Finding the ciliary beating pattern with optimal efficiency

Near-field hydrodynamic interactions determine travelling wave directions of collectively beating cilia

Cilia can beat collectively in the form of a metachronal wave, and we investigate how near-field hydrodynamic interactions between cilia can influence the collective response of the beating cilia. Based on the theoretical framework developed in the work of Meng et al. (Meng et al. 2021 Proc. Natl Acad. Sci. USA 118, e2102828118), we find that the first harmonic mode in the driving force acting on each individual cilium can determine the direction of the metachronal wave after considering the finite size of the beating trajectories, which is confirmed by our agent-based numerical simulations. The stable wave patterns, e.g. the travelling direction, can be controlled by the driving forces acting on the cilia, based on which one can change the flow field generated by the cilia. This work can not only help to understand the role of the hydrodynamic interactions in the collective behaviours of cilia, but can also guide future designs of artificial cilia beating in the desired dynamic mode.

Curved Surfaces Induce Metachronal Motion of Microscopic Magnetic Cilia

Cilia are hair-like organelles present on cell surfaces. They often exhibit a collective wave-like motion that can enhance fluid or particle transportation function, known as metachronal motion. Inspired by nature, researchers have developed artificial cilia capable of inducing metachronal motion, especially magnetic actuation. However, current methods remain intricate, requiring either control of the magnetic or geometrical properties of individual cilia or the generation of a complex magnetic field. In this paper, we present a novel elegant method that eliminates these complexities and induces metachronal motion of arrays of identical microscopic magnetic artificial cilia by applying a simple rotating uniform magnetic field. The key idea of our method is to place arrays of cilia on surfaces with a specially designed curvature. This results in consecutive cilia experiencing different magnetic field directions at each point in time, inducing a phase lag in their motion, thereby causing collective wave-like motion. Moreover, by tuning the surface curvature profile, we can achieve diverse metachronal patterns analogous to symplectic and antiplectic metachronal motion observed in nature, and we can even devise novel combinations thereof. Furthermore, we characterize the local flow patterns generated by the motion of the cilia, revealing the formation of vortical patterns. Our novel approach simplifies the realization of miniaturized metachronal motion in microfluidic systems and opens the possibility of controlling flow pattern generation and transportation, opening avenues for applications such as lab-on-a-chip technologies, organ-on-a-chip platforms, and microscopic object propulsion.

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.

Hydrodynamic synchronization of elastic cilia: How surface effects determine the characteristics of metachronal waves

Cilia are hairlike microactuators whose cyclic motion is specialized to propel extracellular fluids at low Reynolds numbers. Clusters of these organelles can form synchronized beating patterns, called metachronal waves, which presumably arise from hydrodynamic interactions. We model hydrodynamically interacting cilia by microspheres elastically bound to circular orbits, whose inclinations with respect to a no-slip wall model the ciliary power and recovery stroke, resulting in an anisotropy of the viscous flow. We derive a coupled phase-oscillator description by reducing the microsphere dynamics to the slow timescale of synchronization and determine analytical metachronal wave solutions and their stability in a periodic chain setting. In this framework, a simple intuition for the hydrodynamic coupling between phase oscillators is established by relating the geometry of flow near the surface of a cell or tissue to the directionality of the hydrodynamic coupling functions. This intuition naturally explains the properties of the linear stability of metachronal waves. The flow near the surface stabilizes metachronal waves with long wavelengths propagating in the direction of the power stroke and, moreover, metachronal waves with short wavelengths propagating perpendicularly to the power stroke. Performing simulations of phase-oscillator chains with periodic boundary conditions, we indeed find that both wave types emerge with a variety of linearly stable wave numbers. In open chains of phase oscillators, the dynamics of metachronal waves is fundamentally different. Here the elasticity of the model cilia controls the wave direction and selects a particular wave number: At large elasticity, waves traveling in the direction of the power stroke are stable, whereas at smaller elasticity waves in the opposite direction are stable. For intermediate elasticity both wave directions coexist. In this regime, waves propagating towards both ends of the chain form, but only one wave direction prevails, depending on the elasticity and initial conditions.

Metachronal Motion of Biological and Artificial Cilia

Cilia are slender, hair-like cell protrusions that are present ubiquitously in the natural world. They perform essential functions, such as generating fluid flow, propulsion, and feeding, in organisms ranging from protozoa to the human body. The coordinated beating of cilia, which results in wavelike motions known as metachrony, has fascinated researchers for decades for its role in functions such as flow generation and mucus transport. Inspired by nature, researchers have explored diverse materials for the fabrication of artificial cilia and developed several methods to mimic the metachronal motion observed in their biological counterparts. In this review, we will introduce the different types of metachronal motion generated by both biological and artificial cilia, the latter including pneumatically, photonically, electrically, and magnetically driven artificial cilia. Furthermore, we review the possible applications of metachronal motion by artificial cilia, focusing on flow generation, transport of mucus, particles, and droplets, and microrobotic locomotion. The overall aim of this review is to offer a comprehensive overview of the metachronal motions exhibited by diverse artificial cilia and the corresponding practical implementations. Additionally, we identify the potential future directions within this field. These insights present an exciting opportunity for further advancements in this domain.

Programmable metachronal motion of closely packed magnetic artificial cilia

Despite recent advances in artificial cilia technologies, the application of metachrony, which is the collective wavelike motion by cilia moving out-of-phase, has been severely hampered by difficulties in controlling closely packed artificial cilia at micrometer length scales. Moreover, there has been no direct experimental proof yet that a metachronal wave in combination with fully reciprocal ciliary motion can generate significant microfluidic flow on a micrometer scale as theoretically predicted. In this study, using an in-house developed precise micro-molding technique, we have fabricated closely packed magnetic artificial cilia that can generate well-controlled metachronal waves. We studied the effect of pure metachrony on fluid flow by excluding all symmetry-breaking ciliary features. Experimental and simulation results prove that net fluid transport can be generated by metachronal motion alone, and the effectiveness is strongly dependent on cilia spacing. This technique not only offers a biomimetic experimental platform to better understand the mechanisms underlying metachrony, it also opens new pathways towards advanced industrial applications.

