Sedimentation of pairs of hydrodynamically interacting semiflexible filaments

Flexible fibers in shear flow approach attracting periodic solutions

The three-dimensional dynamics of a single non-Brownian flexible fiber in shear flow is evaluated numerically, in the absence of inertia. A wide range of ratios A of bending to hydrodynamic forces and hundreds of initial configurations are considered. We demonstrate that flexible fibers in shear flow exhibit much more complicated evolution patterns than in the case of extensional flow, where transitions to higher-order modes of characteristic shapes are observed when A exceeds consecutive threshold values. In shear flow, we identify the existence of an attracting steady configuration and different attracting periodic motions that are approached by long-lasting rolling, tumbling, and meandering dynamical modes, respectively. We demonstrate that the final stages of the rolling and tumbling modes are effective Jeffery orbits, with the constant parameter C replaced by an exponential function that either decays or increases in time, respectively, corresponding to a systematic drift of the trajectories. In the limit of C→0, the fiber aligns with the vorticity direction and in the limit of C→∞, the fiber periodically tumbles within the shear plane. For moderate values of A, a three-dimensional meandering periodic motion exists, which corresponds to intermediate values of C. Transient, close to periodic oscillations are also detected in the early stages of the modes.

Sedimenting pairs of elastic microfilaments

The dynamics of two identical elastic filaments settling under gravity in a viscous fluid in the low Reynolds number regime is investigated numerically. A large family of initial configurations symmetric with respect to a vertical plane is considered, as well as their non-symmetric perturbations. The behaviour of the filaments is primarily governed by the elasto-gravitational number, which depends on the filament's length and flexibility, and the strength of the external force. Flexible filaments usually converge toward horizontal and parallel orientation. We explain this phenomenon and show that it occurs also for curved rigid particles of similar shapes. Once aligned, the two fibres either converge toward a stationary, flexibility-dependent distance, or tend to collide or continuously repel each other. Rigid and straight rods perform periodic motions while settling down. Apart from very stiff particles, the dynamics is robust to non-symmetric perturbations.

Stokesian dynamics of sedimenting elastic rings

We consider elastic microfilaments which form closed loops. We investigate how the loops change shape and orientation while settling under gravity in a viscous fluid. Loops are circular at the equilibrium. Their dynamics are investigated numerically based on the Stokes equations for the fluid motion and the bead-spring model of the microfilament. The Rotne-Prager approximation for the bead mobility is used. We demonstrate that the relevant dimensionless parameter is the ratio of the bending resistance of the filament to the gravitation force corrected for buoyancy. The inverse of this ratio, called the elasto-gravitation number B, is widely used in the literature for sedimenting elastic linear filaments. We assume that B is of the order of 104-106, which corresponds to easily deformable loops. We find out that initially tilted circles evolve towards different sedimentation modes, depending on B. Very stiff or stiff rings attain almost planar, oval shapes, which are vertical or tilted, respectively. More flexible loops deform significantly and converge towards one of several characteristic periodic motions. These sedimentation modes are also detected when starting from various shapes, and for different loop lengths. In general, multi-stability is observed: an elastic ring converges to one of several sedimentation modes, depending on the initial conditions. This effect is pronounced for very elastic loops. The surprising diversity of long-lasting periodic motions and shapes of elastic rings found in this work gives a new perspective for the dynamics of more complex deformable objects at micrometer and nanometer scales, sedimenting under gravity or rotating in a centrifuge, such as red blood cells, ring polymers or circular DNA.

Periodic Motion of Sedimenting Flexible Knots

We study the dynamics of knotted deformable closed chains sedimenting in a viscous fluid. We show experimentally that trefoil and other torus knots often attain a remarkably regular horizontal toroidal structure while sedimenting, with a number of intertwined loops, oscillating periodically around each other. We then recover this motion numerically and find out that it is accompanied by a very slow rotation around the vertical symmetry axis. We analyze the dependence of the characteristic timescales on the chain flexibility and aspect ratio. It is observed in the experiments that this oscillating mode of the dynamics can spontaneously form even when starting from a qualitatively different initial configuration. In numerical simulations, the oscillating modes are usually present as transients or final stages of the evolution, depending on chain aspect ratio and flexibility, and the number of loops.

