1
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Hu S, Meng F. Multiflagellate Swimming Controlled by Hydrodynamic Interactions. PHYSICAL REVIEW LETTERS 2024; 132:204002. [PMID: 38829103 DOI: 10.1103/physrevlett.132.204002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2023] [Revised: 02/18/2024] [Accepted: 04/16/2024] [Indexed: 06/05/2024]
Abstract
Many eukaryotic microorganisms propelled by multiple flagella can swim very rapidly with distinct gaits. Here, we model a three-dimensional mutiflagellate swimmer, resembling the microalgae. When the flagella are actuated synchronously, the swimming efficiency can be enhanced or reduced by interflagella hydrodynamic interactions (HIs), determined by the intrinsic tilting angle of the flagella. The asynchronous gait with a phase difference between neighboring flagella can reduce oscillatory motion via the basal mechanical coupling. In the presence of a spherical body, simulations taking into account the flagella-body interactions reveal the advantage of anterior configuration compared with posterior configuration, where in the latter case an optimal flagella number arises. Apart from understanding the role of HIs in the multiflagellate microorganisms, this work could also guide laboratory fabrications of novel microswimmers.
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Affiliation(s)
- Shiyuan Hu
- CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Fanlong Meng
- CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, 19A Yuquan Road, Beijing 100049, China
- Wenzhou Institute, University of Chinese Academy of Sciences, Wenzhou, Zhejiang 325000, China
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2
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Wan KY, Poon RN. Mechanisms and functions of multiciliary coordination. Curr Opin Cell Biol 2024; 86:102286. [PMID: 38035649 DOI: 10.1016/j.ceb.2023.102286] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/11/2023] [Revised: 10/20/2023] [Accepted: 11/02/2023] [Indexed: 12/02/2023]
Abstract
Ciliated organisms are present in virtually every branch of the eukaryotic tree of life. In diverse systems, cilia operate in a coordinated manner to drive fluid flows, or even propel entire organisms. How do groups of motile cilia coordinate their activity within a cell or across a tissue to fulfil essential functions of life? In this review, we highlight the latest developments in our understanding of the mechanisms and functions of multiciliary coordination in diverse systems. We explore new and emerging trends in bioimaging, analytical, and computational methods, which together with their application in new model systems, have conspired to deliver important insights into one of the most fundamental questions in cellular dynamics.
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Affiliation(s)
- Kirsty Y Wan
- Living Systems Institute, University of Exeter, Stocker Road, EX4 4QD, UK.
| | - Rebecca N Poon
- Living Systems Institute, University of Exeter, Stocker Road, EX4 4QD, UK
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3
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Faluweki MK, Cammann J, Mazza MG, Goehring L. Active Spaghetti: Collective Organization in Cyanobacteria. PHYSICAL REVIEW LETTERS 2023; 131:158303. [PMID: 37897773 DOI: 10.1103/physrevlett.131.158303] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2023] [Accepted: 08/30/2023] [Indexed: 10/30/2023]
Abstract
Filamentous cyanobacteria can show fascinating examples of nonequilibrium self-organization, which, however, are not well understood from a physical perspective. We investigate the motility and collective organization of colonies of these simple multicellular lifeforms. As their area density increases, linear chains of cells gliding on a substrate show a transition from an isotropic distribution to bundles of filaments arranged in a reticulate pattern. Based on our experimental observations of individual behavior and pairwise interactions, we introduce a nonreciprocal model accounting for the filaments' large aspect ratio, fluctuations in curvature, motility, and nematic interactions. This minimal model of active filaments recapitulates the observations, and rationalizes the appearance of a characteristic length scale in the system, based on the Péclet number of the cyanobacteria filaments.
