Metachronal Wave Research Articles

Mathematical modeling of bio‐magnetic fluid bounded within ciliated walls of wavy channel

AbstractPeristaltic transport of couple stress fluid with heat transfer is investigated through flexible walls of the channel furnished with hair‐like structures. Locomotion of biological fluid is the result of the simultaneous propagation of metachronal waves (MCWs) and peristaltic waves. MCW emerges due to multi‐movement of cilia, while the elastic‐walls of the channel are responsible for the peristaltic wave. The two‐dimensional rheological flow experiences the resistive behavior of applied magnetic fields, besides, the role of viscous dissipation and entropy generation. A closed‐form solution is achieved for the nonlinear coupled differential equation, which is vetted through graphs. Finally, the theoretical analysis introduces the manufacture of a ciliate bronchial tube that can efficiently; remove mucus out of the lungs and bloodstream.

Synchronized oscillations, traveling waves, and jammed clusters induced by steric interactions in active filament arrays.

Autonomous active, elastic filaments that interact with each other to achieve cooperation and synchrony underlie many critical functions in biology. The mechanisms underlying this collective response and the essential ingredients for stable synchronization remain a mystery. Inspired by how these biological entities integrate elasticity with molecular motor activity to generate sustained oscillations, a number of synthetic active filament systems have been developed that mimic oscillations of these biological active filaments. Here, we describe the collective dynamics and stable spatiotemporal patterns that emerge in such biomimetic multi-filament arrays, under conditions where steric interactions may impact or dominate the collective dynamics. To focus on the role of steric interactions, we study the system using Brownian dynamics, without considering long-ranged hydrodynamic interactions. The simulations treat each filament as a connected chain of self-propelling colloids. We demonstrate that short-range steric inter-filament interactions and filament roughness are sufficient - even in the absence of inter-filament hydrodynamic interactions - to generate a rich variety of collective spatiotemporal oscillatory, traveling and static patterns. We first analyze the collective dynamics of two- and three-filament clusters and identify parameter ranges in which steric interactions lead to synchronized oscillations and strongly occluded states. Generalizing these results to large one-dimensional arrays, we find rich emergent behaviors, including traveling metachronal waves, and modulated wavetrains that are controlled by the interplay between the array geometry, filament activity, and filament elasticity. Interestingly, the existence of metachronal waves is non-monotonic with respect to the inter-filament spacing. We also find that the degree of filament roughness significantly affects the dynamics - specifically, filament roughness generates a locking-mechanism that transforms traveling wave patterns into statically stuck and jammed configurations. Taken together, simulations suggest that short-ranged steric inter-filament interactions could combine with complementary hydrodynamic interactions to control the development and regulation of oscillatory collective patterns. Furthermore, roughness and steric interactions may be critical to the development of jammed spatially periodic states; a spatiotemporal feature not observed in purely hydrodynamically interacting systems.

Biomimetic Wave Propagation in Magnetic Soft Actuator

In this paper, a simple but effective method is proposed to generate a bio-mimic wave propagating motion in a magnetic soft actuator. Two examples, a metachronal wave in artificial cilia and a crawling motion of an artificial caterpillar, are shown as demonstrations. These are kinds of wave propagating motions which are popular in natural systems and are also primitive systems of natural creatures. In the proposed system, an elastic material and a magnetic powder were used to fabricate a flexible magnetic structure. There are 2 essential features in the system; one is periodic arrangement of magnetic anisotropy, and the other is application of a rotating magnetic field. Using this method of wave generation, we realized drastic change in flow making behavior of artificial cilia and a crawling motion very similar to living caterpillars. The proposed system would be useful to enhance the field of the bio-mimic soft robotics.

Multi-scale alignment of respiratory cilia and its relation to mucociliary function

