Long-range interactions, wobbles, and phase defects in chains of model cilia

Role of cilia activity and surrounding viscous fluid in properties of metachronal waves

Large groups of active cilia collectively beat in a fluid medium as metachronal waves, essential for some microorganisms motility and for flow generation in mucociliary clearance. Several models can predict the emergence of metachronal waves, but what controls the properties of metachronal waves is still unclear. Here, we numerically investigate the respective impacts of active beating and viscous dissipation on the properties of metachronal waves in a collection of oscillators, using a simple model for cilia in the presence of noise on regular lattices in one and two dimensions. We characterize the wave using spatial correlation and the frequency of collective beating. Our results clearly show that the viscosity of the fluid medium does not affect the wavelength; the activity of the cilia does. These numerical results are supported by a dimensional analysis, which shows that the result of wavelength invariance is robust against the model taken for sustained beating and the structure of hydrodynamic coupling. Interestingly, the enhancement of cilia activity increases the wavelength and decreases the beating frequency, keeping the wave velocity almost unchanged. These results might have significance in understanding paramecium locomotion and mucociliary clearance diseases.

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.

Forceful patterning: theoretical principles of mechanochemical pattern formation

Biological pattern formation is essential for generating and maintaining spatial structures from the scale of a single cell to tissues and even collections of organisms. Besides biochemical interactions, there is an important role for mechanical and geometrical features in the generation of patterns. We review the theoretical principles underlying different types of mechanochemical pattern formation across spatial scales and levels of biological organization.

Lateral contact yields longitudinal cohesion in active undulatory systems

Many animals and robots move using undulatory motion of their bodies. When the bodies are in close proximity undulatory motion can lead to novel collective behavior such as gait synchronization, spatial reconfiguration, and clustering. Here we study the role of contact interactions between model undulatory swimmers: three-link robots in experiment and multilink swimmers in simulation. The undulatory gait of each swimmer is generated through a time-dependent sinusoidal-like waveform which has a fixed phase offset, ϕ. By varying the phase relationship between neighboring swimmers we seek to study how contact forces and planar configurations are governed by the phase difference between neighboring swimmers. We find that undulatory actuation in close proximity drives neighboring swimmers into planar equilibrium configurations that depend on the actuation phase difference. We propose a model for stable planar configurations of nearest-neighbor undulatory swimmers which we call the gait compatibility condition, which is the set of planar and phase configurations in which no collisions occur. Robotic experiments with two, three, and four swimmers exhibit good agreement with the compatibility model. To study the contact forces and the time-averaged equilibrium between undulatory systems we perform simulations. To probe the interaction potential between undulatory swimmers we apply a small force to each swimmer longitudinally to separate them from the compatible configuration and we measure their steady-state displacement. These studies reveal that undulatory swimmers in close proximity exhibit attractive longitudinal interaction forces that drive the swimmers from incompatible to compatible configurations. This system of undulatory swimmers provides new insight into active-matter systems which move through body undulation. In addition to the importance of velocity and orientation coherence in active-matter swarms, we demonstrate that undulatory phase coherence is also important for generating stable, cohesive group configurations.

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.

