Computation of the internal forces in cilia: application to ciliary motion, the effects of viscosity, and cilia interactions.

This paper presents a simple and reasonable method for generating a phenomenological model of the internal mechanism of cilia. The model uses a relatively small number of parameters whose values can be obtained by fitting to ciliary beat shapes. Here, we use beat patterns observed in Paramecium. The forces that generate these beats are computed and fit to a simple functional form called the "engine." This engine is incorporated into a recently developed hydrodynamic model that accounts for interactions between neighboring cilia and between the cilia and the surface from which they emerge. The model results are compared to data on ciliary beat patterns of Paramecium obtained under conditions where the beats are two-dimensional. Many essential features of the motion, including several properties that are not built in explicitly, are shown to be captured. In particular, the model displays a realistic change in beat pattern and frequency in response to increased viscosity and to the presence of neighboring cilia in configurations such as rows of cilia and two-dimensional arrays of cilia. We found that when two adjacent model cilia start beating at different phases they become synchronized within several beat periods, as observed in experiments where two flagella are brought into close proximity. Furthermore, examination of various multiciliary configurations shows that an approximately antiplectic wave pattern evolves autonomously. This modeling evidence supports earlier conjectures that metachronism may occur, at least partially, as a self-organized phenomenon due to hydrodynamic interactions between neighboring cilia.     

Development of a computational fluid dynamics model for mucociliary clearance in the nasal cavity

Intranasal drug delivery has attracted significant attention because of the opportunity to deliver systemic drugs directly to the blood stream. However, the mucociliary clearance poses a challenge in gaining high efficacy of intranasal drug delivery because cilia continuously carry the mucus blanket towards the laryngeal region. To better understand mucus flow behaviour on the human nasal cavity wall, we present computational model development, and evaluation of mucus motion on a realistic nasal cavity model reconstructed from CT-scans. The model development involved two approaches based on the actual nasal cavity geometry namely: (i) unwrapped-surface model in 2D domain and (ii) 3D-shell model. Conservation equations of fluid motion were applied to the domains, where a mucus production source term was used to initiate the mucus motion. The analysis included mucus flow patterns, virtual saccharin tests and quantitative velocity magnitude analysis, which demonstrated that the 3D-shell model results provided better agreement with experimental data. The unwrapped-surface model also suffered from mesh-deformations during the unwrapping stage and this led to higher mucus velocity compared to experimental data. Therefore, the 3D-shell model was recommended for future mucus flow simulations. As a first step towards mucus motion modelling this study provides important information that accurately simulates a mucus velocity field on a human nasal cavity wall, for assessment of toxicology and efficacy of intranasal drug delivery.     

Ciliary motion modeling, and dynamic multicilia interactions

This paper presents a rigorous and accurate modeling tool for ciliary motion. The hydrodynamics analysis, originally suggested by Lighthill (1975), has been modified to remove computational problems. This approach is incorporated into a moment-balance model of ciliary motion in place of the previously used hydrodynamic analyses, known as Resistive Force Theory. The method is also developed to include the effect of a plane surface at the base of the cilium, and the effect of the flow fields produced by neighboring cilia. These extensions were not possible with previous work using the Resistive Force Theory hydrodynamics. Performing reliable simulations of a single cilium as well as modeling multicilia interactions is now possible. The result is a general method which could now be used for detailed modeling of the mechanisms for generating ciliary beat patterns and patterns of metachronal interactions in arrays of cilia. A computer animation technique was designed and applied to display the results.     

Restricted Transport in Small Pores: A Model for Steric Exclusion and Hindered Particle Motion

The basic hydrodynamic equations governing transport in submicron pores are reexamined. Conditions necessary for a simplified, one-dimensional treatment of the diffusion/convection process are established. Steric restrictions and Brownian motion are incorporated directly into the resulting model. Currently available fluid mechanical results are used to evaluate an upper limit on hindered diffusion; this limit is valid for small particle-to-pore ratios. Extensions of the analysis are shown to depend on numerical solutions of the related hydrodynamic problem, that of asymmetrical particle motion in a bounded fluid.     

