Propulsion of microorganisms by a helical flagellum

Propulsion mechanism of artificial flagellated micro-swimmers actuated by acoustic waves-theory and experimental verification

This work examines the acoustically actuated motions of artificial flagellated micro-swimmers (AFMSs) and compares the motility of these micro-swimmers with the predictions based on the corrected resistive force theory (RFT) and the bar-joint model proposed in our previous work. The key ingredient in the theory is the introduction of a correction factorKin drag coefficients to correct the conventional RFT so that the dynamics of an acoustically actuated AFMS with rectangular cross-sections can be accurately modeled. Experimentally, such AFMSs can be easily manufactured based on digital light processing of ultra-violet (UV)-curable resins. We first determined the viscoelastic properties of a UV-cured resin through dynamic mechanical analysis. In particular, the high-frequency storage moduli and loss factors were obtained based on the assumption of time-temperature superposition (TTS), which were then applied in theoretical calculations. Though the extrapolation based on the TTS implied the uncertainty of high-frequency material response and there is limited accuracy in determining head oscillation amplitude, the differences between the measured terminal velocities of the AFMSs and the predicted ones are less than 50%, which, to us, is well acceptable. These results indicate that the motions of acoustic AFMS can be predicted, and thus, designed, which pave the way for their long-awaited applications in targeted therapy.

Geometric phase predicts locomotion performance in undulating living systems across scales

Self-propelling organisms locomote via generation of patterns of self-deformation. Despite the diversity of body plans, internal actuation schemes and environments in limbless vertebrates and invertebrates, such organisms often use similar traveling waves of axial body bending for movement. Delineating how self-deformation parameters lead to locomotor performance (e.g. speed, energy, turning capabilities) remains challenging. We show that a geometric framework, replacing laborious calculation with a diagrammatic scheme, is well-suited to discovery and comparison of effective patterns of wave dynamics in diverse living systems. We focus on a regime of undulatory locomotion, that of highly damped environments, which is applicable not only to small organisms in viscous fluids, but also larger animals in frictional fluids (sand) and on frictional ground. We find that the traveling wave dynamics used by mm-scale nematode worms and cm-scale desert dwelling snakes and lizards can be described by time series of weights associated with two principal modes. The approximately circular closed path trajectories of mode weights in a self-deformation space enclose near-maximal surface integral (geometric phase) for organisms spanning two decades in body length. We hypothesize that such trajectories are targets of control (which we refer to as "serpenoid templates"). Further, the geometric approach reveals how seemingly complex behaviors such as turning in worms and sidewinding snakes can be described as modulations of templates. Thus, the use of differential geometry in the locomotion of living systems generates a common description of locomotion across taxa and provides hypotheses for neuromechanical control schemes at lower levels of organization.

Non-Stokesian dynamics of magnetic helical nanoswimmers under confinement

Electromagnetically propelled helical nanoswimmers offer great potential for nanorobotic applications. Here, the effect of confinement on their propulsion is characterized using lattice-Boltzmann simulations. Two principal mechanisms give rise to their forward motion under confinement: (i) pure swimming and (ii) the thrust created by the differential pressure due to confinement. Under strong confinement, they face greater rotational drag but display a faster propulsion for fixed driving frequency in agreement with experimental findings. This is due to the increased differential pressure created by the boundary walls when they are sufficiently close to each other and the particle. We have proposed two analytical relations (i) for predicting the swimming speed of an unconfined particle as a function of its angular speed and geometrical properties, and (ii) an empirical expression to accurately predict the propulsion speed of a confined swimmer as a function of the degree of confinement and its unconfined swimming speed. At low driving frequencies and degrees of confinement, the systems retain the expected linear behavior consistent with the predictions of the Stokes equation. However, as the driving frequency and/or the degree of confinement increase, their impact on propulsion leads to increasing deviations from the Stokesian regime and emergence of nonlinear behavior.

The nut-and-bolt motion of a bacteriophage sliding along a bacterial flagellum: a complete hydrodynamics model

The 'nut-and-bolt' mechanism of a bacteriophage-bacteria flagellum translocation motion is modelled by numerically integrating the 3D Stokes equations using a Finite-Element Method (FEM). Following the works by Katsamba and Lauga (Phys Rev Fluids 4(1): 013101, 2019), two mechanical models of the flagellum-phage complex are considered. In the first model, the phage fiber wraps around the smooth flagellum surface separated by some distance. In the second model, the phage fiber is partly immersed in the flagellum volume via a helical groove imprinted in the flagellum and replicating the fiber shape. In both cases, the results of the Stokes solution for the translocation speed are compared with the Resistive Force Theory (RFT) solutions (obtained in Katsamba and Lauga Phys Rev Fluids 4(1): 013101, 2019) and the asymptotic theory in a limiting case. The previous RFT solutions of the same mechanical models of the flagellum-phage complex showed opposite trends for how the phage translocation speed depends on the phage tail length. The current work uses complete hydrodynamics solutions, which are free from the RFT assumptions to understand the divergence of the two mechanical models of the same biological system. A parametric investigation is performed by changing pertinent geometrical parameters of the flagellum-phage complex and computing the resulting phage translocation speed. The FEM solutions are compared with the RFT results using insights provided from the velocity field visualisation in the fluid domain.

