Hydrodynamic properties of rigid particles: comparison of different modeling and computational procedures

From an obliquely falling rod in a viscous fluid to the motion of suspended magnetic bead chains that are driven by a gradient magnetic field and that make an arbitrary angle with the magnetic force vector: A Stokes flow study

In view of the growing role of magnetic particles under magnetic field influence in medical and other applications, and perforce the bead chaining, it is important to understand more generally the chain dynamics. As is well known, in the presence of a magnetic field, magnetic beads tend to form chains that are aligned with the magnetic field vector. In addition, if there is a magnetic field gradient, there will be a magnetic force acting on this chain. The main goal of the present research is to study the motion of a magnetic bead chain that makes an arbitrary angle with the magnetic force vector in the Stokes flow limit, that is, in the limit of zero Reynolds number. We used the public-domain computer program HYDRO++ to calculate the mobility matrix, which relates the magnetic force acting on the chain to the velocity of the chain, for a chain of N beads making an arbitrary angle with the magnetic force vector. Because of the presence of off-diagonal elements of the mobility matrix, as the chain is drawn in the direction of the magnetic force, it is also deflected to the side. We derived analytic solutions for this motion. Also, for bead chains moving in directions both parallel and perpendicular to their lengths, we fit three-parameter functions to solutions from HYDRO++. We found the fits to be excellent. Combining these results with the analytic solutions, we obtained expressions for the velocity components for the bead chains that provide excellent fits to HYDRO++ solutions for arbitrary angles. Finally, we apply the methodology used for the bead chain studies to the study of an obliquely falling rod in a viscous fluid and derive analytic solutions for the velocity components of the obliquely falling rod.

First coarse grain then scale: How to estimate diffusion coefficients of confined molecules

An approach for approximating position and orientation dependent translational and rotational diffusion coefficients of rigid molecules of any shape suspended in a viscous fluid under geometric confinement is proposed. It is an extension of the previously developed scheme for evaluating near-wall diffusion of macromolecules, now applied to any geometry of boundaries. The method relies on shape based coarse-graining combined with scaling of mobility matrix components by factors derived based on energy dissipation arguments for Stokes flows. Tests performed for a capsule shaped molecule and its coarse-grained model, a dumbbell, for three different types of boundaries (a sphere, an open cylinder, and two parallel planes) are described. An almost perfect agreement between mobility functions of the detailed and coarse-grained models, even close to boundary surfaces, is obtained. The proposed method can be used to simplify hydrodynamic calculations and reduce errors introduced due to coarse-graining of molecular shapes.

A theory of pitch for the hydrodynamic properties of molecules, helices, and achiral swimmers at low Reynolds number

We present a theory for pitch, a matrix property that is linked to the coupling of rotational and translational motion of rigid bodies at low Reynolds numbers. The pitch matrix is a geometric property of objects in contact with a surrounding fluid, and it can be decomposed into three principal axes of pitch and their associated moments of pitch. The moments of pitch predict the translational motion in a direction parallel to each pitch axis when the object is rotated around that axis and can be used to explain translational drift, particularly for rotating helices. We also provide a symmetrized boundary element model for blocks of the resistance tensor, allowing calculation of the pitch matrix for arbitrary rigid bodies. We analyze a range of chiral objects, including chiral molecules and helices. Chiral objects with a Cn symmetry axis with n > 2 show additional symmetries in their pitch matrices. We also show that some achiral objects have non-vanishing pitch matrices, and we use this result to explain recent observations of achiral microswimmers. We also discuss the small but non-zero pitch of Lord Kelvin's isotropic helicoid.

Determination of 2D Particle Size Distributions in Plasmonic Nanoparticle Colloids via Analytical Ultracentrifugation: Application to Gold Bipyramids

Multidimensional particle properties determine the product properties in numerous advanced applications. Accurate and statistically meaningful measurements of complex particles and their multidimensional distributions are highly challenging but strongly needed. 2D particle size distributions of plasmonic nanoparticles of complex regular shape can be obtained from analytical ultracentrifugation experiments via the optical back coupling method. A workflow for the calculation of frictional properties of arbitrarily shaped nanoparticles was developed based on bead shell models and applied to gold bipyramids with a pentagonal cross-section. The obtained 2D particle length-diameter distributions and the reduced cumulative 1D length and diameter distributions were compared to transmission electron microscopy measurements. While we find very good agreement for most measurements, the obtained length and diameter distributions were shifted by a few nanometers for some samples. Transmission electron microscopy, energy-dispersive X-ray spectroscopy, electron tomography, and finite element modeling indicate that the shift originated from a slight mismatch between the assumed shape of the simulated perfect bipyramids and the real particle shape and composition due to the presence of silver in the particles. This study demonstrates the feasibility of the applied techniques for complex shape analysis of nanoparticle ensembles with unmatched particle count numbers.

Active microrheology of colloidal suspensions of hard cuboids

By performing dynamic Monte Carlo simulations, we investigate the microrheology of isotropic suspensions of hard-core colloidal cuboids. In particular, we infer the local viscoelastic behavior of these fluids by studying the dynamics of a probe spherical particle that is incorporated in the host phase and is dragged by an external force. This technique, known as active microrheology, allows one to characterize the microscopic response of soft materials upon application of a constant force, whose intensity spans here three orders of magnitude. By tuning the geometry of cuboids from oblate to prolate as well as the system density, we observe different responses that are quantified by measuring the effective friction perceived by the probe particle. The resulting friction coefficient exhibits a linear regime at forces that are much weaker and larger than the thermal forces, whereas a nonlinear, force-thinning regime is observed at intermediate force intensities.