Nonreciprocal interactions give rise to fast cilium synchronization in finite systems

Motile cilia beat in an asymmetric fashion in order to propel the surrounding fluid. When many cilia are located on a surface, their beating can synchronize such that their phases form metachronal waves. Here, we computationally study a model where each cilium is represented as a spherical particle, moving along a tilted trajectory with a position-dependent active driving force and a position-dependent internal drag coefficient. The model thus takes into account all the essential broken symmetries of the ciliary beat. We show that taking into account the near-field hydrodynamic interactions, the effective coupling between cilia even over an entire beating cycle can become nonreciprocal: The phase of a cilium is more strongly affected by an adjacent cilium on one side than by a cilium at the same distance in the opposite direction. As a result, synchronization starts from a seed at the edge of a group of cilia and propagates rapidly across the system, leading to a synchronization time that scales proportionally to the linear dimension of the system. We show that a ciliary carpet is characterized by three different velocities: the velocity of fluid transport, the phase velocity of metachronal waves, and the group velocity of order propagation. Unlike in systems with reciprocal coupling, boundary effects are not detrimental for synchronization, but rather enable the formation of the initial seed.

Minimum entropy production by microswimmers with internal dissipation

The energy dissipation and entropy production by self-propelled microswimmers differ profoundly from passive particles pulled by external forces. The difference extends both to the shape of the flow around the swimmer, as well as to the internal dissipation of the propulsion mechanism. Here we derive a general theorem that provides an exact lower bound on the total, external and internal, dissipation by a microswimmer. The problems that can be solved include an active surface-propelled droplet, swimmers with an extended propulsive layer and swimmers with an effective internal dissipation. We apply the theorem to determine the swimmer shapes that minimize the total dissipation while keeping the volume constant. Our results show that the entropy production by active microswimmers is subject to different fundamental limits than the entropy production by externally driven particles.

Miniaturized metachronal magnetic artificial cilia

Biological cilia, hairlike organelles on cell surfaces, often exhibit collective wavelike motion known as metachrony, which helps generating fluid flow. Inspired by nature, researchers have developed artificial cilia as microfluidic actuators, exploring several methods to mimic the metachrony. However, reported methods are difficult to miniaturize because they require either control of individual cilia properties or the generation of a complex external magnetic field. We introduce a concept that generates metachronal motion of magnetic artificial cilia (MAC), even though the MAC are all identical, and the applied external magnetic field is uniform. This is achieved by integrating a paramagnetic substructure in the substrate underneath the MAC. Uniquely, we can create both symplectic and antiplectic metachrony by changing the relative positions of MAC and substructure. We demonstrate the flow generation of the two metachronal motions in both high and low Reynolds number conditions. Our research marks a significant milestone by breaking the size limitation barrier in metachronal artificial cilia. This achievement not only showcases the potential of nature-inspired engineering but also opens up a host of exciting opportunities for designing and optimizing microsystems with enhanced fluid manipulation capabilities.

Theoretical efficiency limits and speed-efficiency trade-off in myosin motors

Muscle myosin is a non-processive molecular motor generates mechanical work when cooperating in large ensembles. During its cyle, each individual motor keeps attaching and detaching from the actin filament. The random nature of attachment and detachment inevitably leads to losses and imposes theoretical limits on the energetic efficiency. Here, we numerically determine the theoretical efficiency limit of a classical myosin model with a given number of mechano-chemical states. All parameters that are not bounded by physical limits (like rate limiting steps) are determined by numerical efficiency optimization. We show that the efficiency is limited by the number of states, the stiffness and the rate-limiting kinetic steps. There is a trade-off between speed and efficiency. Slow motors are optimal when most of the available free energy is allocated to the working stroke and the stiffness of their elastic element is high. Fast motors, on the other hand, work better with a lower and asymmetric stiffness and allocate a larger fraction of free energy to the release of ADP. Overall, many features found in myosins coincide with the findings from the model optimization: there are at least 3 bound states, the largest part of the working stroke takes place during the first transition, the ADP affinity is adapted differently in slow and fast myosins and there is an asymmetry in elastic elements.

Lithium-induced ciliary lengthening sparks Arp2/3 complex-dependent endocytosis

Ciliary length is highly regulated, but can be disrupted by lithium, which causes ciliary elongation across cell types and organisms. We used the algal system Chlamydomonas reinhardtii to investigate the mechanism behind lithium-induced ciliary elongation. Protein synthesis is not required for lengthening, and the target of lithium, GSK3, has substrates that can influence membrane dynamics. Further, ciliary assembly requires a supply of ciliary membrane as well as protein. Lithium-treated cilia elongate normally with brefeldin treatment, but dynasore treatment produced defective lengthening suggesting a source of membrane from the cell surface rather than the Golgi. Genetic or chemical perturbation of the Arp2/3 complex or dynamin, required for endocytosis, blocks lithium-induced ciliary lengthening. Finally, we found an increase in Arp2/3 complex- and endocytosis-dependent actin filaments near the ciliary base upon lithium treatment. Our results identify a mechanism for lithium-mediated cilium lengthening and demonstrate the endocytic pathway for cilium membrane supply in algae is likely a conserved mechanism given lithium's conserved effects across organisms.

3D-printed micrometer-scale wireless magnetic cilia with metachronal programmability

Biological cilia play essential roles in self-propulsion, food capture, and cell transportation by performing coordinated metachronal motions. Experimental studies to emulate the biological cilia metachronal coordination are challenging at the micrometer length scale because of current limitations in fabrication methods and materials. We report on the creation of wirelessly actuated magnetic artificial cilia with biocompatibility and metachronal programmability at the micrometer length scale. Each cilium is fabricated by direct laser printing a silk fibroin hydrogel beam affixed to a hard magnetic FePt Janus microparticle. The 3D-printed cilia show stable actuation performance, high temperature resistance, and high mechanical endurance. Programmable metachronal coordination can be achieved by programming the orientation of the identically magnetized FePt Janus microparticles, which enables the generation of versatile microfluidic patterns. Our platform offers an unprecedented solution to create bioinspired microcilia for programmable microfluidic systems, biomedical engineering, and biocompatible implants.

