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Lim S, Du Y, Lee Y, Panda SK, Tong D, Khalid Jawed M. Fabrication, control, and modeling of robots inspired by flagella and cilia. BIOINSPIRATION & BIOMIMETICS 2022; 18:011003. [PMID: 36533860 DOI: 10.1088/1748-3190/aca63d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/21/2022] [Accepted: 11/25/2022] [Indexed: 06/17/2023]
Abstract
Flagella and cilia are slender structures that serve important functionalities in the microscopic world through their locomotion induced by fluid and structure interaction. With recent developments in microscopy, fabrication, biology, and modeling capability, robots inspired by the locomotion of these organelles in low Reynolds number flow have been manufactured and tested on the micro-and macro-scale, ranging from medicalin vivomicrobots, microfluidics to macro prototypes. We present a collection of modeling theories, control principles, and fabrication methods for flagellated and ciliary robots.
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Affiliation(s)
- Sangmin Lim
- Department of Mechanical & Aerospace Engineering, Westwood Plaza, University of California, Los Angeles, CA 90095, United States of America
| | - Yayun Du
- Department of Mechanical & Aerospace Engineering, Westwood Plaza, University of California, Los Angeles, CA 90095, United States of America
| | - Yongkyu Lee
- Department of Mechanical & Aerospace Engineering, Westwood Plaza, University of California, Los Angeles, CA 90095, United States of America
| | - Shivam Kumar Panda
- Department of Mechanical & Aerospace Engineering, Westwood Plaza, University of California, Los Angeles, CA 90095, United States of America
| | - Dezhong Tong
- Department of Mechanical & Aerospace Engineering, Westwood Plaza, University of California, Los Angeles, CA 90095, United States of America
| | - M Khalid Jawed
- Department of Mechanical & Aerospace Engineering, Westwood Plaza, University of California, Los Angeles, CA 90095, United States of America
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2
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Shen X, Tran PN, Tay BZ, Marcos. Bending stiffness characterization of Bacillus subtilis' flagellar filament. Biophys J 2022; 121:1975-1985. [PMID: 35550881 DOI: 10.1016/j.bpj.2022.05.010] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2021] [Revised: 04/11/2022] [Accepted: 05/06/2022] [Indexed: 11/27/2022] Open
Abstract
The filament of a bacterial flagellum is a tube-like organelle made of single protein - flagellin, and assembled into multiple polymorphic forms. The filament can be further discretized into four subunit domains (D0, D1, D2 and D3) along the radial direction. However, it remains unclear which subunit domain plays an important role in regulating the rigidity of the filament. In this article, we address how the absence of two outer subunit domains (D2 and D3) affects the bending stiffness of the bacterium B. subtilis' flagellar filament. We first shear off flagellar filaments from the cell body, anchor one of its ends to the wall of a microfluidic channel, and correlate the elongation of the filament with the driving background flow. A numerical model is then applied to determine the bending stiffness of the filament. We find that the bending stiffness does not change drastically when the filament transforms from normal to hyperextended forms, which is estimated to be 2-3 pN.μm2. Furthermore, B. subtilis' flagellar filament has similar bending stiffness to Salmonella's, though the radius of the former is almost half of that of the latter, suggesting that the rigidity comes from the inner D0 and D1 subunit domains.
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Affiliation(s)
- Xinhui Shen
- School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore
| | - Phu N Tran
- School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore
| | - Benjamin Z Tay
- School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore
| | - Marcos
- School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore.
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3
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Clopés J, Winkler RG. Flagellar arrangements in elongated peritrichous bacteria: bundle formation and swimming properties. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2021; 44:17. [PMID: 33683543 PMCID: PMC7940165 DOI: 10.1140/epje/s10189-021-00027-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/13/2020] [Accepted: 01/25/2021] [Indexed: 05/26/2023]
Abstract
The surface distribution of flagella in peritrichous bacterial cells has been traditionally assumed to be random. Recently, the presence of a regular grid-like pattern of basal bodies has been suggested. Experimentally, the manipulation of the anchoring points of flagella in the cell membrane is difficult, and thus, elucidation of the consequences of a particular pattern on bacterial locomotion is challenging. We analyze the bundle formation process and swimming properties of Bacillus subtilis-like cells considering random, helical, and ring-like arrangements of flagella by means of mesoscale hydrodynamics simulations. Helical and ring patterns preferentially yield configurations with a single bundle, whereas configurations with no clear bundles are most likely for random anchoring. For any type of pattern, there is an almost equally low probability to form V-shaped bundle configurations with at least two bundles. Variation of the flagellum length yields a clear preference for a single major bundle in helical and ring patterns as soon as the flagellum length exceeds the body length. The average swimming speed of cells with a single or two bundles is rather similar, and approximately [Formula: see text] larger than that of cells of other types of flagellar organization. Considering the various anchoring patterns, rings yield the smallest average swimming speed independent of the type of bundle, followed by helical arrangements, and largest speeds are observed for random anchoring. Hence, a regular pattern provides no advantage in terms of swimming speed compared to random anchoring of flagella, but yields more likely single-bundle configurations.