Estimating near-wall diffusion coefficients of arbitrarily shaped rigid macromolecules

We developed a computationally efficient approach to approximate near-wall diffusion coefficients of arbitrarily shaped rigid macromolecules. The proposed method relies on extremum principles for Stokes flows produced by the motion of rigid bodies. In the presence of the wall, the rate of energy dissipation is decreased relative to the unbounded fluid. In our approach, the position- and orientation-dependent mobility matrix of a body suspended near a no-slip plane is calculated numerically using a coarse-grained molecular model and the Rotne-Prager-Yamakawa description of hydrodynamics. Effects of the boundary are accounted for via Blake's image construction. The matrix components are scaled using ratios of the corresponding bulk values evaluated for the detailed representation of the molecule and its coarse-grained model, leading to accurate values of the near-wall diffusion coefficients. We assess the performance of the approach for two biomolecules at different levels of coarse-graining.

Group formation and collective motion of colloidal rods with an activity triggered by visual perception

We investigate the formation of cohesive groups and the collective diffusion of colloidal spherocylinders with a motility driven by a simple visual perception model. For this, we perform Brownian dynamics simulations without hydrodynamic interactions. The visual perception is based on sight cones attached to the spherocylinders and perception functions quantifying the visual stimuli. If the perception function of a particle reaches a predefined threshold, an active component is added to its motion. We find that, in addition to the opening angle of the cone of sight, the aspect ratio of the particles plays an important role for the formation of cohesive groups. If the elongation of the particles is increased, the maximum angle for which the rods organize themselves into such groups decreases distinctly. After a system forms a cohesive group, it performs a diffusive motion, which can be quantified by an effective diffusion coefficient. For increasing aspect ratios, the spatial expansion of the cohesive groups and the effective diffusion coefficient of the collective motion increase, while the number of active group members decreases. We also find that a larger particle number, a smaller propulsion velocity of the group members, and a smaller threshold for the visual stimulus increase the maximum opening angle for which cohesive groups form. Based on our results, we expect anisotropic particles to be of great relevance for the adjustability of visual perception-dependent motility.

Rotational and translational microrheology from shape-anisotropic particles

In this work, we report measurements of the mean squared angular and translational displacements of a colloidal dumbbell immersed in a viscoelastic fluid using digital microscopy. From the mean squared displacements, we obtain the mechanical properties of the media. Both angular and translational motions provide the same viscoelastic complex modulus and agree with that obtained from the translational motion of a spherical probe particle.

Biophysical characterisation of human LincRNA-p21 sense and antisense Alu inverted repeats

Human Long Intergenic Noncoding RNA-p21 (LincRNA-p21) is a regulatory noncoding RNA that plays an important role in promoting apoptosis. LincRNA-p21 is also critical in down-regulating many p53 target genes through its interaction with a p53 repressive complex. The interaction between LincRNA-p21 and the repressive complex is likely dependent on the RNA tertiary structure. Previous studies have determined the two-dimensional secondary structures of the sense and antisense human LincRNA-p21 AluSx1 IRs using SHAPE. However, there were no insights into its three-dimensional structure. Therefore, we in vitro transcribed the sense and antisense regions of LincRNA-p21 AluSx1 Inverted Repeats (IRs) and performed analytical ultracentrifugation, size exclusion chromatography, light scattering, and small angle X-ray scattering (SAXS) studies. Based on these studies, we determined low-resolution, three-dimensional structures of sense and antisense LincRNA-p21. By adapting previously known two-dimensional information, we calculated their sense and antisense high-resolution models and determined that they agree with the low-resolution structures determined using SAXS. Thus, our integrated approach provides insights into the structure of LincRNA-p21 Alu IRs. Our study also offers a viable pipeline for combining the secondary structure information with biophysical and computational studies to obtain high-resolution atomistic models for long noncoding RNAs.

Rotational and translational diffusion of liquid n-hexane: EFP-based molecular dynamics analysis

Molecular Dynamics (MD) simulations based on the Effective Fragment Potential (EFP) method are utilized to provide a comprehensive assessment of diffusion in liquid n-hexane. We decompose translational diffusion into components along and orthogonal to the long axis of the molecule. Rotational diffusion is decomposed into tumbling and spinning motions about this axis. Our analysis yields four corresponding diffusion coefficients which are related to diagonal entries in the complete 6 × 6 diffusion tensor accounting for the three rotational and three translational degrees of freedom and for the potential coupling between them. However, coupling between different degrees of freedom is expected to be minimal for a natural choice of the molecular body-fixed axis, so then off-diagonal entries in the tensor are negligible. This expectation is supported by a hydrodynamic analysis of the diffusion tensor which treats the liquid surrounding the molecule being tracked as a viscous continuum. Thus, the EFP MD analysis provides a comprehensive characterization of diffusion and also reveals expected shortcomings of the hydrodynamic treatment, particularly for rotational diffusion, when applied to neat liquids.

