics for rate constant calculation

A reaction probability is required to calculate the rate constant of a diffusion-dominated reaction. Due to the complicated geometry and potentially high dimension of the reaction probability problem, it is usually solved by a Brownian dynamics simulation, also known as a random walk or path integral method, instead of solving the equivalent partial differential equation by a discretization method. Building on earlier work, this article completes the development of a robust importance sampling algorithm for Brownian dynamics-i.e., biased Brownian dynamics with weight control-to overcome the high energy and entropy barriers in biomolecular association reactions. The biased Brownian dynamics steers sampling by a bias force, and the weight control algorithm controls sampling by a target weight. This algorithm is optimal if the bias force and the target weight are constructed from the solution of the reaction probability problem. In reality, an approximate reaction probability has to be used to construct the bias force and the target weight. Thus, the performance of the algorithm depends on the quality of the approximation. Given here is a method to calculate a good approximation, which is based on the selection of a reaction coordinate and the variational formulation of the reaction probability problem. The numerically approximated reaction probability is shown by computer experiments to give a factor-of-two speedup over the use of a purely heuristic approximation. Also, the fully developed method is compared to unbiased Brownian dynamics. The tests for human superoxide dismutase, Escherichia coli superoxide dismutase, and antisweetener antibody NC6.8, show speedups of 17, 35, and 39, respectively. The test for reactions between two model proteins with orientations shows speedups of 2578 for one set of configurations and 3341 for another set of configurations.     

Weighted-ensemble Brownian dynamics simulations for protein association reactions.

A new method, weighted-ensemble Brownian dynamics, is proposed for the simulation of protein-association reactions and other events whose frequencies of outcomes are constricted by free energy barriers. The method features a weighted ensemble of trajectories in configuration space with energy levels dictating the proper correspondence between "particles" and probability. Instead of waiting a very long time for an unlikely event to occur, the probability packets are split, and small packets of probability are allowed to diffuse almost immediately into regions of configuration space that are less likely to be sampled. The method has been applied to the Northrup and Erickson (1992) model of docking-type diffusion-limited reactions and yields reaction rate constants in agreement with those obtained by direct Brownian simulation, but at a fraction of the CPU time (10(-4) to 10(-3), depending on the model). Because the method is essentially a variant of standard Brownian dynamics algorithms, it is anticipated that weighted-ensemble Brownian dynamics, in conjunction with biophysical force models, can be applied to a large class of association reactions of interest to the biophysics community.     

Biased Brownian dynamics for rate constant calculation

An enhanced sampling method-biased Brownian dynamics-is developed for the calculation of diffusion-limited biomolecular association reaction rates with high energy or entropy barriers. Biased Brownian dynamics introduces a biasing force in addition to the electrostatic force between the reactants, and it associates a probability weight with each trajectory. A simulation loses weight when movement is along the biasing force and gains weight when movement is against the biasing force. The sampling of trajectories is then biased, but the sampling is unbiased when the trajectory outcomes are multiplied by their weights. With a suitable choice of the biasing force, more reacted trajectories are sampled. As a consequence, the variance of the estimate is reduced. In our test case, biased Brownian dynamics gives a sevenfold improvement in central processing unit (CPU) time with the choice of a simple centripetal biasing force.     

Brownian dynamics simulation of the competitive reactions: binase dimerization and the association of binase and barstar

A comparative study of the competitive reactions-the association reaction of binase with polypeptide inhibitor barstar and the reaction of binase dimerization-has been performed by the Brownian dynamics simulation method. It was shown that three types of the binase dimers could be formed and the dimerization reaction could compete with the inhibition reaction. The first type of the dimers leaves the active centre of binase free. During the formation of the dimers of the second and the third types the active centre of one or both binase molecules is blocked and ribonuclease becomes partially or fully inactive. Brownian dynamics simulation shown, that the ratio of competitive reaction rates depends on pH and ionic strength of solution.     

Brownian dynamics theory for predicting internal and external blockages of tetraethylammonium in the KcsA potassium channel