Hydrodynamic synchronization in strong confinement

Cellular appendages conferring motility, such as flagella and cilia, are known to synchronise their periodic beats. The origin of synchronization is a combination of long-range hydrodynamic interactions with physical mechanisms allowing the phases of these biological oscillators to evolve. Two of such mechanisms have been identified by previous work, the elastic compliance of the periodic orbit or oscillations driven by phase-dependent biological forcing, both of which can lead generically to stable phase locking. In order to help uncover the physical mechanism for hydrodynamic synchronization most essential overall in biology, we theoretically investigate in this paper the effect of strong confinement on the effectiveness of hydrodynamic synchronization. Following past work, we use minimal models of cilia where appendages are modeled as rigid spheres forced to move along circular trajectories near a rigid surface. Strong confinement is modeled by adding a second nearby surface, parallel to the first one, where the distance between the surfaces is much smaller than the typical distance between the cilia, which results in a qualitative change in the nature of hydrodynamic interactions. We calculate separately the impact of hydrodynamic confinement on the synchronization dynamics of the elastic compliance and the force modulation mechanisms and compare our results to the usual case with a single surface. Applying our results to the biologically relevant situation of nodal cilia, we show that force modulation is a mechanism that leads to phase-locked states under strong confinement that are very similar to those without confinement as a difference with the elastic compliance mechanism. Our results point therefore to the robustness of force modulation for synchronization, an important feature for biological dynamics that therefore suggests it could be the most essential physical mechanism overall in arrays of nodal cilia. We further examine the distinct biologically relevant situation of primary cilia and show in that case that the difference in robustness of the mechanisms is not as pronounced but still favors the force modulation.

Minimal model of the hydrodynamical coupling of flagella on a spherical body with application to Volvox

Flagella are hairlike appendages attached to microorganisms that allow the organisms to traverse their fluid environment. The algae Volvox are spherical swimmers with thousands of individual flagella on their surface, and their coordination is not fully understood. In this work, a previously developed minimal model of flagella synchronization is extended to the outer surface of a sphere submerged in a fluid. Each beating flagellum tip is modeled as a small sphere, elastically bound to a circular orbit just above the spherical surface and a regularized image system for Stokes flow outside of a sphere is used to enforce the no-slip condition. Biologically relevant distributions of rotors results in a rapidly developing and robust symplectic metachronal wave traveling from the anterior to the posterior of the spherical Volvox body.

Hydrodynamic synchronization and collective dynamics of colloidal particles driven along a circular path

We study theoretically the collective dynamics of particles driven by an optical vortex along a circular path. Phase equations of N particles are derived by taking into account both hydrodynamic and repulsive interactions between them. For N=2, the particles attract with each other and synchronize, forming a doublet that moves faster than a singlet. For N=3 and 5, we find periodic rearrangement of doublets and a singlet. For N=4 and 6, the system exhibits either a periodic oscillating state or a stable synchronized state depending on the initial conditions. These results reproduce main features of previous experimental findings. We quantitatively discuss the mechanisms governing the nontrivial collective dynamics.

The smallest chimera: Periodicity and chaos in a pair of coupled chemical oscillators

Symmetrically coupled identical oscillators were once believed to support only totally synchronous or totally asynchronous states. More recently, chimera states, in which a subset of oscillators behaves coherently while the other subset exhibits disorder, have been found in large arrays of oscillators, coupled either locally or globally. We demonstrate for the first time the existence of a chimera state with only two diffusively coupled identical oscillators, one behaving nearly periodically (coherently) and the other chaotically (incoherently). We attribute this behavior to a "master-slave" interaction, which arises via a symmetry-breaking canard explosion.

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

Despite evidence for a hydrodynamic origin of flagellar synchronization between different eukaryotic cells, recent experiments have shown that in single multi-flagellated organisms, coordination hinges instead on direct basal body connections. The mechanism by which these connections lead to coordination, however, is currently not understood. Here, we focus on the model biflagellate Chlamydomonas reinhardtii, and propose a minimal model for the synchronization of its two flagella as a result of both hydrodynamic and direct mechanical coupling. A spectrum of different types of coordination can be selected, depending on small changes in the stiffness of intracellular couplings. These include prolonged in-phase and anti-phase synchronization, as well as a range of multi-stable states induced by spontaneous symmetry breaking of the system. Linking synchrony to intracellular stiffness could lead to the use of flagellar dynamics as a probe for the mechanical state of the cell.

