Trapping and wiggling: elastohydrodynamics of driven microfilaments

Multiflagellate Swimming Controlled by Hydrodynamic Interactions

Many eukaryotic microorganisms propelled by multiple flagella can swim very rapidly with distinct gaits. Here, we model a three-dimensional mutiflagellate swimmer, resembling the microalgae. When the flagella are actuated synchronously, the swimming efficiency can be enhanced or reduced by interflagella hydrodynamic interactions (HIs), determined by the intrinsic tilting angle of the flagella. The asynchronous gait with a phase difference between neighboring flagella can reduce oscillatory motion via the basal mechanical coupling. In the presence of a spherical body, simulations taking into account the flagella-body interactions reveal the advantage of anterior configuration compared with posterior configuration, where in the latter case an optimal flagella number arises. Apart from understanding the role of HIs in the multiflagellate microorganisms, this work could also guide laboratory fabrications of novel microswimmers.

Integrating chemistry, fluid flow, and mechanics to drive spontaneous formation of three-dimensional (3D) patterns in anchored microstructures

Enzymatic reactions in solution drive the convection of confined fluids throughout the enclosing chambers and thereby couple the processes of reaction and convection. In these systems, the energy released from the chemical reactions generates a force, which propels the fluids' spontaneous motion. Here, we use theoretical and computational modeling to determine how reaction-convection can be harnessed to tailor and control the dynamic behavior of soft matter immersed in solution. Our model system encompasses an array of surface-anchored, flexible posts in a millimeter-sized, fluid-filled chamber. Selected posts are coated with enzymes, which react with dissolved chemicals to produce buoyancy-driven fluid flows. We show that these chemically generated flows exert a force on both the coated (active) and passive posts and thus produce regular, self-organized patterns. Due to the specificity of enzymatic reactions, the posts display controllable kaleidoscopic behavior where one regular pattern is smoothly morphed into another with the addition of certain reactants. These spatiotemporal patterns also form "fingerprints" that distinctly characterize the system, reflecting the type of enzymes used, placement of the enzyme-coated posts, height of the chamber, and bending modulus of the elastic posts. The results reveal how reaction-convection provides concepts for designing soft matter that readily switches among multiple morphologies. This behavior enables microfluidic devices to be spontaneously reconfigured for specific applications without construction of new chambers and the fabrication of standalone sensors that operate without extraneous power sources.

Non-Markovian Modeling of Nonequilibrium Fluctuations and Dissipation in Active Viscoelastic Biomatter

Based on a Hamiltonian that incorporates the elastic coupling between a tracer particle and the embedding active viscoelastic biomatter, we derive a generalized non-Markovian Langevin model for the nonequilibrium mechanical tracer response. Our analytical expressions for the frequency-dependent tracer response function and the tracer positional autocorrelation function agree quantitatively with experimental data for red blood cells and actomyosin networks with and without adenosine triphosphate over the entire frequency range and in particular reproduce the low-frequency violation of the fluctuation-dissipation theorem. The viscoelastic power laws, the elastic constants and effective friction coefficients extracted from the experimental data allow straightforward physical interpretation.

Self-buckling and self-writhing of semi-flexible microorganisms

The twisting and writhing of a cell body and associated mechanical stresses is an underappreciated constraint on microbial self-propulsion. Multi-flagellated bacteria can even buckle and writhe under their own activity as they swim through a viscous fluid. New equilibrium configurations and steady-state dynamics then emerge which depend on the organism's mechanical properties and on the oriented distribution of flagella along its surface. Modeling the cell body as a semi-flexible Kirchhoff rod and coupling the mechanics to a flagellar orientation field, we derive the Euler-Poincaré equations governing the dynamics of the system, and rationalize experimental observations of buckling and writhing of elongated swarmer cells of the bacterium Proteus mirabilis. A sequence of bifurcations is identified as the body is made more compliant, due to both buckling and torsional instabilities. These studies highlight a practical requirement for the stiffness of bacteria below which self-buckling occurs and cell motility becomes ineffective.

Elastohydrodynamic propulsion of a filament magnetically driven at both ends

The elastohydrodynamic interaction between an elastic filament and its surrounding fluid was exploited to develop the first microswimmers. These flexible microswimmers are typically actuated magnetically at one end and their propulsion behavior is relatively well understood. In this work, we move beyond the traditional single-end actuation setup and explore the propulsion characteristics of an elastic filament driven by magnetic torques at both ends. We report the emergence of new modes of propulsion behaviors in different physical regimes, depending on the balance of elastic and viscous forces as well as the arrangement of the magnetic moments at the filament ends. In particular, under the same magnetic actuation, a filament driven at both ends can propel either forward or backward depending on its relative stiffness. Moreover, this new backward propulsion mode can generate a magnitude of propulsion that is unattainable by the traditional single-end actuation setup. We characterize these new propulsion behaviors and provide some physical insights into how they emerge from the complex interplay between viscous and elastic forces and magnetic actuation in various configurations. Taken together, these findings could guide the development of soft microrobots with enhanced propulsion performance and maneuverability for future biomedical applications.