An operative framework to model mucus clearance in silico by coupling cilia motion with the liquid environment

Mucociliary clearance is the first defense mechanism of the respiratory tract against inhaled particles. This mechanism is based on the collective beating motion of cilia at the surface of epithelial cells. Impaired clearance, either caused by malfunctioning or absent cilia, or mucus defects, is a symptom of many respiratory diseases. Here, by exploiting the lattice Boltzmann particle dynamics technique, we develop a model to simulate the dynamics of multiciliated cells in a two-layer fluid. First, we tuned our model to reproduce the characteristic length- and time-scales of the cilia beating. We then check for the emergence of the metachronal wave as a consequence of hydrodynamic mediated correlations between beating cilia. Finally, we tune the viscosity of the top fluid layer to simulate the mucus flow upon cilia beating, and evaluate the pushing efficiency of a carpet of cilia. With this work, we build a realistic framework that can be used to explore several important physiological aspects of mucociliary clearance.

Mucociliary clearance affected by mucus-periciliary interface stimulations using analytical solution during cough and sneeze

Assessment of mucus velocity variations under different conditions including viscosity variation and boundary conditions is useful to develop mucosal-based medical treatments. This paper deals with the analytical investigation of mucus-periciliary velocities under mucus-periciliary interface movements and mucus viscosity variations. The results for mucus velocity show that there is no difference between the two cases under the free-slip condition. Therefore, power-law mucus can be substituted with a high viscosity Newtonian fluid since the upper boundary of the mucus layer is exposed to the free-slip condition. However, when the upper boundary of the mucus layer is under nonzero shear stress levels, including cough or sneeze, the assumption of a high viscosity Newtonian mucus layer is invalid. Moreover, mucus viscosity variations are investigated for both Newtonian and power-law mucus layers under sneeze and cough to propose a mucosal-based medical treatment. The results indicate by varying mucus viscosity up to a critical value, the direction of mucus movement changes. The critical values of viscosity in sneezing and coughing for Newtonian and power-law mucus layers are 10-4 and 5 × 10-5 and 0.0263 and 006.024 m2 s-1, respectively. Therefore, the pathogen entry into the respiratory system can be prevented by varying mucus viscosity during sneeze and cough.

Two-dimensional hydrodynamic simulation for synchronization in coupled density oscillators

A density oscillator is a fluid system in which oscillatory flow occurs between different density fluids through the pore connecting them. We investigate the synchronization in coupled density oscillators using two-dimensional hydrodynamic simulation and analyze the stability of the synchronous state based on the phase reduction theory. Our results show that the antiphase, three-phase, and 2-2 partial-in-phase synchronization modes spontaneously appear as stable states in two, three, and four coupled oscillators, respectively. The phase dynamics of coupled density oscillators is interpreted with their sufficiently large first Fourier components of the phase coupling function.

Numerical investigation of mucociliary clearance using power law and thixotropic mucus layers under discrete and continuous cilia motion

Mucus layer movement inside the airway system is an important phenomenon as the first defensive mechanism against pathogens. This research deals with the mucus velocity variations inside the nasal cavity using two different power law and thixotropic mucus layers. The cilia movement is replaced with four cyclic velocity profiles at the lower boundary of the mucus layer, while the upper boundary is exposed to the free-slip condition. The effects of boundary conditions and different fluid parameters are evaluated on the mucus flow. Furthermore, the replacement of power law and thixotropic mucus layers with a high viscous Newtonian mucus is examined under the free-slip condition at the mucus upper boundary. The adaptation rate is used as the criteria for replacing fluids instead of each other. The results show the mucus flow has enough time to adjust the changes from the lower boundary and the recovery stroke does not affect the mucus velocity in the effective stroke. Moreover, it is observed that the mucus flow variations are the same under the influence of recovery, breakdown, and breakdown exponent parameters. However, the effects of the exponent parameter on the mucus flow are more than the other two parameters in the recovery stroke. It is concluded that the assumption for replacing the power law mucus with a high viscous Newtonian one is acceptable. However, this assumption leads to the maximum error of 98.5% for thixotropic mucus in the recovery stroke.

Novel analytical tools reveal that local synchronization of cilia coincides with tissue-scale metachronal waves in zebrafish multiciliated epithelia

Motile cilia are hair-like cell extensions that beat periodically to generate fluid flow along various epithelial tissues within the body. In dense multiciliated carpets, cilia were shown to exhibit a remarkable coordination of their beat in the form of traveling metachronal waves, a phenomenon which supposedly enhances fluid transport. Yet, how cilia coordinate their regular beat in multiciliated epithelia to move fluids remains insufficiently understood, particularly due to lack of rigorous quantification. We combine experiments, novel analysis tools, and theory to address this knowledge gap. To investigate collective dynamics of cilia, we studied zebrafish multiciliated epithelia in the nose and the brain. We focused mainly on the zebrafish nose, due to its conserved properties with other ciliated tissues and its superior accessibility for non-invasive imaging. We revealed that cilia are synchronized only locally and that the size of local synchronization domains increases with the viscosity of the surrounding medium. Even though synchronization is local only, we observed global patterns of traveling metachronal waves across the zebrafish multiciliated epithelium. Intriguingly, these global wave direction patterns are conserved across individual fish, but different for left and right noses, unveiling a chiral asymmetry of metachronal coordination. To understand the implications of synchronization for fluid pumping, we used a computational model of a regular array of cilia. We found that local metachronal synchronization prevents steric collisions, i.e., cilia colliding with each other, and improves fluid pumping in dense cilia carpets, but hardly affects the direction of fluid flow. In conclusion, we show that local synchronization together with tissue-scale cilia alignment coincide and generate metachronal wave patterns in multiciliated epithelia, which enhance their physiological function of fluid pumping.

