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In this work, we develop a computational framework that aims at simultaneously optimizing the shape and the slip velocity of an axisymmetric microswimmer suspended in a viscous fluid. We consider shapes of a given reduced volume that maximize the swimming efficiency, i.e., the (size-independent) ratio of the power loss arising from towing the rigid body of the same shape and size at the same translation velocity to the actual power loss incurred by swimming via the slip velocity. The optimal slip and efficiency (with shape fixed) are here given in terms of two Stokes flow solutions, and we then establish shape sensitivity formulas of adjoint-solution that provide objective function derivatives with respect to any set of shape parameters on the sole basis of the above two flow solutions. Our computational treatment relies on a fast and accurate boundary integral solver for solving all Stokes flow problems. We validate our analytic shape derivative formulas via comparisons against finite-difference gradient evaluations, and present several shape optimization examples.

Many micro-swimmers propel themselves by rotating micro-cylindrical organelles such as flagella or cilia. These cylindrical organelles almost never live in free space, yet their motions in a confining geometry can be counter-intuitive. For example, one of the intriguing yet classical results in this regard is that a rotating cylinder next to a plane wall does not generate any net force in Newtonian fluids and therefore does not translate. In this work, we employ analytical and numerical tools to investigate the motions of micro-cylinders under prescribed torques in a confining geometry. We show that a cylinder pair can form four non-trivial hydrodynamic bound states depending on the relative position within the confinement. Our analysis shows that the distinct states are the results of competing effects of the hydrodynamic interactions within the cylinder pair and between the active cylinders and the confinement., Comment: 12 pages, 6 figures

Ciliated tissues such as in the mammalian lungs, brains, and reproductive tracts, are specialized to pump fluid. They generate flows by the collective activity of hundreds of thousands of individual cilia that beat in a striking metachronal wave pattern. Despite progress in analyzing cilia coordination, a general theory that links coordination and fluid pumping in the limit of large arrays of cilia remains lacking. Here, we conduct in-silico experiments with thousands of hydrodynamically-interacting cilia, and we develop a continuum theory in the limit of infinitely-many independently beating cilia by combining tools from active matter and classical Stokes flow. We find, in both simulations and theory, that isotropic and synchronized ciliary states are unstable. Traveling waves emerge regardless of initial conditions, but the characteristics of the wave and net flows depend on cilia and tissue properties. That is, metachronal phase coordination is a stable global attractor in large ciliary carpets, even under finite perturbations to cilia and tissue properties. These results support the notion that functional specificity of ciliated tissues is interlaced with the tissue architecture and cilia beat kinematics and open up the prospect of establishing structure-to-function maps from cilium-level beat to tissue-level coordination and fluid pumping., Comment: 11 pages, 5 figures, 1 SI document (pdf) and 2 video included, revision for publication 09/19/2022

Many biological microswimmers locomote by periodically beating the densely-packed cilia on their cell surface in a wave-like fashion. While the swimming mechanisms of ciliated microswimmers have been extensively studied both from the analytical and the numerical point of view, the optimization of the ciliary motion of microswimmers has received limited attention, especially for non-spherical shapes. In this paper, using an envelope model for the microswimmer, we numerically optimize the ciliary motion of a ciliate with an arbitrary axisymmetric shape. The forward solutions are found using a fast boundary integral method, and the efficiency sensitivities are derived using an adjoint-based method. Our results show that a prolate microswimmer with a 2:1 aspect ratio shares similar optimal ciliary motion as the spherical microswimmer, yet the swimming efficiency can increase two-fold. More interestingly, the optimal ciliary motion of a concave microswimmer can be qualitatively different from that of the spherical microswimmer, and adding a constraint to the ciliary length is found to improve, on average, the efficiency for such swimmers., Comment: 19 pages, 7 figures