The reaction-diffusion basis of animated patterns in eukaryotic flagella

The flagellar beat of bull spermatozoa and C. Reinhardtii are modelled by a minimal, geometrically exact, reaction-diffusion system. Spatio-temporal animated patterns describe flagellar waves, analogous to chemical-patterns from classical reaction-diffusion systems, with sliding-controlled molecular motor reaction-kinetics. The reaction-diffusion system is derived from first principles as a consequence of the high-internal dissipation by the flagellum relative to the external hydrodynamic dissipation. Quantitative comparison with nonlinear, large-amplitude simulations shows that animated reaction-diffusion patterns account for the experimental beating of both bull sperm and C. Reinhardtii. Our results suggest that a unified mechanism may exist for motors controlled by sliding, without requiring curvature-sensing, and uninfluenced by hydrodynamics. High-internal dissipation instigates autonomous travelling waves independently of the external fluid, enabling progressive swimming, otherwise not possible, in low viscosity environments, potentially critical for external fertilizers and aquatic microorganisms. The reaction-diffusion system may prove a powerful tool for studying pattern formation of movement on animated structures.

Durability of Aligned Microtubules Dependent on Persistence Length Determines Phase Transition and Pattern Formation in Collective Motion

Collective motion is a ubiquitous phenomenon in nature. The collective motion of cytoskeleton filaments results mainly from dynamic collisions and alignments; however, the detailed mechanism of pattern formation still needs to be clarified. In particular, the influence of persistence length, which is a measure of filament flexibility, on collective motion is still unclear and lacks experimental verifications although it is likely to directly affect the orientational flexibility of filaments. Here, we investigated the collective motion of microtubules with different persistence lengths using a microtubule-kinesin motility system. We showed that local interactions between microtubules significantly vary depending on their persistence length. We demonstrated that the bundling of microtubules is enhanced by more durable alignment, rather than by greater likelihood of alignment. An agent-based computational model confirmed that the rigidity-dependent durability of microtubule alignment dominates their collective behavior.

Low-Reynolds-number, biflagellated Quincke swimmers with multiple forms of motion

In the limit of zero Reynolds number (Re), swimmers propel themselves exploiting a series of nonreciprocal body motions. For an artificial swimmer, a proper selection of the power source is required to drive its motion, in cooperation with its geometric and mechanical properties. Although various external fields (magnetic, acoustic, optical, etc.) have been introduced, electric fields are rarely utilized to actuate such swimmers experimentally in unbounded space. Here we use uniform and static electric fields to demonstrate locomotion of a biflagellated sphere at low Re via Quincke rotation. These Quincke swimmers exhibit three different forms of motion, including a self-oscillatory state due to elastohydrodynamic-electrohydrodynamic interactions. Each form of motion follows a distinct trajectory in space. Our experiments and numerical results demonstrate a method to generate, and potentially control, the locomotion of artificial flagellated swimmers.

Growth rate-dependent flexural rigidity of microtubules influences pattern formation in collective motion

Background

Microtubules (MTs) are highly dynamic tubular cytoskeleton filaments that are essential for cellular morphology and intracellular transport. In vivo, the flexural rigidity of MTs can be dynamically regulated depending on their intracellular function. In the in vitro reconstructed MT-motor system, flexural rigidity affects MT gliding behaviors and trajectories. Despite the importance of flexural rigidity for both biological functions and in vitro applications, there is no clear interpretation of the regulation of MT flexural rigidity, and the results of many studies are contradictory. These discrepancies impede our understanding of the regulation of MT flexural rigidity, thereby challenging its precise manipulation.

Results

Here, plausible explanations for these discrepancies are provided and a new method to evaluate the MT rigidity is developed. Moreover, a new relationship of the dynamic and mechanic of MTs is revealed that MT flexural rigidity decreases through three phases with the growth rate increases, which offers a method of designing MT flexural rigidity by regulating its growth rate. To test the validity of this method, the gliding performances of MTs with different flexural rigidities polymerized at different growth rates are examined. The growth rate-dependent flexural rigidity of MTs is experimentally found to influence the pattern formation in collective motion using gliding motility assay, which is further validated using machine learning.

Conclusion

Our study establishes a robust quantitative method for measurement and design of MT flexural rigidity to study its influences on MT gliding assays, collective motion, and other biological activities in vitro. The new relationship about the growth rate and rigidity of MTs updates current concepts on the dynamics and mechanics of MTs and provides comparable data for investigating the regulation mechanism of MT rigidity in vivo in the future.

Graphic Abstract

Supplementary Information

The online version contains supplementary material available at 10.1186/s12951-021-00960-y.

Dynamics of undulatory fluctuations of semiflexible filaments in a network

We study the dynamics of a single semiflexible filament coupled to a Hookean spring at its boundary. The spring produces a fluctuating tensile force on the filament, the value of which depends on the filament's instantaneous end-to-end length. The spring thereby introduces a nonlinearity, which mixes the undulatory normal modes of the filament and changes their dynamics. We study these dynamics using the Martin-Siggia-Rose-Janssen-De Dominicis formalism, and compute the time-dependent correlation functions of transverse undulations and of the filament's end-to-end distance. The relaxational dynamics of the modes below a characteristic wavelength sqrt[κ/τ_{R}], set by the filament's bending modulus κ and spring-renormalized tension τ_{R}, are changed by the boundary spring. This occurs near the crossover frequency between tension- and bending-dominated modes of the system. The boundary spring can be used to represent the linear elastic compliance of the rest of the filament network to which the filament is cross linked. As a result, we predict that this nonlinear effect will be observable in the dynamical correlations of constituent filaments of networks and in the networks' collective shear response. The system's dynamic shear modulus is predicted to exhibit the well-known crossover with increasing frequency from ω^{1/2} to ω^{3/4}, but the inclusion of the network's compliance in the analysis of the individual filament dynamics shifts this transition to a higher frequency.

