1
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Dey S, Massiera G, Pitard E. Role of cilia activity and surrounding viscous fluid in properties of metachronal waves. Phys Rev E 2024; 110:014409. [PMID: 39160939 DOI: 10.1103/physreve.110.014409] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2023] [Accepted: 06/27/2024] [Indexed: 08/21/2024]
Abstract
Large groups of active cilia collectively beat in a fluid medium as metachronal waves, essential for some microorganisms motility and for flow generation in mucociliary clearance. Several models can predict the emergence of metachronal waves, but what controls the properties of metachronal waves is still unclear. Here, we numerically investigate the respective impacts of active beating and viscous dissipation on the properties of metachronal waves in a collection of oscillators, using a simple model for cilia in the presence of noise on regular lattices in one and two dimensions. We characterize the wave using spatial correlation and the frequency of collective beating. Our results clearly show that the viscosity of the fluid medium does not affect the wavelength; the activity of the cilia does. These numerical results are supported by a dimensional analysis, which shows that the result of wavelength invariance is robust against the model taken for sustained beating and the structure of hydrodynamic coupling. Interestingly, the enhancement of cilia activity increases the wavelength and decreases the beating frequency, keeping the wave velocity almost unchanged. These results might have significance in understanding paramecium locomotion and mucociliary clearance diseases.
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2
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Wei D, Quaranta G, Aubin-Tam ME, Tam DSW. The younger flagellum sets the beat for Chlamydomonas reinhardtii. eLife 2024; 13:e86102. [PMID: 38752724 PMCID: PMC11098555 DOI: 10.7554/elife.86102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2023] [Accepted: 04/03/2024] [Indexed: 05/18/2024] Open
Abstract
Eukaryotes swim with coordinated flagellar (ciliary) beating and steer by fine-tuning the coordination. The model organism for studying flagellate motility, Chlamydomonas reinhardtii, employs synchronous, breaststroke-like flagellar beating to swim, and it modulates the beating amplitudes differentially to steer. This strategy hinges on both inherent flagellar asymmetries (e.g. different response to chemical messengers) and such asymmetries being effectively coordinated in the synchronous beating. In C. reinhardtii, the synchrony of beating is known to be supported by a mechanical connection between flagella; however, how flagellar asymmetries persist in the synchrony remains elusive. For example, it has been speculated for decades that one flagellum leads the beating, as its dynamic properties (i.e. frequency, waveform, etc.) appear to be copied by the other one. In this study, we combine experiments, computations, and modeling efforts to elucidate the roles played by each flagellum in synchronous beating. With a non-invasive technique to selectively load each flagellum, we show that the coordinated beating essentially only responds to load exerted on the cis flagellum; and that such asymmetry in response derives from a unilateral coupling between the two flagella. Our results highlight a distinct role for each flagellum in coordination and have implication for biflagellates' tactic behaviors.
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Affiliation(s)
- Da Wei
- Department of Bionanoscience, Delft University of TechnologyDelftNetherlands
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of SciencesBeijingChina
| | - Greta Quaranta
- Laboratory for Aero and Hydrodynamics, Delft University of TechnologyDelftNetherlands
| | - Marie-Eve Aubin-Tam
- Department of Bionanoscience, Delft University of TechnologyDelftNetherlands
| | - Daniel SW Tam
- Laboratory for Aero and Hydrodynamics, Delft University of TechnologyDelftNetherlands
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3
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Bondoc-Naumovitz KG, Laeverenz-Schlogelhofer H, Poon RN, Boggon AK, Bentley SA, Cortese D, Wan KY. Methods and Measures for Investigating Microscale Motility. Integr Comp Biol 2023; 63:1485-1508. [PMID: 37336589 PMCID: PMC10755196 DOI: 10.1093/icb/icad075] [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: 02/28/2023] [Revised: 05/31/2023] [Accepted: 06/06/2023] [Indexed: 06/21/2023] Open
Abstract
Motility is an essential factor for an organism's survival and diversification. With the advent of novel single-cell technologies, analytical frameworks, and theoretical methods, we can begin to probe the complex lives of microscopic motile organisms and answer the intertwining biological and physical questions of how these diverse lifeforms navigate their surroundings. Herein, we summarize the main mechanisms of microscale motility and give an overview of different experimental, analytical, and mathematical methods used to study them across different scales encompassing the molecular-, individual-, to population-level. We identify transferable techniques, pressing challenges, and future directions in the field. This review can serve as a starting point for researchers who are interested in exploring and quantifying the movements of organisms in the microscale world.
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Affiliation(s)
| | | | - Rebecca N Poon
- Living Systems Institute, University of Exeter, Stocker Road, EX4 4QD, Exeter, UK
| | - Alexander K Boggon
- Living Systems Institute, University of Exeter, Stocker Road, EX4 4QD, Exeter, UK
| | - Samuel A Bentley
- Living Systems Institute, University of Exeter, Stocker Road, EX4 4QD, Exeter, UK
| | - Dario Cortese
- Living Systems Institute, University of Exeter, Stocker Road, EX4 4QD, Exeter, UK
| | - Kirsty Y Wan
- Living Systems Institute, University of Exeter, Stocker Road, EX4 4QD, Exeter, UK
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4
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Wan KY. Active oscillations in microscale navigation. Anim Cogn 2023; 26:1837-1850. [PMID: 37665482 PMCID: PMC10769930 DOI: 10.1007/s10071-023-01819-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/22/2023] [Revised: 07/27/2023] [Accepted: 08/12/2023] [Indexed: 09/05/2023]
Abstract
Living organisms routinely navigate their surroundings in search of better conditions, more food, or to avoid predators. Typically, animals do so by integrating sensory cues from the environment with their locomotor apparatuses. For single cells or small organisms that possess motility, fundamental physical constraints imposed by their small size have led to alternative navigation strategies that are specific to the microscopic world. Intriguingly, underlying these myriad exploratory behaviours or sensory functions is the onset of periodic activity at multiple scales, such as the undulations of cilia and flagella, the vibrations of hair cells, or the oscillatory shape modes of migrating neutrophils. Here, I explore oscillatory dynamics in basal microeukaryotes and hypothesize that these active oscillations play a critical role in enhancing the fidelity of adaptive sensorimotor integration.
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Affiliation(s)
- Kirsty Y Wan
- Living Systems Institute, University of Exeter, Stocker Road, Exeter, EX4 4QD, UK.
- Department of Mathematics and Statistics, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK.