Beating flagella exhibit a variety of synchronization modes. This synchrony has long been attributed to hydrodynamic coupling between the flagella. However, recent work with flagellated algae indicates that a mechanism internal to the cell, through the contractile fibres connecting the flagella basal bodies, must be at play to actively modulate flagellar synchrony. Exactly how basal coupling mediates flagellar coordination remains unclear. Here, we examine the role of basal coupling in the synchronization of the model biflagellate \textit{Chlamydomonas reinhardtii} using a series of mathematical models of decreasing level of complexity. We report that basal coupling is sufficient to achieve inphase, antiphase, and bistable synchrony, even in the absence of hydrodynamic coupling and flagellar compliance. These modes can be reached by modulating the activity level of the individual flagella or the strength of the basal coupling. We observe a slip mode when allowing for differential flagellar activity, just as in experiments with live cells. We introduce a dimensionless ratio of flagellar activity to basal coupling that is predictive of the mode of synchrony. This ratio allows us to query biological parameters which are not yet directly measurable experimentally. Our work shows a concrete route for cells to actively control the synchronization of their flagella., Comment: 16 pages, 5 figures

This article presents a computational approach for determining the optimal slip velocities on any given shape of an axisymmetric micro-swimmer suspended in a viscous fluid. The objective is to minimize the power loss to maintain a target swimming speed, or equivalently to maximize the efficiency of the micro-swimmer. Owing to the linearity of the Stokes equations governing the fluid motion, we show that this PDE-constrained optimization problem reduces to a simpler quadratic optimization problem, whose solution is found using a high-order accurate boundary integral method. We consider various families of shapes parameterized by the reduced volume and compute their swimming efficiency. {Among those, prolate spheroids were found to be the most efficient micro-swimmer shapes for a given reduced volume. We propose a simple shape-based scalar metric that can determine whether the optimal slip on a given shape makes it a pusher, a puller, or a neutral swimmer.}

Cilia and flagella are self-actuated microtubule-based structures that are present on many cell surfaces, ranging from the outer surface of single-cell organisms to the internal epithelial surfaces in larger animals. A fast and robust numerical method that simulates the coupled hydrodynamics of cilia and the constituent particles in the fluid such as rigid particles, drops or cells would be useful to not only understand several disease and developmental pathologies due to ciliary dysfunction but also to design microfluidic chips with ciliated cultures for some targeted functionality---e.g., maximizing fluid transport or particle mixing. In this paper, we develop a hybrid numerical method that employs a boundary integral method for handling the confining geometries and the constituent rigid particles and the method of regularized Stokeslets for handling the cilia. We provide several examples demonstrating the effects of confining geometries on cilia-generated fluid mixing as well as the cilia-particle hydrodynamics., Comment: 18 pages, 8 figures

Cilia and flagella are highly conserved slender organelles that exhibit a variety of rhythmic beating patterns from non-planar cone-like motions to planar wave-like deformations. Although their internal structure, composed of a microtubule-based axoneme driven by dynein motors, is known, the mechanism responsible for these beating patterns remains elusive. Existing theories suggest that the dynein activity is dynamically regulated, via a geometric feedback from the cilium's mechanical deformation to the dynein force. An alternative, open-loop mechanism based on a 'flutter' instability was recently proven to lead to planar oscillations of elastic filaments under follower forces. Here, we show that an elastic filament in viscous fluid, clamped at one end and acted on by an external distribution of compressive axial forces, exhibits a Hopf bifurcation that leads to non-planar spinning of the buckled filament at a locked curvature. We also show the existence of a second bifurcation, at larger force values, that induces a transition from non-planar spinning to planar wave-like oscillations. We elucidate the nature of these instabilities using a combination of nonlinear numerical analysis, linear stability theory, and low-order bead-spring models. Our results show that away from the transition thresholds, these beating patterns are robust to perturbations in the distribution of axial forces and in the filament configuration. These findings support the theory that an open-loop, instability-driven mechanism could explain both the sustained oscillations and the wide variety of periodic beating patterns observed in cilia and flagella., Comment: 16 pages, 14 figures, accepted by J. R. Soc. Interface