Computer simulation of flagellar movement IX. Oscillation and symmetry breaking in a model for short flagella and nodal cilia

A computer model of flagella in which oscillation results from regulation of active sliding force by sliding velocity can simulate the movements of very short flagella and cilia. Of particular interest are the movements of the short (2-3 microm) nodal cilia of the mammalian embryo, which determine the development of the asymmetry of the internal organs. These cilia must generate a counterclockwise (viewed from base to tip) circling motion. A three-dimensional computer model, with active force generated by a simple mathematical formulation and regulated by sliding velocity, can generate this circling motion if a time delay process is included in the control specification. Without the introduction of a symmetry-breaking mechanism, the computer models start randomly in either direction, and maintain either clockwise or counterclockwise circling. Symmetry can be broken by at least two mechanisms: (1) control of dynein activity on one outer doublet by sliding velocity can be influenced by the sliding velocity experienced on an adjacent outer doublet, or (2) a constant twist of the axoneme caused by an off-axis component of dynein force. This second mechanism appears more reasonable, but its effectiveness is highly dependent upon specifications for the elastic resistances of the model. These symmetry-breaking mechanisms need to be present only at the beginning of circling. With these models, once a circling direction is established, it remains stable even if the symmetry-breaking mechanism is removed.     

Cobblestone HUVECs: a human model system for studying primary ciliogenesis

Primary cilia are microtubule based sensory organelles that play an important role in maintaining cellular homeostasis. Malfunctioning results in a number of abnormalities, diseases (ciliopathies) and certain types of cancer. Morphological and biochemical knowledge on cilia/flagella, (early) ciliogenesis and intraflagellar transport is often obtained from model systems (e.g. Chlamydomonas) or from multi ciliary cells like lung or kidney epithelium.In this study endothelial cells in isolated human umbilical veins (HUVs) and cultured human umbilical vein endothelial cells (HUVECs) are compared and used to study primary ciliogenesis. By combining fluorescence microscopy, SEM, 2D and 3D TEM techniques we found that under the tested culturing conditions 60% of cobblestone endothelial cells form a primary cilium. Only a few of these cilia are present (protruding) on the endothelial cell surface, meaning that most primary cilia are in the cytoplasm (non-protruding). This was also observed in situ in the endothelial cells in the umbilical vein. The exact function(s?) of these non-protruding cilia remains unclear.Ultra-structural analysis of cultured HUVECs and the endothelial layer of the human umbilical veins reveal that there are: vesicles inside the ciliary pocket during the early stages of ciliogenesis; tubules/vesicles from the cytoplasm fuse with the ciliary sheath; irregular axoneme patterns, and two round, membranous vesicles inside the basal body.We conclude that cobblestone cultured HUVECs are comparable to the in vivo epithelial lining of the umbilical veins and therefore provide a well defined, relatively simple human model system with a reproducible number of non-protruding primary cilia for studying ciliogenesis.     

Spontaneous creation of macroscopic flow and metachronal waves in an array of cilia

Cells carrying cilia on their surface show many striking features: alignment of cilia in an array, two-phase asymmetric beating for each cilium, and existence of metachronal coordination with a constant phase difference between two adjacent cilia. We give simple theoretical arguments based on hydrodynamic coupling and an internal mechanism of the cilium derived from the behavior of a collection of molecular motors to account qualitatively for these cooperative features. Hydrodynamic interactions can lead to the alignment of an array of cilia. We study the effect of a transverse external flow and obtain a two-phase asymmetrical beating, faster along the flow and slower against the flow, proceeding around an average curved position. We show that an aligned array of cilia is able to spontaneously break the left-right symmetry and to create a global average flow. Metachronal coordination arises as a consequence of the internal mechanism of the cilia and their hydrodynamic couplings, with a wavelength comparable to that found in experiments. It allows the cilia to start beating at a lower adenosine-triphosphate threshold and at a higher frequency than for a single cilium. It also leads to a rather stationary flow, which might be its major advantage.     

A study of bacterial flagellar bundling

Certain bacteria, such as Escherichia coli (E. coli) and Salmonella typhimurium (S. typhimurium), use multiple flagella often concentrated at one end of their bodies to induce locomotion. Each flagellum is formed in a left-handed helix and has a motor at the base that rotates the flagellum in a corkscrew motion.We present a computational model of the flagellar motion and their hydrodynamic interaction. The model is based on the equations of Stokes flow to describe the fluid motion. The elasticity of the flagella is modeled with a network of elastic springs while the motor is represented by a torque at the base of each flagellum. The fluid velocity due to the forces is described by regularized Stokeslets and the velocity due to the torques by the associated regularized rotlets. Their expressions are derived. The model is used to analyze the swimming motion of a single flagellum and of a group of three flagella in close proximity to one another. When all flagellar motors rotate counterclockwise, the hydrodynamic interaction can lead to bundling. We present an analysis of the flow surrounding the flagella. When at least one of the motors changes its direction of rotation, the same initial conditions lead to a tumbling behavior characterized by the separation of the flagella, changes in their orientation, and no net swimming motion. The analysis of the flow provides some intuition for these processes.     