Separation of Enantiomers through Local Vorticity: A Screw Model Mechanism

We present a model to explain the mechanism behind enantiomeric separation under either shear flow or local rotational motion in a fluid. Local vorticity of the fluid imparts molecular rotation that couples to translational motion, sending enantiomers in opposite directions. Translation-rotation coupling of enantiomers is explored using the molecular hydrodynamic resistance tensor, and a molecular equivalent of the pitch of a screw is introduced to describe the degree of translation-rotation coupling. Molecular pitch is a structural feature of the molecules and can be easily computed, allowing rapid estimation of the pitch of 85 druglike molecules. Simulations of model enantiomers in a range of fluids such as Λ- and Δ-[Ru(bpy)₃]Cl₂ in water and (R, R)- and (S, S)-atorvastatin in methanol support predictions made using molecular pitch values. A competition model and continuum drift-diffusion equations are developed to predict separation of realistic racemic mixtures. We find that enantiomeric separation on a centimeter length scale can be achieved in hours, using experimentally achievable vorticities. Additionally, we find that certain achiral objects can also exhibit a nonzero molecular pitch.

Microscopic diffusion coefficients of dumbbell- and spherocylinder-shaped colloids and their application in simulations of crowded monolayers

We explore the diffusion properties of colloidal particles with dumbbell and spherocylinder shapes using a hydrodynamic bead-shell approach and additional Brownian dynamics (BD) simulations. By applying the bead-shell method, we determine empirical formulas for the microscopic diffusion coefficients. A comparison of these formulas and established experimental and theoretical results shows remarkable agreement. For example, the maximum relative discrepancy found for dumbbells is less than 5%. As an application example of the empirical formulas, we perform two-dimensional (2D) BD simulations based on a single dumbbell or spherocylinder in a suspension of spheres and calculate the resulting effective long-time diffusion coefficients. The performed BD simulations can be compared to quasi-2D systems such as colloids confined at the interface of two fluids. We find that the effective diffusion coefficient of translation mostly depends on the sphere area fraction ϕ, while the effective diffusion coefficient of rotation is influenced by the aspect ratio and ϕ. Furthermore, the effective rotational diffusion constant seems to depend on the particle shape with the corresponding implementation of the interactions. In the resolution limit of our methods, the shape-dependent differences of the microscopic diffusion coefficients and the long-time diffusion constant of translation are negligible in the first approximation. The determined empirical formulas for the microscopic diffusion coefficients add to the knowledge of the diffusion of anisotropic particles, and they can be used in countless future studies.

Physics of the Nuclear Pore Complex: Theory, Modeling and Experiment

The hallmark of eukaryotic cells is the nucleus that contains the genome, enclosed by a physical barrier known as the nuclear envelope (NE). On the one hand, this compartmentalization endows the eukaryotic cells with high regulatory complexity and flexibility. On the other hand, it poses a tremendous logistic and energetic problem of transporting millions of molecules per second across the nuclear envelope, to facilitate their biological function in all compartments of the cell. Therefore, eukaryotes have evolved a molecular "nanomachine" known as the Nuclear Pore Complex (NPC). Embedded in the nuclear envelope, NPCs control and regulate all the bi-directional transport between the cell nucleus and the cytoplasm. NPCs combine high molecular specificity of transport with high throughput and speed, and are highly robust with respect to molecular noise and structural perturbations. Remarkably, the functional mechanisms of NPC transport are highly conserved among eukaryotes, from yeast to humans, despite significant differences in the molecular components among various species. The NPC is the largest macromolecular complex in the cell. Yet, despite its significant complexity, it has become clear that its principles of operation can be largely understood based on fundamental physical concepts, as have emerged from a combination of experimental methods of molecular cell biology, biophysics, nanoscience and theoretical and computational modeling. Indeed, many aspects of NPC function can be recapitulated in artificial mimics with a drastically reduced complexity compared to biological pores. We review the current physical understanding of the NPC architecture and function, with the focus on the critical analysis of experimental studies in cells and artificial NPC mimics through the lens of theoretical and computational models. We also discuss the connections between the emerging concepts of NPC operation and other areas of biophysics and bionanotechnology.

Phase behaviour in 2D assemblies of dumbbell-shaped colloids generated under geometrical confinement

The structure formation and the phase behaviour of monolayers of dumbbell-shaped colloids are explored. For this, we conduct Langmuir-Blodgett experiments at the air/water interface and conventional Brownian dynamic simulations without hydrodynamic interactions. Using Voronoi tessellations and the probability density of the corresponding shape factor of the Voronoi cells p(ζ), the influence of the area fraction φ on the structure of the monolayers is investigated. An increase of the area fraction leads to a higher percentage of domains containing particles with six nearest neighbours and a sharper progression of p(ζ). Especially in dense systems, these domains can consist of aligned particles with uniform Voronoi cells. Thus, the increase of φ enhances the order of the monolayers. Simulations show that a sufficient enhancement of φ also impacts the pair correlation function which develops a substructure in its first maxima. Furthermore, we find that reducing the barrier speed in the Langmuir-Blodgett experiments enhances the final area fraction for a given target surface pressure which, in turn, also increases the percentage of particles with six nearest neighbours and sharpens the progression of p(ζ). Overall, the experiments and simulations show a remarkable qualitative agreement which indicates a versatile way of characterising colloidal monolayers by Brownian dynamics simulations. This opens up perspectives for application to a broad range of nanoparticle-based thin film coatings and devices.

