Session U02: Biological Fluid Dynamics: Locomotion Flagella and Cilia (8:45am - 9:30am CST)

Flow transport of an instability-driven cilium

Cilia are hair-like micro-scale organelles. They usually exhibit self-oscillations which are crucial for flow transport in biological contexts. Some recent studies demonstrate that such motions could result from dynamic instabilities caused by dynein activities inside cilia. However, the capability of such instability-driven cilia in flow transport still remains unclear and this study aims at bridging this gap. Specifically, the cilium is represented by a flexible arc filament. The internal actuation is modelled as a constant follower force imposed at the cilium tip without or with a sinusoidally time-varying perturbation. With a well-established numerical solver based on the immersed boundary lattice Boltzmann method and the nonlinear finite element method, effects of some key parameters, including the follower-force strength and the perturbation magnitude and frequency on flow transport, are systematically explored. Preliminary results show that the flow transport is enhanced with the increasing follower force and that under some conditions the perturbation can trigger a lock-on of the beating frequency on the perturbation frequency. When this occurs, the flow transport can be increased if the perturbed beating frequency is larger than the unperturbed one.   

Symmetry breaking in hydrodynamically-coupled microfilaments

Biological microfilaments exhibit a variety of synchronization modes. Recent experiments revealed that a pair of isolated eukaryotic flagella, coupled solely via the fluid medium, display synchronization with different phase lags. Using an elasto-hydrodynamic filament model in conjunction with numerical simulations and a Floquet-type theoretical analysis, we showed that it is possible to reach synchronization states with multiple phase lags by varying the intrinsic activity of the filament and the strength of hydrodynamic coupling between the two filaments. In particular we found that non-trivial phase lag corresponds to asymmetric synchronization even though the activity of the two filaments is identical. We then derived an evolution equation for the phase difference between the two filaments at weak coupling, and used a Kuramoto-style phase sensitivity analysis to reveal the nature of the bifurcations underlying the transitions between these different synchronized states. Lastly, we analyzed the total hydrodynamic force on the coupled filaments and found that the total propulsive force depends on the filament activity but is independent of the synchronization modes, which could have significant implications in the locomotion of bi-flagellated cells.   

Optimal feeding of ciliated microorganisms in concentration gradients.

The flow field generated by ciliated microorganisms in a viscous fluid can influence the microorganisms' nutrient environment, and the stirred concentration field can correspondingly influence the microorganisms' nutrients uptake. We consider a classic model of ciliated organisms consisting of a spherical body with continuous wave-like surface deformations. This model, known as Blake's spherical envelope model or a squirmer model, emulates the action of tightly-packed distribution of cilia on the organism's surface. Using the ideal solution for the flow field around this spherical body in Stokes flow and solving the advection-diffusion equation numerically using the Legendre polynomial spectral method, we compute the concentration field and nutrient uptake of the microorganism. We analyze the nutrient uptake in a uniform background concentration and in a background concentration with constant gradient. In each case, we discuss the optimal ciliary strokes that maximize nutrient uptake.   

Efficient filament elastohydrodynamics in 3D

The coupled elastohydrodynamics of flexible slender filaments in a viscous fluid have long represented a significant computational challenge, with the inertia-free system exhibiting severe numerical stiffness. Recently, much of this stiffness was circumvented by integrating the governing equations of elasticity and imposing a simple discretisation, yielding a computational framework capable of filament simulation on a laptop computer in a matter of seconds. However, this approach has previously been limited to purely planar motion, reliant on a tangent angle parameterisation in order to be cast as ordinary differential equations that may be efficiently solved. In this talk, we will present an extension of this approach to the motion of inextensible unshearable filaments in three spatial dimensions, numerically avoiding the gimbal lock problem typically associated with Euler angles via adaptive basis selection. We will demonstrate by explicit example that this leads to an efficient framework for filament simulation in 3D, reducing computation times from hours on high performance computing clusters to seconds on a modest laptop, thus enabling thorough and large scale study into a range of biological and biophysical filament problems.   

Swimming sheet in a viscosity-stratified fluid

In this work, we theoretically investigate the motion of a Taylor swimming sheet immersed in a viscosity-stratified fluid. The propulsion of the swimmer disturbs the surrounding fluid which influences the transport of the stratifying agent described by the advection-diffusion equation. We employ a regular perturbation scheme to solve the coupled differential equations of motion up to the second order with the small parameter given by the ratio of the wave amplitude and the wavelength. The expression for the swimming velocity is linear in the magnitude of the viscosity gradient, while depending on the Peclet number in a non-monotonic way. Interestingly, we find that the Peclet number governs the propensity of the sheet to propel towards regions of favorable viscosities. For small Peclet numbers (0\textless Pe\textless 3), the swimmer prefers regions of low viscosity while for high Peclet numbers (Pe\textgreater 3), the swimmer prefers regions of high viscosity. Our analysis shows that purely hydrodynamic effects might be responsible for the experimentally observed accumulation of swimmers near favorable viscosity regions. We find that viscosity gradients influence other motility characteristics of the swimmer such as power expenditure, hydrodynamic efficiency and provide analytical expressions for both.   