Decoding the hydrodynamic properties of microscale helical propellers from Brownian fluctuations

The complex motility of bacteria, ranging from single-swimmer behaviors such as chemotaxis to collective dynamics, including biofilm formation and active matter phenomena, is driven by their microscale propellers. Despite extensive study of swimming flagellated bacteria, the hydrodynamic properties of their helical-shaped propellers have never been directly measured. The primary challenges to directly studying microscale propellers are 1) their small size and fast, correlated motion, 2) the necessity of controlling fluid flow at the microscale, and 3) isolating the influence of a single propeller from a propeller bundle. To solve the outstanding problem of characterizing the hydrodynamic properties of these propellers, we adopt a dual statistical viewpoint that connects to the hydrodynamics through the fluctuation-dissipation theorem (FDT). We regard the propellers as colloidal particles and characterize their Brownian fluctuations, described by 21 diffusion coefficients for translation, rotation, and correlated translation-rotation in a static fluid. To perform this measurement, we applied recent advances in high-resolution oblique plane microscopy to generate high-speed volumetric movies of fluorophore-labeled, freely diffusing Escherichia coli flagella. Analyzing these movies with a bespoke helical single-particle tracking algorithm, we extracted trajectories, calculated the full set of diffusion coefficients, and inferred the average propulsion matrix using a generalized Einstein relation. Our results provide a direct measurement of a microhelix's propulsion matrix and validate proposals that the flagella are highly inefficient propellers, with a maximum propulsion efficiency of less than 3%. Our approach opens broad avenues for studying the motility of particles in complex environments where direct hydrodynamic approaches are not feasible.

Effect of curvature on the diffusion of colloidal bananas

Anisotropic colloidal particles exhibit complex dynamics which play a crucial role in their functionality, transport, and phase behavior. In this Letter, we investigate the two-dimensional diffusion of smoothly curved colloidal rods-also known as colloidal bananas-as a function of their opening angle α. We measure the translational and rotational diffusion coefficients of the particles with opening angles ranging from 0^{∘} (straight rods) to nearly 360^{∘}(closed rings). In particular, we find that the anisotropic diffusion of the particles varies nonmonotonically with their opening angle and that the axis of fastest diffusion switches from the long to the short axis of the particles when α>180^{∘}. We also find that the rotational diffusion coefficient of nearly closed rings is approximately an order of magnitude higher than that of straight rods of the same length. Finally, we show that the experimental results are consistent with slender body theory, indicating that the dynamical behavior of the particles arises primarily from their local drag anisotropy. These results highlight the impact of curvature on the Brownian motion of elongated colloidal particles, which must be taken into account when seeking to understand the behavior of curved colloidal particles.

Bacteria-inspired robotic propulsion from bundling of soft helical filaments at low Reynolds number

The bundling of flagella is known to create a "run" phase, where the bacteria moves in a nearly straight line rather than making changes in direction. Historically, mechanical explanations for the bundling phenomenon intrigued many researchers, and significant advances were made in physical models and experimental methods. Contributing to the field of research, we present a bacteria-inspired centimeter-scale soft robotic hardware platform and a computational framework for a physically plausible simulation model of the multi-flagellated robot under low Reynolds number (∼10^-1). The fluid-structure interaction simulation couples the discrete elastic rods algorithm with the method of regularized Stokeslet segments. Contact between two flagella is handled by a penalty-based method. We present a comparison between our experimental and simulation results and verify that the simulation tool can capture the essential physics of this problem. Preliminary findings on robustness to buckling provided by the bundling phenomenon and the efficiency of a multi-flagellated soft robot are compared with the single-flagellated counterparts. Observations were made on the coupling between geometry and elasticity, which manifests itself in the propulsion of the robot by nonlinear dependency on the rotational speed of the flagella.

A bar-joint model based on the corrected resistive force theory for artificial flagellated micro-swimmers propelled by acoustic waves

In this work, we proposed a bar-joint model based on the corrected resistive force theory (CRFT) for studying artificial flagellated micro-swimmers (AFMSs) propelled by acoustic waves in a two-dimensional (2D) flow field or with a rectangular cross-section. Note that the classical resistive-force theory for 3D cylindrical flagellum leads to over 90% deviation in terminal velocity from those of 2D fluid-structure interaction (FSI) simulations, while the proposed CRFT bar-joint model can reduce the deviation to below 5%; hence, it enables a reliable prediction of the 2D locomotion of an acoustically actuated AFMS with a rectangular cross-section, which is the case in some experiments. Introduced in the CRFT is a single correction factorKdetermined by comparing the linear terminal velocities under acoustic actuation obtained from the CRFT with those from simulations. After the determination ofK, detailed comparisons of trajectories between the CRFT-based bar-joint AFMS model and the FSI simulation were presented, exhibiting an excellent consistency. Finally, a numerical demonstration of the purely acoustic or magneto-acoustic steering of an AFMS based on the CRFT was presented, which can be one of the choices for future AFMS-based precision therapy.

Fabrication, control, and modeling of robots inspired by flagella and cilia

Flagella and cilia are slender structures that serve important functionalities in the microscopic world through their locomotion induced by fluid and structure interaction. With recent developments in microscopy, fabrication, biology, and modeling capability, robots inspired by the locomotion of these organelles in low Reynolds number flow have been manufactured and tested on the micro-and macro-scale, ranging from medicalin vivomicrobots, microfluidics to macro prototypes. We present a collection of modeling theories, control principles, and fabrication methods for flagellated and ciliary robots.

Light-induced swirling and locomotion

The design of remotely activated, untethered devices without onboard power is a continuing challenge in soft robotics. This work describes a method of generating a whirling motion in pre-stressed photomechanical liquid crystal elastomer fibres using steady illumination that can be exploited for propulsion and mixing. Photomechanical liquid crystal elastomers (LCEs) can convert light directly into mechanical deformation, making them attractive candidates for soft actuators capable of remote and multi-mode actuation. We propose a three-dimensional multi-scale model of the nonlinear and non-local dynamics of fibres of photomechanical LCEs under illumination. We use the model to show that pre-stressed helix-like fibres immersed in a fluid can undergo a periodic whirling motion under steady illumination. We analyse the photo-driven spatio-temporal pattern and stability of the whirling deformation, and provide a parametric study. Unlike previous work on photo-driven periodic motion, this whirling motion does not exploit instabilities in the form of snap-through phenomena, or unilateral constraints as in rolling. More broadly, our work provides an unusual example of a physical system capable of periodic motion under steady stimulus that does not exploit instabilities. We finally show that such motion can be exploited in developing remote controlled bioinspired microswimmers and novel micromixers.

