Hydrodynamic calculations on the movements of cilia and flagella. I. Paramecium

Statistical Mobility of Multicellular Colonies of Flagellated Swimming Cells

We study the stochastic hydrodynamics of colonies of flagellated swimming cells, typified by multicellular choanoflagellates, which can form both rosette and chainlike shapes. The objective is to link cell-scale dynamics to colony-scale dynamics for various colonial morphologies. Via autoregressive stochastic models for the cycle-averaged flagellar force dynamics and statistical models for demographic cell-to-cell variability in flagellar properties and placement, we derive effective transport properties of the colonies, including cell-to-cell variability. We provide the most quantitative detail on disclike geometries to model rosettes, but also present formulas for the dynamics of general planar colony morphologies, which includes planar chain-like configurations.

Swimming microorganisms acquire optimal efficiency with multiple cilia

Planktonic microorganisms are ubiquitous in water, and their population dynamics are essential for forecasting the behavior of global aquatic ecosystems. Their population dynamics are strongly affected by these organisms' motility, which is generated by their hair-like organelles, called cilia or flagella. However, because of the complexity of ciliary dynamics, the precise role of ciliary flow in microbial life remains unclear. Here, we have used ciliary hydrodynamics to show that ciliates acquire the optimal propulsion efficiency. We found that ciliary flow highly resists an organism's propulsion and that the swimming velocity rapidly decreases with body size, proportional to the power of minus two. Accordingly, the propulsion efficiency decreases as the cube of body length. By increasing the number of cilia, however, efficiency can be significantly improved, up to 100-fold. We found that there exists an optimal number density of cilia, which provides the maximum propulsion efficiency for all ciliates. The propulsion efficiency in this case decreases inversely proportionally to body length. Our estimated optimal density of cilia corresponds to those of actual microorganisms, including species of ciliates and microalgae, which suggests that now-existing motile ciliates and microalgae have survived by acquiring the optimal propulsion efficiency. These conclusions are helpful for better understanding the ecology of microorganisms, such as the energetic costs and benefits of multicellularity in Volvocaceae, as well as for the optimal design of artificial microswimmers.

On the kinematics-wave motion of living particles in suspension

This work presents theoretical and experimental analyses on the kinematics-wave motion of suspended active particles in a biological fluid. The fluid is an active suspension of nematodes immersed in a gel-like biological structure, moving at a low Reynolds number. The nematode chosen for the study is Caenorhabditis elegans. Its motion is subjected to the time reversibility of creeping flows. We investigate how this worm reacts to this reversibility condition in order to break the flow symmetry and move in the surrounding fluid. We show that the relationship between the length of an individual nematode and the wavelength of its motion is linear and can be fitted by a theoretical prediction proposed in this work. We provide a deep discussion regarding the propulsion mechanics based on a scaling analysis that identifies three major forces acting on an individual nematode. These forces are a viscous force, a yield stress force due to gelification of agar molecules in the gel-like medium, and a bending force associated with the muscular tension imposed by the nematodes in the medium. By the scalings, we identify the most relevant physical parameters of the nematode's motion. In order to examine and quantify the motion, dynamical system tools such as FFT are used in the present analysis. The motion characterization is performed by examining (or studying) two different populations: (i) in the absence of food with starving nematodes and (ii) with well-fed nematodes. In addition, several kinematic quantities of the head, center of mass, and tail for a sample of nematodes are also investigated: their slip velocities, wavelengths, trajectories, frequency spectra, and mean curvatures. The main findings of this work are the confirmation of a linear relationship between the nematode's physical length and its motion wavelength, the identification of secondary movements in high frequencies that helps breaking the time-reversibility in which the worms are bonded, and the observation and interpretation of a systematic difference between the individual motion of well-fed and starving nematodes.

Effect of Cilia Beat Frequency on Muco-ciliary Clearance

BACKGROUND

The airway surface liquid (ASL), which is a fluid layer coating the interior epithelial surface of the bronchi and bronchiolesis, plays an important defensive role against foreign particles and chemicals entering lungs.

OBJECTIVE

Numerical investigation has been employed to solve two-layer model consisting of mucus layer as a viscoelastic fluid and periciliary liquid layer as a Newtonian fluid to study the effects of cilia beat frequency (CBF) at various amounts of mucus properties on muco-ciliary transport problem.

METHODS

Hybrid finite difference-lattice Boltzmann-method (FB-LBM) has been used to solve the momentum equations and to simulate cilia forces, and also the PCL-mucus interface more accurately, immersed boundary method (IBM) has been employed. The main contribution of the current study is to use an Oldroyd-B model as the constitutive equation of mucus.

RESULTS

Our results show that increasing CBF and decreasing mucus viscosity ratio have great effects on mucus flow, but the effect of viscosity ratio is more significant. The results also illustrate that the relation between cilia beat frequency and mean mucus velocity is almost linear and it has similar behavior at different values of viscosity ratio.

CONCLUSION

Numerical investigation based on hybrid IB-FD-LBM has been used to study the effect of CBF at various mounts of mucus viscosity ratio on the muco-ciliary clearance. The results showed that the effect of viscosity ratio on the muco-ciliary transport process is more significant compared with CBF.