Molecular Modeling to Estimate the Diffusion Coefficients of Drugs and Other Small Molecules

Diffusion is a spontaneous process and one of the physicochemical phenomena responsible for molecular transport, the rate of which is governed mainly by the diffusion coefficient; however, few coefficients are available because the measurement of diffusion rates is not straightforward. The translational diffusion coefficient is related by the Stokes–Einstein equation to the approximate radius of the diffusing molecule. Therefore, the stable conformations of small molecules were first calculated by molecular modeling. A simple radius r_s and an effective radius r_e were then proposed and estimated using the stable conformers with the van der Waals radii of atoms. The diffusion coefficients were finally calculated with the Stokes–Einstein equation. The results showed that, for the molecules with strong hydration ability, the diffusion coefficients are best given by r_e and for other compounds, r_s provided the best coefficients, with a reasonably small deviation of ~0.3 × 10⁻⁶ cm²/s from the experimental data. This demonstrates the effectiveness of the theoretical estimation approach, suggesting that diffusion coefficients have potential use as an additional molecular property in drug screening.

The Effect of Number of Arms on the Aggregation Behavior of Thermoresponsive Poly(N-isopropylacrylamide) Star Polymers

The thermoresponsive nature of aqueous solutions of poly(N-isopropylacrylamide) (PNIPAAM) star polymers containing 2, 3, 4, and 6 arms has been investigated by turbidity, dynamic light scattering, rheology, and rheo-SALS. Simulations of the thermosensitive nature of the single star polymers have also been conducted. Some of the samples form aggregates even at temperatures significantly below the lower critical solution temperature (LCST) of PNIPAAM. Increasing concentration and number of arms promotes associations at low temperatures. When the temperature is raised, there is a competition between size increase due to enhanced aggregation and a size reduction caused by contraction. Monte Carlo simulations show that the single stars contract with increasing temperature, and that this contraction is more pronounced when the number of arms is increased. Some samples exhibit a minimum in the turbidity data after the initial increase at the cloud point. The combined rheology and rheo-SALS data suggest that this is due to a fragmentation of the aggregates followed by re-aggregation at even higher temperatures. Although the 6-arm star polymer aggregates more than the other stars at low temperatures, the more compact structure renders it less prone to aggregation at temperatures above the cloud point.

Dynamics of hard colloidal cuboids in nematic liquid crystals

We perform dynamic Monte Carlo simulations to investigate the equilibrium dynamics of hard board-like colloidal particles in oblate and prolate nematic liquid crystals. In particular, we characterize the particles' diffusion along the nematic director and perpendicularly to it, and observe a structural relaxation decay that strongly depends on the particle anisotropy. To assess the Gaussianity of their dynamics and eventual occurrence of collective motion, we calculate two- and four-point correlation functions that incorporate the instantaneous values of the diffusion coefficients parallel and perpendicular to the nematic director. Our simulation results highlight the occurrence of Fickian and Gaussian dynamics at short and long times, locate the minimum diffusivity at the self-dual shape, the particle geometry that would preferentially stabilise biaxial nematics, and exclude the existence of dynamically correlated particles.

Dilute solution structure of bottlebrush polymers

Bottlebrush polymers are a class of macromolecules that have recently found use in a wide variety of materials, ranging from lubricating brushes and nanostructured coatings to elastomeric gels that exhibit structural colors. These polymers are characterized by dense branches extending from a central backbone and thus have properties distinct from linear polymers. It remains a challenge to specifically understand conformational properties of these molecules, due to the wide range of architectural parameters that can be present in a system, and thus there is a need to accurately characterize and model these molecules. In this paper, we use a combination of viscometry, light scattering, and computer simulations to gain insight into the conformational properties of dilute solution bottlebrush polymers. We focus on a series of model bottlebrushes consisting of a poly(norbornene) (PNB) backbone with poly(lactic acid) (PLA) side chains. We demonstrate that intrinsic viscosity and hydrodynamic radius are experimental observations sensitive to molecular architecture, exhibiting distinct differences with different choices of branches and backbone lengths. Informed by the atomistic structure of this PNB-PLA system, we rationalize a coarse-grained simulation model that we evaluate using a combination of Brownian dynamics and Monte Carlo simulations. We show that this exhibits quantitative matching to experimental results, enabling us to characterize the overall shape of the bottlebrush via a number of metrics that can be extended to more general bottlebrush architectures.

Fluctuation-induced hydrodynamic coupling in an asymmetric, anisotropic dumbbell

We recently introduced a model of an asymmetric dumbbell made of two hydrodynamically coupled subunits as a minimal model for a macromolecular complex, in order to explain the observation of enhanced diffusion of catalytically active enzymes. It was shown that internal fluctuations lead to a negative contribution to the overall diffusion coefficient and that the fluctuation-induced contribution is controlled by the strength of the interactions between the subunits and their asymmetry. We develop the model by studying the effect of anisotropy on the diffusion properties of a modular structure. Using a moment expansion method we derive an analytic form for the long-time diffusion coefficient of an asymmetric, anisotropic dumbbell and show systematically its dependence on internal and external symmetry. The method provides a tractable, analytical route for studying the stochastic dynamics of dumbbell models. The present work opens the way to more detailed descriptions of the effect of hydrodynamic interactions on the diffusion and transport properties of biomolecules with complex structures.

Indirect optical trapping using light driven micro-rotors for reconfigurable hydrodynamic manipulation

Optical tweezers are a highly versatile tool for exploration of the mesoscopic world, permitting non-contact manipulation of nanoscale objects. However, direct illumination with intense lasers restricts their use with live biological specimens, and limits the types of materials that can be trapped. Here we demonstrate an indirect optical trapping platform which circumvents these limitations by using hydrodynamic forces to exert nanoscale-precision control over aqueous particles, without directly illuminating them. Our concept is based on optically actuated micro-robotics: closed-loop control enables highly localised flow-fields to be sculpted by precisely piloting the motion of optically-trapped micro-rotors. We demonstrate 2D trapping of absorbing particles which cannot be directly optically trapped, stabilise the position and orientation of yeast cells, and demonstrate independent control over multiple objects simultaneously. Our work expands the capabilities of optical tweezers platforms, and represents a new paradigm for manipulation of aqueous mesoscopic systems.

