Simulation of nitroxide electron paramagnetic resonance spectra from brownian trajectories and molecular dynamics simulations

A simulated continuous wave electron paramagnetic resonance spectrum of a nitroxide spin label can be obtained from the Fourier transform of a free induction decay. It has been previously shown that the free induction decay can be calculated by solving the time-dependent stochastic Liouville equation for a set of Brownian trajectories defining the rotational dynamics of the label. In this work, a quaternion-based Monte Carlo algorithm has been developed to generate Brownian trajectories describing the global rotational diffusion of a spin-labeled protein. Also, molecular dynamics simulations of two spin-labeled mutants of T4 lysozyme, T4L F153R1, and T4L K65R1 have been used to generate trajectories describing the internal dynamics of the protein and the local dynamics of the spin-label side chain. Trajectories from the molecular dynamics simulations combined with trajectories describing the global rotational diffusion of the protein are used to account for all of the dynamics of a spin-labeled protein. Spectra calculated from these combined trajectories correspond well to the experimental spectra for the buried site T4L F153R1 and the helix surface site T4L K65R1. This work provides a framework to further explore the modeling of the dynamics of the spin-label side chain in the wide variety of labeling environments encountered in site-directed spin labeling studies.     

Hydrodynamic study of flexibility in immunoglobulin IgG1 using Brownian dynamics and the Monte Carlo simulations of a simple model

A simple bead model is proposed for the antibody molecule immunoglobulin IgG1. The partial flexibility of the hinge is represented by a quadratic potential associated to the angles between arms. Conformational and hydrodynamic properties are calculated using Monte Carlo (rigid-body) and Brownian dynamics simulations. Comparison of experimental and calculated values for some overall properties allows the assignment of dimensions and other model parameters. The Brownian dynamics technique is used next to simulate a rotational correlation function that is comparable with the decay of fluorescence emission anisotropy. This is done with varying flexibility at the hinge. The longest relaxation time shows a threefold decrease when going from the rigid Y-shaped conformation to the completely flexible case. The calculations are in good agreement with the decay times observed for IgG1. A flexibility analysis of the latter indicates that a variability of +/- 55 degrees (standard deviation) in the angle between the Fab arms.     

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.     

Diffusional channeling in the sulfate-activating complex: combined continuum modeling and coarse-grained brownian dynamics studies

Enzymes required for sulfur metabolism have been suggested to gain efficiency by restricted diffusion (i.e., channeling) of an intermediate APS^2- between active sites. This article describes modeling of the whole channeling process by numerical solution of the Smoluchowski diffusion equation, as well as by coarse-grained Brownian dynamics. The results suggest that electrostatics plays an essential role in the APS^2- channeling. Furthermore, with coarse-grained Brownian dynamics, the substrate channeling process has been studied with reactions in multiple active sites. Our simulations provide a bridge for numerical modeling with Brownian dynamics to simulate the complicated reaction and diffusion and raise important questions relating to the electrostatically mediated substrate channeling in vitro, in situ, and in vivo.     

Simulating nanoscale functional motions of biomolecules

We are describing efficient dynamics simulation methods for the characterization of functional motion of biomolecules on the nanometer scale. Multivariate statistical methods are widely used to extract and enhance functional collective motions from molecular dynamics (MD) simulations. A dimension reduction in MD is often realized through a principal component analysis (PCA) or a singular value decomposition (SVD) of the trajectory. Normal mode analysis (NMA) is a related collective coordinate space approach, which involves the decomposition of the motion into vibration modes based on an elastic model. Using the myosin motor protein as an example we describe a hybrid technique termed amplified collective motions (ACM) that enhances sampling of conformational space through a combination of normal modes with atomic level MD. Unfortunately, the forced orthogonalization of modes in collective coordinate space leads to complex dependencies that are not necessarily consistent with the symmetry of biological macromolecules and assemblies. In many biological molecules, such as HIV-1 protease, reflective or rotational symmetries are present that are broken using standard orthogonal basis functions. We present a method to compute the plane of reflective symmetry or the axis of rotational symmetry from the trajectory frames. Moreover, we develop an SVD that best approximates the given trajectory while respecting the symmetry. Finally, we describe a local feature analysis (LFA) to construct a topographic representation of functional dynamics in terms of local features. The LFA representations are low-dimensional, and provide a reduced basis set for collective motions, but unlike global collective modes they are sparsely distributed and spatially localized. This yields a more reliable assignment of essential dynamics modes across different MD time windows.     

Brownian dynamics study of an open-state KcsA potassium channel.