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Affiliation(s)
- Judit Clopés
- Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425, Jülich, Germany
- Institute for Theoretical Physics, RWTH Aachen University, 52074, Aachen, Germany
| | - Roland G Winkler
- Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425, Jülich, Germany.
- Institute for Theoretical Physics, RWTH Aachen University, 52074, Aachen, Germany.
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4
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Mousavi SM, Gompper G, Winkler RG. Wall entrapment of peritrichous bacteria: a mesoscale hydrodynamics simulation study. SOFT MATTER 2020; 16:4866-4875. [PMID: 32424390 DOI: 10.1039/d0sm00571a] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Microswimmers such as E. coli bacteria accumulate and exhibit an intriguing dynamics near walls, governed by hydrodynamic and steric interactions. Insight into the underlying mechanisms and predominant interactions demand a detailed characterization of the entrapment process. We employ a mesoscale hydrodynamics simulation approach to study entrapment of an E. coli-type cell at a no-slip wall. The cell is modeled by a spherocylindrical body with several explicit helical flagella. Three stages of the entrapment process can be distinguished: the approaching regime, where a cell swims toward a wall on a nearly straight trajectory; a scattering regime, where the cell touches the wall and reorients; and a surface-swimming regime. Our simulations show that steric interactions may dominate the entrapment process, yet, hydrodynamic interactions slow down the adsorption dynamics close to the boundary and imply a circular motion on the wall. The locomotion of the cell is characterized by a strong wobbling dynamics, with cells preferentially pointing toward the wall during surface swimming.
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Affiliation(s)
- S Mahdiyeh Mousavi
- Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, D-52425 Jülich, Germany.
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Danis U, Rasooli R, Chen CY, Dur O, Sitti M, Pekkan K. Thrust and Hydrodynamic Efficiency of the Bundled Flagella. MICROMACHINES 2019; 10:mi10070449. [PMID: 31277385 PMCID: PMC6680724 DOI: 10.3390/mi10070449] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 04/29/2019] [Revised: 05/23/2019] [Accepted: 05/26/2019] [Indexed: 01/09/2023]
Abstract
The motility mechanism of prokaryotic organisms has inspired many untethered microswimmers that could potentially perform minimally invasive medical procedures in stagnant fluid regions inside the human body. Some of these microswimmers are inspired by bacteria with single or multiple helical flagella to propel efficiently and fast. For multiple flagella configurations, the direct measurement of thrust and hydrodynamic propulsion efficiency has been challenging due to the ambiguous mechanical coupling between the flow field and mechanical power input. To address this challenge and to compare alternative micropropulsion designs, a methodology based on volumetric velocity field acquisition is developed to acquire the key propulsive performance parameters from scaled-up swimmer prototypes. A digital particle image velocimetry (PIV) analysis protocol was implemented and experiments were conducted with the aid of computational fluid dynamics (CFD). First, this methodology was validated using a rotating single-flagellum similitude model. In addition to the standard PIV error assessment, validation studies included 2D vs. 3D PIV, axial vs. lateral PIV and simultaneously acquired direct thrust force measurement comparisons. Compatible with typical micropropulsion flow regimes, experiments were conducted both for very low and higher Reynolds (Re) number regimes (up to a Re number = 0.01) than that are reported in the literature. Finally, multiple flagella bundling configurations at 0°, 90° and 180° helical phase-shift angles were studied using scaled-up multiple concentric flagella thrust elements. Thrust generation was found to be maximal for the in-phase (0°) bundling configuration but with ~50% lower hydrodynamic efficiency than the single flagellum. The proposed measurement protocol and static thrust test-bench can be used for bio-inspired microscale propulsion methods, where direct thrust and efficiency measurement are required.