The theory of Brownian dynamics is used to model permeation and the blocking of KcsA potassium channels by tetraethylammonium (TEA). A novel Brownian dynamics simulation algorithm is implemented that comprises two free energy profiles; one profile is seen by the potassium ions and the other by the TEA molecules whose shape is approximated by a sphere. Our simulations reveal that internally applied TEA blocks the passage of K⁺ ions by physically occluding the pore. A TEA molecule in the external reservoir encounters an attractive energy-well created by four tyrosine residues at position 82, in addition to all other attractive and repulsive forces impinging on it. Using Brownian dynamics, we investigate how deep the energy-well needs to be to reproduce the experimentally determined inhibitory constant k_i for the TEA blockade of KcsA or the mutant Shaker T449Y. The one-dimensional free energy profile obtained from molecular dynamics is first converted into a one-dimensional potential energy profile, and is then transformed into a three-dimensional free energy profile in Brownian dynamics by adding the short-range potential from the channel walls. When converted, the free energy profile calculated from molecular dynamics gives a well-depth of ∼10 kT. We systematically alter the depths of the profiles, and then use Brownian dynamics simulations to numerically determine the current versus TEA-concentration curves. We show that the sequence of binding and unbinding events of the TEA molecule to the binding pocket can be modeled by a first-order Markov process. The Brownian dynamics simulations also reveal that the probability of a TEA molecule binding to the binding pocket in KcsA potassium channels increases exponentially with TEA concentration and depends also on the applied potential and the K⁺ concentration in the simulation assembly.     

Concerted simulations reveal how peroxidase compound III formation results in cellular oscillations

A major problem in mathematical modeling of the dynamics of complex biological systems is the frequent lack of knowledge of kinetic parameters. Here, we apply Brownian dynamics simulations, based on protein three-dimensional structures, to estimate a previously undetermined kinetic parameter, which is then used in biochemical network simulations. The peroxidase-oxidase reaction involves many elementary steps and displays oscillatory dynamics important for immune response. Brownian dynamics simulations were performed for three different peroxidases to estimate the rate constant for one of the elementary steps crucial for oscillations in the peroxidase-oxidase reaction, the association of superoxide with peroxidase. Computed second-order rate constants agree well with available experimental data and permit prediction of rate constants at physiological conditions. The simulations show that electrostatic interactions depress the rate of superoxide association with myeloperoxidase, bringing it into the range necessary for oscillatory behavior in activated neutrophils. Such negative electrostatic steering of enzyme-substrate association presents a novel control mechanism and lies in sharp contrast to the electrostatically-steered fast association of superoxide and Cu/Zn superoxide dismutase, which is also simulated here. The results demonstrate the potential of an integrated and concerted application of structure-based simulations and biochemical network simulations in cellular systems biology.     

Brownian dynamics study of the influences of electrostatic interaction and diffusion on protein-protein association kinetics.

A unified model is presented for protein-protein association processes that are under the influences of electrostatic interaction and diffusion (e.g., protein oligomerization, enzyme catalysis, electron and energy transfer). The proteins are modeled as spheres that bear point charges and undergo translational and rotational Brownian motion. Before association can occur the two spheres have to be aligned properly to form a reaction complex via diffusion. The reaction complex can either go on to form the product or it can dissociate into the separate reactants through diffusion. The electrostatic interaction, like diffusion, influences every step except the one that brings the reaction complex into the product. The interaction potential is obtained by extending the Kirkwood-Tanford protein model (Tanford, C., and J. G. Kirkwood. 1957. J. Am. Chem. Soc. 79:5333-5339) to two charge-embedded spheres and solving the consequent equations under a particular basis set. The time-dependent association rate coefficient is then obtained through Brownian dynamics simulations according an algorithm developed earlier (Zhou, H.-X. 1990. J. Phys. Chem. 94:8794-8800). This method is applied to a model system of the cytochrome c and cytochrome c peroxidase association process and the results confirm the experimental dependence of the association rate constant on the solution ionic strength. An important conclusion drawn from this study is that when the product is formed by very specific alignment of the reactants, as is often the case, the effect of the interaction potential is simply to scale the association rate constant by a Boltzmann factor. This explains why mutations in the interface of the reaction complex have strong influences on the association rate constant whereas those away from the interface have minimal effects. It comes about because the former mutations change the interaction potential of the reaction complex significantly and the latter ones do not.     

Brownian dynamics simulation of rigid particles of arbitrary shape in external fields

We have developed a Brownian dynamics simulation algorithm to generate Brownian trajectories of an isolated, rigid particle of arbitrary shape in the presence of electric fields or any other external agents. Starting from the generalized diffusion tensor, which can be calculated with the existing HYDRO software, the new program BROWNRIG (including a case-specific subprogram for the external agent) carries out a simulation that is analyzed later to extract the observable dynamic properties. We provide a variety of examples of utilization of this method, which serve as tests of its performance, and also illustrate its applicability. Examples include free diffusion, transport in an electric field, and diffusion in a restricting environment.     