Three-dimensional Brownian dynamics simulations are used to study conductance of the KcsA potassium channel using the known crystallographic structure. Employing an open-state channel created by molecular dynamics simulations, current-voltage and current-concentration curves broadly consistent with experimental measurements are obtained. In the absence of an applied potential, the channel houses three potassium ions at positions that are in close agreement with X-ray diffraction maps.     

Hydrodynamics of segmentally flexible macromolecules

Segmentally flexible macromolecules are composed of a few rigid subunits linked by joints which are more or less flexible. The dynamics in solution of this type of macromolecule present special aspects that are reviewed here. Three alternative approaches are described. One is the rigid-body treatment, which is shown to be valid for overall dynamic properties such as translational diffusion and intrinsic viscosity. Another approach is the Harvey-Wegener treatment, which is particularly suited for rotational diffusion. The simplest version of this treatment, which ignores hydrodynamic interaction (HI) effects, is found to be quite accurate when compared to a more rigorous version including HI. A third approach is the Brownian dynamics simulation that, albeit at some computational cost, might describe rigorously cases of arbitrary complexity. This technique has been used to test the approximations in the rigid-body and Harvey-Wegener treatments, thus allowing a better understanding of their validity. Brownian trajectories of simplified models such as the trumbbell and the broken rod have been simulated. The comparison of the decay rates of some correlation functions with the predictions of the two treatments leads to a general conclusion: the Harvey-Wegener treatment determines the initial rate, while the long-time behavior is dominated by the rigid-body relaxation time. As an example of application to a specific biological macromolecule, we present a simulation of an immunoglobulin molecule, showing how Brownian Dynamics can be used to predict rotational and internal dynamics. Another typical example is myosin. Literature data of hydrodynamic properties of whole myosin and the myosin rod are compared with predictions from the Harvey-Wegener and rigid-body treatments. The present situation of the problem on myosin flexibility is analyzed, and some indications are given for future experimental and simulation work.     

Semi-Markov models for brownian dynamics permeation in biological ion channels

Constructing accurate computational models that explain how ions permeate through a biological ion channel is an important problem in biophysics and drug design. Brownian dynamics simulations are large-scale interacting particle computer simulations for modeling ion channel permeation but can be computationally prohibitive. In this paper, we show the somewhat surprising result that a small-dimensional semi-Markov model can generate events (such as conduction events and dwell times at binding sites in the protein) that are statistically indistinguishable from brownian dynamics computer simulation. This approach enables the use of extrapolation techniques to predict channel conduction when performing the actual brownian dynamics simulation that is computationally intractable. Numerical studies on the simulation of gramicidin A ion channels are presented.     

Conformation and dynamic properties of a saturated hydrocarbon chain confined in a model membrane: a Brownian dynamics simulation

A Brownian dynamics simulation of a saturated hydrocarbon chain with simple mean-field potentials, namely anchorage, orientation and enclosing, reproducing a biological membrane environment is presented. The simulation was performed for a time equivalent to 1.4 micros thanks to the simplicity of our model. The results are compared with those obtained for a hydrocarbon chain simulated in the absence of the membrane potentials but with confinement. With the appropriate choice of parameters, equilibrium properties, such as deuterium order parameter, chain length, tilt angle and geometry, and dynamic properties, such as dihedral angle transition rate, rotational and translational diffusion, recovered from our simulations, correctly reproduced, are consistent with hydrocarbon-derived molecule experimental results and simulation results obtained from other more complex studies.     

The role of oligomerization and cooperative regulation in protein function: the case of tryptophan synthase

The oligomerization/co-localization of protein complexes and their cooperative regulation in protein function is a key feature in many biological systems. The synergistic regulation in different subunits often enhances the functional properties of the multi-enzyme complex. The present study used molecular dynamics and Brownian dynamics simulations to study the effects of allostery, oligomerization and intermediate channeling on enhancing the protein function of tryptophan synthase (TRPS). TRPS uses a set of α/β-dimeric units to catalyze the last two steps of L-tryptophan biosynthesis, and the rate is remarkably slower in the isolated monomers. Our work shows that without their binding partner, the isolated monomers are stable and more rigid. The substrates can form fairly stable interactions with the protein in both forms when the protein reaches the final ligand-bound conformations. Our simulations also revealed that the α/β-dimeric unit stabilizes the substrate-protein conformation in the ligand binding process, which lowers the conformation transition barrier and helps the protein conformations shift from an open/inactive form to a closed/active form. Brownian dynamics simulations with a coarse-grained model illustrate how protein conformations affect substrate channeling. The results highlight the complex roles of protein oligomerization and the fine balance between rigidity and dynamics in protein function.     

Conducting-state properties of the KcsA potassium channel from molecular and Brownian dynamics simulations.

