Flexible Flagella Research Articles

Eukaryotic swimming cells are shaped by hydrodynamic constraints

Eukaryotic swimming cells such as spermatozoa, algae or protozoa use flagella or cilia to move in viscous fluids. The motion of their flexible appendages in the surrounding fluid induces propulsive forces that balance viscous drag on the cells and lead to a directed swimming motion. Here, we use our recently built database of cell motility (BOSO-Micro) to investigate the extent to which the shapes of eukaryotic swimming cells may be optimal from a hydrodynamic standpoint. We first examine the morphology of flexible flagella undergoing waving deformation and show that their amplitude-to-wavelength ratio is near that predicted theoretically to optimise the propulsive efficiency of active filaments. Next, we consider ciliates, for which locomotion is induced by the collective beating of short cilia covering their surface. We show that the aspect ratios of ciliates are close to that predicted to minimise the viscous drag of the cell body. Both results strongly suggest a key role played by hydrodynamic constraints, in particular viscous drag, in shaping eukaryotic swimming cells.

A numerical method for the locomotion of bi-flagellated bacteria in viscous fluid

Flagellated bacteria propel themselves by rotating flexible flagella driven by independent motors. Depending on the rotation direction of the motors and the handedness of the helical filaments, the flagella either pull or push the cell body. Motivated by experimental observations of Magnetococcus marinus, we develop an elastohydrodynamic model to study the locomotion of a bi-flagellated bacterium with one puller flagellum and one pusher flagellum. In this model, the boundary integral technique and Kirchhoff rod model are employed respectively to calculate the hydrodynamic forces on the swimmer and model the elastic deformations of the flagella. Our numerical results demonstrate that the model bacterium travels along a double helical trajectory, which is consistent with the experimental observations. Varying the stiffness, orientations or positions of the flagella significantly changes the swimming characteristics. Notably, when either the applied torque is higher than a critical value or the flagellum stiffness is lower than a critical stiffness, the pusher flagellum exhibits overwhirling motion, resulting in a more complicated swimming style and a lower swimming speed. For a moderate flagellum stiffness, the swimming speed is insensitive to the rest configuration orientation over a wide range of orientation angles as the flagella deform to maintain alignment with the swimming direction.

Modeling the Locomotion of Articulated Soft Robots in Granular Medium

We introduce a numerical tool for modeling articulated soft robots that couples discrete differential geometry-based simulation of elastic rods, our model for the articulated structure, and other external forces. Parallel to simulations, we build an untethered robot testbed, in the granular medium, comprised of multiple flexible flagella that are rotated about an axis by a motor. Drag from the granules causes the flagella to deform and the deformed shape generates a net forward propulsion. External drag depends on the flagellar shape, while the change in flagellar shape is the result of the competition between the external loading and elastic forces. We find reasonable quantitative agreement between experiments and simulations. Owing to a rod-based kinematic representation of the robot, the simulation can run faster than real-time in some cases, and, therefore, we can use it as a design tool for this class of soft robots. We find that there is an optimal rotational speed at which maximum efficiency is achieved. Moreover, both experiments and simulations show that increasing the number of flagella from two to three decreases the speed of the robot. This indicates that our simulator is potentially applicable for unknown physics exploration. We also gain insight into the mechanics of granular medium - while resistive force theory can successfully describe the propulsion at low number of flagella, it fails when more flagella are added to the robot.

Artificial flexible sperm-like nanorobot based on self-assembly and its bidirectional propulsion in precessing magnetic fields