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5
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Leptos KC, Chioccioli M, Furlan S, Pesci AI, Goldstein RE. Phototaxis of Chlamydomonas arises from a tuned adaptive photoresponse shared with multicellular Volvocine green algae. Phys Rev E 2023; 107:014404. [PMID: 36797913 PMCID: PMC7616094 DOI: 10.1103/physreve.107.014404] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2022] [Accepted: 12/16/2022] [Indexed: 06/18/2023]
Abstract
A fundamental issue in biology is the nature of evolutionary transitions from unicellular to multicellular organisms. Volvocine algae are models for this transition, as they span from the unicellular biflagellate Chlamydomonas to multicellular species of Volvox with up to 50,000 Chlamydomonas-like cells on the surface of a spherical extracellular matrix. The mechanism of phototaxis in these species is of particular interest since they lack a nervous system and intercellular connections; steering is a consequence of the response of individual cells to light. Studies of Volvox and Gonium, a 16-cell organism with a plate-like structure, have shown that the flagellar response to changing illumination of the cellular photosensor is adaptive, with a recovery time tuned to the rotation period of the colony around its primary axis. Here, combining high-resolution studies of the flagellar photoresponse of micropipette-held Chlamydomonas with 3D tracking of freely swimming cells, we show that such tuning also underlies its phototaxis. A mathematical model is developed based on the rotations around an axis perpendicular to the flagellar beat plane that occur through the adaptive response to oscillating light levels as the organism spins. Exploiting a separation of timescales between the flagellar photoresponse and phototurning, we develop an equation of motion that accurately describes the observed photoalignment. In showing that the adaptive timescales in Volvocine algae are tuned to the organisms' rotational periods across three orders of magnitude in cell number, our results suggest a unified picture of phototaxis in green algae in which the asymmetry in torques that produce phototurns arise from the individual flagella of Chlamydomonas, the flagellated edges of Gonium, and the flagellated hemispheres of Volvox.
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Affiliation(s)
- Kyriacos C. Leptos
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
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6
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Ishimoto K, Moreau C, Yasuda K. Self-organized swimming with odd elasticity. Phys Rev E 2022; 105:064603. [PMID: 35854482 DOI: 10.1103/physreve.105.064603] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2021] [Accepted: 03/23/2022] [Indexed: 06/15/2023]
Abstract
We theoretically investigate self-oscillating waves of an active material, which were recently introduced as a nonsymmetric part of the elastic moduli, termed odd elasticity. Using Purcell's three-link swimmer model, we reveal that an odd-elastic filament at low Reynolds number can swim in a self-organized manner and that the time-periodic dynamics are characterized by a stable limit cycle generated by elastohydrodynamic interactions. Also, we consider a noisy shape gait and derive a swimming formula for a general elastic material in the Stokes regime with its elasticity modulus being represented by a nonsymmetric matrix, demonstrating that the odd elasticity produces biased net locomotion from random noise.
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Affiliation(s)
- Kenta Ishimoto
- Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
| | - Clément Moreau
- Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
| | - Kento Yasuda
- Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
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7
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Fleig P, Kramar M, Wilczek M, Alim K. Emergence of behaviour in a self-organized living matter network. eLife 2022; 11:62863. [PMID: 35060901 PMCID: PMC8782570 DOI: 10.7554/elife.62863] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/07/2020] [Accepted: 12/20/2021] [Indexed: 12/16/2022] Open
Abstract
What is the origin of behaviour? Although typically associated with a nervous system, simple organisms also show complex behaviours. Among them, the slime mold Physarum polycephalum, a giant single cell, is ideally suited to study emergence of behaviour. Here, we show how locomotion and morphological adaptation behaviour emerge from self-organized patterns of rhythmic contractions of the actomyosin lining of the tubes making up the network-shaped organism. We quantify the spatio-temporal contraction dynamics by decomposing experimentally recorded contraction patterns into spatial contraction modes. Notably, we find a continuous spectrum of modes, as opposed to a few dominant modes. Our data suggests that the continuous spectrum of modes allows for dynamic transitions between a plethora of specific behaviours with transitions marked by highly irregular contraction states. By mapping specific behaviours to states of active contractions, we provide the basis to understand behaviour’s complexity as a function of biomechanical dynamics.
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Affiliation(s)
- Philipp Fleig
- Department of Physics & Astronomy, University of Pennsylvania
- Max Planck Institute for Dynamics and Self-Organization
| | - Mirna Kramar
- Max Planck Institute for Dynamics and Self-Organization
| | | | - Karen Alim
- Max Planck Institute for Dynamics and Self-Organization
- Physik-Department and Center for Protein Assemblies, Technische Universität München
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8
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Solovev A, Friedrich BM. Synchronization in cilia carpets and the Kuramoto model with local coupling: Breakup of global synchronization in the presence of noise. CHAOS (WOODBURY, N.Y.) 2022; 32:013124. [PMID: 35105113 DOI: 10.1063/5.0075095] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2021] [Accepted: 12/13/2021] [Indexed: 06/14/2023]
Abstract
Carpets of beating cilia represent a paradigmatic example of self-organized synchronization of noisy biological oscillators, characterized by traveling waves of cilia phase. We present a multi-scale model of a cilia carpet that comprises realistic hydrodynamic interactions between cilia computed for a chiral cilia beat pattern from unicellular Paramecium and active noise of the cilia beat. We demonstrate an abrupt loss of global synchronization beyond a characteristic noise strength. We characterize stochastic transitions between synchronized and disordered dynamics, which generalize the notion of phase slips in pairs of coupled noisy phase oscillators. Our theoretical work establishes a link between the two-dimensional Kuramoto model of phase oscillators with mirror-symmetric oscillator coupling and detailed models of biological oscillators with asymmetric, chiral interactions.
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9
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Abstract
Place a drop of pond water under the microscope, and you will likely find an ocean of extraordinary and diverse single-celled organisms called ciliates. This remarkable group of single-celled organisms wield microtubules, active systems, electrical signaling, and chemical sensors to build intricate geometrical structures and perform complex behaviors that can appear indistinguishable from those of macroscopic animals. Advances in computer vision and machine learning are making it possible to completely digitize and track the dynamics of complex ciliates and mine these data for the hidden structure, patterns, and motifs that are responsible for their behaviors. By deconstructing the diversity of ciliate behaviors in the natural world, themes for organizing and controlling matter at the microscale are beginning to take hold, suggesting new modular approaches for the design of autonomous molecular machines that emulate nature’s finest examples.