Spontaneous phase coordination and fluid pumping in model ciliary carpets

Ciliated tissues, such as in the mammalian lungs, brains, and reproductive tracts, are specialized to pump fluid. They generate flows by the collective activity of hundreds of thousands of individual cilia that beat in a striking metachronal wave pattern. Despite progress in analyzing cilia coordination, a general theory that links coordination and fluid pumping in the limit of large arrays of cilia remains lacking. Here, we conduct in silico experiments with thousands of hydrodynamically interacting cilia, and we develop a continuum theory in the limit of infinitely many independently beating cilia by combining tools from active matter and classical Stokes flow. We find, in both simulations and theory, that isotropic and synchronized ciliary states are unstable. Traveling waves emerge regardless of initial conditions, but the characteristics of the wave and net flows depend on cilia and tissue properties. That is, metachronal phase coordination is a stable global attractor in large ciliary carpets, even under finite perturbations to cilia and tissue properties. These results support the notion that functional specificity of ciliated tissues is interlaced with the tissue architecture and cilia beat kinematics and open up the prospect of establishing structure to function maps from cilium-level beat to tissue-level coordination and fluid pumping.

Ciliate cortical organization and dynamics for cell motility: Comparing ciliates and vertebrates

The generation of efficient fluid flow is crucial for organismal development and homeostasis, sexual reproduction, and motility. Multi-ciliated cells possess fields of motile cilia that beat in synchrony to propel fluid. Ciliary arrays are remarkably conserved in their organization and function. Ciliates have polarized multi-ciliary arrays (MCAs) to promote fluid flow for cell motility. The ciliate cortex is decorated with hundreds of basal bodies (BB) forming linear rows along the cell's anterior-posterior axis. BBs scaffold and position cilia to form the organized ciliary array. Nascent BBs assemble at the base of BBs. As nascent BBs mature, they integrate into the cortical BB and cytoskeletal network and nucleate their own cilium. The organization of MCAs is balanced between cortical stability and cortical dynamism. The cortical cytoskeletal network both establishes and maintains a stable organization of the MCA in the face of mechanical forces exerted by ciliary beating. At the same time, MCA organization is plastic, such that it remodels for optimal ciliary mobility during development and in response to environmental conditions. Such plasticity promotes effective feeding and ecological behavior required for these organisms. Together, these properties allow an organism to effectively sense, adapt to, and move through its environment.

Soft-robotic ciliated epidermis for reconfigurable coordinated fluid manipulation

The fluid manipulation capabilities of current artificial cilia are severely handicapped by the inability to reconfigure near-surface flow on various static or dynamically deforming three-dimensional (3D) substrates. To overcome this challenge, we propose an electrically driven soft-robotic ciliated epidermis with multiple independently controlled polypyrrole bending actuators. The beating kinematics and the coordination of multiple actuators can be dynamically reconfigured to control the strength and direction of fluid transportation. We achieve fluid transportation along and perpendicular to the beating directions of the actuator arrays, and toward or away from the substrate. The ciliated epidermises are bendable and stretchable and can be deployed on various static or dynamically deforming 3D surfaces. They enable previously difficult to obtain fluid manipulation functionalities, such as transporting fluid in tubular structures or enhancing fluid transportation near dynamically bending and expanding surfaces.

Flagellar energy costs across the tree of life

Flagellar-driven motility grants unicellular organisms the ability to gather more food and avoid predators, but the energetic costs of construction and operation of flagella are considerable. Paths of flagellar evolution depend on the deviations between fitness gains and energy costs. Using structural data available for all three major flagellar types (bacterial, archaeal, and eukaryotic), flagellar construction costs were determined for Escherichia coli, Pyrococcus furiosus, and Chlamydomonas reinhardtii. Estimates of cell volumes, flagella numbers, and flagellum lengths from the literature yield flagellar costs for another ~200 species. The benefits of flagellar investment were analysed in terms of swimming speed, nutrient collection, and growth rate; showing, among other things, that the cost-effectiveness of bacterial and eukaryotic flagella follows a common trend. However, a comparison of whole-cell costs and flagellum costs across the Tree of Life reveals that only cells with larger cell volumes than the typical bacterium could evolve the more expensive eukaryotic flagellum. These findings provide insight into the unsolved evolutionary question of why the three domains of life each carry their own type of flagellum.

Effects of continuous and discrete boundary conditions on the movement of upper-convected maxwell and Newtonian mucus layers in coughing and sneezing

Mucociliary clearance is an important phenomenon inside the respiratory system as a first defensive mechanism against pathogens. Therefore, any assumption considered for the mucociliary clearance and affects its functionality must be validated. The present research deals with the effects of boundary conditions on the movement of upper-convected Maxwell and high viscosity Newtonian mucus layers, numerically. Furthermore, the validity of replacing the viscoelastic mucus layer with a high viscosity Newtonian layer is evaluated. The airway surface liquid layer is considered a two-layer model including non-Newtonian mucus and Newtonian periciliary layers. Four cyclic boundary conditions are imposed at the mucus-periciliary interface as the cilia movement to obtain variations of mucociliary clearance. The upper boundary of the mucus layer is also exposed to different shear stress levels including free slip, cough, and sneeze conditions. By investigation of velocity variations inside mucus and periciliary layers, it is concluded the differences between viscoelastic and Newtonian mucus are not negligible. The maximum velocity differences between the two fluids are more than 52% and 215% during cough and sneeze, respectively. The results show there is a high order of dependency between the relaxation time and the imposed boundary conditions at the mucus-periciliary interface that leads to the invalidation of replacing two fluids with each other. Moreover, the results show substituting the viscoelastic mucus with a high viscosity Newtonian one depends on the mucus-periciliary interface boundary condition. If an independent time-varying boundary condition is used, the substitution leads to an error less than 7% under different shear stress levels. However, time-varying boundary condition shows 38% and 88% differences between high viscosity Newtonian and viscoelastic mucus layers. Furthermore, neglecting the recovery stroke leads to a velocity underestimation up to 50% by substituting viscoelastic mucus with a high viscosity Newtonian one. Therefore, replacing the viscoelastic mucus with a high viscosity Newtonian one is not acceptable for numerical simulations.

