Inclusion Of Hydrodynamic Interactions Research Articles

Orientationally Averaged Version of the Rotne-Prager-Yamakawa Tensor Provides a Fast but Still Accurate Treatment of Hydrodynamic Interactions in Brownian Dynamics Simulations of Biological Macromolecules.

The Brownian dynamics (BD) simulation technique is widely used to model the diffusive and conformational dynamics of complex systems comprising biological macromolecules. For the diffusive properties of macromolecules to be described correctly by BD simulations, it is necessary to include hydrodynamic interactions (HIs). When modeled at the Rotne-Prager-Yamakawa (RPY) level of theory, for example, the translational and rotational diffusion coefficients of isolated macromolecules can be accurately reproduced; when HIs are neglected, however, diffusion coefficients can be underestimated by an order of magnitude or more. The principal drawback to the inclusion of HIs in BD simulations is their computational expense, and several previous studies have sought to accelerate their modeling by developing fast approximations for the calculation of the correlated random displacements. Here, we explore the use of an alternative way to accelerate the calculation of HIs, i.e., by replacing the full RPY tensor with an orientationally averaged (OA) version which retains the distance dependence of the HIs but averages out their orientational dependence. We seek here to determine whether such an approximation can be justified in application to the modeling of typical proteins and RNAs. We show that the use of an OA-RPY tensor allows translational diffusion of macromolecules to be modeled with very high accuracy at the cost of rotational diffusion being underestimated by ∼25%. We show that this finding is independent of the type of macromolecule simulated and the level of structural resolution employed in the models. We also show, however, that these results are critically dependent on the inclusion of a non-zero term that describes the divergence of the diffusion tensor: when this term is omitted from simulations that use the OA-RPY model, unfolded macromolecules undergo rapid collapse. Our results indicate that the orientationally averaged RPY tensor is likely to be a useful, fast, approximate way of including HIs in BD simulations of intermediate-scale systems.

The rheology of confined colloidal hard disks.

Colloids may be treated as "big atoms" so that they are good models for atomic and molecular systems. Colloidal hard disks are, therefore, good models for 2d materials, and although their phase behavior is well characterized, rheology has received relatively little attention. Here, we exploit a novel, particle-resolved, experimental setup and complementary computer simulations to measure the shear rheology of quasi-hard-disk colloids in extreme confinement. In particular, we confine quasi-2d hard disks in a circular "corral" comprised of 27 particles held in optical traps. Confinement and shear suppress hexagonal ordering that would occur in the bulk and create a layered fluid. We measure the rheology of our system by balancing drag and driving forces on each layer. Given the extreme confinement, it is remarkable that our system exhibits rheological behavior very similar to unconfined 2d and 3d hard particle systems, characterized by a dynamic yield stress and shear-thinning of comparable magnitude. By quantifying particle motion perpendicular to shear, we show that particles become more tightly confined to their layers with no concomitant increase in density upon increasing the shear rate. Shear thinning is, therefore, a consequence of a reduction in dissipation due to weakening in interactions between layers as the shear rate increases. We reproduce our experiments with Brownian dynamics simulations with Hydrodynamic Interactions (HI) included at the level of the Rotne-Prager tensor. That the inclusion of HI is necessary to reproduce our experiments is evidence of their importance in transmission of momentum through the system.

Generalized Rotne-Prager-Yamakawa approximation for Brownian dynamics in shear flow in bounded, unbounded, and periodic domains.

Inclusion of hydrodynamic interactions is essential for a quantitatively accurate Brownian dynamics simulation of colloidal suspensions or polymer solutions. We use the generalized Rotne-Prager-Yamakawa (GRPY) approximation, which takes into account all long-ranged terms in the hydrodynamic interactions, to derive the complete set of hydrodynamic matrices in different geometries: unbounded space, periodic boundary conditions of Lees-Edwards type, and vicinity of a free surface. The construction is carried out both for non-overlapping as well as for overlapping particles. We include the dipolar degrees of freedom, which allows one to use this formalism to simulate the dynamics of suspensions in a shear flow and to study the evolution of their rheological properties. Finally, we provide an open-source numerical package, which implements the GRPY algorithm in Lees-Edwards periodic boundary conditions.