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Affiliation(s)
- Scott M Coyle
- Department of Biochemistry, University of Wisconsin, Madison, Madison, Wisconsin 53706
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10
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Circular swimming motility and disordered hyperuniform state in an algae system. Proc Natl Acad Sci U S A 2021; 118:2100493118. [PMID: 33931505 DOI: 10.1073/pnas.2100493118] [Citation(s) in RCA: 22] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Active matter comprises individually driven units that convert locally stored energy into mechanical motion. Interactions between driven units lead to a variety of nonequilibrium collective phenomena in active matter. One of such phenomena is anomalously large density fluctuations, which have been observed in both experiments and theories. Here we show that, on the contrary, density fluctuations in active matter can also be greatly suppressed. Our experiments are carried out with marine algae ([Formula: see text]), which swim in circles at the air-liquid interfaces with two different eukaryotic flagella. Cell swimming generates fluid flow that leads to effective repulsions between cells in the far field. The long-range nature of such repulsive interactions suppresses density fluctuations and generates disordered hyperuniform states under a wide range of density conditions. Emergence of hyperuniformity and associated scaling exponent are quantitatively reproduced in a numerical model whose main ingredients are effective hydrodynamic interactions and uncorrelated random cell motion. Our results demonstrate the existence of disordered hyperuniform states in active matter and suggest the possibility of using hydrodynamic flow for self-assembly in active matter.
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11
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Jones C, Gomez M, Muoio RM, Vidal A, Mcknight RA, Brubaker ND, Ahmed WW. Stochastic force dynamics of the model microswimmer Chlamydomonas reinhardtii: Active forces and energetics. Phys Rev E 2021; 103:032403. [PMID: 33862745 DOI: 10.1103/physreve.103.032403] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2020] [Accepted: 02/03/2021] [Indexed: 01/26/2023]
Abstract
We study the stochastic force dynamics of a model microswimmer (Chlamydomonas reinhardtii), using a combined experimental, theoretical, and numerical approach. While swimming dynamics have been extensively studied using hydrodynamic approaches, which infer forces from the viscous flow field, we directly measure the stochastic forces generated by the microswimmer using an optical trap via the photon momentum method. We analyze the force dynamics by modeling the microswimmer as a self-propelled particle, à la active matter, and analyze its energetics using methods from stochastic thermodynamics. We find complex oscillatory force dynamics and power dissipation on the order of 10^{6}k_{B}T/s(∼fW).
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Affiliation(s)
- Corbyn Jones
- Department of Physics, California State University Fullerton, California 92831, USA
| | - Mauricio Gomez
- Department of Physics, California State University Fullerton, California 92831, USA
| | - Ryan M Muoio
- Department of Physics, California State University Fullerton, California 92831, USA
| | - Alex Vidal
- Department of Computer Science, California State University Fullerton, California 92831, USA
| | - R Anthony Mcknight
- Department of Biomedical Engineering, University of Virginia, Charlottesville, Virginia 22904, USA
| | - Nicholas D Brubaker
- Department of Mathematics, California State University Fullerton, California 92831, USA
| | - Wylie W Ahmed
- Department of Physics, California State University Fullerton, California 92831, USA
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12
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Solovev A, Friedrich BM. Lagrangian mechanics of active systems. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2021; 44:49. [PMID: 33834308 PMCID: PMC8032648 DOI: 10.1140/epje/s10189-021-00016-x] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/30/2020] [Accepted: 01/12/2021] [Indexed: 05/26/2023]
Abstract
We present a multi-scale modeling and simulation framework for low-Reynolds number hydrodynamics of shape-changing immersed objects, e.g., biological microswimmers and active surfaces. The key idea is to consider principal shape changes as generalized coordinates and define conjugate generalized hydrodynamic friction forces. Conveniently, the corresponding generalized friction coefficients can be pre-computed and subsequently reused to solve dynamic equations of motion fast. This framework extends Lagrangian mechanics of dissipative systems to active surfaces and active microswimmers, whose shape dynamics is driven by internal forces. As an application case, we predict in-phase and anti-phase synchronization in pairs of cilia for an experimentally measured cilia beat pattern.
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13
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Hamilton E, Cicuta P. Changes in geometrical aspects of a simple model of cilia synchronization control the dynamical state, a possible mechanism for switching of swimming gaits in microswimmers. PLoS One 2021; 16:e0249060. [PMID: 33831025 PMCID: PMC8031381 DOI: 10.1371/journal.pone.0249060] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2020] [Accepted: 03/10/2021] [Indexed: 12/23/2022] Open
Abstract
Active oscillators, with purely hydrodynamic coupling, are useful simple models to understand various aspects of motile cilia synchronization. Motile cilia are used by microorganisms to swim and to control the flow fields in their surroundings; the patterns observed in cilia carpets can be remarkably complex, and can be changed over time by the organism. It is often not known to what extent the coupling between cilia is due to just hydrodynamic forces, and neither is it known if it is biological or physical triggers that can change the dynamical collective state. Here we treat this question from a very simplified point of view. We describe three possible mechanisms that enable a switch in the dynamical state, in a simple scenario of a chain of oscillators. We find that shape-change provides the most consistent strategy to control collective dynamics, but also imposing small changes in frequency produces some unique stable states. Demonstrating these effects in the abstract minimal model proves that these could be possible explanations for gait switching seen in ciliated micro organisms like Paramecium and others. Microorganisms with many cilia could in principle be taking advantage of hydrodynamic coupling, to switch their swimming gait through either a shape change that manifests in decreased coupling between groups of cilia, or alterations to the beat style of a small subset of the cilia.
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Affiliation(s)
- Evelyn Hamilton
- Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
| | - Pietro Cicuta
- Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
- * E-mail:
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14
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Abstract
All living cells interact dynamically with a constantly changing world. Eukaryotes, in particular, evolved radically new ways to sense and react to their environment. These advances enabled new and more complex forms of cellular behaviour in eukaryotes, including directional movement, active feeding, mating, and responses to predation. But what are the key events and innovations during eukaryogenesis that made all of this possible? Here we describe the ancestral repertoire of eukaryotic excitability and discuss five major cellular innovations that enabled its evolutionary origin. The innovations include a vastly expanded repertoire of ion channels, the emergence of cilia and pseudopodia, endomembranes as intracellular capacitors, a flexible plasma membrane and the relocation of chemiosmotic ATP synthesis to mitochondria, which liberated the plasma membrane for more complex electrical signalling involved in sensing and reacting. We conjecture that together with an increase in cell size, these new forms of excitability greatly amplified the degrees of freedom associated with cellular responses, allowing eukaryotes to vastly outperform prokaryotes in terms of both speed and accuracy. This comprehensive new perspective on the evolution of excitability enriches our view of eukaryogenesis and emphasizes behaviour and sensing as major contributors to the success of eukaryotes. This article is part of the theme issue 'Basal cognition: conceptual tools and the view from the single cell'.