An experimental and computational analysis of primary cilia deflection under fluid flow

In this study we have developed a novel model of the deflection of primary cilia experiencing fluid flow accounting for phenomena not previously considered. Specifically, we developed a large rotation formulation that accounts for rotation at the base of the cilium, the initial shape of the cilium and fluid drag at high deflection angles. We utilised this model to analyse full 3D data-sets of primary cilia deflecting under fluid flow acquired with high-speed confocal microscopy. We found a wide variety of previously unreported bending shapes and behaviours. We also analysed post-flow relaxation patterns. Results from our combined experimental and theoretical approach suggest that the average flexural rigidity of primary cilia might be higher than previously reported (Schwartz et al. 1997, Am J Physiol. 272(1 Pt 2):F132–F138). In addition our findings indicate that the mechanics of primary cilia are richly varied and mechanisms may exist to alter their mechanical behaviour.     

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.     

Investigation of hydrodynamic focusing in a microfluidic coulter counter device

The Coulter technique enables rapid analysis of particles or cells suspended in a fluid stream. In this technique, the cells are suspended in an electrically conductive solution, which is hydrodynamically focused by nonconducting sheath flows. The cells produce a characteristic voltage signal when they interrupt an electrical path. The population and size of the cells can be obtained through analyzing the voltage signal. In a microfluidic Coulter counter device, the hydrodynamic focusing technique is used to position the conducting sample stream and the cells and also to separate close cells to generate distinct signals for each cell and avoid signal jam. The performance of hydrodynamic focusing depends on the relative flow ratio between the sample stream and sheath stream. We use a numerical approach to study the hydrodynamic focusing in a microfluidic Coulter counter device. In this approach, the flow field is described by solving the incompressible Navier-Stokes equations. The sample stream concentration is modeled by an advection-diffusion equation. The motion of the cells is governed by the Newton-Euler equations of motion. Particle motion through the flow field is handled using an overlapping grid technique. A numerical model for studying a microfluidic Coulter counter has been validated. Using the model, the impact of relative flow rate on the performance of hydrodynamic focusing was studied. Our numerical results show that the position of the sample stream can be controlled by adjusting the relative flow rate. Our simulations also show that particles can be focused into the stream and initially close particles can be separated by the hydrodynamic focusing. From our study, we conclude that hydrodynamic focusing provides an effective way to control the position of the sample stream and cells and it also can be used to separate cells to avoid signal jam.     

Ciliary elongation in blastulae of Arbacia punctulata induced by trypsin

Hatched sea urchin blastulae, which have primarily short 25-μm cilia except for some long 40-to 70-μm cilia at the apical tuft, were induced to form long (40- to 70-μm) cilia around most of their circumference when treated with trypsin (0.008–0.1%) or concanavalin A. Other animalizing agents did not induce the formation of long cilia when applied to the normal blastulae. The formation of long cilia by trypsin was both time and concentration dependent. The long cilia first appeared around the apical tuft after 6–8 hr in trypsin (21°C), and by 18–22 hr most of the blastula was covered with the long cilia. Length distribution studies on cilia isolated at various times showed that the percentage of long cilia increased from approximately 10% in the normal blastula to over 66% in the 22-hr trypsin-treated embryo, and indicated that the long cilia formed by the elongation of the original short cilia. Only the blastulae and gastrulae could be induced to form long cilia; the prisms and plutei could not. Once development was inhibited by the trypsin and the first long cilia appeared, the trypsin effect could not be reversed. When blastulae with long cilia were removed from the trypsin for 10 hr, the cilia remained long; when the long cilia were detached, the blastulae regenerated long cilia in the absence of trypsin. The induced long cilia moved poorly, similar to the long, apical tuft cilia of normal embryos. The formation of long cilia by trypsin treatment of sea urchin blastulae provides a model system for studying the mechanisms of ciliary length control.     

Contractile events in the cilia of Paramecium, Opalina, Mytilus, and Phragmatopoma.

The motion of the abnormal cilia of Opalina and Mytilus can be described by the recently developed model for ciliary motion, provided the activation of the contractility during the effective stroke is reduced by three- to fivefold compared with that in the recovery stroke. The stiffness of the Mytilus cilium during the effective stroke is found several hundred times larger than that predicted by the model, however. The stiffness of the cilia of Paramecium, Opalina, Phragmatopoma, and of Mytilus in the recovery phase, is predicted approximately correctly by the model. The activation of contractility in Mytilus and Phragmatopoma cilia increases with the viscosity of the medium, as the velocity of the ciliary motion slows down. This leads to the equivalent of a force-velocity relation. The velocity of propagation of the bend in the cilia during the recovery stroke is shown to be dependent only on the elastic properties of the ciliary shaft, and to be independent of the contractile activiey.     