Generic flow profiles induced by a beating cilium

We describe a multipole expansion for the low-Reynolds-number fluid flows generated by a localized source embedded in a plane with a no-slip boundary condition. It contains 3 independent terms that fall quadratically with the distance and 6 terms that fall with the third power. Within this framework we discuss the flows induced by a beating cilium described in different ways: a small particle circling on an elliptical trajectory, a thin rod and a general ciliary beating pattern. We identify the flow modes present based on the symmetry properties of the ciliary beat.

Fluid mechanics of nodal flow due to embryonic primary cilia

Breaking of left-right symmetry is crucial in vertebrate development. The role of cilia-driven flow has been the subject of many recent publications, but the underlying mechanisms remain controversial. At approximately 8 days post-fertilization, after the establishment of the dorsal-ventral and anterior-posterior axes, a depressed structure is found on the ventral side of mouse embryos, termed the ventral node. Within the node, 'whirling' primary cilia, tilted towards the posterior, drive a flow implicated in the initial left-right signalling asymmetry. However, the underlying fluid mechanics have not been fully and correctly explained until recently and accurate characterization is required in determining how the flow triggers the downstream signalling cascades. Using the approximation of resistive force theory, we show how the flow is produced and calculate the optimal configuration to cause maximum flow, showing excellent agreement with in vitro measurements and numerical simulation, and paralleling recent analogue experiments. By calculating numerical solutions of the slender body theory equations, we present time-dependent physically based fluid dynamics simulations of particle pathlines in flows generated by large arrays of beating cilia, showing the far-field radial streamlines predicted by the theory.

De novo formation of left-right asymmetry by posterior tilt of nodal cilia

In the developing mouse embryo, leftward fluid flow on the ventral side of the node determines left-right (L-R) asymmetry. However, the mechanism by which the rotational movement of node cilia can generate a unidirectional flow remains hypothetical. Here we have addressed this question by motion and morphological analyses of the node cilia and by fluid dynamic model experiments. We found that the cilia stand, not perpendicular to the node surface, but tilted posteriorly. We further confirmed that such posterior tilt can produce leftward flow in model experiments. These results strongly suggest that L-R asymmetry is not the descendant of pre-existing L-R asymmetry within each cell but is generated de novo by combining three sources of spatial information: antero-posterior and dorso-ventral axes, and the chirality of ciliary movement.

Nodal cilia dynamics and the specification of the left/right axis in early vertebrate embryo development

Nodal cilia dynamics is a key factor for left/right axis determination in mouse embryos through the induction of a leftward fluid flow. So far it has not been clearly established how such dynamics is able to induce the asymmetric leftward flow within the node. Herein we propose that an asymmetric two-phase nonplanar beating cilia dynamics that involves the bending of the ciliar axoneme is responsible for the leftward fluid flow. We support our proposal with a host of hydrodynamic arguments, in silico experiments and in vivo video microscopy data in wild-type embryos and inv mutants. Our phenomenological modeling approach underscores how the asymmetry and speed of the flow depends on different relevant parameters. In addition, we discuss how the combination of internal and external mechanisms might cause the two-phase beating cilia dynamics.

Trifluoperazine-induced changes in swimming behavior of paramecium: evidence for two sites of drug action

Trifluoperazine (TFP), a drug that binds to Ca2+-calmodulin (CaM) complexes, altered swimming behavior not only in living paramecia, but also in reactivated, Triton-extracted "models" of the ciliate. By comparing the responses of living cells and models, we have ascertained that two sites of drug action exist in paramecium cilia. Swimming movements were recorded in darkfield stroboscopic flash photomicrographs; this permitted accurate quantitation of velocities and body-shape parameters. When living paramecia were incubated in a standard buffer containing 10 microM TFP, their speed of forward swimming fell over several minutes and their bodies shortened. Untreated paramecia backed up repeatedly and frequently upon transfer to a solution containing barium ions (the "barium dance"), but cells preincubated in TFP did not "dance." Instead they swam forward slowly for long periods of time without reversing and occasionally then exhibited abnormally prolonged reversals. W7 effects on swimming mimicked low doses of TFP, and the analog W5 did not visibly alter normal swimming patterns. These results suggest that TFP induces a decrease in the intracellular pCa of living paramecia, perhaps by reducing the efficiency of a calmodulin-activated calcium pump in the cell membrane. Paramecia extracted with Triton X-100 and reactivated to swim forward (7 greater than or equal to pCa greater than or equal to 6) were not affected by addition of up to 40 microM TFP to the reactivation medium. We conclude that the main drug effect in living cells is probably not at the axoneme. However, at low pCa, TFP directly affected the ciliary axoneme to shift its behavior to one characteristic of a higher pCa: TFP inhibited backward swimming in models reactivated at pCa less than 6; instead they swam forward or rocked in place. The mechanism of ciliary reversal in paramecium may therefore depend on an axonemal Ca2+-sensor, possibly bound CaM, which is affected by TFP only at low pCa, as has been postulated for other types of cilia.