Multisynchrony in Active Microfilaments

Biological microfilaments exhibit a variety of synchronization modes. Recent experiments observed that a pair of isolated eukaryotic flagella, coupled solely via the fluid medium, display synchrony at nontrivial phase lags in addition to in-phase and antiphase synchrony. Using an elastohydrodynamic filament model in conjunction with numerical simulations and a Floquet-type theoretical analysis, we demonstrate that it is possible to reach multiple synchronization states by varying the intrinsic activity of the filament and the strength of hydrodynamic coupling between the two filaments. Then, we derive an evolution equation for the phase difference between the two filaments at weak coupling, and use a Kuramoto-style phase sensitivity analysis to reveal the nature of the bifurcations underlying the transitions between these different synchronized states.

Human sperm uses asymmetric and anisotropic flagellar controls to regulate swimming symmetry and cell steering

Flagellar beating drives sperm through the female reproductive tract and is vital for reproduction. Flagellar waves are generated by thousands of asymmetric molecular components; yet, paradoxically, forward swimming arises via symmetric side-to-side flagellar movement. This led to the preponderance of symmetric flagellar control hypotheses. However, molecular asymmetries must still dictate the flagellum and be manifested in the beat. Here, we reconcile molecular and microscopic observations, reconnecting structure to function, by showing that human sperm uses asymmetric and anisotropic controls to swim. High-speed three-dimensional (3D) microscopy revealed two coactive transversal controls: An asymmetric traveling wave creates a one-sided stroke, and a pulsating standing wave rotates the sperm to move equally on all sides. Symmetry is thus achieved through asymmetry, creating the optical illusion of bilateral symmetry in 2D microscopy. This shows that the sperm flagellum is asymmetrically controlled and anisotropically regularized by fast-signal transduction. This enables the sperm to swim forward.

Flow-induced buckling dynamics of sperm flagella

The swimming sperm of many external fertilizing marine organisms face complex fluid flows during their search for egg cells. Aided by chemotaxis, relatively weak flows are known to enhance sperm-egg fertilization rates through hydrodynamic guidance. However, strong flows have the potential to mechanically inhibit flagellar motility through elastohydrodynamic interactions-a phenomenon that remains poorly understood. Here we explore the effects of flow on the buckling dynamics of sperm flagella in an extensional flow through detailed numerical simulations, which are informed by microfluidic experiments and high-speed imaging. Compressional fluid forces lead to rich buckling dynamics of the sperm flagellum beyond a critical dimensionless sperm number, Sp, which represents the ratio of viscous force to elastic force. For nonmotile sperm, the maximum buckling curvature and the number of buckling locations, or buckling mode, increase with increasing sperm number. In contrast, motile sperm exhibit a local flagellar curvature due to the propagation of bending waves along the flagellum. In compressional flow, this preexisting curvature acts as a precursor for buckling, which enhances local curvature without creating new buckling modes and leads to asymmetric beating. However, in extensional flow, flagellar beating remains symmetric with a smaller head yawing amplitude due to tensile forces. The flagellar beating frequency also influences the maximum curvature of motile sperm by facilitating sperm reorientation relative to the compressional axis of the flow near stagnation points. These combined simulations and experiments directly illustrate the microscopic elastohydrodynamic mechanisms responsible for inhibiting flagellar motility in flow and have possible implications for our understanding of external fertilization in dynamic marine systems.

Flagellar ultrastructure suppresses buckling instabilities and enables mammalian sperm navigation in high-viscosity media

Eukaryotic flagellar swimming is driven by a slender motile unit, the axoneme, which possesses an internal structure that is essentially conserved in a tremendous diversity of sperm. Mammalian sperm, however, which are internal fertilizers, also exhibit distinctive accessory structures that further dress the axoneme and alter its mechanical response. This raises the following two fundamental questions. What is the functional significance of these structures? How do they affect the flagellar waveform and ultimately cell swimming? Hence we build on previous work to develop a mathematical mechanical model of a virtual human sperm to examine the impact of mammalian sperm accessory structures on flagellar dynamics and motility. Our findings demonstrate that the accessory structures reinforce the flagellum, preventing waveform compression and symmetry-breaking buckling instabilities when the viscosity of the surrounding medium is increased. This is in agreement with previous observations of internal and external fertilizers, such as human and sea urchin spermatozoa. In turn, possession of accessory structures entails that the progressive motion during a flagellar beat cycle can be enhanced as viscosity is increased within physiological bounds. Hence the flagella of internal fertilizers, complete with accessory structures, are predicted to be advantageous in viscous physiological media compared with watery media for the fundamental role of delivering a genetic payload to the egg.

Cilia oscillations

Cilia, or eukaryotic flagella, are microscopic active filaments expressed on the surface of many eukaryotic cells, from single-celled protozoa to mammalian epithelial surfaces. Cilia are characterized by a highly conserved and intricate internal structure in which molecular motors exert forces on microtubule doublets causing cilia oscillations. The spatial and temporal regulations of this molecular machinery are not well understood. Several theories suggest that geometric feedback control from cilium deformations to molecular activity is needed. Here, we implement a recent sliding control model, where the unbinding of molecular motors is dictated by the sliding motion between microtubule doublets. We investigate the waveforms exhibited by the model cilium, as well as the associated molecular motor dynamics, for hinged and clamped boundary conditions. Hinged filaments exhibit base-to-tip oscillations while clamped filaments exhibit both base-to-tip and tip-to-base oscillations. We report the change in oscillation frequencies and amplitudes as a function of motor activity and sperm number, and we discuss the validity of these results in the context of experimental observations of cilia behaviour. This article is part of the Theo Murphy meeting issue 'Unity and diversity of cilia in locomotion and transport'.

