Transitions in synchronization states of model cilia through basal-connection coupling

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.

Biomimetic Synchronization in Biciliated Robots

Direct mechanical coupling is known to be critical for establishing synchronization among cilia. However, the actual role of the connections is still elusive-partly because controlled experiments in living samples are challenging. Here, we employ an artificial ciliary system to address this issue. Two cilia are formed by chains of self-propelling robots and anchored to a shared base so that they are purely mechanically coupled. The system mimics biological ciliary beating but allows fine control over the beating dynamics. With different schemes of mechanical coupling, artificial cilia exhibit rich motility patterns. Particularly, their synchronous beating display two distinct modes-analogous to those observed in C. reinhardtii, the biciliated model organism for studying synchronization. Close examination suggests that the system evolves towards the most dissipative mode. Using this guideline in both simulations and experiments, we are able to direct the system into a desired state by altering the modes' respective dissipation. Our results have significant implications in understanding the synchronization of cilia.

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.

The younger flagellum sets the beat for Chlamydomonas reinhardtii

Eukaryotes swim with coordinated flagellar (ciliary) beating and steer by fine-tuning the coordination. The model organism for studying flagellate motility, Chlamydomonas reinhardtii, employs synchronous, breaststroke-like flagellar beating to swim, and it modulates the beating amplitudes differentially to steer. This strategy hinges on both inherent flagellar asymmetries (e.g. different response to chemical messengers) and such asymmetries being effectively coordinated in the synchronous beating. In C. reinhardtii, the synchrony of beating is known to be supported by a mechanical connection between flagella; however, how flagellar asymmetries persist in the synchrony remains elusive. For example, it has been speculated for decades that one flagellum leads the beating, as its dynamic properties (i.e. frequency, waveform, etc.) appear to be copied by the other one. In this study, we combine experiments, computations, and modeling efforts to elucidate the roles played by each flagellum in synchronous beating. With a non-invasive technique to selectively load each flagellum, we show that the coordinated beating essentially only responds to load exerted on the cis flagellum; and that such asymmetry in response derives from a unilateral coupling between the two flagella. Our results highlight a distinct role for each flagellum in coordination and have implication for biflagellates' tactic behaviors.

Mechanisms and functions of multiciliary coordination

Ciliated organisms are present in virtually every branch of the eukaryotic tree of life. In diverse systems, cilia operate in a coordinated manner to drive fluid flows, or even propel entire organisms. How do groups of motile cilia coordinate their activity within a cell or across a tissue to fulfil essential functions of life? In this review, we highlight the latest developments in our understanding of the mechanisms and functions of multiciliary coordination in diverse systems. We explore new and emerging trends in bioimaging, analytical, and computational methods, which together with their application in new model systems, have conspired to deliver important insights into one of the most fundamental questions in cellular dynamics.

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.

Stochastic transition in synchronized spiking nanooscillators

This work reports that synchronization of Mott material-based nanoscale coupled spiking oscillators can be drastically different from that in conventional harmonic oscillators. We investigated the synchronization of spiking nanooscillators mediated by thermal interactions due to the close physical proximity of the devices. Controlling the driving voltage enables in-phase 1:1 and 2:1 integer synchronization modes between neighboring oscillators. Transition between these two integer modes occurs through an unusual stochastic synchronization regime instead of the loss of spiking coherence. In the stochastic synchronization regime, random length spiking sequences belonging to the 1:1 and 2:1 integer modes are intermixed. The occurrence of this stochasticity is an important factor that must be taken into account in the design of large-scale spiking networks for hardware-level implementation of novel computational paradigms such as neuromorphic and stochastic computing.

Coordination of Cilia Movements in Multi-Ciliated Cells

Multiple motile cilia are formed at the apical surface of multi-ciliated cells in the epithelium of the oviduct or the fallopian tube, the trachea, and the ventricle of the brain. Those cilia beat unidirectionally along the tissue axis, and this provides a driving force for directed movements of ovulated oocytes, mucus, and cerebrospinal fluid in each of these organs. Furthermore, cilia movements show temporal coordination between neighboring cilia. To establish such coordination of cilia movements, cilia need to sense and respond to various cues, including the organ's orientation and movements of neighboring cilia. In this review, we discuss the mechanisms by which cilia movements of multi-ciliated cells are coordinated, focusing on planar cell polarity and the cytoskeleton, and highlight open questions for future research.

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.

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.