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Affiliation(s)
- Kirsty Y. Wan
- Living Systems Institute, University of Exeter, Stocker Road, Exeter EX4 4QD, UK
| | - Gáspár Jékely
- Living Systems Institute, University of Exeter, Stocker Road, Exeter EX4 4QD, UK
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15
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Cortese D, Wan KY. Control of Helical Navigation by Three-Dimensional Flagellar Beating. PHYSICAL REVIEW LETTERS 2021; 126:088003. [PMID: 33709750 DOI: 10.1103/physrevlett.126.088003] [Citation(s) in RCA: 14] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/27/2020] [Revised: 12/10/2020] [Accepted: 01/13/2021] [Indexed: 06/12/2023]
Abstract
Helical swimming is a ubiquitous strategy for motile cells to generate self-gradients for environmental sensing. The model biflagellate Chlamydomonas reinhardtii rotates at a constant 1-2 Hz as it swims, but the mechanism is unclear. Here, we show unequivocally that the rolling motion derives from a persistent, nonplanar flagellar beat pattern. This is revealed by high-speed imaging and micromanipulation of live cells. We construct a fully 3D model to relate flagellar beating directly to the free-swimming trajectories. For realistic geometries, the model reproduces both the sense and magnitude of the axial rotation of live cells. We show that helical swimming requires further symmetry breaking between the two flagella. These functional differences underlie all tactic responses, particularly phototaxis. We propose a control strategy by which cells steer toward or away from light by modulating the sign of biflagellar dominance.
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Affiliation(s)
- Dario Cortese
- Living Systems Institute and College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QD, United Kingdom
| | - Kirsty Y Wan
- Living Systems Institute and College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QD, United Kingdom
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16
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Gadêlha H, Hernández-Herrera P, Montoya F, Darszon A, Corkidi G. Human sperm uses asymmetric and anisotropic flagellar controls to regulate swimming symmetry and cell steering. SCIENCE ADVANCES 2020; 6:eaba5168. [PMID: 32789171 PMCID: PMC7399739 DOI: 10.1126/sciadv.aba5168] [Citation(s) in RCA: 24] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2019] [Accepted: 06/18/2020] [Indexed: 05/21/2023]
Abstract
Flagellar beating drives sperm through the female reproductive tract and is vital for reproduction. Flagellar waves are generated by thousands of asymmetric molecular components; yet, paradoxically, forward swimming arises via symmetric side-to-side flagellar movement. This led to the preponderance of symmetric flagellar control hypotheses. However, molecular asymmetries must still dictate the flagellum and be manifested in the beat. Here, we reconcile molecular and microscopic observations, reconnecting structure to function, by showing that human sperm uses asymmetric and anisotropic controls to swim. High-speed three-dimensional (3D) microscopy revealed two coactive transversal controls: An asymmetric traveling wave creates a one-sided stroke, and a pulsating standing wave rotates the sperm to move equally on all sides. Symmetry is thus achieved through asymmetry, creating the optical illusion of bilateral symmetry in 2D microscopy. This shows that the sperm flagellum is asymmetrically controlled and anisotropically regularized by fast-signal transduction. This enables the sperm to swim forward.
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Affiliation(s)
- Hermes Gadêlha
- Department of Engineering Mathematics, University of Bristol, BS8 1UB Bristol, UK
| | - Paul Hernández-Herrera
- Laboratorio de Imágenes y Visión por Computadora, Departamento de Ingeniería Celular y Biocatálisis, Universidad Nacional Autónoma de México, Cuernavaca, Mexico
| | - Fernando Montoya
- Laboratorio de Imágenes y Visión por Computadora, Departamento de Ingeniería Celular y Biocatálisis, Universidad Nacional Autónoma de México, Cuernavaca, Mexico
| | - Alberto Darszon
- Departamento de Genética del Desarrollo y Fisiología Molecular, Instituto de Biotecnología, Universidad Nacional Autónoma de México, Cuernavaca, Mexico
| | - Gabriel Corkidi
- Laboratorio de Imágenes y Visión por Computadora, Departamento de Ingeniería Celular y Biocatálisis, Universidad Nacional Autónoma de México, Cuernavaca, Mexico
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17
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Walker BJ, Phuyal S, Ishimoto K, Tung CK, Gaffney EA. Computer-assisted beat-pattern analysis and the flagellar waveforms of bovine spermatozoa. ROYAL SOCIETY OPEN SCIENCE 2020; 7:200769. [PMID: 32742702 PMCID: PMC7353979 DOI: 10.1098/rsos.200769] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2020] [Accepted: 05/18/2020] [Indexed: 05/28/2023]
Abstract
Obstructed by hurdles in information extraction, handling and processing, computer-assisted sperm analysis systems have typically not considered in detail the complex flagellar waveforms of spermatozoa, despite their defining role in cell motility. Recent developments in imaging techniques and data processing have produced significantly improved methods of waveform digitization. Here, we use these improvements to demonstrate that near-complete flagellar capture is realizable on the scale of hundreds of cells, and, further, that meaningful statistical comparisons of flagellar waveforms may be readily performed with widely available tools. Representing the advent of high-fidelity computer-assisted beat-pattern analysis, we show how such a statistical approach can distinguish between samples using complex flagellar beating patterns rather than crude summary statistics. Dimensionality-reduction techniques applied to entire samples also reveal qualitatively distinct components of the beat, and a novel data-driven methodology for the generation of representative synthetic waveform data is proposed.
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Affiliation(s)
- Benjamin J. Walker
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
| | - Shiva Phuyal
- Department of Physics, North Carolina A&T State University, Greensboro, NC 27411, USA
| | - Kenta Ishimoto
- Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan
| | - Chih-Kuan Tung
- Department of Physics, North Carolina A&T State University, Greensboro, NC 27411, USA
| | - Eamonn A. Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
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18
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Hamilton E, Pellicciotta N, Feriani L, Cicuta P. Motile cilia hydrodynamics: entrainment versus synchronization when coupling through flow. Philos Trans R Soc Lond B Biol Sci 2020; 375:20190152. [PMID: 31884919 PMCID: PMC7017333 DOI: 10.1098/rstb.2019.0152] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 10/23/2019] [Indexed: 01/16/2023] Open
Abstract
Coordinated motion of cilia is a fascinating and vital aspect of very diverse forms of eukaryotic life, enabling swimming and propulsion of fluid across cellular epithelia. There are many questions still unresolved, and broadly they fall into two classes. (i) The mechanism of how cilia physically transmit forces onto each other. It is not known for many systems if the forces are mainly of hydrodynamical origin, or if elastic forces within the cytoskeleton are important. (ii) In those systems where we know that forces are purely hydrodynamical, we do not have a framework for linking our understanding of how each cilium behaves in isolation to the collective properties of two or more cilia. In this work, we take biological data of cilia dynamics from a variety of organisms as an input for an analytical and numerical study. We calculate the relative importance of external flows versus internal cilia flows on cilia coupling. This study contributes to both the open questions outlined above: firstly, we show that it is, in general, incorrect to infer cilium-cilium coupling strength on the basis of experiments with external flows, and secondly, we show a framework to recapitulate the dynamics of single cilia (the waveform) showing classes that correspond to biological systems with the same physiological activity (swimming by propulsion, versus forming collective waves). This article is part of the Theo Murphy meeting issue 'Unity and diversity of cilia in locomotion and transport'.