Light chain 2 is a Tctex-type related axonemal dynein light chain that regulates directional ciliary motility in Trypanosoma brucei

Flagellar motility is essential for the cell morphology, viability, and virulence of pathogenic kinetoplastids. Trypanosoma brucei flagella beat with a bending wave that propagates from the flagellum's tip to its base, rather than base-to-tip as in other eukaryotes. Thousands of dynein motor proteins coordinate their activity to drive ciliary bending wave propagation. Dynein-associated light and intermediate chains regulate the biophysical mechanisms of axonemal dynein. Tctex-type outer arm dynein light chain 2 (LC2) regulates flagellar bending wave propagation direction, amplitude, and frequency in Chlamydomonas reinhardtii. However, the role of Tctex-type light chains in regulating T. brucei motility is unknown. Here, we used a combination of bioinformatics, in-situ molecular tagging, and immunofluorescence microscopy to identify a Tctex-type light chain in the procyclic form of T. brucei (TbLC2). We knocked down TbLC2 expression using RNAi in both wild-type and FLAM3, a flagellar attachment zone protein, knockdown cells and quantified TbLC2's effects on trypanosome cell biology and biophysics. We found that TbLC2 knockdown reduced the directional persistence of trypanosome cell swimming, induced an asymmetric ciliary bending waveform, modulated the bias between the base-to-tip and tip-to-base beating modes, and increased the beating frequency. Together, our findings are consistent with a model of TbLC2 as a down-regulator of axonemal dynein activity that stabilizes the forward tip-to-base beating ciliary waveform characteristic of trypanosome cells. Our work sheds light on axonemal dynein regulation mechanisms that contribute to pathogenic kinetoplastids' unique tip-to-base ciliary beating nature and how those mechanisms underlie dynein-driven ciliary motility more generally.

Helical micro-swimmer: hierarchical tail design and propulsive motility

Helical micro-swimmers have markedly extended the reach of human beings in numerous fields, ranging from in vitro tasks in lab-on-a-chip to in vivo applications for minimally invasive medicine. The previous studies on the propulsive motility optimization of the micro-swimmers mainly focused on the distinct actuation principles (e.g., chemically powered, magnetic- or ultrasound energy-driven) and paid little attention to the structural design of these swimming machines themselves. The improvements of the structures can assist the externally powered motors in providing propulsion in a tiny scale and satisfy the agile locomotion demands. This paper presents the design, mechanics modeling and available experiments of a novel type of hierarchical helical swimming robot that significantly enhances the motility of the helix-based swimmers. Validated by the resistive force theory, our numerical model can well analyze the mechanical properties with a variety of geometric parameters. The motion performance of the hierarchical and conventional helical structures in low Reynolds regimes is presented, highlighting the advantages of hierarchical swimmers over the existing typical swimmers. In addition, the stability and resilience of the hierarchical swimmers can be maintained at a decent level. Moreover, the variable forward velocity resulting from the combined hierarchical structures is investigated here, which can thereby serve as a reliable design strategy. The proposed hierarchical helical design enables enticing opportunities for various device systems of medical robots and bio-integrated electronics.

Shaping Nanoscale Ribbons into Microhelices of Controllable Radius and Pitch

We report fabrication of highly flexible micron-sized helices from nanometer-thick ribbons. Building upon the helical coiling of such ultrathin ribbons mediated by surface tension, we demonstrate that the enhanced creep properties of highly confined materials can be leveraged to shape helices into the desired geometry with full control of the final shape. The helical radius, total length, and pitch angle are all freely and independently tunable within a wide range: radius within ∼1-100 μm, length within ∼100-3000 μm, and pitch angle within ∼0-70°. This fabrication method is validated for three different materials: poly(methyl methacrylate), poly(dimethylaminoethyl methacrylate), and transition metal chalcogenide quantum dots, each corresponding to a different solid-phase structure: respectively a polymer glass, a cross-linked hydrogel, and a nanoparticle array. This demonstrates excellent versatility with respect to material selection, enabling further control of the helix mechanical properties.

Force and torque-free helical tail robot to study low Reynolds number micro-organism swimming

Helical propulsion is used by many micro-organisms to swim in viscous-dominated environments. Their swimming dynamics are relatively well understood, but a detailed study of the flow fields is still needed to understand wall effects and hydrodynamic interactions among swimmers. In this letter, we describe the development of an autonomous swimming robot with a helical tail that operates in the Stokes regime. The device uses a battery-based power system with a miniature motor that imposes a rotational speed on a helical tail. The speed, direction, and activation are controlled electronically using an infrared remote control. Since the robot is about 5 cm long, we use highly viscous fluids to match the Reynolds number, Re, to be less than 0.1. Measurements of swimming speeds are conducted for a range of helical wavelengths, λ, head geometries, and rotation rates, ω. We provide comparisons of the experimental measurements with analytical predictions derived from resistive force theory. This force and torque-free neutrally buoyant swimmer mimics the swimming strategy of bacteria more closely than previously used designs and offers a lot of potential for future applications.