The tracheobronchial tree is lined by a mucociliary epithelium containing millions of multiciliated cells. Their integrated oscillatory activity continuously propels an overlying pollution-protecting mucus layer in cranial direction, leading to mucociliary clearance – the primary defence mechanism of the airways. Mucociliary transport is commonly thought to co-emerge with the collective ciliary motion pattern under appropriate geometrical and rheological conditions. Proper ciliary alignment is therefore considered essential to establish mucociliary clearance in the respiratory system. Here, we used volume electron microscopy in combination with high-speed reflection contrast microscopy in order to examine ciliary orientation and its spatial organization, as well as to measure the propagation direction of metachronal waves and the direction of mucociliary transport on bovine tracheal epithelia with reference to the tracheal long axis (TLA). Ciliary orientation is measured in terms of the basal body orientation (BBO) and the axonemal orientation (AO), which are commonly considered to coincide, both equivalently indicating the effective stroke as well as the mucociliary transport direction. Our results, however, reveal that only the AO is in line with the mucociliary transport, which was found to run along a left-handed helical trajectory, whereas the BBO was found to be aligned with the TLA. Furthermore, we show that even if ciliary orientation remains consistent between adjacent cells, ciliary orientation exhibits a gradual shift within individual cells. Together with the symplectic beating geometry, this intracellular orientational pattern could provide for the propulsion of highly viscous mucus and likely constitutes a compromise between efficiency and robustness.

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.

Free Interfaces at the Tips of the Cilia in the One-Dimensional Periciliary Layer

Cilia on the surface of ciliated cells in the respiratory system are organelles that beat forward and backward to generate metachronal waves to propel mucus out of lungs. The layer that contains the cilia, coating the interior epithelial surface of the bronchi and bronchiolesis, is called the periciliary layer (PCL). With fluid nourishment, cilia can move efficiently. The fluid in this region is named the PCL fluid and is considered to be an incompressible, viscous, Newtonian fluid. We propose there to be a free boundary at the tips of cilia underlining a gas phase while the cilia are moving forward. The Brinkman equation on a macroscopic scale, in which bundles of cilia are considered rather than individuals, with the Stefan condition was used in the PCL to determine the velocity of the PCL fluid and the height/shape of the free boundary. Regarding the numerical methods, the boundary immobilization technique was applied to immobilize the moving boundaries using coordinate transformation (working with a fixed domain). A finite element method was employed to discretize the mathematical model and a finite difference approach was applied to the Stefan problem to determine the free interface. In this study, an effective stroke is assumed to start when the cilia make a 140∘ angle to the horizontal plane and the velocitiesof cilia increase until the cilia are perpendicular to the horizontal plane. Then, the velocities of the cilia decrease until the cilia make a 40∘ angle with the horizontal plane. From the numerical results, we can see that although the velocities of the cilia increase and then decrease, the free interface at the tips of the cilia continues increasing for the full forward phase. The numerical results are verified and compared with an exact solution and experimental data from the literature. Regarding the fixed boundary, the numerical results converge to the exact solution. Regarding the free interface, the numerical solutions were compared with the average height of the PCL in non-cystic fibrosis (CF) human tissues and were in excellent agreement. This research also proposes possible values of parameters in the mathematical model in order to determine the free interface. Applications of these fluid flows include animal hair, fibers and filter pads, and rice fields.

The role of suction thrust in the metachronal paddles of swimming invertebrates

An abundance of swimming animals have converged upon a common swimming strategy using multiple propulsors coordinated as metachronal waves. The shared kinematics suggest that even morphologically and systematically diverse animals use similar fluid dynamic relationships to generate swimming thrust. We quantified the kinematics and hydrodynamics of a diverse group of small swimming animals who use multiple propulsors, e.g. limbs or ctenes, which move with antiplectic metachronal waves to generate thrust. Here we show that even at these relatively small scales the bending movements of limbs and ctenes conform to the patterns observed for much larger swimming animals. We show that, like other swimming animals, the propulsors of these metachronal swimmers rely on generating negative pressure along their surfaces to generate forward thrust (i.e., suction thrust). Relying on negative pressure, as opposed to high pushing pressure, facilitates metachronal waves and enables these swimmers to exploit readily produced hydrodynamic structures. Understanding the role of negative pressure fields in metachronal swimmers may provide clues about the hydrodynamic traits shared by swimming and flying animals.

Swimming kinematics and hydrodynamics of barnacle larvae throughout development.

Changes in size strongly influence organisms' ecological performances. For aquatic organisms, they can transition from viscosity- to inertia-dominated fluid regimes as they grow. Such transitions are often associated with changes in morphology, swimming speed and kinematics. Barnacles do not fit into this norm as they have two morphologically distinct planktonic larval phases that swim differently but are of comparable sizes and operate in the same fluid regime (Reynolds number 100-101). We quantified the hydrodynamics of the rocky intertidal stalked barnacle Capitulum mitella from the nauplius II to cyprid stage and examined how kinematics and size increases affect its swimming performance. Cyprids beat their appendages in a metachronal wave to swim faster, more smoothly, and with less backwards slip per beat cycle than did all naupliar stages. Micro-particle image velocimetry showed that cyprids generated trailing viscous vortex rings that pushed water backwards for propulsion, contrary to the nauplii's forward suction current for particle capture. Our observations highlight that zooplankton swimming performance can shift via morphological and kinematic modifications without a significant size increase. The divergence in ecological functions through ontogeny in barnacles and the removal of feeding requirement likely contributed to the evolution of the specialized, taxonomically unique cyprid phase.