Sperm cells can move at a high speed in biofluids based on the flexible flagella, which inspire novel flagellar micro-/nanorobots to be designed. Despite progress in fabricating sperm-type robots at micro scale, mass fabrication of vivid sperm-like nanorobots with flagellar flexibility is still challenging. In this work, a facile and efficient strategy is proposed to produce flexible sperm-like nanorobots with self-assembled head-to-tail structure, and its bidirectional propulsion property was studied in detail. The nanorobots were composed of a superparamagnetic head and a flexible Au/PPy flagellum, which were covalently linked via biotin-streptavidin bonding with a high yield. Under precessing magnetic fields, the head drove the flexible tail to rotate and generated undulatory bending waves propagating along the body. Bidirectional locomotion was investigated, and moving velocity as well as direction varied with the actuating conditions (field strength, frequency, direction) and the nanorobot’s structure (tail length). Effective flagellar propulsion was observed near the substrate and high velocities were attained to move back and forth without U-turn. Typical modelling based on elastohydrodynamics and undulatory wave propagation were utilized for propulsion analysis. This research presents novel artificial flexible sperm-like nanorobots with delicate self-assembled head-to-tail structures and remarkable bidirectional locomotion performances, indicating significant potentials for nanorobotic design and future biomedical application.

Energetics of synchronization for model flagella and cilia.

Synchronization is often observed in the swimming of flagellated cells, either for multiple appendages on the same organism or between the flagella of nearby cells. Beating cilia are also seen to synchronize their dynamics. In 1951, Taylor showed that the observed in-phase beating of the flagella of coswimming spermatozoa was consistent with minimization of the energy dissipated in the surrounding fluid. Here we revisit Taylor's hypothesis for three models of flagella and cilia: (1) Taylor's waving sheets with both longitudinal and transverse modes, as relevant for flexible flagella, (2) spheres orbiting above a no-slip surface to model interacting flexible cilia, and (3) whirling rods above a no-slip surface to address the interaction of nodal cilia. By calculating the flow fields explicitly, we show that the rate of working of the model flagella or cilia is minimized in our three models for (1) a phase difference depending on the separation of the sheets and precise waving kinematics, (2) in-phase or opposite-phase motion depending on the relative position and orientation of the spheres, and (3) in-phase whirling of the rods. These results will be useful in future models probing the dynamics of synchronization in these setups.

A Manipulability Criterion for Magnetic Actuation of Miniature Swimmers With Flexible Flagellum

The use of untethered miniature swimmers is a promising trend, especially in biomedical applications. These swimmers are often operated remotely using a magnetic field commonly generated using fixed coils that can suffer from a lack of compactness and heating issues. The analysis of the swimming capabilities is still an ongoing topic of research. In this letter, we focus on the ability of a magnetic actuation system to operate the propulsion of miniature swimmers with flexible flagellum. As a first contribution, we present a new manipulability criterion to assess the ability of a magnetic actuation system to operate a swimming robot, i.e. to ensure a displacement in any desired direction with a fixed minimum speed. This criterion is developed thanks to an analogy with cable-driven parallel robots. As a second contribution, this manipulability criterion is exploited to identify the dexterous swimming workspace which can be used to design of new coil configurations as well as to highlight the possibilities of moving coil systems. A case study for a planar workspace surrounded by three coils is in particular carried out. The accompanying video illustrates the application of the proposed criterion in 3D, for a large number of coils.

Effects of the intrinsic curvature of elastic filaments on the propulsion of a flagellated microrobot

Understanding the propulsion mechanism of swimming microorganisms will facilitate the development of synthetic microswimmers for active cargo deliveries. Herein, we studied, theoretically and numerically, inertialess locomotion of a microswimmer—a spherical body propelled by two symmetrically actuated elastic filaments in the shape of a circular arc at rest, focusing on the effects of their uniform intrinsic curvature κ¯c. Combining the resistive force theory for viscous flow and Euler–Bernoulli beam theory for elastic filaments, the elasto-hydrodynamics was solved asymptotically. Our theory was verified by simulations using regularized Stokeslets posed on the filament centerlines, with and without considering hydrodynamic interactions (HIs) between the body and filaments. The asymptotic and numerical results showed qualitative agreement. Reasonable quantitative agreement between the asymptotic results and the numerical predictions neglecting body–filament HIs was observed, especially for small |κ¯c|. However, they deviated quantitatively from the numerical results with body–filament HIs, especially at a large κ¯c when the HIs became important owing to the short body–filament distance. The propulsive force generated by two arc-shaped filaments significantly depend on their uniform intrinsic curvature κ¯c. An appreciable increase in the thrust can be achieved by adjusting κ¯c, which qualitatively confirms and explains the experimentally reported propulsive enhancement facilitated by intrinsically curved appendages [Z. Ye, S. Régnier, and M. Sitti, “Rotating magnetic miniature swimming robots with multiple flexible flagella,” IEEE Trans. Rob. 30, 3–13 (2014)]. The increase in κ¯c can even change the sign of the thrust, leading to counter-intuitive, backward propulsion. The flow field reveals the hydrodynamic signature of the swimmer that shifts with time between a neutral swimmer, a pusher, and a puller.