Primary cilia have a length-dependent persistence length

The fluctuating position of an optically trapped cilium tip under untreated and Taxol-treated conditions was used to characterize mechanical properties of the cilium axoneme and its basal body by combining experimental, analytical, and computational tools. We provide, for the first time, evidence that the persistence length of a ciliary axoneme is length-dependent; longer cilia are stiffer than shorter cilia. We demonstrate that this apparent length dependence can be understood by a combination of modeling axonemal microtubules as anisotropic elastic shells and including actomyosin-driven stochastic basal body motion. Our results also demonstrate the possibility of using observable ciliary dynamics to probe interior cytoskeletal dynamics. It is hoped that our improved characterization of cilia will result in deeper understanding of the biological function of cellular flow sensing by this organelle.

The asymptotic coarse-graining formulation of slender-rods, bio-filaments and flagella

The inertialess fluid-structure interactions of active and passive inextensible filaments and slender-rods are ubiquitous in nature, from the dynamics of semi-flexible polymers and cytoskeletal filaments to cellular mechanics and flagella. The coupling between the geometry of deformation and the physical interaction governing the dynamics of bio-filaments is complex. Governing equations negotiate elastohydrodynamical interactions with non-holonomic constraints arising from the filament inextensibility. Such elastohydrodynamic systems are structurally convoluted, prone to numerical errors, thus requiring penalization methods and high-order spatio-temporal propagators. The asymptotic coarse-graining formulation presented here exploits the momentum balance in the asymptotic limit of small rod-like elements which are integrated semi-analytically. This greatly simplifies the elastohydrodynamic interactions and overcomes previous numerical instability. The resulting matricial system is straightforward and intuitive to implement, and allows for a fast and efficient computation, more than a hundred times faster than previous schemes. Only basic knowledge of systems of linear equations is required, and implementation achieved with any solver of choice. Generalizations for complex interaction of multiple rods, Brownian polymer dynamics, active filaments and non-local hydrodynamics are also straightforward. We demonstrate these in four examples commonly found in biological systems, including the dynamics of filaments and flagella. Three of these systems are novel in the literature. We additionally provide a Matlab code that can be used as a basis for further generalizations.

Effect of external shear flow on sperm motility

The trajectory of sperm in the presence of background flow is of utmost importance for the success of fertilization, as sperm encounter background flow of different magnitude and direction on their way to the egg. Here, we have studied the effect of an unbounded simple shear flow as well as a Poiseuille flow on the sperm trajectory. In the presence of a simple shear flow, the sperm moves on an elliptical trajectory in the reference frame advecting with the local background flow. The length of the major-axis of this elliptical trajectory decreases with the shear rate. The flexibility of the flagellum and consequently the length of the major axis of the elliptical trajectories increases with the sperm number. The sperm number is a dimensionless number representing the ratio of viscous force to elastic force. The sperm moves downstream or upstream depending on the strength of background Poiseuille flow. In contrast to the simple shear flow, the sperm also moves toward the centerline in a Poiseuille flow. Far away from the centerline, the cross-stream migration velocity of the sperm increases as the transverse distance of the sperm from the centerline decreases. Close to the centerline, on the other hand, the cross-stream migration velocity decreases as the sperm further approaches the center. The cross-stream migration velocity of the sperm also increases with the sperm number.

Morphological transitions of axially-driven microfilaments

The interactions of microtubules with motor proteins are ubiquitous in cellular and sub-cellular processes that involve motility and cargo transport. In vitro motility assays have demonstrated that motor-driven microtubules exhibit rich dynamical behaviors from straight to curved configurations. Here, we theoretically investigate the dynamic instabilities of elastic filaments, with free-ends, driven by single follower forces that emulate the action of molecular motors. Using the resistive force theory at low Reynolds number, and a combination of numerical techniques with linear stability analysis, we show the existence of four distinct regimes of filament behavior, including a novel buckled state with locked curvature. These successive instabilities recapitulate the full range of experimentally-observed microtubule behavior, implying that neither structural nor actuation asymmetry are needed to elicit this rich repertoire of motion.

Spontaneous oscillations of elastic filaments induced by molecular motors

It is known from the wave-like motion of microtubules in motility assays that the piconewton forces that motors produce can be sufficient to bend the filaments. In cellular phenomena such as cytosplasmic streaming, molecular motors translocate along cytoskeletal filaments, carrying cargo which entrains fluid. When large numbers of such forced filaments interact through the surrounding fluid, as in particular stages of oocyte development in Drosophila melanogaster, complex dynamics are observed, but the detailed mechanics underlying them has remained unclear. Motivated by these observations, we study here perhaps the simplest model for these phenomena: an elastic filament, pinned at one end, acted on by a molecular motor treated as a point force. Because the force acts tangential to the filament, no matter what its shape, this 'follower-force' problem is intrinsically non-variational, and thereby differs fundamentally from Euler buckling, where the force has a fixed direction, and which, in the low-Reynolds-number regime, ultimately leads to a stationary, energy-minimizing shape. Through a combination of linear stability theory, analytical study of a solvable simplified 'two-link' model and numerical studies of the full elastohydrodynamic equations of motion, we elucidate the Hopf bifurcation that occurs with increasing forcing of a filament, leading to flapping motion analogous to the high-Reynolds-number oscillations of a garden hose with a free end.

