Discrete Cilia Modelling with Singularity Distributions: Application to the Embryonic Node and the Airway Surface Liquid

We discuss in detail techniques for modelling flows due to finite and infinite arrays of beating cilia. An efficient technique, based on concepts from previous ‘singularity models’ is described, that is accurate in both near and far-fields. Cilia are modelled as curved slender ellipsoidal bodies by distributing Stokeslet and potential source dipole singularities along their centrelines, leading to an integral equation that can be solved using a simple and efficient discretisation. The computed velocity on the cilium surface is found to compare favourably with the boundary condition. We then present results for two topics of current interest in biology. 1) We present the first theoretical results showing the mechanism by which rotating embryonic nodal cilia produce a leftward flow by a ‘posterior tilt,’ and track particle motion in an array of three simulated nodal cilia. We find that, contrary to recent suggestions, there is no continuous layer of negative fluid transport close to the ciliated boundary. The mean leftward particle transport is found to be just over 1 μm/s, within experimentally measured ranges. We also discuss the accuracy of models that represent the action of cilia by steady rotlet arrays, in particular, confirming the importance of image systems in the boundary in establishing the far-field fluid transport. Future modelling may lead to understanding of the mechanisms by which morphogen gradients or mechanosensing cilia convert a directional flow to asymmetric gene expression. 2) We develop a more complex and detailed model of flow patterns in the periciliary layer of the airway surface liquid. Our results confirm that shear flow of the mucous layer drives a significant volume of periciliary liquid in the direction of mucus transport even during the recovery stroke of the cilia. Finally, we discuss the advantages and disadvantages of the singularity technique and outline future theoretical and experimental developments required to apply this technique to various other biological problems, particularly in the reproductive system.     

Ciliary ultrastructure of protobranchs (Mollusca, Bivalvia)

Abstract. The external epithelial cilia and other surface structures of the nuculoid protobranchs Nuculana pernula and Nucula nitidosa were studied. The gill lamellae and labial palps are partly covered with very long cilia. These have a modified slender distal portion, an ordinary metazoan-type basal body, a basal foot. and a single, long cross-striated rootlet. In cilia on the gills of N. nitidosa , the basal foot is thick and attaches to the next basal body directly behind. Unciliated surface areas on the gills, labial palps, and foot are covered with a dense brushborder of microvilli. We observed no specific homologies between the cilia of the protobranchs studied and the epidermal cilia of the enigmatic Xenoturbella bocki , hence the recent hypothesis of a close connection of the latter to the protobranch bivalves is questioned.     

Rhinoglena frontalis (Rotifera,Monogononta): a scanning electron microscopic study

Females and males of Rhinoglena frontalis (Monogononta, Epiphanidae) are observed by SEM and their external morphologies are compared. The two sexes differ in size and shape of the body. The female body is fusiform with a short, conical foot, while the male body is more slender and has a rather long foot. The rotatory apparatus (or corona) of both sexes is similar with only minor differences and consists of rows and tufts of cilia arranged around the mouth opening. The corona is made of two paired lobes lateral to the mouth and of a third prominent dorsal lobe, usually called proboscis. The three lobes are lined externally by dense rows of cilia, which constitute the cingulum, used for swimming. The central surface of the proboscis is covered with numerous longitudinal rows of cilia bent towards the mouth. The lateral lobes show, on their central surfaces, two concentric arcs of cirri (made of tightly packed cilia) bent towards the mouth. The similar organization of the rotatory apparatus of both sexes is related to the fact that the male, in this species, is able to feed and has a developed mastax and digestive system. The trophi of both sexes are illustrated and compared.     

Computational hydrodynamics of animal swimming: boundary element method and three-dimensional vortex wake structure.

The slender body theory, lifting surface theories, and more recently panel methods and Navier-Stokes solvers have been used to study the hydrodynamics of fish swimming. This paper presents progress on swimming hydrodynamics using a boundary integral equation method (or boundary element method) based on potential flow model. The unsteady three-dimensional BEM code 3DynaFS that we developed and used is able to model realistic body geometries, arbitrary movements, and resulting wake evolution. Pressure distribution over the body surface, vorticity in the wake, and the velocity field around the body can be computed. The structure and dynamic behavior of the vortex wakes generated by the swimming body are responsible for the underlying fluid dynamic mechanisms to realize the high-efficiency propulsion and high-agility maneuvering. Three-dimensional vortex wake structures are not well known, although two-dimensional structures termed 'reverse Karman Vortex Street' have been observed and studied. In this paper, simulations about a swimming saithe (Pollachius virens) using our BEM code have demonstrated that undulatory swimming reduces three-dimensional effects due to substantially weakened tail tip vortex, resulting in a reverse Karman Vortex Street as the major flow pattern in the three-dimensional wake of an undulating swimming fish.     