Modeling of an acoustically actuated artificial micro-swimmer

Some recent achievements in microfabrication have demonstrated ultrasound-actuated artificial micro-swimmers for medical applications. However, the theoretical model of actuation and swimming is still lacking. Here we report a theoretical study of an acoustically actuated sperm-like artificial micro-swimmer which consists of a rigid head and a flexible flagellum. We provide the quantitative relation between head oscillation amplitude and acoustic pressure and frequency, and the theoretical account of how the flagellum is whipped, which brings about propulsion. The resistive force theory is employed in our model to relate the dynamic response of a flagellum and the motility of the swimmer. In order to make our theoretical model applicable in a realistic design of sperm-like micro-swimmer, we have involved the inertia term and material damping in the governing equation and considered the variable cross-section of a flagellum. The numerical results reveal that the micro-swimmer actuated by ultrasound can achieve a perceptible velocity, especially at resonance. Influences of non-dimensional parameters, such as the resonance index, sperm number, and material damping coefficient, are discussed and a comparison with experimental results demonstrates the validity of the proposed model.

Controllable switching between planar and helical flagellar swimming of a soft robotic sperm.

Sperm cells undergo a wide variety of swimming patterns by a beating flagellum to maintain high speed regardless of the rheological and physical properties of the background fluid. In this work, we develop and control a soft robotic sperm that undergoes controllable switching between swimming modes like biological sperm cells. The soft robotic sperm consists of a magnetic head and an ultra-thin flexible flagellum, and is actuated using external magnetic fields. We observe that out-of-plane wobbling of the head results in helical wave propagation along the flagellum, whereas in-plane wobbling achieves planar wave propagation. Our theoretical predictions and experimental results show the ability of the soft robotic sperm to change its swimming speed by tuning the beating frequency of its flagellum and the propulsion pattern. The average speed of the soft robotic sperm increases by factors of 2 and 1.2 in fluids with viscosity of 1 Pa.s and 5 Pa.s at relatively low actuation frequencies, respectively, when they switch between planar to helical flagellar propulsion.