Optical tweezing with intense lasers can be harmful to biological specimens and limits the types of materials that can be trapped. Here, the authors demonstrate an indirect optical trapping approach which uses hydrodynamic forces to exert nanoscale-precision control over aqueous particles, without directly illuminating them.

The Effect of FG-Nup Phosphorylation on NPC Selectivity: A One-Bead-Per-Amino-Acid Molecular Dynamics Study

Nuclear pore complexes (NPCs) are large protein complexes embedded in the nuclear envelope separating the cytoplasm from the nucleoplasm in eukaryotic cells. They function as selective gates for the transport of molecules in and out of the nucleus. The inner wall of the NPC is coated with intrinsically disordered proteins rich in phenylalanine-glycine repeats (FG-repeats), which are responsible for the intriguing selectivity of NPCs. The phosphorylation state of the FG-Nups is controlled by kinases and phosphatases. In the current study, we extended our one-bead-per-amino-acid (1BPA) model for intrinsically disordered proteins to account for phosphorylation. With this, we performed molecular dynamics simulations to probe the effect of phosphorylation on the Stokes radius of isolated FG-Nups, and on the structure and transport properties of the NPC. Our results indicate that phosphorylation causes a reduced attraction between the residues, leading to an extension of the FG-Nups and the formation of a significantly less dense FG-network inside the NPC. Furthermore, our simulations show that upon phosphorylation, the transport rate of inert molecules increases, while that of nuclear transport receptors decreases, which can be rationalized in terms of modified hydrophobic, electrostatic, and steric interactions. Altogether, our models provide a molecular framework to explain how extensive phosphorylation of FG-Nups decreases the selectivity of the NPC.

Brownian dynamics assessment of enhanced diffusion exhibited by ‘fluctuating-dumbbell enzymes’

Numerical simulations show moderate diffusion enhancements for fluctuating-dumbbell enzyme models.

Experimental Assessment and Model Validation on How Shape Determines Sedimentation and Diffusion of Colloidal Particles

Understanding the hydrodynamics of colloids with complex shapes is of equal importance to widespread practical applications and fundamental scientific problems, such as gelation, crystallization, and phase behavior. Building on previous work, we present a comprehensive study of sedimentation, diffusion, intrinsic viscosities, and other shape-dependent quantities of clusters built from spherical nanoparticles. Cluster preparation is accomplished by assembling surface-modified polystyrene particles on evaporating emulsion droplets. This results in supracolloids that exhibit well-defined configurations, which are governed by the number of constituent particles. Sorting into uniform cluster fractions is achieved through centrifugation of the cluster mixture in a density gradient. Sedimentation coefficients are elucidated by differential centrifugal sedimentation. Rotational and translational diffusion of the clusters are investigated by polarized and depolarized dynamic light scattering. The experimental results are compared to data obtained via a bead-shell model suitable for predicting hydrodynamic quantities of particles with arbitrary shapes. The experimental data are in excellent agreement with the predictions from hydrodynamic modeling. The variety of investigated shapes shows the robustness of our approach and provides a complete picture of the hydrodynamic behavior of complex particles.

Bead-Based Hydrodynamic Simulations of Rigid Magnetic Micropropellers

The field of synthetic microswimmers, micro-robots moving in aqueous environments, has evolved significantly in the last years. Micro-robots actuated and steered by external magnetic fields are of particular interest because of the biocompatibility of this energy source and the possibility of remote control, features suited for biomedical applications. While initial work has mostly focused on helical shapes, the design space under consideration has widened considerably with recent works, opening up new possibilities for optimization of propellers to meet specific requirements. Understanding the relation between shape on the one hand and targeted actuation and steerability on the other hand requires an understanding of their propulsion behavior. Here we propose hydrodynamic simulations for the characterization of rigid micropropellers of any shape, actuated by rotating external magnetic fields. The method consists of approximating the propellers by rigid clusters of spheres. We characterize the influence of model parameters on the swimming behavior to identify optimal simulation parameters using helical propellers as a test system. We then explore the behavior of randomly shaped propellers that were recently characterized experimentally. The simulations show that the orientation of the magnetic moment with respect to the propeller's internal coordinate system has a strong impact on the propulsion behavior and has to be known with a precision of ≤ 5° to predict the propeller's velocity-frequency curve. This result emphasizes the importance of the magnetic properties of the micropropellers for the design of desired functionalities for potential biomedical applications, and in particular the importance of their orientation within the propeller's structure.

Computational Signaling Protein Dynamics and Geometric Mass Relations in Biomolecular Diffusion

We present an atomistic level computational investigation of the dynamics of a signaling protein, monocyte chemoattractant protein-1 (MCP-1), that explores how simulation geometry and solution ionic strength affect the calculated diffusion coefficient. Using a simple extension of noncubic finite size diffusion correction expressions, it is possible to calculate experimentally comparable diffusion coefficients that are fully consistent with those determined from cubic box simulations. Additionally, increasing the concentration of salt in the solvent environment leads to changes in protein dynamics that are not explainable through changes in solvent viscosity alone. This work in accurate computational determination of protein diffusion coefficients led us to investigate molecular-weight-based predictors for biomolecular diffusion. By introducing protein volume- and protein surface-area-based extensions of traditional statistical relations connecting particle molecular weight to diffusion, we find that protein solvent-excluded surface area rather than volume works as a better geometric property for estimating biomolecule Stokes radii. This work highlights the considerations necessary for accurate computational determination of biomolecule diffusivity and presents insight into molecular weight relations for diffusion that could lead to new routes for estimating protein diffusion beyond the traditional approaches.