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Affiliation(s)
- Umit Danis
- Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA
| | - Reza Rasooli
- Department of Mechanical Engineering, Koc University, Istanbul 34450, Turkey
| | - Chia-Yuan Chen
- Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan
| | - Onur Dur
- Department of Biomedical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA
| | - Metin Sitti
- Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA
- Department of Mechanical Engineering, Koc University, Istanbul 34450, Turkey
- Physical Intelligence Department, Max Planck Institute for Intelligent Systems, Stuttgart 70569, Germany
| | - Kerem Pekkan
- Department of Mechanical Engineering, Koc University, Istanbul 34450, Turkey.
- Department of Biomedical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA.
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Katsamba P, Lauga E. Propulsion by stiff elastic filaments in viscous fluids. Phys Rev E 2019; 99:053107. [PMID: 31212530 DOI: 10.1103/physreve.99.053107] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2018] [Indexed: 01/28/2023]
Abstract
Flexible filaments moving in viscous fluids are ubiquitous in the natural microscopic world. For example, the swimming of bacteria and spermatozoa as well as important physiological functions at organ level, such as the cilia-induced motion of mucus in the lungs, or individual cell level, such as actin filaments or microtubules, all employ flexible filaments moving in viscous fluids. As a result of fluid-structure interactions, a variety of nonlinear phenomena may arise in the dynamics of such moving flexible filaments. In this paper we derive the mathematical tools required to study filament-driven propulsion in the asymptotic limit of stiff filaments. Motion in the rigid limit leads to hydrodynamic loads which deform the filament and impact the filament propulsion. We first derive the general mathematical formulation and then apply it to the case of a helical filament, a situation relevant for the swimming of flagellated bacteria and for the transport of artificial, magnetically actuated motors. We find that, as a result of flexibility, the helical filament is either stretched or compressed (conforming previous studies) and additionally its axis also bends, a result which we interpret physically. We then explore and interpret the dependence of the perturbed propulsion speed due to the deformation on the relevant dimensionless dynamic and geometric parameters.
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Affiliation(s)
- Panayiota Katsamba
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom
| | - 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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Koens L, Zhang H, Moeller M, Mourran A, Lauga E. The swimming of a deforming helix. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2018; 41:119. [PMID: 30302671 DOI: 10.1140/epje/i2018-11728-2] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/29/2018] [Accepted: 09/07/2018] [Indexed: 06/08/2023]
Abstract
Many microorganisms and artificial microswimmers use helical appendages in order to generate locomotion. Though often rotated so as to produce thrust, some species of bacteria such Spiroplasma, Rhodobacter sphaeroides and Spirochetes induce movement by deforming a helical-shaped body. Recently, artificial devices have been created which also generate motion by deforming their helical body in a non-reciprocal way (A. Mourran et al. Adv. Mater. 29, 1604825, 2017). Inspired by these systems, we investigate the transport of a deforming helix within a viscous fluid. Specifically, we consider a swimmer that maintains a helical centreline and a single handedness while changing its helix radius, pitch and wavelength uniformly across the body. We first discuss how a deforming helix can create a non-reciprocal translational and rotational swimming stroke and identify its principle direction of motion. We then determine the leading-order physics for helices with small helix radius before considering the general behaviour for different configuration parameters and how these swimmers can be optimised. Finally, we explore how the presence of walls, gravity, and defects in the centreline allow the helical device to break symmetries, increase its speed, and generate transport in directions not available to helices in bulk fluids.
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Affiliation(s)
- Lyndon Koens
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CB3 0WA, Cambridge, UK.