Flexibility of Micromagnetic Flagella in the Presence of an Oscillating Field

Magnetic microbeads have recently been the subject of much study because of their potential applications in microfluidic systems which facilitate mixing [1], labeling [2], separation [3] and transport [4] in lab-on-achip devices. It has been proposed that rotating superparamagnetic beads chains can enhance the mixing efficiency in microfluidic devices [5] and the magnetic micro-bead chain that is subjected to an oscillating field can be used to design a micro-swimmer [6]. To manipulate the micro-beads chain swimming in the low-Reynolds number environment, various magnetic actuation methods have been employed to obtain the higher propulsive efficiency for the locomotion in the viscous fluid. A flexible flagellum is arguably the simplest mechanism to duplicate as it is a one-dimensional structure. However, a challenge to create the stable planar beating motion for propulsion generation is to fabricate the magnetic flagellum simultaneously flexible and stable structure at a microscale. In order to effectively use an oscillating magnetic flagellum as a tail of a microswimmer, the measurement of the bending rigidity for the flexible micro-chains is essential. In this study, the curvatures and the bending rigidity of distorted chains with different lengths subjected to an external field with various controlling parameters (such as the field intensity and frequency) were probed to evaluate the flexibility of the magnetic flagellum. The magnetic flagellum is obtained by combining the self-assembling ability of micro-sized superparamagnetic particles composed the iron oxide magnetite (Fe 3 O 4 ) embedded in polystyrene microspheres suspended in distilled water. Properties of the commercially available micro-beads are density ρ=1500 kg/m3, diameter d = 4.5 μm, initial magnetic susceptibility χ = 1.6, and without magnetic hysteresis or remanence, which could be magnetized under an applying external field and completely demagnetized when the field is removed. Firstly, the magnetic beads magnetized by a homogeneous static field $H_{d}$ tended to aggregate and form the linear flagellum, and a dynamical sinusoidal field $H_{y}$ with a maximum amplitude $H_{p}$ and an adjustable frequency $f$, that is, $H_{y} = H_{p} { sin(2}\pi {ft)}$, were then applied in a direction perpendicular to $H_{d}$. Thus, the planar beating motion is created to form the dynamic flexible flagellum. In order to accurately obtain the chain flexibility and bending rigidity, we model the waggling chains as continuous elastic flagella and analytically derive their curvatures as a function of magnetic field strength. The maximum dimensionless curvature (denoted as $C_{max})$ of the chain is given as $C_{max}=d/R, R$ represents the flagellum’s radius of curvature. The bending rigidity κ of the waggling chain is measured from the S-shape of the chain by using the equation given as [4]: $C_{max}= {(Bd/4)(}\pi {/3}\kappa\mu _{0})^{0.5}$ Where the $\mu _{0}$ represents the vacuum permeability, $B$ is the overall magnetic field strength. Figure 1 shows the most significant deformed shapes for the flexible flagella of various lengths subjected to the increasing field strength. It can be seen that all the chains bend to S-shape for all values of applying field strength, and the deformation increases slightly with an increase in the perpendicular dynamic field intensity H p for each flagellum. On the other hand, the influences of flagellum strengths are not consistent. By examining figure 1 carefully, we found the increase of the length doesn’t enhance the deformation significantly. On the contrary, the longer length as flagellum consisting of 16 particles (denoted as P16) behaves more insignificant S-shape than the counterpart of the P15 chain because of the more significant constraint from the stronger hydrodynamic drag acting on a longer chain. The further inspection of the interesting finding can be done by the examination of figure 2, which shows the geometrical relation between the flagellum’s radius of curvature $R$ and the diameter $d$ of the magnetic bead. It can be seen that for all lengths of flagella the maximum curvature of the flagellum increases linearly with the intensity of the applied field. In addition, the value of $C_{max}$ of the flagellum firstly gets higher then declines with the increase of the length, which indicates the magnetic flagellum subjected to an oscillating field would have the most flexible structure at the certain length. The further discussions are shown in full paper demonstrating the effects of the other controlling parameters on $C_{max}$ and the bending rigidity κ.

From flagellar undulations to collective motion: predicting the dynamics of sperm suspensions.

Swimming cells and microorganisms are as diverse in their collective dynamics as they are in their individual shapes and propulsion mechanisms. Even for sperm cells, which have a stereotyped shape consisting of a cell body connected to a flexible flagellum, a wide range of collective dynamics is observed spanning from the formation of tightly packed groups to the display of larger-scale, turbulence-like motion. Using a detailed mathematical model that resolves flagellum dynamics, we perform simulations of sperm suspensions containing up to 1000 cells and explore the connection between individual and collective dynamics. We find that depending on the level of variation in individual dynamics from one swimmer to another, the sperm exhibit either a strong tendency to aggregate, or the suspension exhibits large-scale swirling. Hydrodynamic interactions govern the formation and evolution of both states. In addition, a quantitative analysis of the states reveals that the flows generated at the time scale of flagellum undulations contribute significantly to the overall energy in the surrounding fluid, highlighting the importance of resolving these flows.

Highly Efficient Freestyle Magnetic Nanoswimmer.