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Affiliation(s)
- Evelyn Hamilton
- Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK
| | - Nicola Pellicciotta
- Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK
| | - Luigi Feriani
- Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK
- Institute of Clinical Sciences, Imperial College London, London W12 0NN, UK
- MRC London Institute of Medical Sciences, London W12 0NN, UK
| | - Pietro Cicuta
- Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK
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19
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Meyer PG, Anvari M, Kantz H. Identifying characteristic time scales in power grid frequency fluctuations with DFA. CHAOS (WOODBURY, N.Y.) 2020; 30:013130. [PMID: 32013502 DOI: 10.1063/1.5123778] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/09/2019] [Accepted: 12/23/2019] [Indexed: 06/10/2023]
Abstract
Frequency measurements indicate the state of a power grid. In fact, deviations from the nominal frequency determine whether the grid is stable or in a critical situation. We aim to understand the fluctuations of the frequency on multiple time scales with a recently proposed method based on detrended fluctuation analysis. It enables us to infer characteristic time scales and generate stochastic models. We capture and quantify known features of the fluctuations like periodicity due to the trading market, response to variations by control systems, and stability of the long time average. We discuss similarities and differences between the British grid and the continental European grid.
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Affiliation(s)
- Philipp G Meyer
- Max-Planck Institute for the Physics of Complex Systems (MPIPKS), 01187 Dresden, Germany
| | - Mehrnaz Anvari
- Max-Planck Institute for the Physics of Complex Systems (MPIPKS), 01187 Dresden, Germany
| | - Holger Kantz
- Max-Planck Institute for the Physics of Complex Systems (MPIPKS), 01187 Dresden, Germany
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20
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Böddeker TJ, Karpitschka S, Kreis CT, Magdelaine Q, Bäumchen O. Dynamic force measurements on swimming Chlamydomonas cells using micropipette force sensors. J R Soc Interface 2020; 17:20190580. [PMID: 31937233 PMCID: PMC7014799 DOI: 10.1098/rsif.2019.0580] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/16/2019] [Accepted: 12/10/2019] [Indexed: 11/12/2022] Open
Abstract
Flagella and cilia are cellular appendages that inherit essential functions of microbial life including sensing and navigating the environment. In order to propel a swimming microorganism they displace the surrounding fluid by means of periodic motions, while precisely timed modulations of their beating patterns enable the cell to steer towards or away from specific locations. Characterizing the dynamic forces, however, is challenging and typically relies on indirect experimental approaches. Here, we present direct in vivo measurements of the dynamic forces of motile Chlamydomonas reinhardtii cells in controlled environments. The experiments are based on partially aspirating a living microorganism at the tip of a micropipette force sensor and optically recording the micropipette's position fluctuations with high temporal and sub-pixel spatial resolution. Spectral signal analysis allows for isolating the cell-generated dynamic forces caused by the periodic motion of the flagella from background noise. We provide an analytic, elasto-hydrodynamic model for the micropipette force sensor and describe how to obtain the micropipette's full frequency response function from a dynamic force calibration. Using this approach, we measure the amplitude of the oscillatory forces during the swimming activity of individual Chlamydomonas reinhardtii cells of 26 ± 5 pN, resulting from the coordinated flagellar beating with a frequency of 49 ± 5 Hz. This dynamic micropipette force sensor technique generalizes the applicability of micropipettes as force sensors from static to dynamic force measurements, yielding a force sensitivity in the piconewton range. In addition to measurements in bulk liquid environment, we study the dynamic forces of the biflagellated microswimmer in the vicinity of a solid/liquid interface. As we gradually decrease the distance of the swimming microbe to the interface, we measure a significantly enhanced force transduction at distances larger than the maximum extent of the beating flagella, highlighting the importance of hydrodynamic interactions for scenarios in which flagellated microorganisms encounter surfaces.
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Affiliation(s)
| | | | | | | | - Oliver Bäumchen
- Max Planck Institute for Dynamics and Self-Organization (MPIDS), Am Faßberg 17, 37077 Göttingen, Germany
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21
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Wan KY. Coordination of eukaryotic cilia and flagella. Essays Biochem 2018; 62:829-838. [PMID: 30464007 PMCID: PMC6281475 DOI: 10.1042/ebc20180029] [Citation(s) in RCA: 25] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2018] [Revised: 10/04/2018] [Accepted: 10/08/2018] [Indexed: 12/14/2022]
Abstract
Propulsion by slender cellular appendages called cilia and flagella is an ancient means of locomotion. Unicellular organisms evolved myriad strategies to propel themselves in fluid environments, often involving significant differences in flagella number, localisation and modes of actuation. Remarkably, these appendages are highly conserved, occurring in many complex organisms such as humans, where they may be found generating physiological flows when attached to surfaces (e.g. airway epithelial cilia), or else conferring motility to male gametes (e.g. undulations of sperm flagella). Where multiple cilia arise, their movements are often observed to be highly coordinated. Here I review the two main mechanisms for motile cilia coordination, namely, intracellular and hydrodynamic, and discuss their relative importance in different ciliary systems.
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Affiliation(s)
- Kirsty Y Wan
- Living Systems Institute, University of Exeter, Exeter, U.K.
- College of Engineering Mathematics and Physical Sciences, University of Exeter, Exeter, U.K
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22
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Friedrich BM. Load response of shape-changing microswimmers scales with their swimming efficiency. Phys Rev E 2018; 97:042416. [PMID: 29758744 DOI: 10.1103/physreve.97.042416] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2017] [Indexed: 12/15/2022]
Abstract
External forces acting on a microswimmer can feed back on its self-propulsion mechanism. We discuss this load response for a generic microswimmer that swims by cyclic shape changes. We show that the change in cycle frequency is proportional to the Lighthill efficiency of self-propulsion. As a specific example, we consider Najafi's three-sphere swimmer. The force-velocity relation of a microswimmer implies a correction for a formal superposition principle for active and passive motion.
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23
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Wan KY, Goldstein RE. Time Irreversibility and Criticality in the Motility of a Flagellate Microorganism. PHYSICAL REVIEW LETTERS 2018; 121:058103. [PMID: 30118294 PMCID: PMC7616082 DOI: 10.1103/physrevlett.121.058103] [Citation(s) in RCA: 25] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/24/2017] [Revised: 03/27/2018] [Indexed: 06/08/2023]
Abstract
Active living organisms exhibit behavioral variability, partitioning between fast and slow dynamics. Such variability may be key to generating rapid responses in a heterogeneous, unpredictable environment wherein cellular activity effects continual exchanges of energy fluxes. We demonstrate a novel, noninvasive strategy for revealing nonequilibrium control of swimming-specifically, in an octoflagellate microalga. These organisms exhibit surprising features of flagellar excitability and mechanosensitivity, which characterize a novel, time-irreversible "run-stop-shock" motility comprising forward runs, knee-jerk shocks with dramatic beat reversal, and long stops during which cells are quiescent yet continue to exhibit submicron flagellar vibrations. Entropy production, associated with flux cycles arising in a reaction graph representation of the gait-switching dynamics, provides a direct measure of detailed balance violation in this primitive alga.