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Affiliation(s)
- Mixon K Faluweki
- School of Science and Technology, Nottingham Trent University, Nottingham NG11 8NS, United Kingdom
- Malawi Institute of Technology, Malawi University of Science and Technology, S150 Road, Thyolo 310105, Malawi
| | - Jan Cammann
- Interdisciplinary Centre for Mathematical Modelling and Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom
| | - Marco G Mazza
- Interdisciplinary Centre for Mathematical Modelling and Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom
- Max Planck Institute for Dynamics and Self-Organization (MPIDS), Am Faßberg 17, 37077 Göttingen, Germany
| | - Lucas Goehring
- School of Science and Technology, Nottingham Trent University, Nottingham NG11 8NS, United Kingdom
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4
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Hickey DJ, Golestanian R, Vilfan A. Nonreciprocal interactions give rise to fast cilium synchronization in finite systems. Proc Natl Acad Sci U S A 2023; 120:e2307279120. [PMID: 37756336 PMCID: PMC10556628 DOI: 10.1073/pnas.2307279120] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2023] [Accepted: 08/14/2023] [Indexed: 09/29/2023] Open
Abstract
Motile cilia beat in an asymmetric fashion in order to propel the surrounding fluid. When many cilia are located on a surface, their beating can synchronize such that their phases form metachronal waves. Here, we computationally study a model where each cilium is represented as a spherical particle, moving along a tilted trajectory with a position-dependent active driving force and a position-dependent internal drag coefficient. The model thus takes into account all the essential broken symmetries of the ciliary beat. We show that taking into account the near-field hydrodynamic interactions, the effective coupling between cilia even over an entire beating cycle can become nonreciprocal: The phase of a cilium is more strongly affected by an adjacent cilium on one side than by a cilium at the same distance in the opposite direction. As a result, synchronization starts from a seed at the edge of a group of cilia and propagates rapidly across the system, leading to a synchronization time that scales proportionally to the linear dimension of the system. We show that a ciliary carpet is characterized by three different velocities: the velocity of fluid transport, the phase velocity of metachronal waves, and the group velocity of order propagation. Unlike in systems with reciprocal coupling, boundary effects are not detrimental for synchronization, but rather enable the formation of the initial seed.
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Affiliation(s)
- David J. Hickey
- Department of Living Matter Physics, Max Planck Institute for Dynamics and Self-Organization, 37077Göttingen, Germany
| | - Ramin Golestanian
- Department of Living Matter Physics, Max Planck Institute for Dynamics and Self-Organization, 37077Göttingen, Germany
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, OxfordOX1 3PU, United Kingdom
| | - Andrej Vilfan
- Department of Living Matter Physics, Max Planck Institute for Dynamics and Self-Organization, 37077Göttingen, Germany
- Department of Condensed Matter Physics, Jožef Stefan Institute, 1000Ljubljana, Slovenia
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5
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Quillen AC, Peshkov A, Chakrabarti B, Skerrett N, McGaffigan S, Zapiach R. Fluid circulation driven by collectively organized metachronal waves in swimming T. aceti nematodes. Phys Rev E 2022; 106:064401. [PMID: 36671190 DOI: 10.1103/physreve.106.064401] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2022] [Accepted: 11/17/2022] [Indexed: 06/17/2023]
Abstract
Recent experiments have shown that the nematode T. aceti can assemble into collectively undulating groups at the edge of fluid drops. This coordinated state consists of metachronal waves and drives fluid circulation inside the drop. We find that the circulation velocity is about 2 mm/s and nearly half the speed of the metachronal wave. We develop a quasi-two-dimensional hydrodynamics model using the Stokes flow approximation. The periodic motion of the nematodes constitute our moving boundary condition that drives the flow. Our model suggests that large-amplitude excursions of the nematode tails produce the fluid circulation. We discuss the constraints on containers that would enhance fluid motion, which could be used in the future design of on demand flow generating systems.
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Affiliation(s)
- A C Quillen
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
| | - A Peshkov
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
| | - Brato Chakrabarti
- Center for Computational Biology, Flatiron Institute, New York, New York 10010, USA
| | - Nathan Skerrett
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
| | - Sonia McGaffigan
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
| | - Rebeca Zapiach
- Department of Mechanical Engineering, University of Rochester, Rochester, New York 14627, USA
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Tătulea-Codrean M, Lauga E. Elastohydrodynamic Synchronization of Rotating Bacterial Flagella. PHYSICAL REVIEW LETTERS 2022; 128:208101. [PMID: 35657856 DOI: 10.1103/physrevlett.128.208101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/08/2021] [Revised: 02/03/2022] [Accepted: 03/25/2022] [Indexed: 06/15/2023]
Abstract
To rotate continuously without jamming, the flagellar filaments of bacteria need to be locked in phase. While several models have been proposed for eukaryotic flagella, the synchronization of bacterial flagella is less well understood. Starting from a reduced model of flexible and hydrodynamically coupled bacterial flagella, we rigorously coarse grain the equations of motion using the method of multiple scales, and hence show that bacterial flagella generically synchronize to zero phase difference via an elastohydrodynamic mechanism. Remarkably, the far-field rate of synchronization is maximized at an intermediate value of elastic compliance, with surprising implications for bacteria.