Ballooning in spiders using multiple silk threads

In this paper, three-dimensional numerical simulations of ballooning in spiders using multiple silk threads are performed using the discrete elastic rods method. The ballooning of spiders is hypothesized to be caused by the presence of the negative electric charge of the spider silk threads and the positive electric potential field in the Earth's atmosphere. The numerical model presented here is first validated against experimental data from the open literature. After which, two cases are examined, in the first it is assumed that the electric charge is uniformly distributed along the threads while in the second, the electric charge is located at the thread tip. It is shown that the normalized terminal ballooning velocity, i.e., the velocity at which the spiders balloon after they reach steady-state, decrease linearly with the normalized lift force, especially for the tip located charge case. For the uniform electric charge case, this velocity shows a slightly weaker dependence on the normalized lift force. Moreover, it is shown in both cases that the normalized terminal ballooning velocity has no dependence on the normalized elastic bending stiffness of the threads and on the normalized viscous forces. Finally, the multithread bending process shows a three-dimensional conical sheet. Here we show that this behavior is caused by the Coulomb repelling forces owing to the threads electric charge which leads to dispersing the threads apart and thus avoid entanglement.

Using Experimentally Calibrated Regularized Stokeslets to Assess Bacterial Flagellar Motility Near a Surface

The presence of a nearby boundary is likely to be important in the life cycle and evolution of motile flagellate bacteria. This has led many authors to employ numerical simulations to model near-surface bacterial motion and compute hydrodynamic boundary effects. A common choice has been the method of images for regularized Stokeslets (MIRS); however, the method requires discretization sizes and regularization parameters that are not specified by any theory. To determine appropriate regularization parameters for given discretization choices in MIRS, we conducted dynamically similar macroscopic experiments and fit the simulations to the data. In the experiments, we measured the torque on cylinders and helices of different wavelengths as they rotated in a viscous fluid at various distances to a boundary. We found that differences between experiments and optimized simulations were less than 5% when using surface discretizations for cylinders and centerline discretizations for helices. Having determined optimal regularization parameters, we used MIRS to simulate an idealized free-swimming bacterium constructed of a cylindrical cell body and a helical flagellum moving near a boundary. We assessed the swimming performance of many bacterial morphologies by computing swimming speed, motor rotation rate, Purcell’s propulsive efficiency, energy cost per swimming distance, and a new metabolic energy cost defined to be the energy cost per body mass per swimming distance. All five measures predicted that the optimal flagellar wavelength is eight times the helical radius independently of body size and surface proximity. Although the measures disagreed on the optimal body size, they all predicted that body size is an important factor in the energy cost of bacterial motility near and far from a surface.

The Transport Behavior of a Biflagellated Microswimmer before and after Cargo Loading

The changes in the transport behavior of a microswimmer before and after cargo loading are crucial to understanding and control of the motion of a biohybrid microbot. In this work, we show the change in swimming behavior of biflagellated microalgae Chlamydomonas reinhardtii picking up a 4.5 μm polystyrene microbead upon collision. The microswimmer changed from linear forward motion into helical motion upon the attachment of the cargo and swam with a decreased swimming velocity. We revealed the helical motion of the microswimmer upon cargo loading due to suppression of flagella by image analysis of magnified time-lapse images of C. reinhardtii with one microbead attached at the anterior end (between the flagella). Furthered suppression on the flagellum imposed by the loading of the second cargo has led to increased oscillation per displacement traveled and decreased swimming velocity. Moreover, the microswimmer with a microbead attached at the posterior end swam with swimming velocity close to free swimming microalgae and did not exhibit helical swimming behavior. The experimental results and analysis showed that the loading location of the cargo has a great influence over the swimming behavior of the microswimmer. Furthermore, the work balance calculation and mathematical analysis based on Lighthill's model are well consistent with our experimental findings.

The bank of swimming organisms at the micron scale (BOSO-Micro)

Unicellular microscopic organisms living in aqueous environments outnumber all other creatures on Earth. A large proportion of them are able to self-propel in fluids with a vast diversity of swimming gaits and motility patterns. In this paper we present a biophysical survey of the available experimental data produced to date on the characteristics of motile behaviour in unicellular microswimmers. We assemble from the available literature empirical data on the motility of four broad categories of organisms: bacteria (and archaea), flagellated eukaryotes, spermatozoa and ciliates. Whenever possible, we gather the following biological, morphological, kinematic and dynamical parameters: species, geometry and size of the organisms, swimming speeds, actuation frequencies, actuation amplitudes, number of flagella and properties of the surrounding fluid. We then organise the data using the established fluid mechanics principles for propulsion at low Reynolds number. Specifically, we use theoretical biophysical models for the locomotion of cells within the same taxonomic groups of organisms as a means of rationalising the raw material we have assembled, while demonstrating the variability for organisms of different species within the same group. The material gathered in our work is an attempt to summarise the available experimental data in the field, providing a convenient and practical reference point for future studies.

Self-adaptive and efficient propulsion of Ray sperms at different viscosities enabled by heterogeneous dual helixes

We disclose a peculiar rotational propulsion mechanism of Ray sperms enabled by its unusual heterogeneous dual helixes with a rigid spiral head and a soft tail, named Heterogeneous Dual Helixes (HDH) model for short. Different from the conventional beating propulsion of sperm, the propulsion of Ray sperms is from both the rotational motion of the soft helical tail and the rigid spiral head. Such heterogeneous dual helical propulsion style provides the Ray sperm with high adaptability in viscous solutions along with advantages in linearity, straightness, and bidirectional motion. This HDH model is further corroborated by a miniature swimming robot actuated via a rigid spiral head and a soft tail, which demonstrates similar superiorities over conventional ones in terms of adaptability and efficiency under the same power input. Such findings expand our knowledge on microorganisms' motion, motivate further studies on natural fertilization, and inspire engineering designs.