A New Model for the Hydrodynamic Synchronisation of Helical Filaments

Multi-flagellated bacteria swim by rotating rigid helical filaments organised in coherent bundles. The filaments within the bundle rotate in synchrony, and it has long been postulated that hydrodynamic interactions facilitate this synchronisation. Using a combined analytical and computational approach, we derive from first principles a dynamical system that explains the synchronisation of two rotating helical filaments through hydrodynamic interactions. Our model, the first to explain mathematically a phenomenon revealed by computational studies more than a decade ago, brings to light key biophysical features of synchronisation, in particular the dependence on hook stiffness and flagellar geometry. We also consider the effect of polymorphism and number of flagella on the speed of synchronisation.   

Active trapping of microswimmers in a foam

Inspired by the consequences of aquatic foams on planktonic ecosystems, we have studied the sedimentation of a microswimmer in a liquid foam. The model unicellular bi-flagellated \textit{Chlamydomonas reinhardtii} (CR) algae was incorporated in a biocompatible foam, and the dynamics of cell sedimentation out of the foam was measured.\footnote{Roveillo et.al., 2020. J. R. Soc. Interface 17 : 20200077.} Due to gravity, the liquid in a freshly formed foam flows downwards, advecting solid particles. The cells eventually reach the underlying liquid, escaping the foam. Comparing the dynamics of living and dead CR cells in a draining foam, we found that dead cells were totally advected by the flow, as expected for passive solid particles of this size (10 $\mu$m). In contrast, living motile CR cells escape the foam significantly much slowly: two hours after the foam formation, a significant fraction of cells remained blocked in the foam, whereas 95\% of the volume of liquid initially contained in the foam the liquid was released. Microscopic observation of the swimming CR cells in a chamber mimicking the cross-section of a foam channel revealed that the microswimmers accumulate near channels corners, potentially increasing their retention in the foam.   

The hydrodynamically optimal length of a flagellum

Many bacteria use appendages termed flagella for swimming. A flagellum consists of a semi-rigid helical filament passively driven by a rotary motor via a short flexible hook. While the rich elasto-hydrodynamics of the propulsion process and the interactions between multiple flagellar filaments have been investigated theoretically, numerically and experimentally in many studies, the mechanism that determines the length of the flagellar filaments has so far been looked at mainly from a biological point of view, identifying key proteins that control their growth and length. But is there a physical rationale for the observed lengths of flagellar filaments observed in nature? One well-known answer lies in the requirement of a finite-size cell body, relative to flagellar filaments, in order to balance hydrodynamic moments on swimming bacteria. In this talk we propose that a second hydrodynamic mechanism can also set the optimal length of bacterial filaments.   

Hydrodynamic effects of mastigoneme distribution in Cryptomonad flagella

Cryptomonads are aquatic unicellular eukaryotes that inhabit both marine and freshwater environments worldwide and whose photosynthetic forms may be responsible for a large part of primary carbon production. Cryptomonads have flattened, elliptical cells and swim using two flagella that are known to bear rigid mastigonemes, fibrous ultrastructures of a nanometer-scale thickness. Previous studies have shown that the external structure of cryptomonad mastigonemes and their arrangement is relatively uniform. It has been claimed that by affecting the drag of flagella, they can affect swimming behavior --- even reversing swimming direction --- but recent work has not discerned hydrodynamic effects due to mastigonemes in \textit{Chlamydomonas}. In this study, we experimentally investigate flagellar kinematics and mastigoneme geometry of \textit{Chilomonas paramecium} through high-speed imaging of freely swimming cells, and SEM and TEM imaging. We then numerically investigate the potential hydrodynamic effects of \textit{Chilomonas paramecium }mastigonemes using the method of Regularized Stokeslets. We find that hydrodynamic effects are strongly affected by the distribution of mastigonemes; only two-dimensional arrangements of mastigonemes within the beating plane can reverse swimming direction.   