The unique swimming strategies of natural microorganisms have inspired recent development of magnetic micro/nanorobots powered by artificial helical or flexible flagella. However, as artificial nanoswimmers with unique geometries are being developed, it is critical to explore new potential modes for kinetic optimization. For example, the freestyle stroke is the most efficient of the competitive swimming strokes for humans. Here we report a new type of magnetic nanorobot, a symmetric multilinked two-arm nanoswimmer, capable of efficient "freestyle" swimming at low Reynolds numbers. Excellent agreement between the experimental observations and theoretical predictions indicates that the powerful "freestyle" propulsion of the two-arm nanorobot is attributed to synchronized oscillatory deformations of the nanorobot under the combined action of magnetic field and viscous forces. It is demonstrated for the first time that the nonplanar propulsion gait due to the cooperative "freestyle" stroke of the two magnetic arms can be powered by a plane oscillatory magnetic field. These two-arm nanorobots are capable of a powerful propulsion up to 12 body lengths per second, along with on-demand speed regulation and remote navigation. Furthermore, the nonplanar propulsion gait powered by the consecutive swinging of the achiral magnetic arms is more efficient than that of common chiral nanohelical swimmers. This new swimming mechanism and its attractive performance opens new possibilities in designing remotely actuated nanorobots for biomedical operation at the nanoscale.

Flagella bending affects macroscopic properties of bacterial suspensions.

To survive in harsh conditions, motile bacteria swim in complex environments and respond to the surrounding flow. Here, we develop a mathematical model describing how flagella bending affects macroscopic properties of bacterial suspensions. First, we show how the flagella bending contributes to the decrease in the effective viscosity observed in dilute suspension. Our results do not impose tumbling (random reorientation) as was previously done to explain the viscosity reduction. Second, we demonstrate how a bacterium escapes from wall entrapment due to the self-induced buckling of flagella. Our results shed light on the role of flexible bacterial flagella in interactions of bacteria with shear flow and walls or obstacles.

Dynamics of a flexible helical filament rotating in a viscous fluid near a rigid boundary

The effects of a no-slip wall on flagellar propulsion are investigated by experiments with macroscopic-scale model flagella and simulations based on algorithms from computer graphics. A nearby wall shifts the threshold for buckling of the flexible flagellum and increases the propulsive force.

Flexibility of bacterial flagella in external shear results in complex swimming trajectories.

Many bacteria use rotating helical flagella in swimming motility. In the search for food or migration towards a new habitat, bacteria occasionally unbundle their flagellar filaments and tumble, leading to an abrupt change in direction. Flexible flagella can also be easily deformed by external shear flow, leading to complex bacterial trajectories. Here, we examine the effects of flagella flexibility on the navigation of bacteria in two fundamental shear flows: planar shear and Poiseuille flow realized in long channels. On the basis of slender body elastodynamics and numerical analysis, we discovered a variety of non-trivial effects stemming from the interplay of self-propulsion, elasticity and shear-induced flagellar bending. We show that in planar shear flow the bacteria execute periodic motion, whereas in Poiseuille flow, they migrate towards the centre of the channel or converge towards a limit cycle. We also find that even a small amount of random reorientation can induce a strong response of bacteria, leading to overall non-periodic trajectories. Our findings exemplify the sensitive role of flagellar flexibility and shed new light on the navigation of bacteria in complex shear flows.

Rotating Magnetic Miniature Swimming Robots With Multiple Flexible Flagella

Recent studies have been carried out for rotating single flexible flagellum: a possible propelling mechanism that has been adopted by several artificial microswimmers due to its relatively simple structure yet considerable propulsive force generation. In this paper, we introduce a miniature swimming robot design with multiple flexible artificial flagella that benefits from the increased number of flagella. The characteristic length of the robot body is less than 1 mm. Experimental characterization of swimming of the robot shows that swimming speed can be linearly improved solely by increasing the number of attached flagella, suggesting a new way for speed enhancement besides flagellum geometry optimization. In addition, a numerical model modified from the single, straight flexible flagellum case is further established to study propulsive force generation by nonstraight, flexible flagellum. A robot with multiple, sinusoidal flagella design is fabricated to demonstrate the capability of the proposed two-step photolithography-based microfabrication method to handle more complex flagella designs, which may enhance swimming performance.