Efficacy of external tetraethylammonium block of the KcsA potassium channel: Molecular and Brownian dynamics studies

Blockade of the KcsA potassium channel by externally applied tetraethylammonium is investigated using molecular dynamics calculations and Brownian dynamics simulations. In KcsA, the aromatic rings of four tyrosine residues located just external to the selectivity filter create an attractive energy well or a binding cage for a tetraethylammonium molecule. We first investigate the effects of re-orienting the four tyrosine residues such that the centers of the aromatic rings face the tetraethylammonium molecule directly. Then, we systematically move the residues inward in both orientations so that the radius of the binding cage formed by them becomes smaller. For each configuration, we construct a one-dimensional free energy profile by bringing in a tetraethylammonium molecule from the external reservoir toward the selectivity filter. The free energy profile is then converted to a one-dimensional potential energy profile, taking the available space between the tyrosine residues and the tetraethylammonium molecule into account. Incorporating this potential energy profile into the Brownian dynamics algorithm, we determine the conductance properties of the channel under various conditions, construct the current-tetraethylammonium-concentration curve and compare it with the experimentally determined inhibitory constant k_i for externally applied tetraethylammonium. We show that the experimentally determined binding affinity for externally applied tetraethylammonium can be replicated when each of the four tyrosine residues is moved inward by about 0.7 Å, irrespective of orientation of their aromatic rings.     

Brownian dynamics simulation of the association of Bacillus amyloliquefaciens ribonuclease (barnase) and Bacillus intermedius ribonuclease (binase) with barstar

A comparative study of the association of two ribonucleases, barnase and binase, with the polypeptide inhibitor barstar has been performed by the Brownian dynamics simulation method. It was shown that the method adequately reproduced the dependence of the association rate on pH and ionic strength of solution and the influence of mutations of some ribonuclease amino acids. Two types of energetically favorable complexes of binase-barstar encounter were determined. In the type I complex, the amino acids of binase active center take part in the complex formation. In the second complex, the active center is free. It was supposed that the temporary binding of barstar into complex of type II is competitive relative to the inhibition reaction. This can partially explain the decrease in the rate of binase inhibition as compared with the corresponding reaction of barnase.     

Multiple Time Step Brownian Dynamics for Long Time Simulation of Biomolecules

We report a multiple time step algorithm applied to an atomistic Brownian dynamics simulation for simulating the long time scale dynamics of biomolecules. The algorithm was based on the original multiple time step method; a short time step was used to keep faster motions in local equilibrium. When applied to a 28-mer # # ! folded peptide, the simulation gave stable trajectories and the computation time was reduced by a factor of 160 compared to a conventional molecular dynamics simulation using explicit water molecules. We applied it for the folding simulation of a 13-mer ! -helical peptide, giving a successful folding simulation. These results indicate that the Brownian dynamics with the multiple time step algorithm is useful for studies of biomolecular motions by long time simulation.     

Realistic protein-protein association rates from a simple diffusional model neglecting long-range interactions, free energy barriers, and landscape ruggedness

We develop a simple but rigorous model of protein-protein association kinetics based on diffusional association on free energy landscapes obtained by sampling configurations within and surrounding the native complex binding funnels. Guided by results obtained on exactly solvable model problems, we transform the problem of diffusion in a potential into free diffusion in the presence of an absorbing zone spanning the entrance to the binding funnel. The free diffusion problem is solved using a recently derived analytic expression for the rate of association of asymmetrically oriented molecules. Despite the required high steric specificity and the absence of long-range attractive interactions, the computed rates are typically on the order of 10(4)-10(6) M(-1) sec(-1), several orders of magnitude higher than rates obtained using a purely probabilistic model in which the association rate for free diffusion of uniformly reactive molecules is multiplied by the probability of a correct alignment of the two partners in a random collision. As the association rates of many protein-protein complexes are also in the 10(5)-10(6) M(-1) sec(-1) range, our results suggest that free energy barriers arising from desolvation and/or side-chain freezing during complex formation or increased ruggedness within the binding funnel, which are completely neglected in our simple diffusional model, do not contribute significantly to the dynamics of protein-protein association. The transparent physical interpretation of our approach that computes association rates directly from the size and geometry of protein-protein binding funnels makes it a useful complement to Brownian dynamics simulations.     

Brownian dynamics simulations of intramolecular energy transfer

A novel technique for modelling intramolecular energy transfer is presented. Brownian dynamics calculations are used to compute the trajectories of donor and acceptor species, and the instantaneous orientation factor is calculated during each temporal iteration. In this work, several model systems are considered. Trajectories were computed for energy transfer between a flexible donor and a rigidly fixed acceptor. We have considered configurations where the donor is, (1) tethered to a fixed point in space, but free to diffuse rotationally, and (2) constrained to wobble in a cone. The luminescence decay of the donor is ‘measured’, and a non-single-exponential decay is observed for configurations of efficient energy transfer. Luminescence anisotropy measurements of constrained and unconstrained donors reflect the contribution of both energy transfer and rotational diffusion to the shape of the anisotropy decay curve.     