HIERARCHICAL ORTHOGONAL MATRIX GENERATION AND MATRIX-VECTOR MULTIPLICATIONS IN RIGID BODY SIMULATIONS

In this paper, we apply the hierarchical modeling technique and study some numerical linear algebra problems arising from the Brownian dynamics simulations of biomolecular systems where molecules are modeled as ensembles of rigid bodies. Given a rigid body p consisting of n beads, the 6×3n transformation matrix Z that maps the force on each bead to p's translational and rotational forces (a 6 × 1 vector), and V the row space of Z, we show how to explicitly construct the (3n - 6) × 3n matrix Q ˜ consisting of (3n - 6) orthonormal basis vectors of V ⊥ (orthogonal complement of V) using only O ( n log n ) operations and storage. For applications where only the matrix-vector multiplicationsQ ˜ V andQ ˜ T V are needed, we introduce asymptotically optimal O ( n ) hierarchical algorithms without explicitly forming Q ˜ . Preliminary numerical results are presented to demonstrate the performance and accuracy of the numerical algorithms.

From bead to rod: Comparison of theories by measuring translational drag coefficients of micron-sized magnetic bead-chains in Stokes flow

Frictional drag force on an object in Stokes flow follows a linear relationship with the velocity of translation and a translational drag coefficient. This drag coefficient is related to the size, shape, and orientation of the object. For rod-like objects, analytical solutions of the drag coefficients have been proposed based on three rough approximations of the rod geometry, namely the bead model, ellipsoid model, and cylinder model. These theories all agree that translational drag coefficients of rod-like objects are functions of the rod length and aspect ratio, but differ among one another on the correction factor terms in the equations. By tracking the displacement of the particles through stationary fluids of calibrated viscosity in magnetic tweezers setup, we experimentally measured the drag coefficients of micron-sized beads and their bead-chain formations with chain length of 2 to 27. We verified our methodology with analytical solutions of dimers of two touching beads, and compared our measured drag coefficient values of rod-like objects with theoretical calculations. Our comparison reveals several analytical solutions that used more appropriate approximation and derived formulae that agree with our measurement better.

Hydrodynamic Steering in Protein Association Revisited: Surprisingly Minuscule Effects of Considerable Torques

We investigate the previously postulated hydrodynamic steering phenomenon, resulting from complication of molecular shapes, its magnitude and possible relevance for protein-ligand and protein-protein diffusional encounters, and the kinetics of diffusion-controlled association. We consider effects of hydrodynamic interactions in a prototypical model system consisting of a cleft enzyme and an elongated substrate, and real protein-protein complexes, that of barnase and barstar, and human growth hormone and its binding protein. The kinetics of diffusional encounters is evaluated on the basis of rigid-body Brownian dynamics simulations in which hydrodynamic interactions between molecules are modeled using the bead-shell method for detailed description of molecular surfaces, and the first-passage-time approach. We show that magnitudes of steering torques resulting from the hydrodynamic coupling of associating molecules, evaluated for the studied systems on the basis of the Stokes-Einstein type relations for arbitrarily shaped rigid bodies, are comparable with magnitudes of torques resulting from electrostatic interactions of binding partners. Surprisingly, however, unlike in the case of electrostatic torques that strongly affect the diffusional encounter, overall effects of hydrodynamic steering torques on the association kinetics, while clearly discernible in Brownian dynamics simulations, are rather minute. We explain this result as a consequence of the thermal agitation of the binding partners. Our finding is relevant for the general understanding of a wide spectrum of molecular processes in solution but there is also a more practical aspect to it if one considers the low level of shape detail of models that are usually employed to evaluate hydrodynamic interactions in particle-based Stokesian and Brownian dynamics simulations of multicomponent biomolecular systems. Results described in the current work justify, in part at least, such a low-resolution description.

The human guanylate-binding proteins hGBP-1 and hGBP-5 cycle between monomers and dimers only

Belonging to the dynamin superfamily of large GTPases, human guanylate-binding proteins (hGBPs) comprise a family of seven isoforms (hGBP-1 to hGBP-7) that are strongly upregulated in response to interferon-γ and other cytokines. Accordingly, several hGBPs are found to exhibit various cellular functions encompassing inhibitory effects on cell proliferation, tumor suppression as well as antiviral and antibacterial activity; however, their mechanism of action is only poorly understood. Often, cellular functions of dynamin-related proteins are closely linked to their ability to form nucleotide-dependent oligomers, a feature that also applies to hGBP-1 and hGBP-5. hGBPs are described as monomers, dimers, tetramers, and higher oligomeric species, the function of which is not clearly established. Therefore, this work focused on the oligomerization capability of hGBP-1 and hGBP-5, which are reported to assemble to homodimers and homotetramers. Employing independent methods such as size-exclusion chromatography, which relies on the hydrodynamic radius, and multiangle light scattering, which relies on the mass of the protein, revealed that previous interpretations regarding the size of the proteins and their complexes have to be revised. Additional studies using inter- and intramolecular Förster resonance energy transfer demonstrated that nucleotide-triggered intramolecular structural changes lead to a more extended shape of hGBP-1 being responsible for the appearance of larger oligomeric species. Thus, previously reported tetrameric and dimeric species of hGBP-1 and hGBP-5 were unmasked as dimers and monomers, respectively, with their shapes depending on both the bound nucleotide and the ionic strength of the solution.