How Euglena gracilis swims: Flow field reconstruction and analysis

Euglena gracilis is a unicellular organism that swims by beating a single anterior flagellum. We study the nonplanar waveforms spanned by the flagellum during a swimming stroke and the three-dimensional flows that they generate in the surrounding fluid. Starting from a small set of time-indexed images obtained by optical microscopy on a swimming Euglena cell, we construct a numerical interpolation of the stroke. We define an optimal interpolation (which we call synthetic stroke) by minimizing the discrepancy between experimentally measured velocities (of the swimmer) and those computed by solving numerically the equations of motion of the swimmer driven by the trial interpolated stroke. The good match we obtain between experimentally measured and numerically computed trajectories provides a first validation of our synthetic stroke. We further validate the procedure by studying the flow velocities induced in the surrounding fluid. We compare the experimentally measured flow fields with the corresponding quantities computed by solving numerically the Stokes equations for the fluid flow, in which the forcing is provided by the synthetic stroke, and find good matching. Finally, we use the synthetic stroke to derive a coarse-grained model of the flow field resolved in terms of a few dominant singularities. The far field is well approximated by a time-varying Stresslet, and we show that the average behavior of Euglena during one stroke is that of an off-axis puller. The reconstruction of the flow field closer to the swimmer body requires a more complex system of singularities. A system of two Stokeslets and one Rotlet, that can be loosely associated with the force exerted by the flagellum, the drag of the body, and a torque to guarantee rotational equilibrium, provides a good approximation.

Rods coiling about a rigid constraint: helices and perversions

Mechanical instabilities can be exploited to design innovative structures, able to change their shape in the presence of external stimuli. In this work, we derive a mathematical model of an elastic beam subjected to an axial force and constrained to smoothly slide along a rigid support, where the distance between the rod midline and the constraint is fixed and finite. Using both theoretical and computational techniques, we characterize the bifurcations of such a mechanical system, in which the axial force and the natural curvature of the beam are used as control parameters. We show that, in the presence of a straight support, the rod can deform into shapes exhibiting helices and perversions, namely transition zones connecting together two helices with opposite chirality. The mathematical predictions of the proposed model are also compared with some experiments, showing a good quantitative agreement. In particular, we find that the buckled configurations may exhibit multiple perversions and we propose a possible explanation for this phenomenon based on the energy landscape of the mechanical system.

Numerical Modeling of Sperm Swimming

Due to rising human infertility, sperm motility has been an important subject. Among the hundreds of millions of sperms on the journey up the oviducts, only a few excellent travelers will reach the eggs. This journey is affected by many factors, some of which include sperm quality, sperm density, fluid rheology and chemotaxis. In addition, the sperm swimming through different body tracks and fluids involves complex sperm flagellar, complex fluid environment, and multi-sperm and sperm-wall interactions. Therefore, this topic has generated substantial research interest. In this paper, we present a review of computational studies on sperm swimming from an engineering perspective with focus on both simplified theoretical methods and fluid–structure interaction methods. Several open issues in this field are highlighted.

Comparative study of snake lateral undulation kinematics in model heterogeneous terrain

Terrestrial organisms that use traveling waves to locomote must leverage heterogeneities to overcome drag on the elongate body. While previous studies illuminated how habitat generalist snakes self-deform to use rigid obstacles in the surroundings, control strategies for multi-component terrain are largely unknown. We compared the sand-specialist Chionactis occipitalis to a habitat generalist, Pantherophis guttatus, navigating a model terrestrial terrain-rigid post arrays on a low-friction substrate. We found the waveshapes used by the generalist were more variable than the specialist. Principal component analysis revealed that while the specialized sand-swimming waveform was always present on C. occipitalis, the generalist did not have a similarly pervasive low-dimensional waveshape. We expected the generalist to thus outperform the specialist in the arrays, but body slip of both species was comparable on level ground and in all trials the snakes successfully traversed the arena. When we further challenged the snakes to ascend an inclined lattice, the sand-specialist had difficulty maintaining contact with the obstacles and was unable to progress up the steepest inclines in the largest lattice spacings. Our results suggest that species adapted to different habitats use different control modalities-the specialist is primarily controlling its kinematics to achieve a target shape while, consistent with previous research, the generalist is using force control and self-deforms in response to terrain contacts. While both strategies allowed progress on the uninclined low-friction terrain with posts, the more variable waveshapes of the generalist may be necessary when faced with more challenging locomotor tasks like climbing inclines.

Helical locomotion in a porous medium

Microorganisms and artificial microswimmers often need to swim through environments that are more complex than purely viscous liquids in their natural habitats or operational environments, such as gel-like mucus, wet soil, and aquifers. The question of how the properties of these complex environments affect locomotion has attracted considerable recent attention. In this paper, we present a theoretical model to examine how the additional resistance due to the network of stationary obstacles in a porous medium affects helical locomotion. Here, we focus on helical locomotion for its ubiquity as a propulsion mechanism adopted by many swimming bacteria and artificial microswimmers. We show that the additional resistance can have qualitatively different effects on various scenarios of helical locomotion: (1) a helical propeller driven by an external torque, (2) a free swimming bacterium consisting of a helical flagellum and a head, and (3) a cargo-carrying helical propeller driven by an external torque. Our results elucidate the subtle and significant differences between torqued helical propulsion versus force-free and torque-free swimming in a porous medium. We also remark on the limitations as well as potential connections of our results with experimental measurements of bacterial swimming speeds in polymeric solutions.

Mitigating memory effects during undulatory locomotion on hysteretic materials

While terrestrial locomotors often contend with permanently deformable substrates like sand, soil, and mud, principles of motion on such materials are lacking. We study the desert-specialist shovel-nosed snake traversing a model sand and find body inertia is negligible despite rapid transit and speed dependent granular reaction forces. New surface resistive force theory (RFT) calculation reveals how wave shape in these snakes minimizes material memory effects and optimizes escape performance given physiological power limitations. RFT explains the morphology and waveform-dependent performance of a diversity of non-sand-specialist snakes but overestimates the capability of those snakes which suffer high lateral slipping of the body. Robophysical experiments recapitulate aspects of these failure-prone snakes and elucidate how re-encountering previously deformed material hinders performance. This study reveals how memory effects stymied the locomotion of a diversity of snakes in our previous studies (Marvi et al., 2014) and indicates avenues to improve all-terrain robots.