| | - Hang Zhang
- DWI-Leibniz Institute for Interactive Materials RWTH Aachen University, Forckenbeck str. 50, D-52056, Aachen, Germany
| | - Martin Moeller
- DWI-Leibniz Institute for Interactive Materials RWTH Aachen University, Forckenbeck str. 50, D-52056, Aachen, Germany
| | - Ahmed Mourran
- DWI-Leibniz Institute for Interactive Materials RWTH Aachen University, Forckenbeck str. 50, D-52056, Aachen, Germany
| | - Eric Lauga
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CB3 0WA, Cambridge, UK
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8
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Hintsche M, Waljor V, Großmann R, Kühn MJ, Thormann KM, Peruani F, Beta C. A polar bundle of flagella can drive bacterial swimming by pushing, pulling, or coiling around the cell body. Sci Rep 2017; 7:16771. [PMID: 29196650 PMCID: PMC5711944 DOI: 10.1038/s41598-017-16428-9] [Citation(s) in RCA: 31] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2017] [Accepted: 11/06/2017] [Indexed: 01/30/2023] Open
Abstract
Bacteria swim in sequences of straight runs that are interrupted by turning events. They drive their swimming locomotion with the help of rotating helical flagella. Depending on the number of flagella and their arrangement across the cell body, different run-and-turn patterns can be observed. Here, we present fluorescence microscopy recordings showing that cells of the soil bacterium Pseudomonas putida that are decorated with a polar tuft of helical flagella, can alternate between two distinct swimming patterns. On the one hand, they can undergo a classical push-pull-push cycle that is well known from monopolarly flagellated bacteria but has not been reported for species with a polar bundle of multiple flagella. Alternatively, upon leaving the pulling mode, they can enter a third slow swimming phase, where they propel themselves with their helical bundle wrapped around the cell body. A theoretical estimate based on a random-walk model shows that the spreading of a population of swimmers is strongly enhanced when cycling through a sequence of pushing, pulling, and wrapped flagellar configurations as compared to the simple push-pull-push pattern.
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Affiliation(s)
- Marius Hintsche
- University of Potsdam, Institute of Physics and Astronomy, 14476, Potsdam, Germany
| | - Veronika Waljor
- University of Potsdam, Institute of Physics and Astronomy, 14476, Potsdam, Germany
| | - Robert Großmann
- Université Côte d'Azur, Laboratoire J. A. Dieudonné, UMR 7351 CNRS, F-06108, Nice Cedex 02, France
| | - Marco J Kühn
- Institut für Mikrobiologie und Molekularbiologie, Justus-Liebig-Universität Giessen, 35392, Giessen, Germany
| | - Kai M Thormann
- Institut für Mikrobiologie und Molekularbiologie, Justus-Liebig-Universität Giessen, 35392, Giessen, Germany
| | - Fernando Peruani
- Université Côte d'Azur, Laboratoire J. A. Dieudonné, UMR 7351 CNRS, F-06108, Nice Cedex 02, France
| | - Carsten Beta
- University of Potsdam, Institute of Physics and Astronomy, 14476, Potsdam, Germany.
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9
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Ko W, Lim S, Lee W, Kim Y, Berg HC, Peskin CS. Modeling polymorphic transformation of rotating bacterial flagella in a viscous fluid. Phys Rev E 2017; 95:063106. [PMID: 28709256 PMCID: PMC5656015 DOI: 10.1103/physreve.95.063106] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/21/2017] [Indexed: 11/07/2022]
Abstract
The helical flagella that are attached to the cell body of bacteria such as Escherichia coli and Salmonella typhimurium allow the cell to swim in a fluid environment. These flagella are capable of polymorphic transformation in that they take on various helical shapes that differ in helical pitch, radius, and chirality. We present a mathematical model of a single flagellum described by Kirchhoff rod theory that is immersed in a fluid governed by Stokes equations. We perform numerical simulations to demonstrate two mechanisms by which polymorphic transformation can occur, as observed in experiments. First, we consider a flagellar filament attached to a rotary motor in which transformations are triggered by a reversal of the direction of motor rotation [L. Turner et al., J. Bacteriol. 182, 2793 (2000)10.1128/JB.182.10.2793-2801.2000]. We then consider a filament that is fixed on one end and immersed in an external fluid flow [H. Hotani, J. Mol. Biol. 156, 791 (1982)10.1016/0022-2836(82)90142-5]. The detailed dynamics of the helical flagellum interacting with a viscous fluid is discussed and comparisons with experimental and theoretical results are provided.