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Affiliation(s)
- Kirsty Y. Wan
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom
- Living Systems Institute, University of Exeter, Exeter EX4 4QD, United Kingdom
| | - Raymond E. Goldstein
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom
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24
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Kawamura Y, Tsubaki R. Phase reduction approach to elastohydrodynamic synchronization of beating flagella. Phys Rev E 2018; 97:022212. [PMID: 29548174 DOI: 10.1103/physreve.97.022212] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/12/2017] [Indexed: 06/08/2023]
Abstract
We formulate a theory for the phase reduction of a beating flagellum. The theory enables us to describe the dynamics of a beating flagellum in a systematic manner using a single variable called the phase. The theory can also be considered as a phase reduction method for the limit-cycle solutions in infinite-dimensional dynamical systems, namely, the limit-cycle solutions to partial differential equations representing beating flagella. We derive the phase sensitivity function, which quantifies the phase response of a beating flagellum to weak perturbations applied at each point and at each time. Using the phase sensitivity function, we analyze the phase synchronization between a pair of beating flagella through hydrodynamic interactions at a low Reynolds number.
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Affiliation(s)
- Yoji Kawamura
- Department of Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan
- Research and Development Center for Marine Biosciences, Japan Agency for Marine-Earth Science and Technology, Yokosuka 237-0061, Japan
| | - Remi Tsubaki
- Research and Development Center for Marine Biosciences, Japan Agency for Marine-Earth Science and Technology, Yokosuka 237-0061, Japan
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25
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Saggiorato G, Alvarez L, Jikeli JF, Kaupp UB, Gompper G, Elgeti J. Human sperm steer with second harmonics of the flagellar beat. Nat Commun 2017; 8:1415. [PMID: 29123094 PMCID: PMC5680276 DOI: 10.1038/s41467-017-01462-y] [Citation(s) in RCA: 55] [Impact Index Per Article: 7.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/14/2017] [Accepted: 09/20/2017] [Indexed: 12/21/2022] Open
Abstract
Sperm are propelled by bending waves traveling along their flagellum. For steering in gradients of sensory cues, sperm adjust the flagellar waveform. Symmetric and asymmetric waveforms result in straight and curved swimming paths, respectively. Two mechanisms causing spatially asymmetric waveforms have been proposed: an average flagellar curvature and buckling. We image flagella of human sperm tethered with the head to a surface. The waveform is characterized by a fundamental beat frequency and its second harmonic. The superposition of harmonics breaks the beat symmetry temporally rather than spatially. As a result, sperm rotate around the tethering point. The rotation velocity is determined by the second-harmonic amplitude and phase. Stimulation with the female sex hormone progesterone enhances the second-harmonic contribution and, thereby, modulates sperm rotation. Higher beat frequency components exist in other flagellated cells; therefore, this steering mechanism might be widespread and could inspire the design of synthetic microswimmers.
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Affiliation(s)
- Guglielmo Saggiorato
- Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, 52425, Jülich, Germany
- Department of Molecular Sensory Systems, Center of Advanced European Studies and Research (CAESAR), 53175, Bonn, Germany
| | - Luis Alvarez
- Department of Molecular Sensory Systems, Center of Advanced European Studies and Research (CAESAR), 53175, Bonn, Germany.
| | - Jan F Jikeli
- Laboratoire de Physique Théorique et Modèles Statistiques, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405, Orsay, France
- Biophysical Imaging, Institute of Innate Immunity, University Hospital Bonn, 53127, Bonn, Germany
| | - U Benjamin Kaupp
- Department of Molecular Sensory Systems, Center of Advanced European Studies and Research (CAESAR), 53175, Bonn, Germany
| | - Gerhard Gompper
- Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, 52425, Jülich, Germany
| | - Jens Elgeti
- Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, 52425, Jülich, Germany.
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26
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Wheeler RJ. Use of chiral cell shape to ensure highly directional swimming in trypanosomes. PLoS Comput Biol 2017; 13:e1005353. [PMID: 28141804 PMCID: PMC5308837 DOI: 10.1371/journal.pcbi.1005353] [Citation(s) in RCA: 42] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/19/2016] [Revised: 02/14/2017] [Accepted: 01/10/2017] [Indexed: 11/23/2022] Open
Abstract
Swimming cells typically move along a helical path or undergo longitudinal rotation as they swim, arising from chiral asymmetry in hydrodynamic drag or propulsion bending the swimming path into a helix. Helical paths are beneficial for some forms of chemotaxis, but why asymmetric shape is so prevalent when a symmetric shape would also allow highly directional swimming is unclear. Here, I analyse the swimming of the insect life cycle stages of two human parasites; Trypanosoma brucei and Leishmania mexicana. This showed quantitatively how chirality in T. brucei cell shape confers highly directional swimming. High speed videomicrographs showed that T. brucei, L. mexicana and a T. brucei RNAi morphology mutant have a range of shape asymmetries, from wild-type T. brucei (highly chiral) to L. mexicana (near-axial symmetry). The chiral cells underwent longitudinal rotation while swimming, with more rapid longitudinal rotation correlating with swimming path directionality. Simulation indicated hydrodynamic drag on the chiral cell shape caused rotation, and the predicted geometry of the resulting swimming path matched the directionality of the observed swimming paths. This simulation of swimming path geometry showed that highly chiral cell shape is a robust mechanism through which microscale swimmers can achieve highly directional swimming at low Reynolds number. It is insensitive to random variation in shape or propulsion (biological noise). Highly symmetric cell shape can give highly directional swimming but is at risk of giving futile circular swimming paths in the presence of biological noise. This suggests the chiral T. brucei cell shape (associated with the lateral attachment of the flagellum) may be an adaptation associated with the bloodstream-inhabiting lifestyle of this parasite for robust highly directional swimming. It also provides a plausible general explanation for why swimming cells tend to have strong asymmetries in cell shape or propulsion. Swimming cells often follow a helical swimming path, however the advantage of helical paths over a simple straight line path is not clear. To analyse this phenomenon, I analysed the swimming of the human parasites Trypanosoma brucei (which causes sleeping sickness/trypanosomiasis) and Leishmania mexicana (which causes leishmaniasis). Using new computational methods to determine the three dimensional shape of swimming cells I showed that T. brucei have a helical shape which causes rotation as the cell swims, and the geometry of the resulting swimming path makes the cell movement highly directional. In contrast, L. mexicana are symmetrical, do not rotate, and their swimming paths are curved and have low directionality. Using a T. brucei mutant I showed that the cell structure responsible for the helical shape while swimming is the flagellum attachment zone. This explains a key function of this structure. Finally, simulations showed the phenomenon of rotation while swimming is a way cells can ensure highly directional swimming along a controlled helical path, overcoming random variation in cell shape or propulsion. This provides a general explanation for why swimming cells are often asymmetric and tend to follow helical paths.