A numerical study of muco-ciliary transport under the condition of diseased cilia

Structural and functional disorders of pulmonary cilia may result from genetic disorders and acquired insults. A two-dimensional numerical model based on the immersed boundary method coupled with the projection method is used to study the flow physics of muco-ciliary transport of the human respiratory tract under various abnormalities of cilia. The effects of the cilia beat pattern (CBP), ciliary length, immotile cilia, beating amplitude and uncoordinated beating of cilia are investigated. As expected, the mucus velocity decreases as the beating amplitude reduces. The windscreen wiper motion and rigid planar motion, which are two abnormal CBPs owing to genetic disorders, greatly reduce or almost stop the mucus transport. If the ciliary length varies from its standard length, the mucus velocity would decrease. The mucus velocity decreases rather linearly if the number of uniformly distributed immotile cilia increases. The numerical results show that the mucus velocity would be further reduced marginally when the uniformly distributed immotile cilia are rearranged as a cluster of immotile cilia. Furthermore, if half of the cilia are immotile and uniformly distributed and motile cilia beat at reduced amplitude, the incoordination between the active motile cilia would not significantly affect the mucus velocity.     

Fluid transport in a thick layer above an active ciliated surface.

The discrete cilia approach is employed to describe the flow field above an active ciliated surface in a layer of finite depth. The infinite-size cilia surface model predicts a uniform flow above the cilia surface. Modifications as a result of a finitely extending cilia surface inside a finite-size dish predict a backward parabolic profile above an active cilia surface. Experiments are described which demonstrate a good fit between theoretical predictions and observations. The results provide a sound physical basis for the proper interpretation of fluid-flow observations close to an active ciliary field.     

The role of the periciliary fluid in mucociliary flows: flow velocity profiles in frog palate mucus

Observations of flow velocity profiles over frog mucociliated palate are used to estimate viscosity, shear rate and shear stress in the periciliary flow field. The ability of cilia to generate significant shear stress at long distances and their utility as rhoeometers are examined. It is proposed that the depth of significant ciliary shear penetration into the periciliary fluid is sufficient to move mucus masses well beyond the ciliary tips, obviating the need for tip penetration where anchoring phenomena are sufficiently reduced.     

Primary cilia regulate mTORC1 activity and cell size through Lkb1

The mTOR pathway is the central regulator of cell size. External signals from growth factors and nutrients converge on the mTORC1 multi-protein complex to modulate downstream targets, but how the different inputs are integrated and translated into specific cellular responses is incompletely understood. Deregulation of the mTOR pathway occurs in polycystic kidney disease (PKD), where cilia (filiform sensory organelles) fail to sense urine flow because of inherited mutations in ciliary proteins. We therefore investigated if cilia have a role in mTOR regulation. Here, we show that ablation of cilia in transgenic mice results in enlarged cells when compared with control animals. In vitro analysis demonstrated that bending of the cilia by flow is required for mTOR downregulation and cell-size control. Surprisingly, regulation of cell size by cilia is independent of flow-induced calcium transients, or Akt. However, the tumour-suppressor protein Lkb1 localises in the cilium, and flow results in increased AMPK phosphorylation at the basal body. Conversely, knockdown of Lkb1 prevents normal cell-size regulation under flow conditions. Our results demonstrate that the cilium regulates mTOR signalling and cell size, and identify the cilium-basal body compartment as a spatially restricted activation site for Lkb1 signalling.     

Analysis of the flow field induced by the sessile peritrichous ciliate Opercularia asymmetrica