From indication to decision: A hierarchical approach to model the chemotactic behavior of Escherichia coli

Reducing the complex behavior of living entities to its underlying physical and chemical processes is a formidable task in biology. Complex behaviors can be characterized as decision making: the ability to process the incoming information via an intracellular network and act upon this information to choose appropriate strategies. Motility is one such behavior that has been the focus many modeling efforts in the past. Our aim is to reduce the chemotactic behavior in Escherichia coli to its molecular constituents in order to paint a comprehensive and end-to-end picture of this intricate behavior. We utilize a hierarchical approach, consisting of three layers, to achieve this goal: at the first level, chemical reactions involved in chemotaxis are simulated. In the second level, the chemical reactions give rise to the mechanical movement of six independent flagella. At the last layer, the two lower layers are combined to allow a digital bacterium to receive information from its environment and swim through it with verve. Our results are in concert with the experimental studies concerning the motility of E.coli cells. In addition, we show that our detailed model of chemotaxis is reducible to a non-homogeneous Markov process.

Nanopumping of water via rotation of graphene nanoribbons

In this paper, we perform molecular dynamics simulations to propose a novel bio-inspired nanopumping mechanism that is achieved through the rotation of graphene nanoribbons. Due to the rotation and interaction with water, the graphene nanoribbons undergo morphological transformation. It is shown that with appropriate geometrical and spatial parameters, the resulting morphology is twisted ribbon, which is efficient in pumping of water through a channel. This mimics the propulsive behavior of bacterial flagella through continual rotation at the base and causing morphology of the geometry into twisted ribbons, thus driving flow. It was observed that the maximum flux rate decreases upon reaching the optimal configuration even with increasing rotational speed and graphene width. This is due to the development of cavitation near the region of the nanoribbon with tip velocities approaching the speed of sound in water. The simulation shows promising results where the flux rate of the driven flow outperforms various nanopump configurations that have been reported in recent literature by more than one order.

Bioinspired reorientation strategies for application in micro/nanorobotic control

Engineers have recently been inspired by swimming methodologies of microorganisms in creating micro-/nanorobots for biomedical applications. Future medicine may be revolutionized by the application of these small machines in diagnosing, monitoring, and treating diseases. Studies over the past decade have often concentrated on propulsion generation. However, there are many other challenges to address before the practical use of robots at the micro-/nanoscale. The control and reorientation ability of such robots remain as some of these challenges. This paper reviews the strategies of swimming microorganisms for reorientation, including tumbling, reverse and flick, direction control of helical-path swimmers, by speed modulation, using complex flagella, and the help of mastigonemes. Then, inspired by direction change in microorganisms, methods for orientation control for microrobots and possible directions for future studies are discussed. Further, the effects of solid boundaries on the swimming trajectories of microorganisms and microrobots are examined. In addition to propulsion systems for artificial microswimmers, swimming microorganisms are promising sources of control methodologies at the micro-/nanoscale.

Brownian fluctuations and hydrodynamics of a microhelix near a solid wall

We combine two-photon lithography and optical tweezers to investigate the Brownian fluctuations and propeller characteristics of a microfabricated helix. From the analysis of mean squared displacements and time correlation functions we recover the components of the full mobility tensor. We find that Brownian motion displays correlations between angular and translational fluctuations from which we can directly measure the hydrodynamic coupling coefficient that is responsible for thrust generation. By varying the distance of the microhelices from a no-slip boundary we can systematically measure the effects of a nearby wall on the resistance matrix. Our results indicate that a rotated helix moves faster when a nearby no-slip boundary is present, providing a quantitative insight on thrust enhancement in confined geometries for both synthetic and biological microswimmers.

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.

Tuning the torque-speed characteristics of the bacterial flagellar motor to enhance swimming speed

In a classic paper, Purcell [Proc. Natl. Acad. Sci. U. S. A. 94, 11307 (1997)10.1073/pnas.94.21.11307] analyzed the dynamics of flagellated bacterial swimmers and derived a geometrical relationship which maximizes the propulsion efficiency. Experimental measurements for wild-type bacterial species E. coli have revealed that they closely satisfy this geometric optimality. However, dependence of the flagellar motor speed on the load and more generally the role of the torque-speed characteristics of the flagellar motor are not considered in Purcell's original analysis. Here we derive a tuned condition representing a match between the flagella geometry and the torque-speed characteristics of the flagellar motor to maximize the bacterial swimming speed for a given load. This condition is independent of the geometric optimality condition derived by Purcell. Interestingly, this condition is not satisfied by wild-type E. coli which swims 2-3 times slower than the maximum possible speed given the amount of available motor torque. Finally, we present experimental data on swimming dynamics of a cargo laden bacterial system which follows our analytical model. Our analysis also reveals the existence of an anomalous propulsion regime where the swim speed increases with increasing load (drag).

Thrust and Hydrodynamic Efficiency of the Bundled Flagella

The motility mechanism of prokaryotic organisms has inspired many untethered microswimmers that could potentially perform minimally invasive medical procedures in stagnant fluid regions inside the human body. Some of these microswimmers are inspired by bacteria with single or multiple helical flagella to propel efficiently and fast. For multiple flagella configurations, the direct measurement of thrust and hydrodynamic propulsion efficiency has been challenging due to the ambiguous mechanical coupling between the flow field and mechanical power input. To address this challenge and to compare alternative micropropulsion designs, a methodology based on volumetric velocity field acquisition is developed to acquire the key propulsive performance parameters from scaled-up swimmer prototypes. A digital particle image velocimetry (PIV) analysis protocol was implemented and experiments were conducted with the aid of computational fluid dynamics (CFD). First, this methodology was validated using a rotating single-flagellum similitude model. In addition to the standard PIV error assessment, validation studies included 2D vs. 3D PIV, axial vs. lateral PIV and simultaneously acquired direct thrust force measurement comparisons. Compatible with typical micropropulsion flow regimes, experiments were conducted both for very low and higher Reynolds (Re) number regimes (up to a Re number = 0.01) than that are reported in the literature. Finally, multiple flagella bundling configurations at 0°, 90° and 180° helical phase-shift angles were studied using scaled-up multiple concentric flagella thrust elements. Thrust generation was found to be maximal for the in-phase (0°) bundling configuration but with ~50% lower hydrodynamic efficiency than the single flagellum. The proposed measurement protocol and static thrust test-bench can be used for bio-inspired microscale propulsion methods, where direct thrust and efficiency measurement are required.