The non-Gaussian tops and tails of diffusing boomerangs

Experiments involving the two-dimensional passive diffusion of colloidal boomerangs tracked off their centre of mobility have shown striking non-Gaussian tails in their probability distribution function [Chakrabarty et al., Soft Matter, 2016, 12, 4318]. This in turn can lead to anomalous diffusion characteristics, including mean drift. In this paper, we develop a general theoretical explanation for these measurements. The idea relies on calculating the two-dimensional probability densities at the centre of mobility of the particle, where all distributions are Gaussian, and then transforming them to a different reference point. Our model clearly captures the experimental results, without any fitting parameters, and demonstrates that the one-dimensional probability distributions may also exhibit strongly non-Gaussian tops. These results indicate that the choice of tracking point can cause a considerable departure from Gaussian statistics, potentially causing some common modelling techniques to fail.

Efficient shapes for microswimming: From three-body swimmers to helical flagella

We combine a general formulation of microswimmer equations of motion with a numerical bead-shell model to calculate the hydrodynamic interactions with the fluid, from which the swimming speed, power, and efficiency are extracted. From this framework, a generalized Scallop theorem emerges. The applicability to arbitrary shapes allows for the optimization of the efficiency with respect to the swimmer geometry. We apply this scheme to "three-body swimmers" of various shapes and find that the efficiency is characterized by the single-body friction coefficient in the long-arm regime, while in the short-arm regime the minimal approachable distance becomes the determining factor. Next, we apply this scheme to a biologically inspired set of swimmers that propel using a rotating helical flagellum. Interestingly, we find two distinct optimal shapes, one of which is fundamentally different from the shapes observed in nature (e.g., bacteria).

Extending calibration-free force measurements to optically-trapped rod-shaped samples

Optical trapping has become an optimal choice for biological research at the microscale due to its non-invasive performance and accessibility for quantitative studies, especially on the forces involved in biological processes. However, reliable force measurements depend on the calibration of the optical traps, which is different for each experiment and hence requires high control of the local variables, especially of the trapped object geometry. Many biological samples have an elongated, rod-like shape, such as chromosomes, intracellular organelles (e.g., peroxisomes), membrane tubules, certain microalgae, and a wide variety of bacteria and parasites. This type of samples often requires several optical traps to stabilize and orient them in the correct spatial direction, making it more difficult to determine the total force applied. Here, we manipulate glass microcylinders with holographic optical tweezers and show the accurate measurement of drag forces by calibration-free direct detection of beam momentum. The agreement between our results and slender-body hydrodynamic theoretical calculations indicates potential for this force-sensing method in studying protracted, rod-shaped specimens.

Free Diffusivity of Icosahedral and Tailed Bacteriophages: Experiments, Modeling, and Implications for Virus Behavior in Media Filtration and Flocculation

The aqueous bulk diffusivities of several near-spherical (icosahedral) and nonspherical (tailed) bacterial viruses were experimentally determined by measuring their flux across large pore membranes and using dynamic light scattering, with excellent agreement between values measured using the two techniques. For the icosahedral viruses, good agreement was also found between measured diffusivity values and values predicted with the Stokes-Einstein equation. However, when the tailed viruses were approximated as spheres, poor agreement was found between measured values and Stokes-Einstein predictions. The shape of the tailed organisms was incorporated into two modeling approaches used to predict diffusivity. Model predictions were found to be in good agreement with measured values, demonstrating the importance of the tail in the diffusive transport of these viruses. Our calculations also show that inaccurate estimates of virus diffusion can lead to significant errors when predicting diffusive contributions to flocculation and to single collector efficiency in media filtration.

Exploring the Structural Properties of Positively Charged Peptide Dendrimers

We report a combined experimental and computational approach to study the structural behavior of positively charged peptide dendrimers. Third-generation dendrimers containing combinations of positive/neutral amino acid residues in the different dendrimer generations were synthesized and their overall size evaluated using diffusion NMR. Molecular dynamics simulations were performed to obtain a comprehensive description of the molecular-level phenomena substantiating the structural differences observed. Comparison of the results presented with previous findings reveals a striking charge-dependent tendency in these systems, where the simple number and placement of charged amino acids in the sequence allows an extensive control over the exhibited structural features. Indeed, we observe that peptide dendrimers bearing progressively higher amounts of charged residues are characterized by an increasing structural plasticity, with a myriad of conformational states equally accessible to them. On the other hand, dendrimers containing only small amounts of charged residues evidence, to some extent, a characteristic structural rigidity.

Perspective: On the importance of hydrodynamic interactions in the subcellular dynamics of macromolecules

An outstanding challenge in computational biophysics is the simulation of a living cell at molecular detail. Over the past several years, using Stokesian dynamics, progress has been made in simulating coarse grained molecular models of the cytoplasm. Since macromolecules comprise 20%-40% of the volume of a cell, one would expect that steric interactions dominate macromolecular diffusion. However, the reduction in cellular diffusion rates relative to infinite dilution is due, roughly equally, to steric and hydrodynamic interactions, HI, with nonspecific attractive interactions likely playing rather a minor role. HI not only serve to slow down long time diffusion rates but also cause a considerable reduction in the magnitude of the short time diffusion coefficient relative to that at infinite dilution. More importantly, the long range contribution of the Rotne-Prager-Yamakawa diffusion tensor results in temporal and spatial correlations that persist up to microseconds and for intermolecular distances on the order of protein radii. While HI slow down the bimolecular association rate in the early stages of lipid bilayer formation, they accelerate the rate of large scale assembly of lipid aggregates. This is suggestive of an important role for HI in the self-assembly kinetics of large macromolecular complexes such as tubulin. Since HI are important, questions as to whether continuum models of HI are adequate as well as improved simulation methodologies that will make simulations of more complex cellular processes practical need to be addressed. Nevertheless, the stage is set for the molecular simulations of ever more complex subcellular processes.