Different bending models predict different dynamics of sedimenting elastic trumbbells

The main goal of this paper is to examine theoretically and numerically the impact of a chosen bending model on the dynamics of elastic filaments settling in a viscous fluid under gravity at low-Reynolds-number. We use the bead-spring approximation of a filament and the Rotne-Prager mobility matrix to describe hydrodynamic interactions between the beads. We analyze the dynamics of trumbbells, for which bending angles are typically larger than for thin and long filaments. Each trumbbell is made of three beads connected by springs and it exhibits a bending resistance, described by the harmonic or - alternatively - by the 'cosine' (also called the Kratky-Porod) bending models, both often used in the literature. Using the harmonic bending potential, and coupling it to the spring potential by the Young's modulus, we find simple benchmark solutions: stable stationary configurations of a single elastic trumbbell and attraction of two elastic trumbbells towards a periodic long-lasting orbit. As the most significant result of this paper, we show that for very elastic trumbbells at the same initial conditions, the Kratky-Porod bending potential can lead to qualitatively and quantitatively different spurious dynamics, with artificially large bending angles and unrealistic shapes. We point out that for the bead models of an elastic filament, the range of applicability of the Kratky-Porod model might not go beyond bending angles smaller than π/2 for touching beads and beyond an even much lower value for beads well-separated from each other. The existence of stable stationary configurations of elastic trumbbells and a family of periodic oscillations of two elastic trumbbells are very important findings on their own.

Possible mechanism for aligning microscopic flexible filaments predicted using "caterpillar" hydrodynamics

We use the "caterpillar" model for accurately calculating the inhomogeneous hydrodynamic friction along a microscopic slender cylindrical filaments using Oseen level hydrodynamics. The methodology is applied to study the motion of a flexible filament in a circularly polarized field. Our results predict that in dilute solution alignment occurs along the axis of the field. For electric fields, the strengths and frequencies required are deduced. These are experimentally accessible. We therefore propose that this is a practical method for aligning filaments such as microtubules and functionalized carbon nanotubes.

Transient Pinning and Pulling: A Mechanism for Bending Microtubules

Microtubules have a persistence length of the order of millimeters in vitro, but inside cells they bend over length scales of microns. It has been proposed that polymerization forces bend microtubules in the vicinity of the cell boundary or other obstacles, yet bends develop even when microtubules are polymerizing freely, unaffected by obstacles and cell boundaries. How these bends are formed remains unclear. By tracking the motions of microtubules marked by photobleaching, we found that in LLC-PK1 epithelial cells local bends develop primarily by plus-end directed transport of portions of the microtubule contour towards stationary locations (termed pinning points) along the length of the microtubule. The pinning points were transient in nature, and their eventual release allowed the bends to relax. The directionality of the transport as well as the overall incidence of local bends decreased when dynein was inhibited, while myosin inhibition had no observable effect. This suggests that dynein generates a tangential force that bends microtubules against stationary pinning points. Simulations of microtubule motion and polymerization accounting for filament mechanics and dynein forces predict the development of bends of size and shape similar to those observed in cells. Furthermore, simulations show that dynein-generated bends at a pinning point near the plus end can cause a persistent rotation of the tip consistent with the observation that bend formation near the tip can change the direction of microtubule growth. Collectively, these results suggest a simple physical mechanism for the bending of growing microtubules by dynein forces accumulating at pinning points.