Shape of optimal active flagella

AbstractMany eukaryotic cells use the active waving motion of flexible flagella to self-propel in viscous fluids. However, the criterion governing the selection of particular flagellar waveforms among all possible shapes has proved elusive so far. To address this question, we derive computationally the optimal shape of an internally forced periodic planar flagellum deforming as a travelling wave. The optimum is here defined as the shape leading to a given swimming speed with minimum energetic cost. To calculate the energetic cost, we consider the irreversible internal power expended by the molecular motors forcing the flagellum, only a portion of which is dissipated in the fluid. This optimization approach allows us to derive a family of shapes depending on a single dimensionless number quantifying the relative importance of elastic to viscous effects: the Sperm number. The computed optimal shapes are found to agree with the waveforms observed on spermatozoon of marine organisms, thus suggesting that these eukaryotic flagella might have evolved to be mechanically optimal.

Flagella as a novel alignment medium for the measurement of residual dipolar couplings in proteins

The two flexible rod-like flagella (∼500 nm in diameter and 5-15 μm long) of Chlamydomonas reinhardtii, a unicellular green alga, can weakly align molecules in an external magnetic field, thereby enabling the measurement of various residual dipolar couplings in solution NMR spectroscopy.

Numerical study on the propulsion of a bacterial flagellum in a viscous fluid using an immersed boundary method

The propulsion of a bacterial flagellum in a viscous fluid has gained much attention in the field of biological fluid dynamics. Inspired by the bacterial propulsion, we present a three-dimensional computational model based on an immersed boundary (IB) method to study the propulsive and fluid dynamic features of a solid flexible flagellum in a viscous fluid driven at one side by an external torque. The helical flagellum is modelled by a series of triangular cross-sections with three IB points on each cross-section. Three types of elastic links are created to connect the IB points of one cross-section to the IB points of the next cross-section to obtain an elastic network model of the flagellum. An external torque is applied at the center of the first cross-section modelled as the flagellum motor. The elastic forces are computed based on elastic energy approach and the motor forces are obtained from the magnitude and direction of the applied torque. The Stokes equations governing the flow are solved on a staggered Cartesian grid system using the fractional-step based finite-volume method. The computational model is validated by comparing the swimming speed of the flagellum obtained from the numerical results with that of the existing numerical results. The interplay of propulsive and hydrodynamic features of a left handed helical flagellum driven by an external torque is well captured using the developed model. The effect of elasticity of various types of links on the swimming speed of the flagellum is investigated. It is revealed that among the three types of elastic links, the diagonal links have dominant role in the propulsion behavior of the flagellum. The effect of pitch on the propulsion speed of the flagellum is analyzed by computing its optimum value for a constant flagellum length based on the maximum swimming speed. Numerical simulations are performed to demonstrate the forward to backward swimming behavior of the flagellum and it is concluded that the backward swimming speed is higher than the forward swimming speed. The effect of the channel wall on the forward and backward propulsion of the flagellum is examined. It is observed that the flagellum swims faster in a channel than that in an unbounded fluid domain. Further, it is found that the swimming speed of the flagellum near to a channel wall is considerably higher than that away from the walls.

Fluid-mechanical interaction of flexible bacterial flagella by the immersed boundary method

Flagellar bundling is an important aspect of locomotion in bacteria such as Escherichia coli. To study the hydrodynamic behavior of helical flagella, we present a computational model that is based on the geometry of the bacterial flagellar filament at the micrometer scale. We consider two model flagella, each of which has a rotary motor at its base with the rotation rate of the motor set at 100 Hz. Bundling occurs when both flagella are left-handed helices turning counterclockwise (when viewed from the nonmotor end of the flagellum looking back toward the motor) or when both flagella are right-handed helices turning clockwise. Helical flagella of the other combinations of handedness and rotation direction do not bundle.