Flow patterns and defect dynamics of active nematic liquid crystals under an electric field.

The effects of an electric field on the flow patterns and defect dynamics of two-dimensional active nematic liquid crystals are numerically investigated. We found that field-induced director reorientation causes anisotropic active turbulence characterized by enhanced flow perpendicular to the electric field. The average flow speed and its anisotropy are maximized at an intermediate field strength. Topological defects in the anisotropic active turbulence are localized and show characteristic dynamics with simultaneous creation of two pairs of defects. A laning state characterized by stripe domains with alternating flow directions is found at a larger field strength near the transition to the uniformly aligned state. We obtained periodic oscillations between the laning state and active turbulence, which resembles an experimental observation of active nematics subject to anisotropic friction.

Effect of mobility on collective phase dynamics of nonlocally coupled oscillators with a phase lag.

Nonlocally coupled oscillators with a phase lag self-organize into various patterns, such as global synchronization, the twisted state, and the chimera state. In this paper, we consider nonlocally coupled oscillators that move on a ring by randomly exchanging their positions with the neighbors and investigate the combined effects of phase lag and mobility on the collective phase dynamics. Spanning the whole range of phase lag and mobility, we show that mobility promotes synchronization for an attractive coupling, whereas it destroys coherence for a repulsive coupling. The transition behaviors are discussed in terms of the timescales of synchronization and diffusion of the oscillators. We also find a novel spatiotemporal pattern at the border between coherent and incoherent states.

Gait switching with phase reversal of locomotory waves in the centipedeScolopocryptops rubiginosus.

Crawling using locomotory waves is a common method of locomotion for limbless and many-legged invertebrates and stimulates the biomimetic engineering of flexible locomotion. It is generally believed that the direction of locomotory waves is fixed for a given species. However, we found that a centipede,Scolopocryptops rubiginosus, flexibly generated its gait to allow for locomotory waves that varied in direction, depending on (i) locomotion speed and (ii) the physical conditions of terrain. We also found a new type of centipede's swimming gait unlike eel-like way known so far which is using posteriorly traveling waves of horizontal body undulation. The gait patterns of the centipede were examined in various conditions and analyzed how the waves switched in detailed. We showed that gait patterns were associated with control of stride length rather than stride frequency. Discussion was made on a possible scenario of the gait transition in the centipede compatible with our observations. This finding may give a hint at bio-inspired control of flexible gait switching in response to irregular terrain.

Large-scale spatiotemporal patterns in a ring of nonlocally coupled oscillators with a repulsive coupling.

Nonlocally coupled oscillators with a phase lag exhibit various nontrivial spatiotemporal patterns such as the chimera states and the multitwisted states. We numerically study large-scale spatiotemporal patterns in a ring of oscillators with a repulsive coupling with a phase delay parameter α. We find that the multichimera state disappears when α exceeds a critical value. Analysis of the fraction of incoherent regions shows that the transition is analogous to that of directed percolation with two absorbing states but that their critical behaviors are different. The multichimera state reappears when α is further increased, exhibiting nontrivial spatiotemporal patterns with a plateau in the fraction of incoherent regions. A transition from the multichimera to multitwisted states follows at a larger value of α, resulting in five collective phases in total.

Conditions for metachronal coordination in arrays of model cilia.

On surfaces with many motile cilia, beats of the individual cilia coordinate to form metachronal waves. We present a theoretical framework that connects the dynamics of an individual cilium to the collective dynamics of a ciliary carpet via systematic coarse graining. We uncover the criteria that control the selection of frequency and wave vector of stable metachronal waves of the cilia and examine how they depend on the geometric and dynamical characteristics of a single cilium, as well as the geometric properties of the array. We perform agent-based numerical simulations of arrays of cilia with hydrodynamic interactions and find quantitative agreement with the predictions of the analytical framework. Our work sheds light on the question of how the collective properties of beating cilia can be determined using information about the individual units and, as such, exemplifies a bottom-up study of a rich active matter system.

Hydrodynamic synchronization and collective dynamics of colloidal particles driven along a circular path.

We study theoretically the collective dynamics of particles driven by an optical vortex along a circular path. Phase equations of N particles are derived by taking into account both hydrodynamic and repulsive interactions between them. For N=2, the particles attract with each other and synchronize, forming a doublet that moves faster than a singlet. For N=3 and 5, we find periodic rearrangement of doublets and a singlet. For N=4 and 6, the system exhibits either a periodic oscillating state or a stable synchronized state depending on the initial conditions. These results reproduce main features of previous experimental findings. We quantitatively discuss the mechanisms governing the nontrivial collective dynamics.