The mechanisms underlying transport of ions across the potassium channel are examined using electrostatic calculations and three-dimensional Brownian dynamics simulations. We first build open-state configurations of the channel with molecular dynamics simulations, by pulling the transmembrane helices outward until the channel attains the desired interior radius. To gain insights into ion permeation, we construct potential energy profiles experienced by an ion traversing the channel in the presence of other resident ions. These profiles reveal that in the absence of an applied field the channel accommodates three potassium ions in a stable equilibrium, two in the selectivity filter and one in the central cavity. In the presence of a driving potential, this three-ion state becomes unstable, and ion permeation across the channel is observed. These qualitative explanations are confirmed by the results of three-dimensional Brownian dynamics simulations. We find that the channel conducts when the ionizable residues near the extracellular entrance are fully charged and those near the intracellular side are partially charged. The conductance increases steeply as the radius of the intracellular mouth of the channel is increased from 2 A to 5 A. Our simulation results reproduce several experimental observations, including the current-voltage curves, conductance-concentration relationships, and outward rectification of currents.     

Preparation and properties of asymmetric large unilamellar vesicles: interleaflet coupling in asymmetric vesicles is dependent on temperature but not curvature

Using both Brownian and molecular dynamics, we replicate many of the salient features of Kv1.2, including the current-voltage-concentration profiles and the binding affinity and binding mechanisms of charybdotoxin, a scorpion venom. We also elucidate how structural differences in the inner vestibule can give rise to significant differences in its permeation characteristics. Current-voltage-concentration profiles are constructed using Brownian dynamics simulations, based on the crystal structure 2A79. The results are compatible with experimental data, showing similar conductance, rectification, and saturation with current. Unlike KcsA, for example, the inner pore of Kv1.2 is mainly hydrophobic and neutral, and to explore the consequences of this, we investigate the effect of mutating neutral proline residues at the mouth of the inner vestibule to charged aspartate residues. We find an increased conductance, less inward rectification, and quicker saturation of the current-voltage profile. Our simulations use modifications to our Brownian dynamics program that extend the range of channels that can be usefully modeled. Using molecular dynamics, we investigate the binding of the charybdotoxin scorpion venom to the outer vestibule of the channel. A potential of mean force is derived using umbrella sampling, giving a dissociation constant within a factor of ∼2 to experimentally derived constants. The residues involved in the toxin binding are in agreement with experimental mutagenesis studies. We thus show that the experimental observations on the voltage-gated channel, including the toxin-channel interaction, can reliably be replicated by using the two widely used computational tools.     

Hierarchical approach to predicting permeation in ion channels

A hierarchical computational strategy combining molecular modeling, electrostatics calculations, molecular dynamics, and Brownian dynamics simulations is developed and implemented to compute electrophysiologically measurable properties of the KcsA potassium channel. Models for a series of channels with different pore sizes are developed from the known x-ray structure, using insights into the gating conformational changes as suggested by a variety of published experiments. Information on the pH dependence of the channel gating is incorporated into the calculation of potential profiles for K(+) ions inside the channel, which are then combined with K(+) ion mobilities inside the channel, as computed by molecular dynamics simulations, to provide inputs into Brownian dynamics simulations for computing ion fluxes. The open model structure has a conductance of approximately 110 pS under symmetric 250 mM K(+) conditions, in reasonable agreement with experiments for the largest conducting substate. The dimensions of this channel are consistent with electrophysiologically determined size dependence of quaternary ammonium ion blocking from the intracellular end of this channel as well as with direct structural evidence that tetrabutylammonium ions can enter into the interior cavity of the channel. Realistic values of Ussing flux ratio exponents, distribution of ions within the channel, and shapes of the current-voltage and current-concentration curves are obtained. The Brownian dynamics calculations suggest passage of ions through the selectivity filter proceeds by a "knock-off" mechanism involving three ions, as has been previously inferred from functional and structural studies of barium ion blocking. These results suggest that the present calculations capture the essential nature of K(+) ion permeation in the KcsA channel and provide a proof-of-concept for the integrated microscopic/mesoscopic multitiered approach for predicting ion channel function from structure, which can be applied to other channel structures.     

Mechanisms of permeation and selectivity in calcium channels

The mechanisms underlying ion transport and selectivity in calcium channels are examined using electrostatic calculations and Brownian dynamics simulations. We model the channel as a rigid structure with fixed charges in the walls, representing glutamate residues thought to be responsible for ion selectivity. Potential energy profiles obtained from multi-ion electrostatic calculations provide insights into ion permeation and many other observed features of L-type calcium channels. These qualitative explanations are confirmed by the results of Brownian dynamics simulations, which closely reproduce several experimental observations. These include the current-voltage curves, current-concentration relationship, block of monovalent currents by divalent ions, the anomalous mole fraction effect between sodium and calcium ions, attenuation of calcium current by external sodium ions, and the effects of mutating glutamate residues in the amino acid sequence.     