Equations of Interdoublet Separation during Flagella Motion Reveal Mechanisms of Wave Propagation and Instability

The motion of flagella and cilia arises from the coordinated activity of dynein motor protein molecules arrayed along microtubule doublets that span the length of axoneme (the flagellar cytoskeleton). Dynein activity causes relative sliding between the doublets, which generates propulsive bending of the flagellum. The mechanism of dynein coordination remains incompletely understood, although it has been the focus of many studies, both theoretical and experimental. In one leading hypothesis, known as the geometric clutch (GC) model, local dynein activity is thought to be controlled by interdoublet separation. The GC model has been implemented as a numerical simulation in which the behavior of a discrete set of rigid links in viscous fluid, driven by active elements, was approximated using a simplified time-marching scheme. A continuum mechanical model and associated partial differential equations of the GC model have remained lacking. Such equations would provide insight into the underlying biophysics, enable mathematical analysis of the behavior, and facilitate rigorous comparison to other models. In this article, the equations of motion for the flagellum and its doublets are derived from mechanical equilibrium principles and simple constitutive models. These equations are analyzed to reveal mechanisms of wave propagation and instability in the GC model. With parameter values in the range expected for Chlamydomonas flagella, solutions to the fully nonlinear equations closely resemble observed waveforms. These results support the ability of the GC hypothesis to explain dynein coordination in flagella and provide a mathematical foundation for comparison to other leading models.     

The active and passive ciliary motion in the embryo node: A computational fluid dynamics model

The breaking of left–right symmetry in the mammalian embryo is believed to occur in a transient embryonic structure, the node, when cilia create a leftward flow of liquid. The two-cilia hypothesis proposes that the node contains two kinds of primary cilia: motile cilia that rotate autonomously to generate the leftward fluid flow and passive cilia that act as mechano-sensors, responding to flow. While studies support this hypothesis, the mechanism by which the sensory cilia respond to the fluid flow is still unclear. In this paper, we present a computational model of two cilia, one active and one passive. By employing computational fluid dynamics, deformable mesh computational techniques and fluid–structure interaction analysis, and solving the three-dimensional unsteady transport equations, we study the flow pattern produced by the movement of the active cilium and the response of the passive cilium to this flow. Our results reveal that clockwise rotation of the active cilium can generate a counter-clockwise elliptical rotation and overall lateral displacement for its neighboring passive one, of measurable magnitude and consistent pattern. This supports the plausibility of the two-cilia hypothesis and helps quantify the motion pattern for the passive cilium induced by this regional flow.     

Equations of Interdoublet Separation during Flagella Motion Reveal Mechanisms of Wave Propagation and Instability

The motion of flagella and cilia arises from the coordinated activity of dynein motor protein molecules arrayed along microtubule doublets that span the length of axoneme (the flagellar cytoskeleton). Dynein activity causes relative sliding between the doublets, which generates propulsive bending of the flagellum. The mechanism of dynein coordination remains incompletely understood, although it has been the focus of many studies, both theoretical and experimental. In one leading hypothesis, known as the geometric clutch (GC) model, local dynein activity is thought to be controlled by interdoublet separation. The GC model has been implemented as a numerical simulation in which the behavior of a discrete set of rigid links in viscous fluid, driven by active elements, was approximated using a simplified time-marching scheme. A continuum mechanical model and associated partial differential equations of the GC model have remained lacking. Such equations would provide insight into the underlying biophysics, enable mathematical analysis of the behavior, and facilitate rigorous comparison to other models. In this article, the equations of motion for the flagellum and its doublets are derived from mechanical equilibrium principles and simple constitutive models. These equations are analyzed to reveal mechanisms of wave propagation and instability in the GC model. With parameter values in the range expected for Chlamydomonas flagella, solutions to the fully nonlinear equations closely resemble observed waveforms. These results support the ability of the GC hypothesis to explain dynein coordination in flagella and provide a mathematical foundation for comparison to other leading models.     