Role of spatial heterogeneity in the collective dynamics of cilia beating in a minimal one-dimensional model

Cilia are elastic hairlike protuberances of the cell membrane found in various unicellular organisms and in several tissues of most living organisms. In some tissues such as the airway tissues of the lung, the coordinated beating of cilia induces a fluid flow of crucial importance as it allows the continuous cleaning of our bronchia, known as mucociliary clearance. While most of the models addressing the question of collective dynamics and metachronal wave consider homogeneous carpets of cilia, experimental observations rather show that cilia clusters are heterogeneously distributed over the tissue surface. The purpose of this paper is to investigate the role of spatial heterogeneity on the coherent beating of cilia using a very simple one-dimensional model for cilia known as the rower model. We systematically study systems consisting of a few rowers to hundreds of rowers and we investigate the conditions for the emergence of collective beating. When considering a small number of rowers, a phase drift occurs, hence, a bifurcation in beating frequency is observed as the distance between rower clusters is changed. In the case of many rowers, a distribution of frequencies is observed. We found in particular the pattern of the patchy structure that shows the best robustness in collective beating behavior, as the density of cilia is varied over a wide range.

The chimera state in colloidal phase oscillators with hydrodynamic interaction

The chimera state is the incongruous situation where coherent and incoherent populations coexist in sets of identical oscillators. Using driven non-linear oscillators interacting purely through hydrodynamic forces at low Reynolds number, previously studied as a simple model of motile cilia supporting waves, we find concurrent incoherent and synchronised subsets in small arrays. The chimeras seen in simulation display a "breathing" aspect, reminiscent of uniformly interacting phase oscillators. In contrast to other systems where chimera has been observed, this system has a well-defined interaction metric, and we know that the emergent dynamics inherit the symmetry of the underlying Oseen tensor eigenmodes. The chimera state can thus be connected to a superposition of eigenstates, whilst considering the mean interaction strength within and across subsystems allows us to make a connection to more generic (and abstract) chimeras in populations of Kuramoto phase oscillators. From this work, we expect the chimera state to emerge in experimental observations of oscillators coupled through hydrodynamic forces.

Assessing the Collective Dynamics of Motile Cilia in Cultures of Human Airway Cells by Multiscale DDM

The technique of differential dynamic microscopy is extended here, showing that it can provide a powerful and objective method of video analysis for optical microscopy videos of in vitro samples of live human bronchial epithelial ciliated cells. These cells are multiciliated, with motile cilia that play key physiological roles. It is shown that the ciliary beat frequency can be recovered to match conventional analysis, but in a fully automated fashion. Furthermore, it is shown that the properties of spatial and temporal coherence of cilia beat can be recovered and distinguished, and that if a collective traveling wave (the metachronal wave) is present, this has a distinct signature and its wavelength and direction can be measured.

Generalized synchronization between chimera states

Networks of coupled oscillators in chimera states are characterized by an intriguing interplay of synchronous and asynchronous motion. While chimera states were initially discovered in mathematical model systems, there is growing experimental and conceptual evidence that they manifest themselves also in natural and man-made networks. In real-world systems, however, synchronization and desynchronization are not only important within individual networks but also across different interacting networks. It is therefore essential to investigate if chimera states can be synchronized across networks. To address this open problem, we use the classical setting of ring networks of non-locally coupled identical phase oscillators. We apply diffusive drive-response couplings between pairs of such networks that individually show chimera states when there is no coupling between them. The drive and response networks are either identical or they differ by a variable mismatch in their phase lag parameters. In both cases, already for weak couplings, the coherent domain of the response network aligns its position to the one of the driver networks. For identical networks, a sufficiently strong coupling leads to identical synchronization between the drive and response. For non-identical networks, we use the auxiliary system approach to demonstrate that generalized synchronization is established instead. In this case, the response network continues to show a chimera dynamics which however remains distinct from the one of the driver. Hence, segregated synchronized and desynchronized domains in individual networks congregate in generalized synchronization across networks.

Symplectic and antiplectic waves in an array of beating cilia attached to a closed body

By taking into account the hydrodynamic interactions in a one dimensional array of model cilia attached to a no-slip cylinderical surface, we investigate their synchronized motion. We show how the emergence of metachronal waves depends on the initial state of the system and investigate the conditions under which the formation of symplectic and antiplectic waves are possible.