Longitudinal to Transverse Metachronal Wave Transitions in an In Vitro Model of Ciliated Bronchial Epithelium

Myriads of cilia beat on ciliated epithelia, which are ubiquitous in life. When ciliary beats are synchronized, metachronal waves emerge, whose direction of propagation depends on the living system in an unexplained way. We show on a reconstructed human bronchial epithelium in vitro that the direction of propagation is determined by the ability of mucus to be transported at the epithelial surface. Numerical simulations show that longitudinal waves maximize the transport of mucus while transverse waves, observed when the mucus is rigid and still, minimize the energy dissipated by the cilia.

Fluid-structure interaction of bio-inspired flexible slender structures: a review of selected topics

Flexible slender structures are ubiquitous in biological systems and engineering applications. Fluid-structure interaction (FSI) plays a key role in the dynamics of such structures immersed in fluids. Here, we survey recent studies on highly simplified bio-inspired models (either mathematical or mechanical) that aim to revealthe flow physics associated with FSI. Various models from different sources of biological inspiration are included, namely flexible flapping foil inspired by fish and insects, deformable membrane inspired by jellyfish and cephalopods, beating filaments inspired by flagella and cilia of microorganisms, and flexible wall-mounted filaments inspired by terrestrial and aquatic plants. Suggestions on directions for future research are also provided.

Metachronal patterns by magnetically-programmable artificial cilia surfaces for low Reynolds number fluid transport and mixing

Motile cilia can produce net fluid flows at low Reynolds number because of their asymmetric motion and metachrony of collective beating. Mimicking this with artificial cilia can find application in microfluidic devices for fluid transport and mixing. Here, we study the metachronal beating of nonidentical, magnetically-programmed artificial cilia whose individual non-reciprocal motion and collective metachronal beating pattern can be independently controlled. We use a finite element method that accounts for magnetic forces, cilia deformation and fluid flow in a fully coupled manner. Mimicking biological cilia, we study magnetic cilia subject to a full range of metachronal driving patterns, including antiplectic, symplectic, laeoplectic and diaplectic waves. We analyse the induced primary flow, secondary flow and mixing rate as a function of the phase lag between cilia and explore the underlying physical mechanism. Our results show that shielding effects between neighboring cilia lead to a primary flow that is larger for antiplectic than for symplectic metachronal waves. The secondary flow can be fully explained by the propagation direction of the metachronal wave. Finally, we show that the mixing rate can be strongly enhanced by laeoplectic and diaplectic metachrony resulting in large velocity gradients and vortex-like flow patterns.

Enhanced clamshell swimming with asymmetric beating at low Reynolds number

A single flexible filament can be actuated to escape from the scallop theorem and generate net propulsion at low Reynolds number. In this work, we study the dynamics of a simple boundary-driven multi-filament swimmer, a two-arm clamshell actuated at the hinged point, using a nonlocal slender body approximation with hydrodynamic interactions. We first consider an elastic clamshell consisted of flexible filaments with intrinsic curvature, and then build segmental models consisted of rigid segments connected by different mechanical joints with different forms of response torques. The simplicity of the system allows us to fully explore the effect of various parameters on the swimming performance. Optimal included angles and elastoviscous numbers are identified. The segmental models capture the characteristic dynamics of the elastic clamshell. We further demonstrate how the swimming performance can be significantly enhanced by the asymmetric beating patterns induced by biased torques.

Microscopic artificial cilia - a review

Cilia are microscopic hair-like external cell organelles that are ubiquitously present in nature, also within the human body. They fulfill crucial biological functions: motile cilia provide transportation of fluids and cells, and immotile cilia sense shear stress and concentrations of chemical species. Inspired by nature, scientists have developed artificial cilia mimicking the functions of biological cilia, aiming at application in microfluidic devices like lab-on-chip or organ-on-chip. By actuating the artificial cilia, for example by a magnetic field, an electric field, or pneumatics, microfluidic flow can be generated and particles can be transported. Other functions that have been explored are anti-biofouling and flow sensing. We provide a critical review of the progress in artificial cilia research and development as well as an evaluation of its future potential. We cover all aspects from fabrication approaches, actuation principles, artificial cilia functions - flow generation, particle transport and flow sensing - to applications. In addition to in-depth analyses of the current state of knowledge, we provide classifications of the different approaches and quantitative comparisons of the results obtained. We conclude that artificial cilia research is very much alive, with some concepts close to industrial implementation, and other developments just starting to open novel scientific opportunities.

Hydrodynamics of chiral squirmers

Many microorganisms take a chiral path while swimming in an ambient fluid. In this paper we study the combined behavior of two chiral swimmers using the well-known squirmer model taking into account chiral asymmetries. In contrast to the simple squirmer model, which has an axisymmetric distribution of slip velocity, the chiral squirmer has additional asymmetries in the surface slip, which contribute to both translations and rotations of the motion. As a result, swimming trajectories can become helical and chiral asymmetries arise in the flow patterns. We study the swimming trajectories of a pair of chiral squirmers that interact hydrodynamically. This interaction can lead to attraction and repulsion, and in some cases even to bounded states where the swimmers continue to periodically orbit around a common average trajectory. Such bound states are a signature of the chiral nature of the swimmers. Our study could be relevant to the collective movements of ciliated microorganisms.

A multiscale biophysical model gives quantized metachronal waves in a lattice of beating cilia

Motile cilia are slender, hair-like cellular appendages that spontaneously oscillate under the action of internal molecular motors and are typically found in dense arrays. These active filaments coordinate their beating to generate metachronal waves that drive long-range fluid transport and locomotion. Until now, our understanding of their collective behavior largely comes from the study of minimal models that coarse grain the relevant biophysics and the hydrodynamics of slender structures. Here we build on a detailed biophysical model to elucidate the emergence of metachronal waves on millimeter scales from nanometer-scale motor activity inside individual cilia. Our study of a one-dimensional lattice of cilia in the presence of hydrodynamic and steric interactions reveals how metachronal waves are formed and maintained. We find that, in homogeneous beds of cilia, these interactions lead to multiple attracting states, all of which are characterized by an integer charge that is conserved. This even allows us to design initial conditions that lead to predictable emergent states. Finally, and very importantly, we show that, in nonuniform ciliary tissues, boundaries and inhomogeneities provide a robust route to metachronal waves.