Shape Analysis of DNA-Au Hybrid Particles by Analytical Ultracentrifugation

Current developments in nanotechnology have increased the demand for nanocrystal assemblies with well-defined shapes and tunable sizes. DNA is a particularly well-suited building block in nanoscale assemblies because of its scalable sizes, conformational variability, and convenient self-assembly capabilities via base pairing. In hybrid materials, gold nanoparticles (AuNPs) can be assembled into nanoparticle structures with programmable interparticle distances by applying appropriate DNA sequences. However, the development of stoichiometrically defined DNA/NP structures is still challenging since product mixtures are frequently obtained and their purification and characterization is the rate-limiting step in the development of DNA-NP hybrid assemblies. Improvements in nanostructure fractionation and characterization techniques offer great potential for nanotechnology applications in general. This study reports the application of analytical ultracentrifugation (AUC) for the characterization of anisotropic DNA-linked metal-crystal assemblies. On the basis of transmission electron microscopy data and the DNA primary sequence, hydrodynamic bead models are set up for the interpretation of the measured frictional ratios and sedimentation coefficients. We demonstrate that the presence of single DNA strands on particle surfaces as well as the shape factors of multiparticle structures in mixtures can be quantitatively described by AUC. This study will significantly broaden the possibilities to analyze mixtures of shape-anisotropic nanoparticle assemblies. By establishing insights into the analysis of nanostructure mixtures based on fundamental principles of sedimentation, a wide range of potential applications in basic research and industry become accessible.

Effects of Spatially Dependent Mobilities on the Kinetics of the Diffusion-Controlled Association Derived from the First-Passage-Time Approach

Brownian dynamics (BD) simulations and the first-passage-time approach are applied to investigate diffusion-controlled association in a biologically relevant model system consisting of a fixed receptor with an elongated cavity and a capsule-like ligand that fits this cavity precisely. Before the binding at the receptor cavity, the ligand undergoes translational and rotational diffusion, either free or under the influence of electrostatic interactions with the receptor. The spatial dependence of the translational and rotational mobilities of the ligand resulting from its hydrodynamic interactions (HIs) with the receptor is accounted for in BD simulations, and an accurate numerical approach is applied for the evaluation of the spatially dependent mobility tensor of the ligand. Different magnitudes of electrostatic interactions (either attraction or repulsion) between the ligand and receptor are considered. The effective range of receptor-ligand electrostatic interactions is varied to account for their screening under different conditions of ionic strength. The effects of HIs on the kinetics of the diffusion-controlled association, evaluated for different electrostatic properties of binding partners, are thoroughly analyzed and discussed.

Molecular dynamics of protein A and a WW domain with a united-residue model including hydrodynamic interaction

The folding of the N-terminal part of the B-domain of staphylococcal protein A (PDB ID: 1BDD, a 46-residue three-α-helix bundle) and the formin-binding protein 28 WW domain (PDB ID: 1E0L, a 37-residue three-stranded anti-parallel β protein) was studied by means of Langevin dynamics with the coarse-grained UNRES force field to assess the influence of hydrodynamic interactions on protein-folding pathways and kinetics. The unfolded, intermediate, and native-like structures were identified by cluster analysis, and multi-exponential functions were fitted to the time dependence of the fractions of native and intermediate structures, respectively, to determine bulk kinetics. It was found that introducing hydrodynamic interactions slows down both the formation of an intermediate state and the transition from the collapsed structures to the final native-like structures by creating multiple kinetic traps. Therefore, introducing hydrodynamic interactions considerably slows the folding, as opposed to the results obtained from earlier studies with the use of Gō-like models.

Hydrodynamic interactions between two forced objects of arbitrary shape. II. Relative translation

We study the relative translation of two arbitrarily shaped objects, caused by their hydrodynamic interaction as they are forced through a viscous fluid in the limit of zero Reynolds number. It is well known that in the case of two rigid spheres in an unbounded fluid, the hydrodynamic interaction does not produce relative translation. More generally, such an effective pair-interaction vanishes in configurations with spatial inversion symmetry; for example, an enantiomorphic pair in mirror image positions has no relative translation. We show that the breaking of inversion symmetry by boundaries of the system accounts for the interactions between two spheres in confined geometries, as observed in experiments. The same general principle also provides new predictions for interactions in other object configurations near obstacles. We examine the time-dependent relative translation of two self-aligning objects, extending the numerical analysis of our preceding publication [Goldfriend, Diamant, and Witten, Phys. Fluids 27, 123303 (2015)]PHFLE61070-663110.1063/1.4936894. The interplay between the orientational interaction and the translational one, in most cases, leads over time to repulsion between the two objects. The repulsion is qualitatively different for self-aligning objects compared to the more symmetric case of uniform prolate spheroids. The separation between the two objects increases with time t as t^{1/3} in the former case, and more strongly, as t, in the latter.