A Comparison of Collective Coordinates for Analyzing Protein Dynamics

The internal dynamics of a protein is commonly analyzed by examining the fluctuations along the protein's alpha-carbons. However, a vast majority of an N-residue protein's internal dynamics can be described successfully using less than the complete (3N-6)-dimensional space spanned by the protein's internal fluctuations. This dimensionality reduction is achieved by transforming from the lab-frame coordinates to a set of judiciously-selected collective coordinates. A common technique, called principal component analysis (PCA), is to select the collective coordinates along which the variance of the original (3N-6)-dimensional coordinates is maximized. These statistically-independent, collective coordinates are termed principal modes (PMs) and are ordered in terms of descending variance, with the first PM possessing the maximum variance. However, these PMs contain no inherent information regarding the timescale over which the motion described by the PM occurs. We present an alternative set of independent, collective coordinates generated using the Langevin Equation for Protein Dynamics (LE4PD), a method for analyzing the internal fluctuations of proteins along its alpha-carbon backbone. The LE4PD accounts for the presence of hydrodynamic interactions between each residue and barriers on each mode's free-energy surface. In contrast to PMs, the LE4PD normal modes are ordered by timescales instead of variance; thus, the first normal mode from the LE4PD describes longest timescale internal motion. The fluctuations predicted by the corresponding PMs and LE4PD normal modes are in good agreement without including hydrodynamics or barriers in the LE4PD, but inclusion of these phenomena have a significant impact on the location and magnitude of the predicted fluctuations along the protein's backbone. The results suggest the LE4PD is an effective method for describing the internal fluctuations of proteins along collective coordinates, and the inclusion of hydrodynamic interactions and free-energy barriers significantly alters the fluctuations predicted along each normal mode.

Importance of Hydrodynamic Interactions in the Stepping Kinetics of Kinesin.

Conventional kinesin walks by a hand-over-hand mechanism on the microtubule (MT) by taking ∼8 nm discrete steps and consumes one ATP molecule per step. The time needed to complete a single step is on the order of 20 μs. We show, using simulations of a coarse-grained model of the complex containing the two motor heads, the MT and the coiled coil, that to obtain quantitative agreement with experiments for the stepping kinetics hydrodynamic interactions (HIs) have to be included. In simulations without hydrodynamic interactions, spanning nearly 20 μs, not a single step was completed in one hundred trajectories. In sharp contrast, nearly 14% of the steps reached the target binding site within 6 μs when HIs were included. Somewhat surprisingly, there are qualitative differences in the diffusion pathways in simulations with and without HI. The extent of movement of the trailing head of kinesin on the MT during the diffusion stage of stepping is considerably greater in simulations with HI than in those without HI. It is likely that inclusion of HI is crucial in the accurate description of motility of other motors as well.

Molecular based prediction of the extensional rheology of high molecular weight polystyrene dilute solutions: A hi-fidelity Brownian dynamics approach

A highly efficient Brownian dynamics algorithm for simulation of bead-spring chain micromechanical models that utilizes the Krylov framework and the semi-implicit predictor-corrector scheme is used to study the behavior of dilute solutions of high molecular weight polystyrene in uniaxial extensional flow. The influence of key parameters, namely, appropriate inclusion of hydrodynamic interactions (HI) and excluded volume (EV), the level of fine-graining, and the flow strength on the observed extensional hardening of the dilute solutions over a broad molecular weight range is considered. Specifically, it is demonstrated that the combination of HI and successive fine-graining results in very good predictions of rheological properties of solutions containing 1.95, 3.9, and 10.2 × 106 molecular weight macromolecules. However, for the highest molecular weight system, namely, 20 × 106, some level of discrepancy between experiment and the simulation results is observed. Finally, the incorporation of EV in order to predict material functions that are more consistent with the measurements for the highest molecular weight macromolecule solution is discussed.