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Affiliation(s)
- Maria Tătulea-Codrean
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom
- Collège de France, 11 place Marcelin Berthelot, 75005 Paris, France
| | - Eric Lauga
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom
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7
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Zhou W, Peralta JD, Hao Z, Gravish N. Lateral contact yields longitudinal cohesion in active undulatory systems. Phys Rev E 2022; 105:054604. [PMID: 35706245 DOI: 10.1103/physreve.105.054604] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/10/2021] [Accepted: 04/11/2022] [Indexed: 06/15/2023]
Abstract
Many animals and robots move using undulatory motion of their bodies. When the bodies are in close proximity undulatory motion can lead to novel collective behavior such as gait synchronization, spatial reconfiguration, and clustering. Here we study the role of contact interactions between model undulatory swimmers: three-link robots in experiment and multilink swimmers in simulation. The undulatory gait of each swimmer is generated through a time-dependent sinusoidal-like waveform which has a fixed phase offset, ϕ. By varying the phase relationship between neighboring swimmers we seek to study how contact forces and planar configurations are governed by the phase difference between neighboring swimmers. We find that undulatory actuation in close proximity drives neighboring swimmers into planar equilibrium configurations that depend on the actuation phase difference. We propose a model for stable planar configurations of nearest-neighbor undulatory swimmers which we call the gait compatibility condition, which is the set of planar and phase configurations in which no collisions occur. Robotic experiments with two, three, and four swimmers exhibit good agreement with the compatibility model. To study the contact forces and the time-averaged equilibrium between undulatory systems we perform simulations. To probe the interaction potential between undulatory swimmers we apply a small force to each swimmer longitudinally to separate them from the compatible configuration and we measure their steady-state displacement. These studies reveal that undulatory swimmers in close proximity exhibit attractive longitudinal interaction forces that drive the swimmers from incompatible to compatible configurations. This system of undulatory swimmers provides new insight into active-matter systems which move through body undulation. In addition to the importance of velocity and orientation coherence in active-matter swarms, we demonstrate that undulatory phase coherence is also important for generating stable, cohesive group configurations.
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Affiliation(s)
- Wei Zhou
- Mechanical and Aerospace Engineering, University of California, San Diego, California 92093, USA
| | - Jaquelin Dezha Peralta
- Mechanical and Aerospace Engineering, University of California, San Diego, California 92093, USA
| | - Zhuonan Hao
- Mechanical and Aerospace Engineering, University of California, San Diego, California 92093, USA
| | - Nick Gravish
- Mechanical and Aerospace Engineering, University of California, San Diego, California 92093, USA
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8
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Hao Z, Zhou W, Gravish N. Proprioceptive feedback design for gait synchronization in collective undulatory robots. Adv Robot 2022. [DOI: 10.1080/01691864.2022.2050810] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/01/2022]
Affiliation(s)
- Zhuonan Hao
- Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA, USA
| | - Wei Zhou
- Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA, USA
| | - Nick Gravish
- Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA, USA
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Chakrabarti B, Fürthauer S, Shelley MJ. A multiscale biophysical model gives quantized metachronal waves in a lattice of beating cilia. Proc Natl Acad Sci U S A 2022; 119:e2113539119. [PMID: 35046031 PMCID: PMC8795537 DOI: 10.1073/pnas.2113539119] [Citation(s) in RCA: 15] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2021] [Accepted: 12/02/2021] [Indexed: 11/18/2022] Open
Abstract
Motile cilia are slender, hair-like cellular appendages that spontaneously oscillate under the action of internal molecular motors and are typically found in dense arrays. These active filaments coordinate their beating to generate metachronal waves that drive long-range fluid transport and locomotion. Until now, our understanding of their collective behavior largely comes from the study of minimal models that coarse grain the relevant biophysics and the hydrodynamics of slender structures. Here we build on a detailed biophysical model to elucidate the emergence of metachronal waves on millimeter scales from nanometer-scale motor activity inside individual cilia. Our study of a one-dimensional lattice of cilia in the presence of hydrodynamic and steric interactions reveals how metachronal waves are formed and maintained. We find that, in homogeneous beds of cilia, these interactions lead to multiple attracting states, all of which are characterized by an integer charge that is conserved. This even allows us to design initial conditions that lead to predictable emergent states. Finally, and very importantly, we show that, in nonuniform ciliary tissues, boundaries and inhomogeneities provide a robust route to metachronal waves.