Proximal-to-distal Molecular Motor Asymmetry Controls Flagellar Wave Reversal

Beating eukaryotic flagella exhibit a wide range of waveforms despite their conserved `9+2' axonemal structure. While most spermatozoa propel head-first using base-to-tip flagellar waves, other organisms like the parasite Trypanosomes primarily move with forward-pointing flagella using tip-to-base flagellar waves. In addition, certain Trypanosomes can actively reverse the direction of propagation of their flagellar wave for fast reorientation. Although experiments on genetic mutants linked reversed beating to structural asymmetries of dynein motors along the flagellum length, the underlying physical mechanism of the active switching is not fully understood. By introducing proximal-to-distal asymmetry to molecular motor activities of a known geometric feedback model, we show such flagellar wave reversals are only possible if the dominant feedback mechanism is based on sliding-control. We conclude by commenting on the implications of our results to flagellar waveforms in other organisms, as well as the feasibility of a universal geometric feedback mechanism for explaining the diverse waveforms in cilia oscillations.   

Anisotropic bending rigidities in eukaryotic flagella can lead to stable planar beating dynamics

Flagella and cilia are common features of biological cells and play important roles in locomotion and feeding at the microscale. Flagellar beating is controlled by molecular motors that exert forces along the length of the flagellum and are regulated by a feedback mechanism coupled to the flagellar motion. Built on previous work on 2-dimensional (2D) flagella beating models, we develop a three-dimensional (3D) flagellum beating model on sliding-controlled motor feedback, accounting for both bending and twist --- the latter a feature that is absent in 2D models. We show that anisotropic bending rigidities in the flagellum lead to stable planar beating dynamics out of a 3D beating machinery, a feature shown in many eukaryotic flagella such as mammalian spermatozoa. Perturbation analysis also reveals the parameter regimes of stable non-beating, planar or spiral beating dynamics. Finally we show that with isotropic bending, stable planar beating is no more feasible and only spiral beating can be spontaneously generated beyond a critical molecular activity, with either retrograde or anterograde wave propagation depending on the activity.   

Synchronised states of hydrodynamically coupled filaments and their stability

Cilia and flagella are organelles central to fluid transport around tissues, unicellular locomotion, and in early mammalian development. They are observed to undulate, rotate, and beat symmetrically in pairs or even in large numbers via metachronal waves. Inspired by biflagellate swimmers like Chlamydomonas, we analyse regions of bistable synchrony exhibited in Stokes flow by a filament pair tethered to a rigid planar surface. In this study, we use a base-driven, geometric switch model to generate filament motion and establish the existence of two stable and two unstable branches of synchrony. An unstable anti-phase branch is characterised using Floquet analysis, while an edge state between two basins of attraction is found via a bisection algorithm to track the edge behaviour over time. We fully characterise a bifurcation diagram, the nature of the bifurcation points, and further find that the observed dynamical system can be captured by the development of a modified Adler equation.   

Hydrodynamic performance of macrorobots inspired by quadriflagellate algae

Animals can coordinate their appendages in rhythmic patterns known as gaits. While appendage coordination is often thought to be exclusive to macroscopic systems, microscopic quadriflagellate algae (body length of ~10 μm) have been found to coordinate their flagella to various patterns reminiscent of gaits seen in quadrupeds (Wan & Goldstein, 2016). To study appendage coordination of quadriflagellates, we developed a macroscopic robophysical model (body length of 3.87 cm) that swims in viscous fluid (mineral oil, 1,000 cSt), replicating low Reynolds number swimming . We focus on three gaits, the pronk, the trot, the gallop, and studied the effects of flagellar orientation. When the flagella were oriented perpendicular to the body, the robot achieved a speed of 0.020-0.1 body lengths per second depending on the gait. Results are comparable to microorganisms’ performance, where using the trot enables a higher speed than the pronk and the gallop. When the flagella were oriented parallel to the body, hydrodynamic performance decreased significantly for all gaits. The results show that hydrodynamic performance is sensitive to swimming gait and flagellar orientation, and suggest that diverse gaits may have evolved across different phyla as adaptations to distinct ecological habitats.   

Flagellated Janus particles for swimming and catalytic propulsion

Catalytic Janus particles will become a vital tool for developing new medical applications involving drug delivery and cellular penetration. Hydrogen peroxide decomposition enables catalytic Janus particles with platinum hemispheres and magnetic cores to self-propel while being guided by an externally applied magnetic field. While effective, having alternative propulsion mechanisms available would increase the utility of Janus particles in fuel-less environments. The flagellated Janus particles (FJPs) presented here were propelled using both catalysis and swimming locomotion induced by rotating magnetic fields. The FJPs consist of a magnetic core, a platinum hemisphere, and a flagellated hemisphere composed of bacterial flagella that were isolated from \textit{Salmonella typhimurium. }FJPs were suspended inside Newtonian fluids and actuated under both motion modes to characterize their velocity profiles. Responses to varying magnetic fields, mean square displacements, and trajectory following abilities using a proportional closed loop controller were also explored. Both motion modes were found to be similarly effective at propulsion and navigation. These are one of the first Janus particles developed to propel under two distinct motion modes and will be explored further for \textit{in vivo} medical applications.