How colloid–colloid interactions and hydrodynamic effects influence the percolation threshold: A simulation study in alumina suspensions

The percolation behavior of alumina suspensions is studied by computer simulations. The percolation threshold ϕc is calculated, determining the key factors that affect its magnitude: the strength of colloid–colloid attraction and the presence of hydrodynamic interactions (HIs). To isolate the effects of HIs, we compare the results of Brownian Dynamics, which do not include hydrodynamics, with those of Stochastic Rotation Dynamics–Molecular Dynamics, which include hydrodynamics. Our results show that ϕc decreases with the increase of the attraction between the colloids. The inclusion of HIs always leads to more elongated structures during the aggregation process, producing a sizable decrease of ϕc when the colloid–colloid attraction is not too strong. On the other hand, the effects of HIs on ϕc tend to become negligible with increasing attraction strength. Our ϕc values are in good agreement with those estimated by the yield stress model by Flatt and Bowen.

Non-spherical osmotic motor: chemical sailing

AbstractThe behaviour of a non-spherical osmotic motor – an axisymmetric catalytic particle self-propelling in a dilute dispersion of reactant particles – is considered. In contrast to a conventional osmotic motor that creates differences in concentration, and hence in osmotic pressure, due to asymmetry in reaction rate along its surface (e.g. a Janus particle with reactive and non-reactive patches), a non-spherical particle is able to move even with uniform chemical activity on its surface. For small departures from a sphere the velocity of self-propulsion is proportional to the square of the non-sphericity or distortion of the particle shape. It is shown that the inclusion of hydrodynamic interactions (HI) may drastically change the self-propulsion. Except for very slow chemical reactions, even the direction of self-propulsion changes with and without HI. Numerical calculations at finite non-sphericity suggest that the maximum velocity of self-propulsion is obtained by a sail-like motor shape, leading to the name ‘chemical sailing’. Moreover, no saturation in the speed of propulsion is found; the motor velocity increases as the area of this ‘sail’ grows and its thickness decreases. The self-propulsion of a non-spherical particle releasing products of a chemical reaction – a constant flux motor – is also considered.

Molecule-Centered Method for Accelerating the Calculation of Hydrodynamic Interactions in Brownian Dynamics Simulations Containing Many Flexible Biomolecules

Inclusion of hydrodynamic interactions (HIs) is essential in simulations of biological macromolecules that treat the solvent implicitly if the macromolecules are to exhibit correct translational and rotational diffusion. The present work describes the development and testing of a simple approach aimed at allowing more rapid computation of HIs in coarse-grained Brownian dynamics simulations of systems that contain large numbers of flexible macromolecules. The method combines a complete treatment of intramolecular HIs with an approximate treatment of the intermolecular HIs which assumes that the molecules are effectively spherical; all of the HIs are calculated at the Rotne-Prager-Yamakawa level of theory. When combined with Fixman's Chebyshev polynomial method for calculating correlated random displacements, the proposed method provides an approach that is simple to program but sufficiently fast that it makes it computationally viable to include HIs in large-scale simulations. Test calculations performed on very coarse-grained models of the pyruvate dehydrogenase (PDH) E2 complex and on oligomers of ParM (ranging in size from 1 to 20 monomers) indicate that the method reproduces the translational diffusion behavior seen in more complete HI simulations surprisingly well; the method performs less well at capturing rotational diffusion but its discrepancies diminish with increasing size of the simulated assembly. Simulations of residue-level models of two tetrameric protein models demonstrate that the method also works well when more structurally detailed models are used in the simulations. Finally, test simulations of systems containing up to 1024 coarse-grained PDH molecules indicate that the proposed method rapidly becomes more efficient than the conventional BD approach in which correlated random displacements are obtained via a Cholesky decomposition of the complete diffusion tensor.