Numerical simulation of oscillating plates at the visco-inertial regime for bio-inspired pumping and mixing applications

In this numerical study, flow driven by oscillating plates in a channel is investigated at the intermediate Reynolds regime by means of the arbitrary Lagrangian–Eulerian method. The effects of temporal asymmetry, Reynolds number, channel height, phase differences between adjacent plates, and orientation angle on pumping performance, which are unclear under temporally asymmetric linear plate kinematics, are studied. By examining efficiency measures, new insight is gained from energy, mixing, and combined flowrate/energy standpoints. Furthermore, the mixing performance of oscillating plates, which is to a large extent unknown at the visco-inertial regime, is scrutinized. By studying a finite number of plates, end effects that are not apparent in the more common studies on infinite plate/cilia arrays are accounted for. For a single plate, results show an almost threefold increase in the average flow rate between Reynolds numbers of 5 and 40 but a fall thereafter caused by the restriction of the flow by a region of circulation. The average flow rate and energy conversion efficiency increase by 20% when the orientation angle is increased from −4° to 12°, and a point of maximum flow generation is found at a plate length to channel height ratio of 0.7. For an array of five plates, a phase difference of 90° between adjacent plates (antiplectic metachronal wave) generates the largest flow, while a phase difference of 270° (symplectic metachronal wave) brings about the best mixing efficacy. This suggests that the optimal phase difference depends on the intended use of the device.

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.

Simulating squirmers with volumetric solvers

Squirmers are models of a class of microswimmers that self-propel in fluids without significant deformation of their body shape, such as ciliated organisms and phoretic particles. Available techniques for their simulation are based on the boundary element method and do not contemplate nonlinearities such as those arising from the inertia of the fluid or non-Newtonian rheology. This article describes a methodology to simulate squirmers that overcomes these limitations by using volumetric numerical methods, such as finite elements or finite volumes. It deals with interface conditions at the surface of the squirmer that generalize those in the published literature, which are generally restricted to the imposition of slip velocities. The actual procedures to be performed on a fluid solver to implement the proposed methodology are provided, including the treatment of metachronal surface waves. Among the several numerical examples, a two-dimensional simulation is shown of the hydrodynamic interaction of two individuals of Opalina ranarum.

Rapid Ex-Vivo Ciliogenesis and Dose-Dependent Effect of Notch Inhibition on Ciliogenesis of Respiratory Epithelia.

Background: Cilia are actin based cellular protrusions conserved from algae to complex multicellular organisms like Homo sapiens. Respiratory motile cilia line epithelial cells of the tracheobronchial tree, beat in a synchronous, metachronal wave, moving inhaled pollutants and pathogens cephalad. Their role in both congenital disorders like primary ciliary dyskinesia (PCD) to acquired disorders like chronic obstructive pulmonary disease (COPD) continues to evolve. In this current body of work we outline a protocol optimized to reciliate human nasal epithelial cells and mouse tracheal cells in vitro. Using this protocol, we knocked down known cilia genes, as well as use a small molecule inhibitor of Notch, N-[N-(3,5-Difluorophenacetyl)-L-alanyl]-S-phenylglycine t-butyl Ester (DAPT), to assess the effect of these on ciliogenesis in order to show the validity of our protocol. Methods: Tracheas were harvested from wild-type, adult C57B6 mice, pronase digested and sloughed off epithelial cells grown to confluence in stationary culture on rat-tail collagen coated wells. Upon reaching confluence, collagen was digested and cells placed suspension culture protocol to reciliate the cells. Using this suspension culture protocol, we employed siRNA gene knockdown to assay gene functions required for airway ciliogenesis. Knock down of Dynein axonemal heavy chain 5 (Dnah5), a ciliary structural protein, was confirmed using immunostaining. Mouse tracheal cells were treated in suspension with varying doses of DAPT, an inhibitor of Notch, with the purpose of evaluating its effect and dose response on ciliogenesis. The optimum dose was then used on reciliating human nasal epithelial cells. Results: siRNA knockdown of Foxj1 prevented ciliation, consistent with its role as a master regulator of motile cilia. Knockdown of Dnai1 and Dnah5 resulted in immotile cilia, and Cand1 knockdown, a centrosome protein known to regulate centrosome amplification, inhibited airway ciliogenesis. Dnah5 knockdown was confirmed with significantly decreased immunostaining of cilia for this protein. Inhibiting Notch signaling by inhibiting gamma secretase with DAPT enhanced the percentage of ciliation, and resulted in longer cilia that beat with higher frequency in both mouse and human airway epithelia. Conclusions: Modifying existing reciliation protocols to suit both human nasal epithelial and mouse tracheal tissue, we have shown that knockdown of known cilia-related genes have the expected effects. Additionally, we have demonstrated the optimal dosage for significantly improving reciliation of airway epithelia using DAPT. Given that cilia length and function are significantly compromised in COPD, these findings open up interesting avenues for further exploration.