Conformations, hydrodynamic interactions, and instabilities of sedimenting semiflexible filaments

The conformations and dynamics of semiflexible filaments subject to a homogeneous external (gravitational) field, e.g., in a centrifuge, are studied numerically and analytically. The competition between hydrodynamic drag and bending elasticity generates new shapes and dynamical features. We show that the shape of a semiflexible filament undergoes instabilities as the external field increases. We identify two transitions that correspond to the excitation of higher bending modes. In particular, for strong fields the filament stabilizes in a non-planar shape, resulting in a sideways drift or in helical trajectories. For two interacting filaments, we find the same transitions, with the important consequence that the new non-planar shapes have an effective hydrodynamic repulsion, in contrast to the planar shapes which attract themselves even when their osculating planes are rotated with respect to each other. For the case of planar filaments, we show analytically and numerically that the relative velocity is not necessarily due to a different drag of the individual filaments, but to the hydrodynamic interactions induced by their shape asymmetry.

FLUCTUATING MOTOR FORCES BEND GROWING MICROTUBULES

Despite their rigidity, microtubules in living cells bend significantly during polymerization resulting in greater curvature than can be explained by thermal forces alone. However, the source of the non-thermal forces that bend growing microtubules remains obscure. We analyzed the motion of microtubule tips in NIH-3T3 fibroblasts expressing EGFP-EB1, a fluorescent +TIP protein that specifically binds to the growing ends of microtubules. We found that dynein inhibition significantly reduced the deviation of the growing tip from its initial trajectory. Inhibiting myosin modestly reduced tip fluctuations, while simultaneous myosin and dynein inhibition caused no further decrease in fluctuations compared to dynein inhibition alone. Our results can be interpreted with a model in which dynein linkages play a key role in generating and transmitting fluctuating forces that bend growing microtubules.

Accurate simulation dynamics of microscopic filaments using "caterpillar" Oseen hydrodynamics

Microscopic semiflexible filaments suspended in a viscous fluid are widely encountered in biophysical problems. The classic example is the flagella used by microorganisms to generate propulsion. Simulating the dynamics of these filaments numerically is complicated because of the coupling between the motion of the filament and that of the surrounding fluid. An attractive idea is to simplify this coupling by modeling the fluid motion by using Stokeslets distributed at equal intervals along the model filament. We show that, with an appropriate choice of the hydrodynamic radii, one can recover accurate hydrodynamic behavior of a filament with a finite cross section without requiring an explicit surface. This is true, however, only if the hydrodynamic radii take specific values and that they differ in the parallel and perpendicular directions leading to a caterpillarlike hydrodynamic shape. Having demonstrated this, we use the model to compare with analytic theory of filament deformation and rotation in the small deformation limit. Generalization of the methodology, including application to simulations using the Rotne-Prager tensor, is discussed.

A symplectic integration method for elastic filaments

A new method is proposed for integrating the equations of motion of an elastic filament. In the standard finite-difference and finite-element formulations the continuum equations of motion are discretized in space and time, but it is then difficult to ensure that the Hamiltonian structure of the exact equations is preserved. Here we discretize the Hamiltonian itself, expressed as a line integral over the contour of the filament. This discrete representation of the continuum filament can then be integrated by one of the explicit symplectic integrators frequently used in molecular dynamics. The model systematically approximates the continuum partial differential equations, but has the same level of computational complexity as molecular dynamics and is constraint-free. Numerical tests show that the algorithm is much more stable than a finite-difference formulation and can be used for high aspect ratio filaments, such as actin.

Hydrodynamic interactions between two semiflexible inextensible filaments in Stokes flow

Hydrodynamic interactions between two semiflexible inextensible filaments are shown to have a significant impact on filament buckling and their subsequent motion in Stokesian fluids. In linear shear flow, hydrodynamic interactions lead to filament shear dispersion that depends on the filament aspect ratio and the initial filament separation. In linear extensional flow, hydrodynamic interactions lead to complex filament dynamics around the stagnation point. These results suggest that hydrodynamic interactions need to be taken into account to determine the self-diffusion of non-Brownian semiflexible filaments in a cellular flow [Y.-N. Young and M. J. Shelley, Phys. Rev. Lett. 99, 058303 (2007)].