Numerical and experimental study of a rotating magnetic particle chain in a viscous fluid

A simple and fast numerical method is developed capable of accurately determining the 3D rotational dynamics of a magnetic particle chain in an infinite fluid domain. The focus is to control the alternating breakup and reformation of the bead chain which we believe is essential to achieve effective fluid mixing at small scales. The numerical scheme makes use of magnetic dipole moments and extended forms of the Oseen-Burgers tensor to account for both the magnetic and hydrodynamic interactions between the particles. It is shown that the inclusion of hydrodynamic interaction between the particles is crucial to obtain a good description of the particle dynamics. Only a small error of deviation is observed when benchmarking the numerical scheme against a more computationally intensive method, the direct simulation method. The numerical results are compared with experiments and the simulated rotational dynamics correspond well with those obtained from video-microscopy experiments qualitatively and quantitatively. In addition, a dimensionless number (R(T)) is derived as the sole control parameter for the rotational bead chain dynamics. Numerically and experimentally, R(T)≈ 1 is the boundary between rigid "rod" and dynamic "breaking and reformation" behaviors.

Relaxation of an initially-stretched, tethered polymer under shear flow: a Brownian dynamics simulation

The relaxation behaviour of an initially-stretched, tethered flexible polymer chain is studied using the bead–spring model and the Brownian dynamics simulation method. Simulations are performed with and without the inclusion of hydrodynamic interactions. Relaxation is studied in the presence of a shear flow field, characterised by Weissenberg numbers in the range 0 ⩽ Wi ⩽ 55. The presence of a substrate is observed to slow the relaxation of the polymer chain markedly. When shear flow is applied two types of relaxation behaviour are observed, for timescales which are short compared with the longest relaxation time of the chain. For Wi smaller than a critical value, the behaviour is dominated by the entropic collapse of the chain, accompanied by a degree of rotation in the flow. However, above the critical value of Wi, the polymer chain behaves as an almost rigid rod, initially relaxing slightly, and then stretching as it rotates in the flow. The chain attains a maximum value of extension after which an approximately linear relaxation is observed. At long timescales a steady-state extension is reached which varies with the value of Wi. The inclusion of hydrodynamic interactions does not qualitatively affect the observed behaviour. However, it is found that hydrodynamic interactions accelerate the chain relaxation in the absence of flow, while leading to a marked slowing of chain relaxation when shear flows are applied.

Dynamics of rotating paramagnetic particle chains simulated by particle dynamics, Stokesian dynamics and lattice Boltzmann methods

Paramagnetic particles, when subjected to external unidirectional rotating magnetic fields, form chains which rotate along with the magnetic field. In this paper three simulation methods, particle dynamics (PD), Stokesian dynamics (SD) and lattice Boltzmann (LB) methods, are used to study the dynamics of these rotating chains. SD simulations with two different levels of approximations—additivity of forces (AF) and additivity of velocities (AV)—for hydrodynamic interactions have been carried out. The effect of hydrodynamic interactions between paramagnetic particles under the effect of a rotating magnetic field is analyzed by comparing the LB and SD simulations, both of which include hydrodynamic interactions, with PD simulations in which hydrodynamic interactions are neglected. It was determined that for macroscopically observable properties like average chain length as a function of Mason number, reasonable agreement is found between all the three methods. For microscopic properties like the force distribution on each particle along the chain, inclusion of hydrodynamic interaction becomes important to understand the underlying physics of chain formation.

Diffusion of spheres in crowded suspensions of rods

Translational tracer diffusion of spherical macromolecules in crowded suspensions of rodlike colloids is investigated. Experiments are done using several kinds of spherical tracers in fd-virus suspensions. A wide range of size ratios L/2a of the length L of the rods and the diameter 2a of the tracer sphere is covered by combining several experimental methods: fluorescence correlation spectroscopy for small tracer spheres, dynamic light scattering for intermediate sized spheres, and video microscopy for large spheres. Fluorescence correlation spectroscopy is shown to measure long-time diffusion only for relatively small tracer spheres. Scaling of diffusion coefficients with a/xi, predicted for static networks, is not found for our dynamical network of rods (with xi the mesh size of the network). Self-diffusion of tracer spheres in the dynamical network of freely suspended rods is thus fundamentally different as compared to cross-linked networks. A theory is developed for the rod-concentration dependence of the translational diffusion coefficient at low rod concentrations for freely suspended rods. The proposed theory is based on a variational solution of the appropriate Smoluchowski equation without hydrodynamic interactions. The theory can, in principle, be further developed to describe diffusion through dynamical networks at higher rod concentrations with the inclusion of hydrodynamic interactions. Quantitative agreement with the experiments is found for large tracer spheres, and qualitative agreement for smaller spheres. This is probably due to the increasing importance of hydrodynamic interactions as compared to direct interactions as the size of the tracer sphere decreases.