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Affiliation(s)
- Brato Chakrabarti
- Center for Computational Biology, Flatiron Institute, New York, NY 10010
| | - Sebastian Fürthauer
- Center for Computational Biology, Flatiron Institute, New York, NY 10010;
- Institute of Applied Physics, TU Wien, Vienna 1040, Austria
| | - Michael J Shelley
- Center for Computational Biology, Flatiron Institute, New York, NY 10010;
- Courant Institute, New York University, New York, NY 10012
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10
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Quillen AC, Peshkov A, Wright E, McGaffigan S. Metachronal waves in concentrations of swimming Turbatrix aceti nematodes and an oscillator chain model for their coordinated motions. Phys Rev E 2021; 104:014412. [PMID: 34412226 DOI: 10.1103/physreve.104.014412] [Citation(s) in RCA: 14] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/04/2021] [Accepted: 05/14/2021] [Indexed: 01/23/2023]
Abstract
At high concentration, free swimming nematodes known as vinegar eels (Turbatrix aceti), collectively exhibit metachronal waves near a boundary. We find that the frequency of the collective traveling wave is lower than that of the freely swimming organisms. We explore models based on a chain of oscillators with nearest-neighbor interactions that inhibit oscillator phase velocity. The phase of each oscillator represents the phase of the motion of the eel's head back and forth about its mean position. A strongly interacting directed chain model mimicking steric repulsion between organisms robustly gives traveling wave states and can approximately match the observed wavelength and oscillation frequency of the observed traveling wave. We predict body shapes assuming that waves propagate down the eel body at a constant speed. The phase oscillator model that impedes eel head overlaps also reduces close interactions throughout the eel bodies.
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Affiliation(s)
- A C Quillen
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
| | - A Peshkov
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
| | - Esteban Wright
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
| | - Sonia McGaffigan
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
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11
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Fatehiboroujeni S, Gopinath A, Goyal S. Three-dimensional nonlinear dynamics of prestressed active filaments: Flapping, swirling, and flipping. Phys Rev E 2021; 103:013005. [PMID: 33601644 DOI: 10.1103/physreve.103.013005] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/20/2020] [Accepted: 12/24/2020] [Indexed: 11/07/2022]
Abstract
Initially straight slender elastic filaments or rods with constrained ends buckle and form stable two-dimensional shapes when prestressed by bringing the ends together. Beyond a critical value of this prestress, rods can also deform off plane and form twisted three-dimensional equilibrium shapes. Here, we analyze the three-dimensional instabilities and dynamics of such deformed filaments subject to nonconservative active follower forces and fluid drag. We find that softly constrained filaments that are clamped at one end and pinned at the other exhibit stable two-dimensional planar flapping oscillations when active forces are directed toward the clamped end. Reversing the directionality of the forces quenches the instability. For strongly constrained filaments with both ends clamped, computations reveal an instability arising from the twist-bend-activity coupling. Planar oscillations are destabilized by off-planar perturbations resulting in twisted three-dimensional swirling patterns interspersed with periodic flipping or reversal of the swirling direction. These striking swirl-flip transitions are characterized by two distinct timescales: the time period for a swirl (rotation) and the time between flipping events. We interpret these reversals as relaxation oscillation events driven by accumulation of torsional energy. Each cycle is initiated by a fast jump in torsional deformation with a subsequent slow decrease in net torsion until the next cycle. Our work reveals the rich tapestry of spatiotemporal patterns when weakly inertial strongly damped rods are deformed by nonconservative active forces. Taken together, our results suggest avenues by which prestress, elasticity, and activity may be used to design synthetic macroscale pumps or mixers.
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Affiliation(s)
- Soheil Fatehiboroujeni
- Department of Mechanical Engineering, University of California, Merced, California 95343, USA
| | - Arvind Gopinath
- Department of Bioengineering, University of California, Merced, California 95343, USA
| | - Sachin Goyal
- Department of Mechanical Engineering, University of California, Merced, California 95343, USA and Health Sciences Research Institute, University of California, Merced, California 95343, USA
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