Ultrasound-activated ciliary bands for microrobotic systems inspired by starfish

Cilia are short, hair-like appendages ubiquitous in various biological systems, which have evolved to manipulate and gather food in liquids at regimes where viscosity dominates inertia. Inspired by these natural systems, synthetic cilia have been developed and utilized in microfluidics and microrobotics to achieve functionalities such as propulsion, liquid pumping and mixing, and particle manipulation. Here, we demonstrate ultrasound-activated synthetic ciliary bands that mimic the natural arrangements of ciliary bands on the surface of starfish larva. Our system leverages nonlinear acoustics at microscales to drive bulk fluid motion via acoustically actuated small-amplitude oscillations of synthetic cilia. By arranging the planar ciliary bands angled towards (+) or away (-) from each other, we achieve bulk fluid motion akin to a flow source or sink. We further combine these flow characteristics with a physical principle to circumvent the scallop theorem and realize acoustic-based propulsion at microscales. Finally, inspired by the feeding mechanism of a starfish larva, we demonstrate an analogous microparticle trap by arranging + and - ciliary bands adjacent to each other.

Fluid Mechanics of Mosaic Ciliated Tissues

In tissues as diverse as amphibian skin and the human airway, the cilia that propel fluid are grouped in sparsely distributed multiciliated cells (MCCs). We investigate fluid transport in this "mosaic" architecture, with emphasis on the trade-offs that may have been responsible for its evolutionary selection. Live imaging of MCCs in embryos of the frog Xenopus laevis shows that cilia bundles behave as active vortices that produce a flow field accurately represented by a local force applied to the fluid. A coarse-grained model that self-consistently couples bundles to the ambient flow reveals that hydrodynamic interactions between MCCs limit their rate of work so that they best shear the tissue at a finite but low area coverage, a result that mirrors findings for other sparse distributions such as cell receptors and leaf stomata.

Conditions for metachronal coordination in arrays of model cilia

Motile cilia can coordinate with each other to beat in the form of a metachronal wave, which can facilitate the self-propulsion of microorganisms such as Paramecium and can also be used for fluid transport such as mucus removal in trachea. How can we predict the collective behavior of arrays of many cilia coordinated by hydrodynamic interactions, and in particular, the properties of the emerging metachronal waves, from the single-cilium characteristics? We address this question using a bottom-up coarse-graining approach and present results that contribute to understanding how the dynamical self-organization of ciliary arrays can be controlled, which can have significant biological, medical, and engineering implications.

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

Multi-ciliated microswimmers-metachronal coordination and helical swimming

The dynamics and motion of multi-ciliated microswimmers with a spherical body and a small number N (with [Formula: see text]) of cilia with length comparable to the body radius, is investigated by mesoscale hydrodynamics simulations. A metachronal wave is imposed for the cilia beat, for which the wave vector has both a longitudinal and a latitudinal component. The dynamics and motion is characterized by the swimming velocity, its variation over the beat cycle, the spinning velocity around the main body axis, as well as the parameters of the helical trajectory. Our simulation results show that the microswimmer motion strongly depends on the latitudinal wave number and the longitudinal phase lag. The microswimmers are found to swim smoothly and usually spin around their own axis. Chirality of the metachronal beat pattern generically generates helical trajectories. In most cases, the helices are thin and stretched, i.e., the helix radius is about an order of magnitude smaller than the pitch. The rotational diffusion of the microswimmer is significantly smaller than the passive rotational diffusion of the body alone, which indicates that the extended cilia contribute strongly to the hydrodynamic radius. The swimming velocity is found to increase with the cilia number N with a slightly sublinear power law, consistent with the behavior expected from the dependence of the transport velocity of planar cilia arrays on the cilia separation.

Metachronal μ-Cilia for On-Chip Integrated Pumps and Climbing Robots

Biological cilia often perform metachronal motion, that is, neighboring cilia move out of phase creating a travelling wave, which enables highly efficient fluid pumping and body locomotion. Current methods for creating metachronal artificial cilia suffer from the complex design and sophisticated actuation schemes. This paper demonstrates a simple method to realize metachronal microscopic magnetic artificial cilia (μMAC) through control over the paramagnetic particle distribution within the μMAC based on their tendency to align with an applied magnetic field. Actuated by a 2D rotating uniform magnetic field, the metachronal μMAC enable strong microfluidic pumping and soft robot locomotion. The metachronal μMAC induce twice the pumping efficiency and 3 times the locomotion speed of synchronously moving μMAC. The ciliated soft robots show an unprecedented slope climbing ability (0 to 180°), and they display strong cargo-carrying capacity (>10 times their own weight) in both dry and wet conditions. These findings advance the design of on-chip integrated pumps and versatile soft robots, among others.

Integrative Neuroscience of Paramecium, a "Swimming Neuron"

Paramecium is a unicellular organism that swims in fresh water by beating thousands of cilia. When it is stimulated (mechanically, chemically, optically, thermally…), it often swims backward then turns and swims forward again. This "avoiding reaction" is triggered by a calcium-based action potential. For this reason, some authors have called Paramecium a "swimming neuron." This review summarizes current knowledge about the physiological basis of behavior of Paramecium.

Changes in geometrical aspects of a simple model of cilia synchronization control the dynamical state, a possible mechanism for switching of swimming gaits in microswimmers

Active oscillators, with purely hydrodynamic coupling, are useful simple models to understand various aspects of motile cilia synchronization. Motile cilia are used by microorganisms to swim and to control the flow fields in their surroundings; the patterns observed in cilia carpets can be remarkably complex, and can be changed over time by the organism. It is often not known to what extent the coupling between cilia is due to just hydrodynamic forces, and neither is it known if it is biological or physical triggers that can change the dynamical collective state. Here we treat this question from a very simplified point of view. We describe three possible mechanisms that enable a switch in the dynamical state, in a simple scenario of a chain of oscillators. We find that shape-change provides the most consistent strategy to control collective dynamics, but also imposing small changes in frequency produces some unique stable states. Demonstrating these effects in the abstract minimal model proves that these could be possible explanations for gait switching seen in ciliated micro organisms like Paramecium and others. Microorganisms with many cilia could in principle be taking advantage of hydrodynamic coupling, to switch their swimming gait through either a shape change that manifests in decreased coupling between groups of cilia, or alterations to the beat style of a small subset of the cilia.