Micro rheology of Jeffrey nanofluid through cilia beating subject to the surrounding temperature

The objective of this study is to discuss the micro rheology of mucus (Jeffrey nanofluid) which is a complex biological fluid that protects lungs from pollutants, bacteria, and allergens that can be inhaled during the breathing process. To see the insight of pollutants and effect of surrounding temperature in the mucus, momentum, energy, and concentration equations are modeled with the help of metachronal wave formed by cilia beating. The governing system of equations are modeled in the wave and fixed frame and simplified by the lubrication approach. The velocity profile for recovery and effective stroke is compared and it is analyzed that effective stroke possess high magnitude of velocity when compared with the recovery stroke. The flow of mucus with pollutants and surrounding temperature is calculated with homotopy perturbation method and software “MATHEMATICA.” The results within the given domain are convergent under the different parameters appearing into the system of equations. The physical interpretation of involved parameters is explained through graphs.

Magnetically powered metachronal waves induce locomotion in self-assemblies

When tiny soft ferromagnetic particles are placed along a liquid interface and exposed to a vertical magnetic field, the balance between capillary attraction and magnetic repulsion leads to self-organization into well-defined patterns. Here, we demonstrate experimentally that precessing magnetic fields induce metachronal waves on the periphery of these assemblies, similar to the ones observed in ciliates and some arthropods. The outermost layer of particles behaves like an array of cilia or legs whose sequential movement causes a net and controllable locomotion. This bioinspired many-particle swimming strategy is effective even at low Reynolds number, using only spatially uniform fields to generate the waves.

Multi-scale spatial heterogeneity enhances particle clearance in airway ciliary arrays.

Mucus clearance constitutes the primary defence of the respiratory system against viruses, bacteria and environmental insults [1]. This transport across the entire airway emerges from the integrated activity of thousands of multiciliated cells, each containing hundreds of cilia, which together must coordinate their spatial arrangement, alignment and motility [2, 3]. The mechanisms of fluid transport have been studied extensively at the level of an individual cilium [4, 5], collectively moving metachronal waves [6-10], and more generally the hydrodynamics of active matter [11, 12]. However, the connection between local cilia architecture and the topology of the flows they generate remains largely unexplored. Here, we image the mouse airway from the sub-cellular (nm) to the organ scales (mm), characterising quantitatively its ciliary arrangement and the generated flows. Locally we measure heterogeneity in both cilia organisation and flow structure, but across the trachea fluid transport is coherent. To examine this result, a hydrodynamic model was developed for a systematic exploration of different tissue architectures. Surprisingly, we find that disorder enhances particle clearance, whether it originates from fluctuations, heterogeneity in multiciliated cell arrangement or ciliary misalignment. This resembles elements of 'stochastic resonance' [13-15], in the sense that noise can improve the function of the system. Taken together, our results shed light on how the microstructure of an active carpet [16, 17] determines its emergent dynamics. Furthermore, this work is also directly applicable to human airway pathologies [1], which are the third leading cause of deaths worldwide [18].