The feeding mechanism of the sessile protozoon Opercularia asymmetrica (Oligohymenophorea, Peritrichia) relies on the cilia beat generating a flow field that convectively transports suspended particles and dissolved substances to the oral cavity of the organism. By use of optical micro-flow measurement and theoretical methods the flow environment of two neighbouring peritrichous ciliate cells is studied. Both, yeast cells (Saccharomyces cerevisiae) and artificial flow tracers are used for the visualisation of the flow field. Artificial tracers are rejected by the protozoa and deviate from the fluid path lines, while yeast cells follow the flow almost perfectly. This is shown through a dimensional analysis of the involved hydrodynamic forces on the tracers. The measured flow field exhibits maximum velocities of 25 microm/s at around 20 microm distance ahead of an individual ciliate. The flow field extends 200 microm from the location of the ciliate. A nicking motion of the protozoon is observed and found not to obey any periodic law. Multiples of protozoa exhibit most commonly an alternating cilia beat regime generating a non-stationary flow field. It can be shown through theoretical methods that fluid exchange is enhanced in this alternating regime compared to a flow field generated by a single ciliate. Fluid exchange depends on the distance of the ciliates from each other and on the alteration frequency of the cilia beat. The comparison of an analytical Stokes' flow solution with the observed fluid flow serves to determine the force required to maintain the flow field against viscous dissipation. The force magnitude is in the order of magnitude of 10-100 pN.     

Force-response considerations in ciliary mechanosensation

Considerable experimental evidence indicates that the primary, nonmotile cilium is a mechanosensory organelle in several epithelial cell types. As the relationship between cellular responses and nature and magnitude of applied forces is not well understood, we have investigated the effects of exposure of monolayers of renal collecting duct chief cells to orbital shaking and quantified the forces incident on cilia. An exposure of 24 h of these cells to orbital shaking resulted in a decrease of amiloride-sensitive sodium current by approximately 60% and ciliary length by approximately 30%. The sensitivity of the sodium current to shaking was dependent on intact cilia. The drag force on cilia due to induced fluid flow during orbital shaking was estimated at maximally 5.2x10(-3) pN at 2 Hz, approximately 4 times that of thermal noise. The major structural feature of cilia contributing to their sensitivity appears to be ciliary length. As more than half of the total drag force is exerted on the ciliary cap, one function of the slender stalk may be to expose the cap to greater drag force. Regardless, the findings indicate that the cilium is a mechanosensory organelle with a sensitivity much lower than previously recognized.     

HIF Stabilization Weakens Primary Cilia

Although solitary or sensory cilia are present in most cells of the body and their existence has been known since the sixties, very little is known about their functions. One suspected function is fluid flow sensing- physical bending of cilia produces an influx of Ca⁺⁺, which can then result in a variety of activated signaling pathways. Defective cilia and ciliary-associated proteins have been shown to result in cystic diseases. Autosomal Dominant Polycystic Kidney Disease (ADPKD) is a progressive disease, typically appearing in the 5^th decade of life and is one of the most common monogenetic inherited human diseases, affecting approximately 600,000 people in the United States. Because the mechanical properties of cilia impact their response to applied flow, we asked how the stiffness of cilia can be controlled pharmacologically. We performed an experiment subjecting cilia to Taxol (a microtubule stabilizer) and CoCl₂ (a HIF stabilizer to model hypoxia). Madin-Darby Canine Kidney (MDCK) cells were selected as our model system. After incubation with a selected pharmacological agent, cilia were optically trapped and the bending modulus measured. We found that HIF stabilization significantly weakens cilia. These results illustrate a method to alter the mechanical properties of primary cilia and potentially alter the flow sensing properties of cilia.     

Analysis of the flow field of the krill,Euphausia pacifica

Velocity measurements were performed for the flow field generated by tethered krill Euphausia pacifica. The particle image velocimetry (PIV) technique was used to measure the velocity field in vertical planes aligned with the krill body axis. The krill generates a narrow jet-like flow behind and below the pleopods (roughly 25° below horizontal). The volume of fluid moving at greater than 10% of the maximum velocity near the pleopods is roughly 18 times larger than the volume of the krill. Thus, the hydrodynamic disturbance occupies a significantly larger region than the animal body. Other krill, sensing the flow disturbance, may take advantage of the flow induced by a neighbor to locate a mate or to draft for efficient propulsion.     

Analysis of the flow field of the krill, Euphausia pacifica

Velocity measurements were performed for the flow field generated by tethered krill Euphausia pacifica. The particle image velocimetry (PIV) technique was used to measure the velocity field in vertical planes aligned with the krill body axis. The krill generates a narrow jet-like flow behind and below the pleopods (roughly 25° below horizontal). The volume of fluid moving at greater than 10% of the maximum velocity near the pleopods is roughly 18 times larger than the volume of the krill. Thus, the hydrodynamic disturbance occupies a significantly larger region than the animal body. Other krill, sensing the flow disturbance, may take advantage of the flow induced by a neighbor to locate a mate or to draft for efficient propulsion.     