Microfluidic viscometry using magnetically actuated micropost arrays

Here we describe development of a microfluidic viscometer based on arrays of magnetically actuated micro-posts. Quantitative viscosities over a range of three orders of magnitude were determined for samples of less than 20 μL. This represents the first demonstration of quantitative viscometry using driven flexible micropost arrays. Critical to the success of our system is a comprehensive analytical model that includes the mechanical and magnetic properties of the actuating posts, the optical readout, and fluid-structure interactions. We found that alterations of the actuator beat shape as parameterized by the dimensionless “sperm number” must be taken into account to determine the fluid properties from the measured actuator dynamics. Beyond our particular system, the model described here can provide dynamics predictions for a broad class of flexible microactuator designs. We also show how the model can guide the design of new arrays that expand the accessible range of measurements.

The counterbend dynamics of cross-linked filament bundles and flagella

Cross-linked filament bundles, such as in cilia and flagella, are ubiquitous in biology. They are considered in textbooks as simple filaments with larger stiffness. Recent observations of flagellar counterbend, however, show that induction of curvature in one section of a passive flagellum instigates a compensatory counter-curvature elsewhere, exposing the intricate role of the diminutive cross-linking proteins at large scales. We show that this effect, a material property of the cross-linking mechanics, modifies the bundle dynamics and induces a bimodal L² - L³ length-dependent material response that departs from the Euler-Bernoulli theory. Hence, the use of simpler theories to analyse experiments can result in paradoxical interpretations. Remarkably, the counterbend dynamics instigates counter-waves in opposition to driven oscillations in distant parts of the bundle, with potential impact on the regulation of flagellar bending waves. These results have a range of physical and biological applications, including the empirical disentanglement of material quantities via counterbend dynamics.

Acoustic Actuation of in situ Fabricated Artificial Cilia

We present on-chip acoustic actuation of in situ fabricated artificial cilia. Arrays of cilia structures are UV polymerized inside a microfluidic channel using a photocurable polyethylene glycol (PEG) polymer solution and photomasks. During polymerization, cilia structures are attached to a silane treated glass surface inside the microchannel. Then, the cilia structures are actuated using acoustic vibrations at 4.6 kHz generated by piezo transducers. As a demonstration of a practical application, DI water and fluorescein dye solutions are mixed inside a microfluidic channel. Using pulses of acoustic excitations, and locally fabricated cilia structures within a certain region of the microchannel, a waveform of mixing behavior is obtained. This result illustrates one potential application wherein researchers can achieve spatiotemporal control of biological microenvironments in cell stimulation studies. These acoustically actuated, in situ fabricated, cilia structures can be used in many on-chip applications in biological, chemical and engineering studies.

Single Actin Bundle Rheology

Bundled actin structures play an essential role in the mechanical response of the actin cytoskeleton in eukaryotic cells. Although responsible for crucial cellular processes, they are rarely investigated in comparison to single filaments and isotropic networks. Presenting a highly anisotropic structure, the determination of the mechanical properties of individual bundles was previously achieved through passive approaches observing bending deformations induced by thermal fluctuations. We present a new method to determine the bending stiffness of individual bundles, by measuring the decay of an actively induced oscillation. This approach allows us to systematically test anisotropic, bundled structures. Our experiments revealed that thin, depletion force-induced bundles behave as semiflexible polymers and obey the theoretical predictions determined by the wormlike chain model. Thickening an individual bundle by merging it with other bundles enabled us to study effects that are solely based on the number of involved filaments. These thicker bundles showed a frequency-dependent bending stiffness, a behavior that is inconsistent with the predictions of the wormlike chain model. We attribute this effect to internal processes and give a possible explanation with regard to the wormlike bundle theory.

Asymmetric motion of magnetically actuated artificial cilia

Most microorganisms use hair-like cilia with asymmetric beating to perform vital bio-physical processes. In this paper, we demonstrate a novel fabrication method for creating magnetic artificial cilia capable of such a biologically inspired asymmetric beating pattern essential for inducing microfluidic transport at low Reynolds number. The cilia are fabricated using a lithographic process in conjunction with deposition of magnetic nickel-iron permalloy to create flexible filaments that can be manipulated by varying an external magnetic field. A rotating permanent magnet is used to actuate the cilia. We examine the kinematics of a cilium and demonstrate that the cilium motion is defined by an interplay among elastic, magnetic, and viscous forces. Specifically, the forward stroke is induced by the rotation of the magnet which bends the cilium, whereas the recovery stroke is defined by the straightening of the deformed cilium, releasing accumulated elastic potential energy. This difference in dominating forces acting during the forward stroke and the recovery stroke leads to an asymmetric beating pattern of the cilium. Such magnetic cilia can find applications in microfluidic pumping, mixing, and other fluid handling processes.

Dynamic Wrinkling and Strengthening of an Elastic Filament in a Viscous Fluid

We investigate the wrinkling dynamics of an elastic filament immersed in a viscous fluid submitted to compression at a finite rate with experiments and by combining geometric nonlinearities, elasticity, and slender body theory. The drag induces a dynamic lateral reinforcement of the filament leading to growth of wrinkles that coarsen over time. We discover a new dynamical regime characterized by a time scale with a nontrivial dependence on the loading rate, where the growth of the instability is superexponential and the wave number is an increasing function of the loading rate. We find that this time scale can be interpreted as the characteristic time over which the filament transitions from the extensible to the inextensible regime. In contrast with our analysis with moving boundary conditions, Biot's analysis in the limit of infinitely fast loading leads to rate independent exponential growth and wavelength.