Motor torque measurement of Halobacterium salinarum archaellar suggests a general model for ATP-driven rotary motors.

It is unknown how the archaellum-the rotary propeller used by Archaea for motility-works. To further understand the molecular mechanism by which the hexameric ATPase motor protein FlaI drives rotation of the membrane-embedded archaellar motor, we determined motor torque by imposition of various loads on Halobacterium salinarum archaella. Markers of different sizes were attached to single archaella, and their trajectories were quantified using three-dimensional tracking and high-speed recording. We show that rotation slows as the viscous drag of markers increases, but torque remains constant at 160 pN·nm independent of rotation speed. Notably, the estimated work done in a single rotation is twice the expected energy that would come from hydrolysis of six ATP molecules in the hexamer, indicating that more ATP molecules are required for one rotation of archaellum. To reconcile the apparent contradiction, we suggest a new and general model for the mechanism of ATP-driven rotary motors.

Three-dimensional tracking of microbeads attached to the tip of single isolated tracheal cilia beating under external load.

To study the properties of tracheal cilia beating under various conditions, we developed a method to monitor the movement of the ciliary tip. One end of a demembranated cilium was immobilized on the glass surface, while the other end was capped with a polystyrene bead and tracked in three dimensions. The cilium, when activated by ATP, stably repeated asymmetric beating as in vivo. The tip of a cilium in effective and recovery strokes moved in discrete trajectories that differed in height. The trajectory remained asymmetric in highly viscous solutions. Model calculation showed that cilia maintained a constant net flux during one beat cycle irrespective of the medium viscosity. When the bead attached to the end was trapped with optical tweezers, it came to display linear oscillation only in the longitudinal direction. Such a beating-mode transition may be an inherent nature of movement-restricted cilia.

Direct observation of rotation and steps of the archaellum in the swimming halophilic archaeon Halobacterium salinarum.

Motile archaea swim using a rotary filament, the archaellum, a surface appendage that resembles bacterial flagella structurally, but is homologous to bacterial type IV pili. Little is known about the mechanism by which archaella produce motility. To gain insights into this mechanism, we characterized archaellar function in the model organism Halobacterium salinarum. Three-dimensional tracking of quantum dots enabled visualization of the left-handed corkscrewing of archaea in detail. An advanced analysis method combined with total internal reflection fluorescence microscopy, termed cross-kymography, was developed and revealed a right-handed helical structure of archaella with a rotation speed of 23 ± 5 Hz. Using these structural and kinetic parameters, we computationally reproduced the swimming and precession motion with a hydrodynamic model and estimated the archaellar motor torque to be 50 pN nm. Finally, in a tethered-cell assay, we observed intermittent pauses during rotation with ∼36° or 60° intervals, which we speculate may be a unitary step consuming a single adenosine triphosphate molecule, which supplies chemical energy of 80 pN nm when hydrolysed. From an estimate of the energy input as ten or six adenosine triphosphates per revolution, the efficiency of the motor is calculated to be ∼6-10%.

Hydrodynamic synchronization between objects with cyclic rigid trajectories.

Synchronization induced by long-range hydrodynamic interactions is attracting attention as a candidate mechanism behind coordinated beating of cilia and flagella. Here we consider a minimal model of hydrodynamic synchronization in the low Reynolds number limit. The model consists of rotors, each of which assumed to be a rigid bead making a fixed trajectory under periodically varying driving force. By a linear analysis, we derive the necessary and sufficient conditions for a pair of rotors to synchronize in phase. We also derive a non-linear evolution equation for their phase difference, which is reduced to minimization of an effective potential. The effective potential is calculated for a variety of trajectory shapes and geometries (either bulk or substrated), for which the stable and metastable states of the system are identified. Finite size of the trajectory induces asymmetry of the potential, which also depends sensitively on the tilt of the trajectory. Our results show that flexibility of cilia or flagella is not a requisite for their synchronized motion, in contrast to previous expectations. We discuss the possibility to directly implement the model and verify our results by optically driven colloids.

Generic conditions for hydrodynamic synchronization.