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Affiliation(s)
- Richard John Wheeler
- Sir William Dunn School of Pathology, University of Oxford, Oxford, United Kingdom
- Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany
- * E-mail:
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27
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Goldstein RE, Lauga E, Pesci AI, Proctor MRE. Elastohydrodynamic Synchronization of Adjacent Beating Flagella. PHYSICAL REVIEW FLUIDS 2016; 1:073201. [PMID: 29750206 PMCID: PMC5939993 DOI: 10.1103/physrevfluids.1.073201] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
It is now well established that nearby beating pairs of eukaryotic flagella or cilia typically synchronize in phase. A substantial body of evidence supports the hypothesis that hydrodynamic coupling between the active filaments, combined with waveform compliance, provides a robust mechanism for synchrony. This elastohydrodynamic mechanism has been incorporated into 'bead-spring' models in which the beating flagella are represented by microspheres tethered by radial springs as they are driven about orbits by internal forces. While these low-dimensional models reproduce the phenomenon of synchrony, their parameters are not readily relatable to those of the filaments they represent. More realistic models which reflect the underlying elasticity of the axonemes and the active force generation, take the form of fourth-order nonlinear PDEs. While computational studies have shown the occurrence of synchrony, the effects of hydrodynamic coupling between nearby filaments governed by such continuum models have been theoretically examined only in the regime of interflagellar distances d large compared to flagellar length L. Yet, in many biological situations d/L ≪ 1. Here, we first present an asymptotic analysis of the hydrodynamic coupling between two extended filaments in the regime d/L ≪ 1, and find that the form of the coupling is independent of the microscopic details of the internal forces that govern the motion of the individual filaments. The analysis is analogous to that yielding the localized induction approximation for vortex filament motion, extended to the case of mutual induction. In order to understand how the elastohydrodynamic coupling mechanism leads to synchrony of extended objects, we introduce a heuristic model of flagellar beating. The model takes the form of a single fourth-order nonlinear PDE whose form is derived from symmetry considerations, the physics of elasticity, and the overdamped nature of the dynamics. Analytical and numerical studies of this model illustrate how synchrony between a pair of filaments is achieved through the asymptotic coupling.
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Affiliation(s)
- Raymond E Goldstein
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
| | - Eric Lauga
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
| | - Adriana I Pesci
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
| | - Michael R E Proctor
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
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28
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Mathijssen AJTM, Doostmohammadi A, Yeomans JM, Shendruk TN. Hotspots of boundary accumulation: dynamics and statistics of micro-swimmers in flowing films. J R Soc Interface 2016; 13:20150936. [PMID: 26841796 DOI: 10.1098/rsif.2015.0936] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/14/2023] Open
Abstract
Biological flows over surfaces and interfaces can result in accumulation hotspots or depleted voids of microorganisms in natural environments. Apprehending the mechanisms that lead to such distributions is essential for understanding biofilm initiation. Using a systematic framework, we resolve the dynamics and statistics of swimming microbes within flowing films, considering the impact of confinement through steric and hydrodynamic interactions, flow and motility, along with Brownian and run-tumble fluctuations. Micro-swimmers can be peeled off the solid wall above a critical flow strength. However, the interplay of flow and fluctuations causes organisms to migrate back towards the wall above a secondary critical value. Hence, faster flows may not always be the most efficacious strategy to discourage biofilm initiation. Moreover, we find run-tumble dynamics commonly used by flagellated microbes to be an intrinsically more successful strategy to escape from boundaries than equivalent levels of enhanced Brownian noise in ciliated organisms.
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Affiliation(s)
| | - Amin Doostmohammadi
- The Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK
| | - Julia M Yeomans
- The Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK
| | - Tyler N Shendruk
- The Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK
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29
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Wan KY, Goldstein RE. Coordinated beating of algal flagella is mediated by basal coupling. Proc Natl Acad Sci U S A 2016; 113:E2784-93. [PMID: 27140605 PMCID: PMC4878519 DOI: 10.1073/pnas.1518527113] [Citation(s) in RCA: 81] [Impact Index Per Article: 10.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022] Open
Abstract
Cilia and flagella often exhibit synchronized behavior; this includes phase locking, as seen in Chlamydomonas, and metachronal wave formation in the respiratory cilia of higher organisms. Since the observations by Gray and Rothschild of phase synchrony of nearby swimming spermatozoa, it has been a working hypothesis that synchrony arises from hydrodynamic interactions between beating filaments. Recent work on the dynamics of physically separated pairs of flagella isolated from the multicellular alga Volvox has shown that hydrodynamic coupling alone is sufficient to produce synchrony. However, the situation is more complex in unicellular organisms bearing few flagella. We show that flagella of Chlamydomonas mutants deficient in filamentary connections between basal bodies display markedly different synchronization from the wild type. We perform micromanipulation on configurations of flagella and conclude that a mechanism, internal to the cell, must provide an additional flagellar coupling. In naturally occurring species with 4, 8, or even 16 flagella, we find diverse symmetries of basal body positioning and of the flagellar apparatus that are coincident with specific gaits of flagellar actuation, suggesting that it is a competition between intracellular coupling and hydrodynamic interactions that ultimately determines the precise form of flagellar coordination in unicellular algae.
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Affiliation(s)
- Kirsty Y Wan
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom
| | - Raymond E Goldstein
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom
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30
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Battle C, Broedersz CP, Fakhri N, Geyer VF, Howard J, Schmidt CF, MacKintosh FC. Broken detailed balance at mesoscopic scales in active biological systems. Science 2016; 352:604-7. [PMID: 27126047 DOI: 10.1126/science.aac8167] [Citation(s) in RCA: 164] [Impact Index Per Article: 20.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2015] [Accepted: 01/26/2016] [Indexed: 12/28/2022]
Abstract
Systems in thermodynamic equilibrium are not only characterized by time-independent macroscopic properties, but also satisfy the principle of detailed balance in the transitions between microscopic configurations. Living systems function out of equilibrium and are characterized by directed fluxes through chemical states, which violate detailed balance at the molecular scale. Here we introduce a method to probe for broken detailed balance and demonstrate how such nonequilibrium dynamics are manifest at the mesosopic scale. The periodic beating of an isolated flagellum from Chlamydomonas reinhardtii exhibits probability flux in the phase space of shapes. With a model, we show how the breaking of detailed balance can also be quantified in stationary, nonequilibrium stochastic systems in the absence of periodic motion. We further demonstrate such broken detailed balance in the nonperiodic fluctuations of primary cilia of epithelial cells. Our analysis provides a general tool to identify nonequilibrium dynamics in cells and tissues.