Propulsion by stiff elastic filaments in viscous fluids

Flexible filaments moving in viscous fluids are ubiquitous in the natural microscopic world. For example, the swimming of bacteria and spermatozoa as well as important physiological functions at organ level, such as the cilia-induced motion of mucus in the lungs, or individual cell level, such as actin filaments or microtubules, all employ flexible filaments moving in viscous fluids. As a result of fluid-structure interactions, a variety of nonlinear phenomena may arise in the dynamics of such moving flexible filaments. In this paper we derive the mathematical tools required to study filament-driven propulsion in the asymptotic limit of stiff filaments. Motion in the rigid limit leads to hydrodynamic loads which deform the filament and impact the filament propulsion. We first derive the general mathematical formulation and then apply it to the case of a helical filament, a situation relevant for the swimming of flagellated bacteria and for the transport of artificial, magnetically actuated motors. We find that, as a result of flexibility, the helical filament is either stretched or compressed (conforming previous studies) and additionally its axis also bends, a result which we interpret physically. We then explore and interpret the dependence of the perturbed propulsion speed due to the deformation on the relevant dimensionless dynamic and geometric parameters.

Hydrophobicity Influence on Swimming Performance of Magnetically Driven Miniature Helical Swimmers

Helical microswimmers have been involved in a wide variety of applications, ranging from in vivo tasks such as targeted drug delivery to in vitro tasks such as transporting micro objects. Over the past decades, a number of studies have been established on the swimming performance of helical microswimmers and geometrical factors influencing their swimming performance. However, limited studies have focused on the influence of the hydrophobicity of swimmers' surface on their swimming performance. In this paper, we first demonstrated through theoretical analysis that the hydrophobicity of swimmer's surface material of the swimmer does affect its swimming performance: the swimmer with more hydrophobic surface is exerted less friction drag torque, and should therefore exhibit a higher step-out frequency, indicating that the swimmer with more hydrophobic surface should have better swimming performance. Then a series of experiments were conducted to verify the theoretical analysis. As a result, the main contribution of this paper is to demonstrate that one potential approach to improve the helical microswimmers' swimming performance could be making its surface more hydrophobic.

Strictures of a microchannel impose fierce competition to select for highly motile sperm

Investigating sperm locomotion in the presence of external fluid flow and geometries simulating the female reproductive tract can lead to a better understanding of sperm motion during fertilization. Using a microfluidic device featuring a stricture that simulates the fluid mechanical properties of narrow junctions inside the female reproductive tract, we documented the gate-like role played by the stricture in preventing sperm with motilities below a certain threshold from advancing through the stricture to the other side (i.e., fertilization site). All the slower sperm accumulate below (i.e., in front of) the stricture and swim in a butterfly-shaped path between the channel walls, thus maintaining the potential for penetrating the stricture and ultimately advancing toward the fertilization site. Accumulation below the stricture occurs in a hierarchical manner so that dense concentrations of sperm with higher velocities remain closer to the stricture, with more sparsely distributed arrays of lower-velocity sperm lagging behind.

On the limitations of some popular numerical models of flagellated microswimmers: importance of long-range forces and flagellum waveform

For a sperm-cell-like flagellated swimmer in an unbounded domain, several numerical models of different fidelity are considered based on the Stokes flow approximation. The models include a regularized Stokeslet method and a three-dimensional finite-element method, which serve as the benchmark solutions for several approximate models considered. The latter include the resistive force theory versions of Lighthill, and Gray and Hancock, as well as a simplified approximation based on computing the hydrodynamic forces exerted on the head and the flagellum separately. It is shown how none of the simplified models is robust enough with regards to predicting the effect of the swimmer head shape change on the swimmer dynamics. For a range of swimmer motions considered, the resulting solutions for the swimmer force and velocities are analysed and the applicability of the Stokes model for the swimmers in question is probed.

Nonmotile Single-Cell Migration as a Random Walk in Nonuniformity: The "Extreme Dumping Limit" for Cell-to-Cell Communications

In the present work, we model single-cell movement as a random walk in an external potential observed within the extreme dumping limit, which we define herein as the extreme nonuniform behavior observed for cell responses and cell-to-cell communications. Starting from the Newton–Langevin equation of motion, we solve the corresponding Fokker–Planck equation to compute higher moments of the displacement of the cell, and then we build certain quantities that can be measurable experimentally. We show that, each time, the dynamics depend on the external force applied, leading to predictions distinct from the standard results of a free Brownian particle. Our findings demonstrate that cell migration viewed as a stochastic process is still compatible with biological and experimental observations without the need to rely on more complicated or sophisticated models proposed previously in the literature.