Pulsatile flow of blood through a stenosed tube: effect of periodic body acceleration and a magnetic field

A proper understanding of the interactions of body acceleration and a magnetic field with blood flow could be useful in the diagnosis and treatment of some health problems. In the work reported in this paper we studied the pulsatile flow of blood through stenosed arteries, including the effects of body acceleration and a magnetic field. Blood is regarded as an electrically conducting, incompressible, couple-stress fluid in the presence of a magnetic field along the radius of the tube. The effects of the body acceleration and the magnetic field on the axial velocity, flow rate, and fluid acceleration were obtained analytically by use of the Hankel transform and the Laplace transform. Velocity variations under different conditions are shown graphically. The results have been compared with those from other theoretical models, and are in good agreement. Finally, our mathematical model gives a simple velocity expression for blood flow so it will help not only in the field of physiological fluid dynamics but will also help medical practitioners with elementary knowledge of mathematics.     

An analysis of the flow of a power law fluid due to ciliary motion in an infinite channel

The propulsion mechanics of cilia-induced flow is studied through a mathematical model. The problem of two-dimensional motion of a power law fluid inside a channel with ciliated walls is considered. The characteristics of ciliary systems are determined by the dominance of viscous effects over inertial effects using the long-wavelength approximation. Solutions for the longitudinal, transverse, and resultant velocities are obtained. The pressure gradient and volume flow rate for different values of the power law index are also calculated. The flow properties for the power law fluid are determined as a function of the cilia and metachronal wave velocity. The viscous and power law fluid are compared and discussed graphically.     

Mechanistic basis of otolith formation during teleost inner ear development

Otoliths, which are connected to stereociliary bundles in the inner ear, serve as inertial sensors for balance. In teleostei, otolith development is critically dependent on flow forces generated by beating cilia; however, the mechanism by which flow controls otolith formation remains unclear. Here, we have developed a noninvasive flow probe using optical tweezers and a viscous flow model in order to demonstrate how the observed hydrodynamics influence otolith assembly. We show that rotational flow stirs and suppresses precursor agglomeration in the core of the cilia-driven vortex. The velocity field correlates with the shape of the otolith and we provide evidence that hydrodynamics is actively involved in controlling otolith morphogenesis. An implication of this hydrodynamic effect is that otolith self-assembly is mediated by the balance between Brownian motion and cilia-driven flow. More generally, this flow feature highlights an alternative biological strategy for controlling particle localization in solution.     

Flow profiles and wall shear stress distribution at a hemodialysis venous anastomosis: preliminary study

The phasic velocity field in the vicinity of the venous anastomosis in a hemodialysis angioaccess arteriovenous fistula loop graft (AVLG) is investigated employing a laser Doppler anemometer (LDA) system. Detailed LDA velocity profiles are obtained by sectional survey performed in a transparent, elastic flow model which was fabricated to represent the geometry of the AVLG system under physiological pressure and flow waveforms. The geometry of the flow model was based on a silicone rubber cast obtained from an experimental dog model. In the present study, detailed distribution of velocity profiles is obtained. The distribution of wall shear stress in the model is computed from the slope of the local velocity profiles near the wall. The relationship between the results obtained by flow visualization and the LDA measurement is discussed.     

Ciliary behavior of a negatively phototactic Chlamydomonas reinhardtii

With an instrument that can record the motion of both cilia of the unicellular alga Chlamydomonas reinhardtii for many hours, the behavioral differences of its two cilia have been studied to determine their specific role in phototaxis. The organism was held on a fixed micropipette with the plane of ciliary beating rotated into the imaging plane of a quadrant photodetector. The responses to square-wave light patterns of a wide range of temporal frequencies were used to characterize the responses of each cilium. Eighty-one cells were examined showing an unexpectedly diverse range of responses. Plausible common signals for the linear and nonlinear signals from the cell body are suggested. Three independent ciliary measures--the beat frequency, stroke velocity, and phasing of the two cilia--have been identified. The cell body communicates to the cilia the direction of phototaxis the cell desires to go, the absolute light intensity, and the appropriate graded transient response for tracking the light source. The complexity revealed by each measure of the ciliary response indicates many independent variables are involved in the net phototactic response. In spite of their morphological similarity, the two cilia of Chlamydomonas respond uniquely. Probably the signals from the cell body fan out to independent pathways in the cilia. Each cilium modifies the input in its own way. The change in the pattern of the effective and recovery strokes of each cilium associated with negative phototaxis has been demonstrated and its involvement in phototactic turning is described.