Cilia and flagella are essential building blocks for biological fluid transport and locomotion at the micron scale. They often beat in synchrony and may transition between different synchronization modes in the same cell type. Here, we investigate the behavior of elastic microfilaments, protruding from a surface and driven at their base by a configuration-dependent torque. We consider full hydrodynamic interactions among and within filaments and no slip at the surface. Isolated filaments exhibit periodic deformations, with increasing waviness and frequency as the magnitude of the driving torque increases. Two nearby but independently-driven filaments synchronize their beating in-phase or anti-phase. This synchrony arises autonomously via the interplay between hydrodynamic coupling and filament elasticity. Importantly, in-phase and anti-phase synchronization modes are bistable and co-exist for a range of driving torques and separation distances. These findings are consistent with experimental observations of in-phase and anti-phase synchronization in the biflagellate \textit{Chlamydomonas reinhardtii} and could have important implications on understanding the biophysical mechanisms underlying transitions between multiple synchronization modes., Comment: 26 pages, 11 figures

Mucociliary clearance in the lung is the primary defense mechanism that protects the airways from inhaled toxicants and infectious agents. The system consists of a viscoelastic mucus layer driven by motile cilia in a periciliary layer underneath the mucus layer. Under healthy conditions, the thickness of the periciliary layer is comparable to cilia length. Perturbations to this system, whether due to a genetic disorder or acquired causes, are directly linked to infection and disease. For example, depletion of the periciliary layer is typically observed in diseases such as chronic obstructive pulmonary disease and cystic fibrosis. Clinical evidence connects the periciliary layer depletion to reduced rates of mucus clearance. In this work, we develop a novel computational model to study mucociliary transport in a microfluidic channel consisting of a mucus layer (viscoelastic fluid) atop a periciliary layer (nearly-viscous fluid). We systematically vary the viscoelastic properties and thickness of the mucus layer to emulate healthy and diseased conditions. We assess cilia performance in terms of three metrics: flow transport, internal power expended by the cilia, and transport efficiency. We find that, compared to a control case with no mucus, a healthy mucus layer enhances cilia performance in all three metrics. That is to say, a healthy mucus layer not only improves flow transport, resulting in better clearance of harmful substances, but it does so at an energetic advantage to the cilia. In healthy states, an increase in the mucus elastic properties enhances transport efficiency. In contrast, under diseased conditions where the periciliary layer is depleted, mucus hinders transport... (trimmed by length restriction), Comment: 14 pages, 9 figures

Motile cilia are used by many eukaryotic cells to transport flow. Cilia-driven flows are important to many physiological functions, yet a deep understanding of the interplay between the mechanical structure of cilia and their physiological functions in healthy and diseased conditions remains elusive. For developing such understanding, one needs a quantitative framework for assessing cilia performance and robustness when subject to perturbations in the cilia apparatus. Here, we link cilia design (beating patterns) to function (flow transport) in the context of experimentally- and theoretically-derived cilia models. We particularly examine the optimality and robustness of cilia design. Optimality refers to efficiency of flow transport, while robustness is defined as low sensitivity to variations in the design parameters. We find that suboptimal designs can be more robust than optimal ones. That is, designing for the most efficient cilium does not guarantee robustness. These findings have significant implications on the understanding of cilia design in artificial and biological systems., Comment: 10 pages, 8 figures

We examine the hydrodynamic performance of two cilia beating patterns reconstructed from experimental data. In their respective natural systems, the two beating patterns correspond to: (A) pumping-specialized cilia, and (B) swimming-specialized cilia. We compare the performance of these two cilia beating patterns as a function of the metachronal coordination in the context of two model systems: the swimming of a ciliated cylinder and the fluid pumping by a ciliated carpet. Three performance measures are used for this comparison: (i) average swimming speed/pumping flow rate; (ii) maximum internal moments generated by the cilia; and (iii) swimming/pumping efficiencies. We found that, in both models, pattern (B) outperforms pattern (A) in almost all three measures, including hydrodynamic efficiency. These results challenge the notion that hydrodynamic efficiency dictates the cilia beating kinematics, and suggest that other biological functions and constraints play a role in explaining the wide variety of cilia beating patterns observed in biological systems., Comment: 15 pages, 8 figures