Flagellar number governs bacterial spreading and transport efficiency

Peritrichous bacteria synchronize and bundle their flagella to actively swim, while disruption of the bundle leads to a slow motility phase with a weak propulsion. It is still not known whether the number of flagella represents an evolutionary adaptation toward optimizing bacterial navigation. We study the swimming dynamics of differentially flagellated Bacillus subtilis strains in a quasi-two-dimensional system. We find that decreasing the number of flagella N f reduces the average turning angle between two successive run phases and enhances the run time and the directional persistence of the run phase. As a result, having fewer flagella is beneficial for long-distance transport and fast spreading, while having a lot of flagella is advantageous for the processes that require a slower spreading, such as biofilm formation. We develop a two-state random walk model that incorporates spontaneous switchings between the states and yields exact analytical expressions for transport properties, in remarkable agreement with experiments. The results of numerical simulations based on our two-state model suggest that the efficiency of searching and exploring the environment is optimized at intermediate values of N f. The optimal choice of N f, for which the search time is minimized, decreases with increasing the size of the environment in which the bacteria swim.

Dynamics of a magnetically rotated micro swimmer inspired by paramecium metachronal wave

In the past few years, a significant body of research has been devoted to designing magnetic micron scale robotic systems for minimally invasive medicine. The motion of different microorganisms is the nature's solution for efficient propulsion of these swimmers. So far, there has been a considerable effort in designing micro swimmers based on the propulsion of bacteria while the motion of numerous other microorganisms has not been a source of inspiration for designing micro swimmers yet. Inspired by propulsion of Paramecium which is a ciliate microorganism, a novel micro swimmer is proposed in this article which is capable of cargo transport. This novel swimmer is composed of multiple equally spaced rigid loxodromic rods spanning the surface of a sphere which can carry a cargo placed inside it. The propulsion of this swimmer is influenced by the geometry of the swimmer (diameter, number of rods, cargo size), therefore, CFD simulations have been performed to investigate it. Finally, the dynamics of this swimmer is investigated analytically which sheds light into the complex dynamics of a swimmer with this geometry.

Swimming of peritrichous bacteria is enabled by an elastohydrodynamic instability

Peritrichously-flagellated bacteria, such as Escherichia coli, self-propel in fluids by using specialised motors to rotate multiple helical filaments. The rotation of each motor is transmitted to a short flexible segment called the hook which in turn transmits it to a flagellar filament, enabling swimming of the whole cell. Since multiple motors are spatially distributed on the body of the organism, one would expect the propulsive forces from the filaments to push against each other leading to negligible swimming. We use a combination of computations and theory to show that the swimming of peritrichous bacteria is enabled by an elastohydrodynamic bending instability occurring for hooks more flexible than a critical threshold. Using past measurements of hook bending stiffness, we demonstrate how real bacteria are safely on the side of the instability that promotes systematic swimming.

Mixing and pumping by pairs of helices in a viscous fluid

Here, we study the fluid dynamics of a pair of rigid helices rotating at a constant velocity, tethered at their bases, in a viscous fluid. Our computations use a regularized Stokeslet framework, both with and without a bounding plane, so we are able to discern precisely what flow features are unaccounted for in studies that ignore the surface from which the helices emanate. We examine how the spacing and phase difference between identical rotating helices affects their pumping ability, axial thrust, and power requirements. We also find that optimal mixing of the fluid around two helices is achieved when they rotate in opposite phase, and that the mixing is enhanced as the distance between the helices decreases.

Predicting and Optimizing Microswimmer Performance from the Hydrodynamics of Its Components: The Relevance of Interactions

Interest in the design of bioinspired robotic microswimmers is growing rapidly, motivated by the spectacular capabilities of their unicellular biological templates. Predicting the swimming speed and efficiency of such devices in a reliable way is essential for their rational design, and to optimize their performance. The hydrodynamic simulations needed for this purpose are demanding and simplified models that neglect nonlocal hydrodynamic interactions (e.g., resistive force theory for slender, filament-like objects that are the typical propulsive apparatus for unicellular swimmers) are commonly used. We show through a detailed case study of a model robotic system consisting of a spherical head powered by a rotating helical flagellum that (a) the errors one makes in the prediction of swimming speed and efficiency by neglecting hydrodynamic interactions are never quite acceptable and (b) there are simple ways to correct the predictions of the simplified theories to make them more accurate. We also formulate optimal design problems for the length of the helical flagellum giving maximal energetic efficiency, maximal distance traveled per motor turn, or maximal distance traveled per unit of work expended, and exhibit optimal solutions.

Transition and formation of the torque pattern of undulatory locomotion in resistive force dominated media

In undulatory locomotion, torques along the body are required to overcome external forces from the environment and bend the body. These torques are usually generated by muscles in animals and closely related to muscle activations. In previous studies, researchers observed a single traveling wave pattern of the torque or muscle activation, but the formation of the torque pattern is still not well understood. To elucidate the formation of the torque pattern required by external resistive forces and the transition as kinematic parameters vary, we use simplistic resistive force theory models of self-propelled, steady undulatory locomotors and examine the spatio-temporal variation of the internal torque. We find that the internal torque has a traveling wave pattern with a decreasing speed normalized by the curvature speed as the wave number (the number of wavelengths on the locomotor's body) increases from 0.5 to 1.8. As the wave number increases to 2 and greater values, the torque transitions into a two-wave-like pattern and complex patterns. Using phasor diagram analysis, we reveal that the formation and transitions of the pattern are consequences of the integration and cancellation of force phasors.

Optimal propulsion of an undulating slender body with anisotropic friction

This study investigates theoretically and numerically the propulsive sliding of a slender body. The body sustains a transverse and propagative wave along its main axis, and undergoes anisotropic friction caused by its surface texture sliding on the floor. A model accounting for the anisotropy of frictional forces acting on the body is implemented. This describes the propulsive force and gives the optimal undulating parameters for efficient forward propulsion. The optimal wave characteristics are effectively compared to the undulating motion of a slithering snakes, as well as with the motion of sandfish lizards swimming through the sand. Furthermore, numerical simulations have indicated the existence of certain specialized segments along the body that are highly efficient for propulsion, explaining why snakes lift parts of their body while slithering. Finally, the inefficiency of slithering as a form of locomotion to ascend a slope is discussed.