Mechanical compression enhances ciliary beating through cytoskeleton remodeling in human nasal epithelial cells

Nasal inflammatory diseases, including nasal polyps and acute/chronic sinusitis, are characterized by impaired mucociliary clearance and eventually inflammation and infection. Contact of nasal polyps with adjacent nasal mucosa or stagnated mucus within the maxillary sinus produces compressive mechanical stresses on the apical surface of epithelium which can induce cytoskeleton remodeling in epithelial cells. In this study, we hypothesized that compressive stress modulates ciliary beating by altering the mechanical properties of the cytoskeleton of ciliated cell basal bodies. For the primary human nasal epithelial cells, we found that the applied compressive stress higher than the critical value of 1.0 kPa increased the stroke speed of cilia leading to the enhancement of ciliary beating frequency and mucociliary transportability. Immunostained images of the cytoskeleton showed reorganization and compactness of the actin filaments in the presence of compressive stress. Analysis of beating trajectory with the computational modeling for ciliary beating revealed that the stroke speed of cilium increased as the relative elasticity to viscosity of the surrounding cytoskeleton increases. These results suggest that the compressive stress on epithelial cells increases the ciliary beating speed through cytoskeleton remodeling to prevent mucus stagnation at the early stage of airway obstruction. Our study provides an insight into the defensive mechanism of airway epithelium against pathological conditions. STATEMENT OF SIGNIFICANCE: Cilia dynamics of the nasal epithelium is critical for not only maintaining normal breathing but preventing inflammatory diseases. It has been shown that mechanical compressive stresses can alter the shape and phenotype of epithelial cells. However, the effect of compressive stress on cilia dynamics is unclear. In this study, we demonstrated that the oscillation speed of cilia in human nasal epithelial cells was increased by the applied compressive stress experimentally. The computational simulation revealed that the change of ciliary beating dynamics was attributed to the viscoelastic properties of the reorganized cytoskeleton in response to compressive stress. Our results will be beneficial in understanding the defensive mechanism of airway epithelium against pathological conditions.

Intracellular coupling modulates biflagellar synchrony

Beating flagella exhibit a variety of synchronization modes. This synchrony has long been attributed to hydrodynamic coupling between the flagella. However, recent work with flagellated algae indicates that a mechanism internal to the cell, through the contractile fibres connecting the flagella basal bodies, must be at play to actively modulate flagellar synchrony. Exactly how basal coupling mediates flagellar coordination remains unclear. Here, we examine the role of basal coupling in the synchronization of the model biflagellate Chlamydomonas reinhardtii using a series of mathematical models of decreasing levels of complexity. We report that basal coupling is sufficient to achieve inphase, antiphase and bistable synchrony, even in the absence of hydrodynamic coupling and flagellar compliance. These modes can be reached by modulating the activity level of the individual flagella or the strength of the basal coupling. We observe a slip mode when allowing for differential flagellar activity, just as in experiments with live cells. We introduce a dimensionless ratio of flagellar activity to basal coupling that is predictive of the mode of synchrony. This ratio allows us to query biological parameters which are not yet directly measurable experimentally. Our work shows a concrete route for cells to actively control the synchronization of their flagella.

Multisynchrony in Active Microfilaments

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

First-order synchronization transition in a large population of strongly coupled relaxation oscillators

Onset and loss of synchronization in coupled oscillators are of fundamental importance in understanding emergent behavior in natural and man-made systems, which range from neural networks to power grids. We report on experiments with hundreds of strongly coupled photochemical relaxation oscillators that exhibit a discontinuous synchronization transition with hysteresis, as opposed to the paradigmatic continuous transition expected from the widely used weak coupling theory. The resulting first-order transition is robust with respect to changes in network connectivity and natural frequency distribution. This allows us to identify the relaxation character of the oscillators as the essential parameter that determines the nature of the synchronization transition. We further support this hypothesis by revealing the mechanism of the transition, which cannot be accounted for by standard phase reduction techniques.

Entrainment of mammalian motile cilia in the brain with hydrodynamic forces

Motile cilia are widespread across the animal and plant kingdoms, displaying complex collective dynamics central to their physiology. Their coordination mechanism is not generally understood, with previous work mainly focusing on algae and protists. We study here the entrainment of cilia beat in multiciliated cells from brain ventricles. The response to controlled oscillatory external flows shows that flows at a similar frequency to the actively beating cilia can entrain cilia oscillations. We find that the hydrodynamic forces required for this entrainment strongly depend on the number of cilia per cell. Cells with few cilia (up to five) can be entrained at flows comparable to cilia-driven flows, in contrast with what was recently observed in Chlamydomonas Experimental trends are quantitatively described by a model that accounts for hydrodynamic screening of packed cilia and the chemomechanical energy efficiency of the flagellar beat. Simulations of a minimal model of cilia interacting hydrodynamically show the same trends observed in cilia.

Synchrony and symmetry-breaking in active flagellar coordination

Living creatures exhibit a remarkable diversity of locomotion mechanisms, evolving structures specialized for interacting with their environment. In the vast majority of cases, locomotor behaviours such as flying, crawling and running are orchestrated by nervous systems. Surprisingly, microorganisms can enact analogous movement gaits for swimming using multiple, fast-moving cellular protrusions called cilia and flagella. Here, I demonstrate intermittency, reversible rhythmogenesis and gait mechanosensitivity in algal flagella, to reveal the active nature of locomotor patterning. In addition to maintaining free-swimming gaits, I show that the algal flagellar apparatus functions as a central pattern generator that encodes the beating of each flagellum in a network in a distinguishable manner. The latter provides a novel symmetry-breaking mechanism for cell reorientation. These findings imply that the capacity to generate and coordinate complex locomotor patterns does not require neural circuitry but rather the minimal ingredients are present in simple unicellular organisms. This article is part of the Theo Murphy meeting issue 'Unity and diversity of cilia in locomotion and transport'.