GPR88 Reveals a Discrete Function of Primary Cilia as Selective Insulators of GPCR Cross-Talk

A number of G protein-coupled receptors (GPCRs) localize to primary cilia but the functional significance of cilia to GPCR signaling remains incompletely understood. We investigated this question by focusing on the D1 dopamine receptor (D1R) and beta-2 adrenergic receptor (B2AR), closely related catecholamine receptors that signal by stimulating production of the diffusible second messenger cyclic AMP (cAMP) but differ in localization relative to cilia. D1Rs robustly concentrate on cilia of IMCD3 cells, as shown previously in other ciliated cell types, but disrupting cilia did not affect D1R surface expression or ability to mediate a concentration-dependent cAMP response. By developing a FRET-based biosensor suitable for resolving intra- from extra- ciliary cAMP changes, we found that the D1R-mediated cAMP response is not restricted to cilia and extends into the extra-ciliary cytoplasm. Conversely the B2AR, which we show here is effectively excluded from cilia, also generated a cAMP response in both ciliary and extra-ciliary compartments. We identified a distinct signaling effect of primary cilia through investigating GPR88, an orphan GPCR that is co-expressed with the D1R in brain, and which we show here is targeted to cilia similarly to the D1R. In ciliated cells, mutational activation of GPR88 strongly reduced the D1R-mediated cAMP response but did not affect the B2AR-mediated response. In marked contrast, in non-ciliated cells, GPR88 was distributed throughout the plasma membrane and inhibited the B2AR response. These results identify a discrete ‘insulating’ function of primary cilia in conferring selectivity on integrated catecholamine signaling through lateral segregation of receptors, and suggest a cellular activity of GPR88 that might underlie its effects on dopamine-dependent behaviors.     

Lack of Dopaminergic Inputs Elongates the Primary Cilia of Striatal Neurons

In the rodent brain, certain G protein-coupled receptors and adenylyl cyclase type 3 are known to localize to the neuronal primary cilium, a primitive sensory organelle protruding singly from almost all neurons. A recent chemical screening study demonstrated that many compounds targeting dopamine receptors regulate the assembly of Chlamydomonas reinhardtii flagella, structures which are analogous to vertebrate cilia. Here we investigated the effects of dopaminergic inputs loss on the architecture of neuronal primary cilia in the rodent striatum, a brain region that receives major dopaminergic projections from the midbrain. We first analyzed the lengths of neuronal cilia in the dorsolateral striatum of hemi-parkinsonian rats with unilateral lesions of the nigrostriatal dopamine pathway. In these rats, the striatal neuronal cilia were significantly longer on the lesioned side than on the non-lesioned side. In mice, the repeated injection of reserpine, a dopamine-depleting agent, elongated neuronal cilia in the striatum. The combined administration of agonists for dopamine receptor type 2 (D2) with reserpine attenuated the elongation of striatal neuronal cilia. Repeated treatment with an antagonist of D2, but not of dopamine receptor type 1 (D1), elongated the striatal neuronal cilia. In addition, D2-null mice displayed longer neuronal cilia in the striatum compared to wild-type controls. Reserpine treatment elongated the striatal neuronal cilia in D1-null mice but not in D2-null mice. Repeated treatment with a D2 agonist suppressed the elongation of striatal neuronal cilia on the lesioned side of hemi-parkinsonian rats. These results suggest that the elongation of striatal neuronal cilia following the lack of dopaminergic inputs is attributable to the absence of dopaminergic transmission via D2 receptors. Our results provide the first evidence that the length of neuronal cilia can be modified by the lack of a neurotransmitter''s input.     

Hydrodynamic model of temperature change in open ionic channels.

Most theories of open ionic channels ignore heat generated by current flow, but that heat is known to be significant when analogous currents flow in semiconductors, so a generalization of the Poisson-Nernst-Planck theory of channels, called the hydrodynamic model, is needed. The hydrodynamic theory is a combination of the Poisson and Euler field equations of electrostatics and fluid dynamics, conservation laws that describe diffusive and convective flow of mass, heat, and charge (i.e., current), and their coupling. That is to say, it is a kinetic theory of solute and solvent flow, allowing heat and current flow as well, taking into account density changes, temperature changes, and electrical potential gradients. We integrate the equations with an essentially nonoscillatory shock-capturing numerical scheme previously shown to be stable and accurate. Our calculations show that 1) a significant amount of electrical energy is exchanged with the permeating ions; 2) the local temperature of the ions rises some tens of degrees, and this temperature rise significantly alters for ionic flux in a channel 25 A long, such as gramicidin-A; and 3) a critical parameter, called the saturation velocity, determines whether ionic motion is overdamped (Poisson-Nernst-Planck theory), is an intermediate regime (called the adiabatic approximation in semiconductor theory), or is altogether unrestricted (requiring the full hydrodynamic model). It seems that significant temperature changes are likely to accompany current flow in the open ionic channel.