Effect of Hydrodynamic Interactions on Chaos Control in Semi-Dilute Suspensions of Periodically Forced Slender Rods in Simple Shear Flow

In earlier work we demonstrated that in the presence of an external periodic force, the rheological parameters of slender rods in simple shear flow vary chaotically for a certain range of parameters. Algorithms are available in the literature to control the chaotic dynamics of these rheological parameters onto periodic solutions. In earlier work, we also showed that upon including hydrodynamic interactions between the particles, we can extend the range of parameters of the suspension in which the rheological parameters vary chaotically. This result suggested that we try to determine the effect of hydrodynamic interactions in the semi-dilute limit on control of chaos in the rheological parameters. We present numerical evidence that inclusion of hydrodynamic interactions in the semi-dilute regime greatly increases the difficulty of control of chaos. This suggests that inclusion of hydrodynamic interactions could lead to considerably different results when compared to dilute suspensions and thus it may be necessary to include these effects in simulations of moderately concentrated suspensions.

Modeling hydrodynamic interaction in Brownian dynamics: simulations of extensional flows of dilute solutions of DNA and polystyrene

We describe a method to include full hydrodynamic interactions (HI) using the Rotne–Prager tensor into the bead–spring model without excluded-volume interactions in a way suitable for Brownian dynamics (BD) simulations of the transient nonlinear rheological properties of dilute polymer solutions. First, we develop a scheme to determine the HI parameter h ∗ and bead radius “ a” that keeps the number of beads N modest at high molecular weight and yet matches the bead–spring model to the “real” polymer both in its longest relaxation time or diffusivity near equilibrium and in the drag on the chain at full extension, the latter being estimated by a formula of Batchelor. Second, we compare three different numerical integration methods, namely an explicit Euler’s method, a semi-implicit Newton’s method and a semi-implicit predictor–corrector method, and conclude that the predictor–corrector method is the best one available now, because of its ability to use larger time-steps and the relatively low computational expense for each step. Third, we perform simulations for two different macromolecules, namely λ-phage DNA and high molecular weight polystyrene (PS). We find that we can model λ-phage DNA with full HI with only 10 beads, and find that HI has negligible effect on extensional-flow behavior because of DNA’s expanded configuration even at rest, and therefore, its small value of h ∗=0.03 . For PS in a theta solvent, however, molecular configurations are much more compact, and to avoid bead overlap we must keep h ∗<0.5 . This requires the use of more beads, at least N=20 for a molecular weight of 2 million, even if we drop the requirement that we match the longest relaxation time. Surprisingly, despite our inability to match the experimental longest relaxation time with a limited number of beads, excellent agreement with experimental filament-stretching strain–stress data is obtained except for the plateau Trouton ratio for PS of molecular weight of 2 million at various Weissenberg numbers. The inclusion of HI eliminates the “lag”, the delay in growth of the Trouton ratio relative to experimental data, seen in earlier simulations.