Entropy analysis in ciliated inclined channel filled with hydromagnetic Williamson fluid flow induced by metachronal waves

This study reveals the entropy analysis of hydromagnetic pumping flow of Williamson fluid through a 2-D symmetric channel carrying cilia. Propulsive metachronal waves are mobilized by whipping and beating of uniformly distributed cilia which follow elliptic trajectory movements in the parallel direction of flow. The flow is resisted by a uniform transverse magnetic field. The entire study is carried out in wave frame of reference. After implying lubrication approximations, the governing equations of the present flow problem are solved by perturbation method. Effects of physical parameters of interest on various flow quantities, the total entropy generation number and the Bejan number are plotted and discussed. It is observed that fluid velocity and temperature is enhanced in the core channel region for small values of Hartmann number and cilia length. It is also noticed that the entropy generation and the Bejan number are decreasing function of magnetic field. Near the channel center, irreversibility due to fluid friction is dominant but at the channel wall heat transfer irreversibility effects are observed to be substantial. The confined bolus reduces in size for small values of cilia length parameter and large values of Hartmann number.

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.

Simulating cilia-driven mixing and transport in complex geometries

An efficient hybrid numerical method for ciliary flow is proposed, combining the boundary integral method and the method of regularized stokeslets. It is capable of simulating ciliary flows and particle transport in complex 2D confined geometries. The effects of metachronal waves and geometries on ciliary transport as well as fluid mixing are analyzed. Size effect of submerged rigid particles inside the confined geometries is also studied.

Respiratory Cilia as a Therapeutic Target of Phosphodiesterase Inhibitors

Mucociliary clearance is an essential airway defense mechanism dependent predominantly on the proper ciliary function and mucus rheology. The crucial role of cilia is evident in `a variety of respiratory diseases, as the ciliary dysfunction is associated with a progressive decline in lung function over time. The activity of cilia is under supervision of multiple physiological regulators, including second messengers. Their role is to enable a movement in coordinated metachronal waves at certain beat frequency. Ciliary function can be modulated by various stimuli, including agents from the group of beta2 agonists, cholinergic drugs, and adenosine triphosphate (ATP). They trigger cilia to move faster in response to elevated cytoplasmic Ca2+ originated from intracellular sources or replenished from extracellular space. Well-known cilia-stimulatory effect of Ca2+ ions can be abolished or even reversed by modulating the phosphodiesterase (PDE)-mediated breakdown of cyclic adenosine monophosphate (cAMP) since the overall change in ciliary beating has been dependent on the balance between Ca2+ ions and cAMP. Moreover, in chronic respiratory diseases, high ATP levels may contribute to cAMP hydrolysis and thus to a decrease in the ciliary beat frequency (CBF). The role of PDE inhibitors in airway cilia-driven transport may help in prevention of progressive loss of pulmonary function often observed despite current therapy. Furthermore, administration of selective PDE inhibitors by inhalation lowers the risk of their systemic effects. Based on this review we may conclude that selective (PDE1, PDE4) or dual PDE inhibitors (PDE3/4) increase the intracellular level of cyclic nucleotides in airway epithelial cells and thus may be an important target in the development of new inhaled mucokinetic agents. Further research is required to provide evidence of their effectiveness and feasibility regarding their cilia-modulating properties.

Axisymmetric mixed convective propulsion of a non-Newtonian fluid through a ciliated tubule

Thousands of chemical reactions occur in the human body when certain biological fluids, such as blood, semen, mucus, and synovial joint materials, move in various organs. These reactions play a vital role in regulating the life sustaining metabolic processes in the body. Analysis of thermal effects on these chemical reactions is relatively a new area in modern clinical medications. The present study investigates a simulation of the combined response due to heat and mass transport mechanisms taking place in the human body during the flow of physiological fluids. In particular, we focus our attention on the human male reproductive system, wherein the semen transports through the ductus efferentes due to metachronal waves of cilia. The constitutive relations of the robust Jeffrey viscoelastic fluid are used to model the human semen. The mathematical model of the present problem constitutes the axisymmetric flow of a Jeffrey fluid inside a vertical tubule under the influence of mixed convective heat and mass transfers. The inner side of the tubule is covered with ciliated structures. The influence of thermal behaviors of various metabolic processes in the human body due to an external heat source or sink is also taken into account. The mathematical formulation consists of using the approach of lubrication theory approximation; the nonlinear momentum, energy, and concentration equations are simplified to get analytical solutions. Explicit expressions for temperature, concentration, velocity, pressure gradient, and volume flow rate of the proposed bodily fluid (i.e., human semen) are formulated. The expression for the volume flow rate is used to estimate the volume flux of the semen under the influence of various parameters. A comparison between the theoretical and experimentally obtained values of the flow rate of the human semen is also made. It is noted that our calculated values are very close to the estimated values. Industrial applications of the present results are obvious in the fabrication of artificial cilia pumping systems for microfluidic flow systems.