Very high frequency electron paramagnetic resonance of 2,2,6,6-tetramethyl-1-piperidinyloxy in 1,2-dipalmitoyl-sn-glycero-3-phosphatidylcholine liposomes: partitioning and molecular dynamics.

Partitioning and molecular dynamics of 2,2,6,6,-tetramethylpiperedine-1-oxyl (TEMPO) nitroxide radicals in large unilamellar liposomes (LUV) composed from 1,2-dipalmitoyl-sn-glycero-3-phosphatidylcholine were investigated by using very high frequency electron paramagnetic resonance (EPR) spectroscopy. Experiments carried out at a microwave frequency of 94.3 GHz completely resolved the TEMPO EPR spectrum in the aqueous and hydrocarbon phases. An accurate computer simulation method combined with Levenberg-Marquardt optimization was used to analyze the TEMPO EPR spectra in both phases. Spectral parameters extracted from the simulations gave the actual partitioning of the TEMPO probe between the LUV hydrocarbon and aqueous phases and allowed analysis of picosecond rotational dynamics of the probe in the LUV hydrocarbon phase. In very high frequency EPR experiments, phase transitions in the LUV-TEMPO system were observed as sharp changes in both partitioning and rotational correlation times of the TEMPO probe. The phase transition temperatures (40.5 +/- 0.2 and 32.7 +/- 0.5 degrees C) are in agreement with previously reported differential scanning microcalorimetry data. Spectral line widths were analyzed by using existing theoretical expressions for motionally narrowed nitroxide spectra. It was found that the motion of the small, nearly spherical, TEMPO probe can be well described by anisotropic Brownian diffusion in isotropic media and is not restricted by the much larger hydrocarbon chains existing in ripple structure (P beta') or fluid bilayer structure (L alpha) phases.     

An energy-efficient gating mechanism in the acetylcholine receptor channel suggested by molecular and Brownian dynamics

Acetylcholine receptors mediate electrical signaling between nerve and muscle by opening and closing a transmembrane ion conductive pore. Molecular and Brownian dynamics simulations are used to shed light on the location and mechanism of the channel gate. Four separate 5 ns molecular dynamics simulations are carried out on the imaged structure of the channel, a hypothetical open structure with a slightly wider pore and a mutant structure in which a central ring of hydrophobic residues is replaced by polar groups. Water is found to partially evacuate the pore during molecular simulations of the imaged structure, whereas ions face a large energy barrier and do not conduct through the channel in Brownian dynamics simulations. The pore appears to be in a closed configuration despite containing an unobstructed pathway across the membrane as a series of hydrophobic residues in the center of the channel provide an unfavorable home to water and ions. When the channel is widened slightly, water floods into the channel and ions conduct at a rate comparable to the currents measured experimentally in open channels. The pore remains permeable to ions provided the extracellular end of the pore-lining helix is restrained near the putative open configuration to mimic the presence of the ligand binding domain. Replacing some of the hydrophobic residues with polar ones decreases the barrier for ion permeation but does not result in significant currents. The channel is posited to utilize an energy efficient gating mechanism in which only minor conformational changes of the hydrophobic region of the pore are required to create macroscopic changes in conductance.     

Kinetics of nanochain formation in a simplified model of amelogenin biomacromolecules

We show that the kinetics of nanochain formation of amelogenin molecules is well described by a combination of translational and rotational diffusion of a simplified anisotropic bipolar model consisting of hydrophobic spherical colloid particles and a point charge located on each particle surface. The colloid particles interact via a standard depletion attraction whereas the point charges interact through a screened Coulomb repulsion. We study the kinetics via a Brownian dynamics simulation of both translational and rotational motions and show that the anisotropy brought in by the charge dramatically affects the kinetic pathway of cluster formation and our simple model captures the main features of the experimental observations.     

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.     

Reservoir boundaries in Brownian dynamics simulations of ion channels

Brownian dynamics (BD) simulations provide a practical method for the calculation of ion channel conductance from a given structure. There has been much debate about the implementation of reservoir boundaries in BD simulations in recent years, with claims that the use of improper boundaries could have large effects on the calculated conductance values. Here we compare the simple stochastic boundary that we have been using in our BD simulations with the recently proposed grand canonical Monte Carlo method. We also compare different methods of creating transmembrane potentials. Our results confirm that the treatment of the reservoir boundaries is mostly irrelevant to the conductance properties of an ion channel as long as the reservoirs are large enough.