Minimum Dissipation Theorem for Microswimmers

We derive a theorem for the lower bound on the energy dissipation rate by a rigid surface-driven active microswimmer of arbitrary shape in a fluid at a low Reynolds number. We show that, for any swimmer, the minimum dissipation at a given velocity can be expressed in terms of the resistance tensors of two passive bodies of the same shape with a no-slip and perfect-slip boundary. To achieve the absolute minimum dissipation, the optimal swimmer needs a surface velocity profile that corresponds to the flow around the perfect-slip body, and a propulsive force density that corresponds to the no-slip body. Using this theorem, we propose an alternative definition of the energetic efficiency of microswimmers that, unlike the commonly used Lighthill efficiency, can never exceed unity. We validate the theory by calculating the efficiency limits of spheroidal swimmers.

Metachronal patterns in artificial cilia for low Reynolds number fluid propulsion

Cilia are hair-like organelles, present in arrays that collectively beat to generate flow. Given their small size and consequent low Reynolds numbers, asymmetric motions are necessary to create a net flow. Here, we developed an array of six soft robotic cilia, which are individually addressable, to both mimic nature's symmetry-breaking mechanisms and control asymmetries to study their influence on fluid propulsion. Our experimental tests are corroborated with fluid dynamics simulations, where we find a good agreement between both and show how the kymographs of the flow are related to the phase shift of the metachronal waves. Compared to synchronous beating, we report a 50% increase of net flow speed when cilia move in an antiplectic wave with phase shift of -π/3 and a decrease for symplectic waves. Furthermore, we observe the formation of traveling vortices in the direction of the wave when metachrony is applied.

Bioinspired cilia arrays with programmable nonreciprocal motion and metachronal coordination

Coordinated nonreciprocal dynamics in biological cilia is essential to many living systems, where the emergentmetachronal waves of cilia have been hypothesized to enhance net fluid flows at low Reynolds numbers (Re). Experimental investigation of this hypothesis is critical but remains challenging. Here, we report soft miniature devices with both ciliary nonreciprocal motion and metachronal coordination and use them to investigate the quantitative relationship between metachronal coordination and the induced fluid flow. We found that only antiplectic metachronal waves with specific wave vectors could enhance fluid flows compared with the synchronized case. These findings further enable various bioinspired cilia arrays with unique functionalities of pumping and mixing viscous synthetic and biological complex fluids at low Re Our design method and developed soft miniature devices provide unprecedented opportunities for studying ciliary biomechanics and creating cilia-inspired wireless microfluidic pumping, object manipulation and lab- and organ-on-a-chip devices, mobile microrobots, and bioengineering systems.

Metachronal Actuation of Microscale Magnetic Artificial Cilia

Biological cells often interact with the environment through carpets of microscopic hair-like cilia. These elastic structures are known to beat in a synchronized wavy fashion called metachronal motion to produce fluid transport. Metachronal motion emerges due to a phase difference between beating cycles of neighboring cilia and appears as traveling waves propagating along the ciliary carpet. We demonstrate submerged in water microscale magnetic cilia that are externally actuated to beat in a metachronal fashion. Two approaches are used to induce coordinated phase differences among the beating cilia. In the first case, we fabricate cilia with an imposed gradient of geometrical properties that are subject to a rotating uniform magnetic field. In the second scenario, a ciliary array is composed of identical cilia that experience a magnetic field that varies spatiotemporally. We demonstrate that magnetic cilia can achieve symplectic, antiplectic, and leoplectic metachrony.

Metachronal actuation of microscopic magnetic artificial cilia generates strong microfluidic pumping

Biological cilia that generate fluid flow or propulsion are often found to exhibit a collective wavelike metachronal motion, i.e. neighboring cilia beat slightly out-of-phase rather than synchronously. Inspired by this observation, this article experimentally demonstrates that microscopic magnetic artificial cilia (μMAC) performing a metachronal motion can generate strong microfluidic flows, though, interestingly, the mechanism is different from that in biological cilia, as is found through a systematic experimental study. The μMAC are actuated by a facile magnetic setup, consisting of an array of rod-shaped magnets. This arrangement imposes a time-dependent non-uniform magnetic field on the μMAC array, resulting in a phase difference between the beatings of adjacent μMAC, while each cilium exhibits a two-dimensional whip-like motion. By performing the metachronal 2D motion, the μMAC are able to generate a strong flow in a microfluidic chip, with velocities of up to 3000 μm s^-1 in water, which, different from biological cilia, is found to be a result of combined metachronal and inertial effects, in addition to the effect of asymmetric beating. The pumping performance of the metachronal μMAC outperforms all previously reported microscopic artificial cilia, and is competitive with that of most of the existing microfluidic pumping methods, while the proposed platform requires no physical connection to peripheral equipment, reduces the usage of reagents by minimizing "dead volumes", avoids undesirable electrical effects, and accommodates a wide range of different fluids. The 2D metachronal motion can also generate a flow with velocities up to 60 μm s^-1 in pure glycerol, where Reynolds number is less than 0.05 and the flow is primarily caused by the metachronal motion of the μMAC. These findings offer a novel solution to not only create on-chip integrated micropumps, but also design swimming and walking microrobots, as well as self-cleaning and antifouling surfaces.

Propulsion and energetics of a minimal magnetic microswimmer

In this manuscript we describe the realization of a minimal hybrid microswimmer, composed of a ferromagnetic nanorod and a paramagnetic microsphere. The unbounded pair is propelled in water upon application of a swinging magnetic field that induces a periodic relative movement of the two composing elements, where the nanorod rotates and slides on the surface of the paramagnetic sphere. When taken together, the processes of rotation and sliding describe a finite area in the parameter space, which increases with the frequency of the applied field. We develop a theoretical approach and combine it with numerical simulations, which allow us to understand the dynamics of the propeller and explain the experimental observations. Furthermore, we demonstrate a reversal of the microswimmer velocity by varying the length of the nanorod, as predicted by the model. Finally, we determine theoretically and in experiments the Lighthill's energetic efficiency of this minimal magnetic microswimmer.