Propulsion via flexible flapping in granular media

Biological locomotion in nature is often achieved by the interaction between a flexible body and its surrounding medium. The interaction of a flexible body with granular media is less understood compared with viscous fluids partially due to its complex rheological properties. In this work, we explore the effect of flexibility on granular propulsion by considering a simple mechanical model in which a rigid rod is connected to a torsional spring that is under a displacement actuation using a granular resistive force theory. Through a combined numerical and asymptotic investigation, we characterize the propulsive dynamics of such a flexible flapper in relation to the actuation amplitude and spring stiffness, and we compare these dynamics with those observed in a viscous fluid. In addition, we demonstrate that the maximum possible propulsive force can be obtained in the steady propulsion limit with a finite spring stiffness and large actuation amplitude. These results may apply to the development of synthetic locomotive systems that exploit flexibility to move through complex terrestrial media.

Maximizing propulsive thrust of a driven filament at low Reynolds number via variable flexibility

At low Reynolds numbers the locomotive capability of a body can be dramatically hindered by the absence of inertia. In this work, we show how propulsive performance in this regime can be significantly enhanced by employing spatially varying flexibility. As a prototypical example, we consider the propulsive thrust generated by a filament periodically driven at one end. The rigid case leads to zero propulsion, as so constrained by Purcell's scallop theorem, while for uniform filaments there exists a bending stiffness maximizing the propulsive force at a given frequency; here we demonstrate explicitly how considerable further improvement can be achieved by simply varying the stiffness along the filament. The optimal flexibility distribution is strongly configuration-dependent: while increasing the flexibility towards the tail-end enhances the propulsion of a clamped filament, for a hinged filament decreasing the flexibility towards the tail-end is instead favorable. The results reveal new design principles for maximizing propulsion at low Reynolds numbers, potentially useful for developing synthetic micro-swimmers requiring large propulsive force for various biomedical applications.

Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods

The rigidity of numerous biological filaments and crafted microrods has been conveniently deduced from the analysis of their thermal fluctuations. However, the difficulty of measuring nanometric displacements with an optical microscope has so far limited such studies to sufficiently flexible rods, of which the persistence length ([Formula: see text]) rarely exceeds 1 m at room temperature. Here, we demonstrate the possibility to probe 10-fold stiffer rods by a combination of superresolutive optical methods and a statistical analysis of the data based on a recent theoretical model that predicts the amplitude of the fluctuations at any location of the rod [Benetatos P, Frey E (2003) Phys Rev E Stat Nonlin Soft Matter Phys 67(5):051108]. Using this approach, we report measures of [Formula: see text] up to 0.5 km. We obtained these measurements on recently designed superparamagnetic [Formula: see text]40-[Formula: see text]m-long microrods containing iron-oxide nanoparticles connected by a polymer mesh. Using their magnetic properties, we provide an alternative proof of validity of these thermal measurements: For each individual studied rod, we performed a second measure of its rigidity by deflecting it with a uniform magnetic field. The agreement between the thermal and the magnetoelastic measures was realized with more than a decade of values of [Formula: see text] from 5.1 m to 129 m, corresponding to a bending modulus ranging from 2.2 to 54 (×[Formula: see text] Jm). Despite the apparent homogeneity of the analyzed microrods, their Young modulus follows a broad distribution from 1.9 MPa to 59 MPa and up to 200 MPa, depending on the size of the nanoparticles.

Identification of internal properties of fibres and micro-swimmers

In this paper, we address the identifiability of constitutive parameters of passive or active micro-swimmers. We first present a general framework for describing fibres or micro-swimmers using a bead-model description. Using a kinematic constraint formulation to describe fibres, flagellum or cilia, we find explicit linear relationship between elastic constitutive parameters and generalized velocities from computing contact forces. This linear formulation then permits one to address explicitly identifiability conditions and solve for parameter identification. We show that both active forcing and passive parameters are both identifiable independently but not simultaneously. We also provide unbiased estimators for generalized elastic parameters in the presence of Langevin-like forcing with Gaussian noise using a Bayesian approach. These theoretical results are illustrated in various configurations showing the efficiency of the proposed approach for direct parameter identification. The convergence of the proposed estimators is successfully tested numerically.

Motion planning and motility maps for flagellar microswimmers

We study two microswimmers consisting of a spherical rigid head and a passive elastic tail. In the first one the tail is clamped to the head, and the system oscillates under the action of an external torque. In the second one, head and tail are connected by a joint allowing the angle between them to vary periodically, as a result of an oscillating internal torque. Previous studies on these models were restricted to sinusoidal actuations, showing that the swimmers can propel while moving on average along a straight line, in the direction given by the symmetry axis around which beating takes place. We extend these results to motions produced by generic (non-sinusoidal) periodic actuations within the regime of small compliance of the tail. We find that modulation in the velocity of actuation can provide a mechanism to select different directions of motion. With velocity-modulated inputs, the externally actuated swimmer can translate laterally with respect to the symmetry axis of beating, while the internally actuated one is able to move along curved trajectories. The governing equations are analysed with an asymptotic perturbation scheme, providing explicit formulas, whose results are expressed through motility maps. Asymptotic approximations are further validated by numerical simulations.

A nonequilibrium power balance relation for analyzing dissipative filament dynamics

Biofilaments like F-actin or microtubules, as well as cilia, flagella, or filament bundles, are often deformed by distributed and time-dependent external forces. It is highly desirable to characterize these filaments' mechanics in an efficient way, either using a single experiment or a high throughput method. We here propose a dynamic power balance approach to study nonequilibrium filament dynamics and exemplify it both experimentally and theoretically by applying it to microtubule gliding assay dynamics. Its usefulness is highlighted by the experimental determination of the lateral friction coefficient for microtubules on kinesins. In contrast to what is usually assumed, friction is anisotropic, in a similar fashion as hydrodynamic friction. We also exemplify, by considering a microtubule buckling event, that if at least one parameter is known in advance, all other parameters can be determined by analyzing a single time-dependent experiment.