Colloidal electrolyte friction and sedimentation: A mode-coupling approach

Using a simplified mode-coupling scheme (MCS) for multicomponent Brownian dispersions, we calculate the effect of mobile salt and counterions on the tracer diffusion and sedimentation of a suspended colloidal macroion. In this scheme, potential forces and also the leading far-field hydrodynamic interactions (HI) between macroion and small electrolyte ions forming its ion atmosphere are accounted for on the basis of the many-body Smoluchowski equation. The static pair correlation functions, which are needed as input, are treated in a first step within the Debye–Hückel approximation. In this way, analytical results are derived for the time-dependent friction function, the long-time tracer diffusion coefficient, and the sedimentation velocity of the macroion as functions of the electrolyte concentration, electrolyte ion mobilities, and macroion charge. Onsager’s limiting law for simple electrolytes is recovered from the MCS in the limit of vanishing macroion radius. The results of our theoretical treatment are compared with experimental data on electrolyte friction and predictions of other existing theories. Good agreement with experimental data on tracer diffusion and with Booth’s theory of electrolyte friction is observed. The inclusion of HI between macroion and microions is shown to be essential for a proper description of the electrolyte friction effect.

Brownian dynamics simulation of a multi-subunit deformable particle in simple shear flow

The structure and dynamics of a multi-subunit deformable particle are investigated by Brownian dynamics simulations. Pairwise hydrodynamic interactions between spherical subunits are described by the Rotne–Prager diffusion tensor and non-hydrodynamic interactions by means of an attractive energy parameter, E. In the absence of shear flow, the average simulated particle structure ranges from ‘stringy’(E≲kT) to compact, with a transition region in between. Time-dependent numerical results are presented for the average subunit coordination number and the average particle perimeter area as a function of increasing shear rate. Particle elongation in the flow field leads to a reduction in the effective fractal dimension of the particle. The greatest sensitivity to shear rate occurs in the stringy–compact transition regime. Substantially different dynamic behaviour is obtained when the simulation is repeated without the inclusion of hydrodynamic interactions.

Dynamics of multicomponent polymer mixtures via the random phase approximation including hydrodynamic interactions

A new and extended formulation of the random phase approximation (RPA) in the study of statics and dynamics of multicomponent polymer mixtures is presented. The new formulation simplifies the implementation of the RPA in both compressible and incompressible mixtures and allows the inclusion of hydrodynamic interaction in the dynamics of polymer melts in the RPA. The dynamics of copolymer melts with hydrodynamic interaction is studied in detail as an illustration of the extended formulation, and the variation of the first cumulant as a function of the wavenumber and interaction parameter is obtained. In this paper we present a generalization of the above elimination procedure by allowing the monomers of the matrix component to interact or to be connected with the monomers in other components, so that the assumption xjo(q,s) = 0, for j = 1, ..., n is no longer needed. This generalization not only simplifies the implementation of the RPA in incompressible multicomponent mixtures in which none of the components consists of homopolymers, such as in melts of copolymers, stars, etc., but also it allows more flexibility in the choice of the bare system in the implementation of the RPA, even in the case of ho-

Flow birefringence of flexible polymer chains in steady shear flow: a Brownian dynamics simulation

The steady-shear flow birefringence of flexible polymers modeled as Gaussian chains has been studied. The influence of flow rate end chain length on the two important parameters in flow birefringence measurements, the extinction angle χ end the birefringence Δn, has been investigated, end the power laws obtained have been compared with theoretical predictions. Hydrodynamic interaction (HI) has been included by means of the Rotne-Prager-Yamakawa tensor, and the simulation results have been compared with the no-HI case. For low shear rates inclusion of HI reduces the influence of flow on Δn and tan 2χ, while for high flow rates the chain behaves as in the no-HI case due to the large separation between beads and thereby elimination of flow interaction

First Cumulant for Chains with Constraints

AbstractThe Fixman-Kovac formulation of chain dynamics with constraints is used to calculate the first cumulant Q(q) of the dynamic scattering function. This general formalism is applied to the case of freely jointed chains. It is shown that the large q limit (q being the scattering wavenumber) of Q(q) for a chain of N bonds in the absence of hydrodynamic interaction is proportional to the ratio (2N+3)/3(N+l) representing the fraction of unconstrained degrees of freedom of the chain. The inclusion of hydrodynamic interaction seems to enhance the apparent segmental diffusion. The use of constrained chain dynamics has no appreciable effects, however, on the behavior of Q(q) in the small and intermediate q regions for long enough chains. This formalism can be used to interpret neutron (spin echo) scattering experiments from semiflexible polymers in solution.