Synchronization of actively oscillating organelles such as cilia and flagella facilitates self-propulsion of cells and pumping fluid in low Reynolds number environments. To understand the key mechanism behind synchronization induced by hydrodynamic interaction, we study a model of rigid-body rotors making fixed trajectories of arbitrary shape under driving forces that are arbitrary functions of the phase. For a wide class of geometries, we obtain the necessary and sufficient conditions for synchronization of a pair of rotors. We also find a novel synchronized pattern with an oscillating phase shift. Our results shed light on the role of hydrodynamic interactions in biological systems, and could help in developing efficient mixing and transport strategies in microfluidic devices.

Many-body theory of synchronization by long-range interactions.

Synchronization of coupled oscillators on a d-dimensional lattice with the power-law coupling G(r) = g0/rα and randomly distributed intrinsic frequency is analyzed. A systematic perturbation theory is developed to calculate the order parameter profile and correlation functions in powers of ϵ = α/d-1. For α ≤ d, the system exhibits a sharp synchronization transition as described by the conventional mean-field theory. For α > d, the transition is smeared by the quenched disorder, and the macroscopic order parameter ψ decays slowly with g0 as |ψ| ∝ g(0)(2).

Synchronization and collective dynamics in a carpet of microfluidic rotors.

We study synchronization of an array of rotors on a substrate that are coupled by hydrodynamic interaction. Each rotor, which is modeled by an effective rigid body, is driven by an internal torque and exerts an active force on the surrounding fluid. The long-ranged nature of the hydrodynamic interaction between the rotors causes a rich pattern of dynamical behaviors including phase ordering and self-proliferating spiral waves. Our results suggest strategies for designing controllable microfluidic mixers using the emergent behavior of hydrodynamically coupled active components.

Viscoelasticity and primitive path analysis of entangled polymer liquids: from F-actin to polyethylene.

We combine computer simulations and scaling arguments to develop a unified view of polymer entanglement based on the primitive path analysis of the microscopic topological state. Our results agree with experimentally measured plateau moduli for three different polymer classes over a wide range of reduced polymer densities: (i) semidilute theta solutions of synthetic polymers, (ii) the corresponding dense melts above the glass transition or crystallization temperature, and (iii) solutions of semiflexible (bio)polymers such as F-actin or suspensions of rodlike viruses. Together, these systems cover the entire range from loosely to tightly entangled polymers. In particular, we argue that the primitive path analysis renormalizes a loosely to a tightly entangled system and provide a new explanation of the successful Lin-Noolandi packing conjecture for polymer melts.

Numerical simulation of the twist-grain-boundary phase of chiral liquid crystals.

We study the structure of the twist-grain-boundary phase of chiral liquid crystals by numerically minimizing the Landau-de Gennes free energy. We analyze the morphology of layers at the grain boundary, to better understand the mechanism of frustration between the smectic layer order and chirality. As the chirality increases, the layer compression energy strongly increases while the effective layer bending rigidity is reduced due to unlocking of the layer orientation and the director. This results in large deviation of the layer morphology from that of Scherk's first minimal surface and linear stack of screw dislocations.

Boundary conditions for equilibrating incommensurate periodic patterns.

Simulation of periodic patterns often suffer from artifacts due to incommensurability of the intrinsic length scale and the system size. We introduce a simple numerical scheme to avoid this problem in finding equilibrium domain morphologies from a Ginzburg-Landau-type free energy. In this scheme, the boundary values are determined only by the local equilibrium condition at the adjacent bulk sites. The scheme is especially advantageous in equilibrating patterns that have two or more characteristic lengths. We demonstrate it using a model of lamellar-lamellar coexistence in block copolymer blends.

Dynamics of orientational ordering in fluid membranes.

We study the dynamics of orientational phase ordering in fluid membranes. Through numerical simulation we find an unusually slow coarsening of topological texture, which is limited by subdiffusive propagation of membrane curvature. The growth of the orientational correlation length xi obeys a power law xi proportional, variant t(w) with w<1/4 in the late stage. We will also discuss defect profiles and correlation patterns in terms of long-range interaction mediated by curvature elasticity.

Pattern formation in phase-separating gels with spontaneous shear.

We study pattern formation in gels undergoing simultaneous phase separation and orientational ordering. A 2D numerical simulation is performed using a minimal model of nonlinear elasticity with density-anisotropy coupling. For strong positive coupling, the collapsed phase elongates along the phase boundary and buckle, creating a folded structure with paired topological defects. For negative coupling, soft elasticity of the swollen phase causes a droplet morphology as in liquid-liquid phase separation. Their possible realizations in nematic liquid-crystalline gels are discussed.