Degenerate states, emergent dynamics and fluid mixing by magnetic rotors

We investigate the collective motion of magnetic rotors suspended in a viscous fluid under a uniform rotating magnetic field. The rotors are positioned on a square lattice, and low Reynolds hydrodynamics is assumed. For a 3 × 3 array of magnets, we observe three characteristic dynamical patterns as the external field strength is varied: a synchronized pattern, an oscillating pattern, and a chessboard pattern. The relative stability of these depends on the competition between the energy due to the external magnetic field and the energy of the magnetic dipole-dipole interactions among the rotors. We argue that the chessboard pattern can be understood as an alternation in the stability of two degenerate states, characterized by striped and spin-ice configurations, as the applied magnetic field rotates. For larger arrays, we observe propagation of slip waves that are similar to metachronal waves. The rotor arrays have potential as microfluidic devices that can mix fluids and create vortices of different sizes.

Magnetic cilia carpets with programmable metachronal waves

Metachronal waves commonly exist in natural cilia carpets. These emergent phenomena, which originate from phase differences between neighbouring self-beating cilia, are essential for biological transport processes including locomotion, liquid pumping, feeding, and cell delivery. However, studies of such complex active systems are limited, particularly from the experimental side. Here we report magnetically actuated, soft, artificial cilia carpets. By stretching and folding onto curved templates, programmable magnetization patterns can be encoded into artificial cilia carpets, which exhibit metachronal waves in dynamic magnetic fields. We have tested both the transport capabilities in a fluid environment and the locomotion capabilities on a solid surface. This robotic system provides a highly customizable experimental platform that not only assists in understanding fundamental rules of natural cilia carpets, but also paves a path to cilia-inspired soft robots for future biomedical applications.

Why antiplectic metachronal cilia waves are optimal to transport bronchial mucus

The coordinated beating of epithelial cilia in human lungs is a fascinating problem from the hydrodynamics perspective. The phase lag between neighboring cilia is able to generate collective cilia motions, known as metachronal waves. Different kinds of waves can occur, antiplectic or symplectic, depending on the direction of the wave with respect to the flow direction. It is shown here, using a coupled lattice Boltzmann-immersed boundary solver, that the key mechanism responsible for their transport efficiency is a blowing-suction effect that displaces the interface between the periciliary liquid and the mucus phase. The contribution of this mechanism on the average flow generated by the cilia is compared to the contribution of the lubrication effect. The results reveal that the interface displacement is the main mechanism responsible for the better efficiency of antiplectic metachronal waves over symplectic ones to transport bronchial mucus. The conclusions drawn here can be extended to any two-layer fluid configuration having different viscosities, and put into motion by cilia-shaped or comb-plate structures, having a back-and-forth motion with phase lags.

Cilia and centrosomes: Ultrastructural and mechanical perspectives

Cilia and centrosomes of eukaryotic cells play important roles in cell movement, fluid transport, extracellular sensing, and chromosome division. The physiological functions of cilia and centrosomes are generated by their dynamics, motions, and forces controlled by the physical, chemical, and biological environments. How an individual cilium achieves its beat pattern and induces fluid flow is governed by its ultrastructure as well as the coordination of associated molecular motors. Thus, a bottom-up understanding of the physiological functions of cilia and centrosomes from the molecular to tissue levels is required. Correlations between the structure and motion can be understood in terms of mechanics. This review first focuses on cilia and centrosomes at the molecular level, introducing their ultrastructure. We then shift to the organelle level and introduce the kinematics and mechanics of cilia and centrosomes. Next, at the tissue level, we introduce nodal ciliary dynamics and nodal flow, which play crucial roles in the organogenetic process of left-right asymmetry. We also introduce respiratory ciliary dynamics and mucous flow, which are critical for protecting the epithelium from drying and exposure to harmful particles and viruses, i.e., respiratory clearance function. Finally, we discuss the future research directions in this field.

Living System Adapts Harmonics of Peristaltic Wave for Cost-Efficient Optimization of Pumping Performance

Wavelike patterns driving transport are ubiquitous in life. Peristaltic pumps are a paradigm of efficient mass transport by contraction driven flows-often limited by energetic constraints. We show that a cost-efficient increase in pumping performance can be achieved by modulating the phase difference between harmonics to increase occlusion. In experiments we find a phase difference shift in the living peristalsis model P. polycephalum as dynamic response to forced mass transport. Our findings provide a novel metric for wavelike patterns and demonstrate the crucial role of nonlinearities in life.

Motile cilia hydrodynamics: entrainment versus synchronization when coupling through flow

Coordinated motion of cilia is a fascinating and vital aspect of very diverse forms of eukaryotic life, enabling swimming and propulsion of fluid across cellular epithelia. There are many questions still unresolved, and broadly they fall into two classes. (i) The mechanism of how cilia physically transmit forces onto each other. It is not known for many systems if the forces are mainly of hydrodynamical origin, or if elastic forces within the cytoskeleton are important. (ii) In those systems where we know that forces are purely hydrodynamical, we do not have a framework for linking our understanding of how each cilium behaves in isolation to the collective properties of two or more cilia. In this work, we take biological data of cilia dynamics from a variety of organisms as an input for an analytical and numerical study. We calculate the relative importance of external flows versus internal cilia flows on cilia coupling. This study contributes to both the open questions outlined above: firstly, we show that it is, in general, incorrect to infer cilium-cilium coupling strength on the basis of experiments with external flows, and secondly, we show a framework to recapitulate the dynamics of single cilia (the waveform) showing classes that correspond to biological systems with the same physiological activity (swimming by propulsion, versus forming collective waves). This article is part of the Theo Murphy meeting issue 'Unity and diversity of cilia in locomotion and transport'.