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.

Effects of thermal noise on the transitional dynamics of an inextensible elastic filament in stagnation flow

We investigate the dynamics of a single inextensible elastic filament subject to anisotropic friction in a viscous stagnation-point flow, by employing both a continuum model represented by Langevin type stochastic partial differential equations (SPDEs) and a dissipative particle dynamics (DPD) method. Unlike previous works, the filament is free to rotate and the tension along the filament is determined by the local inextensible constraint. The kinematics of the filament is recorded and studied with normal modes analysis. The results show that the filament displays an instability induced by negative tension, which is analogous to Euler buckling of a beam. Symmetry breaking of normal modes dynamics and stretch-coil transitions are observed above the threshold of the buckling instability point. Furthermore, both temporal and spatial noise are amplified resulting from the interaction of thermal fluctuations and nonlinear filament dynamics. Specifically, the spatial noise is amplified with even normal modes being excited due to symmetry breaking, while the temporal noise is amplified with increasing time correlation length and variance.

Intracellular and extracellular forces drive primary cilia movement

Primary cilia are ubiquitous, microtubule-based organelles that play diverse roles in sensory transduction in many eukaryotic cells. They interrogate the cellular environment through chemosensing, osmosensing, and mechanosensing using receptors and ion channels in the ciliary membrane. Little is known about the mechanical and structural properties of the cilium and how these properties contribute to ciliary perception. We probed the mechanical responses of primary cilia from kidney epithelial cells [Madin-Darby canine kidney-II (MDCK-II)], which sense fluid flow in renal ducts. We found that, on manipulation with an optical trap, cilia deflect by bending along their length and pivoting around an effective hinge located below the basal body. The calculated bending rigidity indicates weak microtubule doublet coupling. Primary cilia of MDCK cells lack interdoublet dynein motors. Nevertheless, we found that the organelles display active motility. 3D tracking showed correlated fluctuations of the cilium and basal body. These angular movements seemed random but were dependent on ATP and cytoplasmic myosin-II in the cell cortex. We conclude that force generation by the actin cytoskeleton surrounding the basal body results in active ciliary movement. We speculate that actin-driven ciliary movement might tune and calibrate ciliary sensory functions.

A model of filamentous cyanobacteria leading to reticulate pattern formation

The filamentous cyanobacterium, Pseudanabaena, has been shown to produce reticulate patterns that are thought to be the result of its gliding motility. Similar fossilized structures found in the geological record constitute some of the earliest signs of life on Earth. It is difficult to tie these fossils, which are billions of years old, directly to the specific microorganisms that built them. Identifying the physicochemical conditions and microorganism properties that lead microbial mats to form macroscopic structures can lead to a better understanding of the conditions on Earth at the dawn of life. In this article, a cell-based model is used to simulate the formation of reticulate patterns in cultures of Pseudanabaena. A minimal system of long and flexible trichomes capable of gliding motility is shown to be sufficient to produce stable patterns consisting of a network of streams. Varying model parameters indicate that systems with little to no cohesion, high trichome density and persistent movement are conducive to reticulate pattern formation, in conformance with experimental observations.

Filaments in curved streamlines: Rapid formation of Staphylococcus aureus biofilm streamers

Biofilms are surface-associated conglomerates of bacteria that are highly resistant to antibiotics. These bacterial communities can cause chronic infections in humans by colonizing, for example, medical implants, heart valves, or lungs. Staphylococcus aureus, a notorious human pathogen, causes some of the most common biofilm-related infections. Despite the clinical importance of S. aureus biofilms, it remains mostly unknown how physical effects, in particular flow, and surface structure influence biofilm dynamics. Here we use model microfluidic systems to investigate how environmental factors, such as surface geometry, surface chemistry, and fluid flow affect biofilm development in S. aureus. We discovered that S. aureus rapidly forms flow-induced, filamentous biofilm streamers, and furthermore if surfaces are coated with human blood plasma, streamers appear within minutes and clog the channels more rapidly than if the channels are uncoated. To understand how biofilm streamer filaments reorient in flows with curved streamlines to bridge the distances between corners, we developed a mathematical model based on resistive force theory of slender filaments. Understanding physical aspects of biofilm formation in S. aureus may lead to new approaches for interrupting biofilm formation of this pathogen.

Torque correlation length and stochastic twist dynamics of DNA

We introduce a short correlation length for torque in twisting-stiff biomolecules, which is necessary for the physical property that torque fluctuations be finite in amplitude. We develop a nonequilibrium theory of dynamics of DNA twisting which predicts two crossover time scales for temporal torque correlations in single-molecule experiments. Bending fluctuations can be included, and at linear order we find that they do not affect the twist dynamics. However, twist fluctuations affect bending, and we predict the spatial inhomogeneity of twist, torque, and buckling arising in nonequilibrium "rotor-bead" experiments.

Simulation of swimming of a flexible filament using the generalized lattice-spring lattice-Boltzmann method

A generalized lattice-spring lattice-Boltzmann model (GLLM) is introduced by adding a three-body force in the traditional lattice-spring model. This method is able to deal with bending deformation of flexible biological bodies in fluids. The interactions between elastic solids and fluid are treated with the immersed boundary-lattice Boltzmann method. GLLM is validated by comparing the present results with the existing theoretical and simulation results. As an application of GLLM, swimming of flagellum in fluid is simulated and propulsive force as a function of driven frequency and fluid structures at various Reynolds numbers 0.15-5.1 are presented in this paper.