Simulation of the diffusion-controlled reaction between superoxide and superoxide dismutase. I. Simple models

A Brownian dynamics simulation method is used to study the diffusion-influenced bimolecular reaction between superoxide and superoxide dismutase (SOD). Using simple models, the details of which are based on the crystallographic structure of SOD, it is found that the electrostatic charge distribution of SOD serves to guide superoxide into the active site and enhance the diffusion-controlled rate constant by about 40%.     

Computer simulations of glycolytic enzyme interactions with F-actin

Muscle actin and fructose-1,6-bisphosphate aldolase (aldolase) were chemically crosslinked to produce an 80 kDa product representing one subunit of aldolase linked to one subunit of actin. Hydroxylamine digestion of the crosslinked product resulted in two 40.5 kDa fragments, one that was aldolase linked to the 12 N-terminal residues of actin. Brownian dynamics simulations of muscle aldolase and GAPDH with F-actin (muscle, yeast, and various mutants) estimated the association free energy. Mutations of residues 1-4 of muscle actin to Ala individually or two in combination of the first four residues reduced the estimated binding free energy. Simulations showed that muscle aldolase binds with the same affinity to the yeast actin as to the double mutated muscle actin; these mutations make the N-terminal of muscle actin identical to yeast, supporting the conclusion that the actin N-terminus participates in binding. Because the depth of free energy wells for yeast and the double mutants is less than for native rabbit actin, the simulations support experimental findings that muscle aldolase and GAPDH have a higher affinity for muscle actin than for yeast actin. Furthermore, Brownian dynamics revealed that the lower affinity of yeast actin for aldolase and GAPDH compared to muscle actin, was directly related to the acidic residues at the N-terminus of actin.     

Brownian dynamics study of a multiply-occupied cation channel: application to understanding permeation in potassium channels.

The behavior of a multiply-occupied cation-selective channel has been computed by Brownian dynamics. The length, cross-section, ion-ion repulsion force, and ionic mobility within the channel are all estimated from data and physical reasoning. The only free parameter is a partition energy at the mouth of the channel, defining the free energy of an ion in the channel compared to the bath. It is presumed that this partition energy is associated with the energetics of exchanging a bulk hydration environment for a channel hydration environment. Varying the partition energy alone, keeping all other parameters fixed, gives approximately the full range of magnitudes of single channel conductances seen experimentally for K channels. Setting the partition energy at -11 kT makes the computed channel look similar to a squid axon K channel with respect to magnitude of conductance, shape of the I-V curve, non-unity of Ussing flux ratio exponents, decrease of current and increase of conductance with extracellular ion accumulation, and saturation at high ion concentration in the bathing solution. The model includes no preferred binding sites (local free energy minima) for ions in the channel. Therefore it follows that none of the above-mentioned properties of K channels are strong evidence for the existence of such sites. The model does not show supersaturation of current at very high bathing concentrations nor any pronounced voltage-dependence of the Ussing flux ratio exponent, suggesting that these features would require additional details not included in the model presented herein.     

Configurational Temperature for Brownian Dynamics

Abstract

Expressions for the configurational temperature are evaluated in Brownian dynamics simulations of a Lennard-Jones fluid and compared with the input temperature which is used to generate the random displacements. It is found that the two temperatures agree in the limit of large numbers of particles, and even for moderate system sizes the configurational temperature is a useful check on the correctness of the simulation algorithm. Investigation of the autocorrelation functions shows that for Lennard-Jones and power-law fluids, the correlation time of the configurational temperature is shorter than other typical thermodynamic quantities, and it generally increases with the range of the potential.     

Anomalous versus Slowed-Down Brownian Diffusion in the Ligand-Binding Equilibrium

Measurements of protein motion in living cells and membranes consistently report transient anomalous diffusion (subdiffusion) that converges back to a Brownian motion with reduced diffusion coefficient at long times after the anomalous diffusion regime. Therefore, slowed-down Brownian motion could be considered the macroscopic limit of transient anomalous diffusion. On the other hand, membranes are also heterogeneous media in which Brownian motion may be locally slowed down due to variations in lipid composition. Here, we investigate whether both situations lead to a similar behavior for the reversible ligand-binding reaction in two dimensions. We compare the (long-time) equilibrium properties obtained with transient anomalous diffusion due to obstacle hindrance or power-law-distributed residence times (continuous-time random walks) to those obtained with space-dependent slowed-down Brownian motion. Using theoretical arguments and Monte Carlo simulations, we show that these three scenarios have distinctive effects on the apparent affinity of the reaction. Whereas continuous-time random walks decrease the apparent affinity of the reaction, locally slowed-down Brownian motion and local hindrance by obstacles both improve it. However, only in the case of slowed-down Brownian motion is the affinity maximal when the slowdown is restricted to a subregion of the available space. Hence, even at long times (equilibrium), these processes are different and exhibit irreconcilable behaviors when the area fraction of reduced mobility changes.