Antiplectic Waves Research Articles

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.

Swimming mediated by ciliary beating: comparison with a squirmer model

The squirmer model of Lighthill and Blake has been widely used to analyse swimming ciliates. However, real ciliates are covered by hair-like organelles, called cilia; the differences between the squirmer model and real ciliates remain unclear. Here, we developed a ciliate model incorporating the distinct ciliary apparatus, and analysed motion using a boundary element–slender-body coupling method. This methodology allows us to accurately calculate hydrodynamic interactions between cilia and the cell body under free-swimming conditions. Results showed that an antiplectic metachronal wave was optimal in the swimming speed with various cell-body aspect ratios, which is consistent with former theoretical studies. Exploiting oblique wave propagation, we reproduced a helical trajectory, likeParamecium, although the cell body was spherical. We confirmed that the swimming velocity of model ciliates was well represented by the squirmer model. However, squirmer modelling outside the envelope failed to estimate the energy costs of swimming; over 90 % of energy was dissipated inside the ciliary envelope. The optimal swimming efficiency was given by the antiplectic wave; the value was 6.7 times larger than in-phase beating. Our findings provide a fundamental basis for modelling swimming micro-organisms.

Fractional Burgers Fluid Flow Due to Metachronal Ciliary Motion in an Inclined Tube.

Cilia-induced flow of fractional Burgers fluid is studied in an inclined tube for both symplectic and antiplectic wave patterns. The solution of the problem is persued under the long wave length limitation. The fractional Adomian decomposition method is employed to evaluate the pressure gradient. Mathematical expressions for the axial velocity, frictional force, pressure gradient, and stream function are obtained and the influence of the main operating parameters is discussed in detail. It is noted that the velocity profile is more dominant in the case of antiplectic metachronal waves compared to symplectic ones, which confirms former results on the better capability of antiplectic waves to transport mucus, obtained with more complex numerical solvers.

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

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

Optimizing the Energetic Efficiency of Ciliated Microorganisms

Energetic efficiency of swimming has long been considered a non-issue in microorganisms, but newer studies show that ciliates can use more than half of their energy for propulsion. To estimate how close the ciliates are to the theoretically optimal way of swimming we address the following problems: i) we determine the optimal stroke of a cilium, ii) we determine the optimal beating pattern of a ciliated surface and iii) we calculate the optimal shape of a ciliated swimmer.For a single cilium we define the efficiency in a scale-invariant way and show that the optimal stroke consists of a working stroke with a stretched cilium and a recovery stroke where the cilium bends and moves closer to the surface. When optimizing an array of cilia we additionally show that metachronal waves improve the efficiency and that the optimal efficiency is achieved for antiplectic waves. The resulting beating patterns, as well as the optimal ciliary density, show remarkable similarity with those observed in ciliated microorganisms. In order to optimize the shape of the whole swimmer we use a simplified description where we replace the ciliated layer with a surface slip velocity. Depending on the curvature we allow, the optimal swimmer has an elongated shape, possibly with two protrusions along the symmetry axis. These optimal shapes again resemble those of different ciliates.If we combine the results of our optimization with experimental efficiency estimates we can show that Paramecium has a propulsion efficiency that is within a factor of 2 of the theoretical optimum. This result suggests that ciliates have evolved for efficient swimming at different scales.[1] N. Osterman and A. Vilfan, Proc. Natl. Acad. Sci. USA 108, 15727-15732 (2011)[2] A. Vilfan, Phys. Rev. Lett. 109, 128105 (2012)

Finding the ciliary beating pattern with optimal efficiency

We introduce a measure for energetic efficiency of biological cilia acting individually or collectively and numerically determine the optimal beating patterns according to this criterion. Maximizing the efficiency of a single cilium leads to curly, often symmetric, and somewhat counterintuitive patterns. However, when looking at a densely ciliated surface, the optimal patterns become remarkably similar to what is observed in microorganisms like Paramecium. The optimal beating pattern then consists of a fast effective stroke and a slow sweeping recovery stroke. Metachronal coordination is essential for efficient pumping and the highest efficiency is achieved with antiplectic waves. Efficiency also increases with an increasing density of cilia up to the point where crowding becomes a problem. We finally relate the pumping efficiency of cilia to the swimming efficiency of a spherical microorganism and show that the experimentally estimated efficiency of Paramecium is surprisingly close to the theoretically possible optimum.

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.