Methods for modeling cytoskeletal and DNA filaments

This review summarizes the models that researchers use to represent the conformations and dynamics of cytoskeletal and DNA filaments. It focuses on models that address individual filaments in continuous space. Conformation models include the freely jointed, Gaussian, angle-biased chain (ABC), and wormlike chain (WLC) models, of which the first three bend at discrete joints and the last bends continuously. Predictions from the WLC model generally agree well with experiment. Dynamics models include the Rouse, Zimm, stiff rod, dynamic WLC, and reptation models, of which the first four apply to isolated filaments and the last to entangled filaments. Experiments show that the dynamic WLC and reptation models are most accurate. They also show that biological filaments typically experience strong hydrodynamic coupling and/or constrained motion. Computer simulation methods that address filament dynamics typically compute filament segment velocities from local forces using the Langevin equation and then integrate these velocities with explicit or implicit methods; the former are more versatile and the latter are more efficient. Much remains to be discovered in biological filament modeling. In particular, filament dynamics in living cells are not well understood, and current computational methods are too slow and not sufficiently versatile. Although primarily a review, this paper also presents new statistical calculations for the ABC and WLC models. Additionally, it corrects several discrepancies in the literature about bending and torsional persistence length definitions, and their relations to flexural and torsional rigidities.

Swimming dynamics of bidirectional artificial flagella

We study magnetic artificial flagella whose swimming speed and direction can be controlled using light and magnetic field as external triggers. The dependence of the swimming velocity on the system parameters (e.g., length, stiffness, fluid viscosity, and magnetic field) is explored using a computational framework in which the magnetostatic, fluid dynamic, and solid mechanics equations are solved simultaneously. A dimensionless analysis is carried out to obtain an optimal combination of system parameters for which the swimming velocity is maximal. The swimming direction reversal is addressed by incorporating photoresponsive materials, which in the photoactuated state can mimic natural mastigonemes.

Passive elastic mechanism to mimic fish-muscle action in anguilliform swimming

Swimmers in nature use body undulations to generate propulsive and manoeuvring forces. The anguilliform kinematics is driven by muscular actions all along the body, involving a complex temporal and spatial coordination of all the local actuations. Such swimming kinematics can be reproduced artificially, in a simpler way, by using the elasticity of the body passively. Here, we present experiments on self-propelled elastic swimmers at a free surface in the inertial regime. By addressing the fluid-structure interaction problem of anguilliform swimming, we show that our artificial swimmers are well described by coupling a beam theory with the potential flow model of Lighthill. In particular, we show that the propagative nature of the elastic wave producing the propulsive force is strongly dependent on the dissipation of energy along the body of the swimmer.

FINITE ELEMENT MODELING OF THE THERMAL FLUCTUATIONS OF A SINGLE ANISOTROPIC POLYMER

Thermal fluctuations of microtubules (MTs) and other cytoskeletal filaments govern to a great extent the complex rheological properties of the cytoskeleton in eukaryotic cells. In recent years, much effort has been put into capturing the dynamics of these fluctuations by means of analytical and numerical models. These attempts have been very successful for, but also remain limited to, isotropic polymers. To correctly interpret experimental work on (strongly) anisotropic semiflexible polymers, there is a need for a numerical modeling tool that accurately captures the dynamics of polymers with anisotropic material properties. In the current study, we present a finite element (FE) framework for simulating the thermal dynamics of a single anisotropic semiflexible polymer. First, we demonstrated the accuracy of our framework by comparison of the simulated mean square displacement (MSD) of the end-to-end distance with analytical predictions based on the worm-like chain model. Then, we implemented a transversely isotropic material model, characteristic for biopolymers such as MTs, and studied the persistence length for various ratios between the longitudinal shear modulus, G12, and corresponding Young's modulus, E1. Finally, we put our findings in context by addressing a recent experimental work on grafted transversely isotropic MTs. In that research, a simplified static mechanical model was used to deduce a very high level of MT anisotropy to explain the observation that the persistence length of grafted MTs increases as contour length increases. We showed, by means of our FE framework, that the anisotropic properties cannot account for the reported length-dependent persistence length.

Viscous damping of vibrations in microtubules

Pokorný et al. have recently suggested that metabolic processes drivemicrotubules in a cell to vibrate at Megahertz frequencies, but the theorydoes not explicitly consider dissipative effects which will tend to damp outthe vibrations. To examine the effects of viscous damping on the structure,we determine viscous forces and rate of energy loss in a cylinderundergoing longitudinal oscillations in water. A nondimensional expressionis obtained for the viscous drag on the cylinder. When applied to amicrotubule, the results indicate that viscous damping is several orders ofmagnitude too large to allow resonant vibrations.

Dynamics of the primary cilium in shear flow

In this work, the equilibrium shape and dynamics of a primary cilium under flow are investigated by using both theoretical modeling and experiment. The cilium is modeled as an elastic beam that may undergo large deflection due to the hydrodynamic load. Equilibrium results show that the anchoring effects of the basal body on the cilium axoneme behave as a nonlinear rotational spring. Details of the rotational spring are elucidated by coupling the elastic beam with an elastic shell. We further study the dynamics of cilium under shear flow with the cilium base angle determined from the nonlinear rotational spring, and obtain good agreement in cilium bending and relaxing dynamics when comparing between modeling and experimental results. These results potentially shed light on the physics underlying the mechanosensitive ion channel transport through the ciliary membrane.