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Affiliation(s)
- Christopher Battle
- Drittes Physikalisches Institut, Georg-August-Universität, 37077 Göttingen, Germany. The Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
| | - Chase P Broedersz
- The Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA. Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333 München, Germany. Lewis-Sigler Institute for Integrative Genomics and Joseph Henry Laboratories of Physics, Princeton University, Princeton, NJ 08544, USA
| | - Nikta Fakhri
- Drittes Physikalisches Institut, Georg-August-Universität, 37077 Göttingen, Germany. The Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA. Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
| | - Veikko F Geyer
- Department of Molecular Biophysics and Biochemistry, Yale University, New Haven, CT, USA
| | - Jonathon Howard
- Department of Molecular Biophysics and Biochemistry, Yale University, New Haven, CT, USA
| | - Christoph F Schmidt
- Drittes Physikalisches Institut, Georg-August-Universität, 37077 Göttingen, Germany. The Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA.
| | - Fred C MacKintosh
- The Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA. Department of Physics and Astronomy, Vrije Universiteit, Amsterdam, Netherlands.
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31
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Barsanti L, Coltelli P, Evangelista V, Frassanito AM, Gualtieri P. Swimming patterns of the quadriflagellate Tetraflagellochloris mauritanica (Chlamydomonadales, Chlorophyceae). JOURNAL OF PHYCOLOGY 2016; 52:209-218. [PMID: 27037586 DOI: 10.1111/jpy.12384] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2015] [Accepted: 12/04/2015] [Indexed: 06/05/2023]
Abstract
Chlamydomonadales are elective subjects for the investigation of the problems related to locomotion and transport in biological fluid dynamics, whose resolution could enhance searching efficiency and assist in the avoidance of dangerous environments. In this paper, we elucidate the swimming behavior of Tetraflagellochloris mauritanica, a unicellular-multicellular alga belonging to the order Chlamydomonadales. This quadriflagellate alga has a complex swimming motion consisting of alternating swimming phases connected by in-place random reorientations and resting phases. It is capable of both forward and backward swimming, both being normal modes of swimming. The complex swimming behavior resembles the run-and-tumble motion of peritrichous bacteria, with in-place reorientation taking the place of tumbles. In the forward swimming, T. mauritanica shows a very efficient flagellar beat, with undulatory retrograde waves that run along the flagella to their tip. In the backward swimming, the flagella show a nonstereotypical synchronization mode, with a pattern that does not fit any of the modes present in the other Chlamydomonadales so far investigated.
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Affiliation(s)
- Laura Barsanti
- Istituto di Biofisica, CNR, Via Moruzzi 1, Pisa, 56124, Italy
| | - Primo Coltelli
- Istituto di Scienza e Tecnologia Informazione, CNR, Via Moruzzi 1, Pisa, 56124, Italy
| | | | | | - Paolo Gualtieri
- Istituto di Biofisica, CNR, Via Moruzzi 1, Pisa, 56124, Italy
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32
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Nasouri B, Elfring GJ. Hydrodynamic interactions of cilia on a spherical body. Phys Rev E 2016; 93:033111. [PMID: 27078451 DOI: 10.1103/physreve.93.033111] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/07/2015] [Indexed: 12/15/2022]
Abstract
Microorganisms develop coordinated beating patterns on surfaces lined with cilia known as metachronal waves. For a chain of cilia attached to a flat ciliate, it has been shown that hydrodynamic interactions alone can lead the system to synchronize. However, several microorganisms possess a curve-shaped ciliate body and so to understand the effect of this geometry on the formation of metachronal waves, we evaluate the hydrodynamic interactions of cilia near a large spherical body. Using a minimal model, we show that for a chain of cilia around the sphere, the natural periodicity in the geometry leads the system to synchronize. We also report an emergent wavelike behavior when an asymmetry is introduced to the system.
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Affiliation(s)
- Babak Nasouri
- Department of Mechanical Engineering, University of British Columbia, Vancouver, British Columbia, V6T 1Z4, Canada
| | - Gwynn J Elfring
- Department of Mechanical Engineering, University of British Columbia, Vancouver, British Columbia, V6T 1Z4, Canada
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Kirkegaard JB, Marron AO, Goldstein RE. Motility of Colonial Choanoflagellates and the Statistics of Aggregate Random Walkers. PHYSICAL REVIEW LETTERS 2016; 116:038102. [PMID: 26849616 PMCID: PMC7616083 DOI: 10.1103/physrevlett.116.038102] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/16/2015] [Indexed: 06/05/2023]
Abstract
We illuminate the nature of the three-dimensional random walks of microorganisms composed of individual organisms adhered together. Such aggregate random walkers are typified by choanoflagellates, eukaryotes that are the closest living relatives of animals. In the colony-forming species Salpingoeca rosetta we show that the beating of each flagellum is stochastic and uncorrelated with others, and the vectorial sum of the flagellar propulsion manifests as stochastic helical swimming. A quantitative theory for these results is presented and species variability discussed.
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Affiliation(s)
- Julius B Kirkegaard
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
| | - Alan O Marron
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
| | - Raymond E Goldstein
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
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Quaranta G, Aubin-Tam ME, Tam D. Hydrodynamics Versus Intracellular Coupling in the Synchronization of Eukaryotic Flagella. PHYSICAL REVIEW LETTERS 2015; 115:238101. [PMID: 26684142 DOI: 10.1103/physrevlett.115.238101] [Citation(s) in RCA: 36] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2015] [Indexed: 05/27/2023]
Abstract
The influence of hydrodynamic forces on eukaryotic flagella synchronization is investigated by triggering phase locking between a controlled external flow and the flagella of C. reinhardtii. Hydrodynamic forces required for synchronization are over an order of magnitude larger than hydrodynamic forces experienced in physiological conditions. Our results suggest that synchronization is due instead to coupling through cell internal fibers connecting the flagella. This conclusion is confirmed by observations of the vfl3 mutant, with impaired mechanical connection between the flagella.
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Affiliation(s)
- Greta Quaranta
- Laboratory for Aero and Hydrodynamics, Delft University of Technology, 2628CD Delft, Netherlands
| | - Marie-Eve Aubin-Tam
- Department of Bionanoscience, Delft University of Technology, 2628CJ Delft, Netherlands
| | - Daniel Tam
- Laboratory for Aero and Hydrodynamics, Delft University of Technology, 2628CD Delft, Netherlands
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