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Affiliation(s)
- William Ko
- Department of Mathematical Sciences, University of Cincinnati, 4199 French Hall West, Cincinnati, Ohio 45221, USA
| | - Sookkyung Lim
- Department of Mathematical Sciences, University of Cincinnati, 4199 French Hall West, Cincinnati, Ohio 45221, USA
| | - Wanho Lee
- National Institute for Mathematical Sciences, KT Daeduk 2 Research Center, 70, Yuseong-daero 1689-gil, Yuseong-gu, Daejeon 305-811, Republic of Korea
| | - Yongsam Kim
- Department of Mathematics, Chung-Ang University, Dongjakgu, Heukseokdong, Seoul 156-756, Republic of Korea
| | - Howard C Berg
- Rowland Institute at Harvard, 100 Edwin H. Land Boulevard, Cambridge, Massachusetts 02142, USA
| | - Charles S Peskin
- Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012, USA
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10
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Bacteria exploit a polymorphic instability of the flagellar filament to escape from traps. Proc Natl Acad Sci U S A 2017; 114:6340-6345. [PMID: 28559324 DOI: 10.1073/pnas.1701644114] [Citation(s) in RCA: 61] [Impact Index Per Article: 8.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Many bacterial species swim by rotating single polar helical flagella. Depending on the direction of rotation, they can swim forward or backward and change directions to move along chemical gradients but also to navigate their obstructed natural environment in soils, sediments, or mucus. When they get stuck, they naturally try to back out, but they can also resort to a radically different flagellar mode, which we discovered here. Using high-speed microscopy, we monitored the swimming behavior of the monopolarly flagellated species Shewanella putrefaciens with fluorescently labeled flagellar filaments at an agarose-glass interface. We show that, when a cell gets stuck, the polar flagellar filament executes a polymorphic change into a spiral-like form that wraps around the cell body in a spiral-like fashion and enables the cell to escape by a screw-like backward motion. Microscopy and modeling suggest that this propagation mode is triggered by an instability of the flagellum under reversal of the rotation and the applied torque. The switch is reversible and bacteria that have escaped the trap can return to their normal swimming mode by another reversal of motor direction. The screw-type flagellar arrangement enables a unique mode of propagation and, given the large number of polarly flagellated bacteria, we expect it to be a common and widespread escape or motility mode in complex and structured environments.
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Qi M, Gong X, Wu B, Zhang G. Landing Dynamics of Swimming Bacteria on a Polymeric Surface: Effect of Surface Properties. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2017; 33:3525-3533. [PMID: 28298087 DOI: 10.1021/acs.langmuir.7b00439] [Citation(s) in RCA: 33] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Landing of bacteria for adhesion on a surface is a common phenomenon in our life. However, how surface properties are involved in this process remains largely unclear. Using digital holographic microscopy, we investigated the three-dimensional motions of flagellate Escherichia coli swimming near polymeric surfaces with different properties in aqueous solution before adhesion. We monitored the bacteria landing dynamics, which shows that the density distribution, the probability, and the orientation for collisions of the bacteria are determined by their motility but are slightly affected by the surface properties. However, surface hydrophobicity reduces the near-wall velocity of the bacteria through collisions and slightly increases the collision duration. This promotes the landing and adhesion of bacteria. By contrast, most bacteria collide with the surface using their flagella, which resist adhesion.
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Affiliation(s)
- Meng Qi
- Faculty of Materials Science and Engineering, South China University of Technology , Guangzhou 510640, P. R. China
| | - Xiangjun Gong
- Faculty of Materials Science and Engineering, South China University of Technology , Guangzhou 510640, P. R. China
| | - Bo Wu
- Faculty of Materials Science and Engineering, South China University of Technology , Guangzhou 510640, P. R. China
| | - Guangzhao Zhang
- Faculty of Materials Science and Engineering, South China University of Technology , Guangzhou 510640, P. R. China
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Eisenstecken T, Hu J, Winkler RG. Bacterial swarmer cells in confinement: a mesoscale hydrodynamic simulation study. SOFT MATTER 2016; 12:8316-8326. [PMID: 27714355 DOI: 10.1039/c6sm01532h] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
A wide spectrum of Peritrichous bacteria undergo considerable physiological changes when they are inoculated onto nutrition-rich surfaces and exhibit a rapid and collective migration denoted as swarming. Thereby, the length of such swarmer cells and their number of flagella increases substantially. In this article, we investigated the properties of individual E. coli-type swarmer cells confined between two parallel walls via mesoscale hydrodynamic simulations, combining molecular dynamics simulations of the swarmer cell with the multiparticle particle collision dynamics approach for the embedding fluid. E. coli-type swarmer cells are three-times longer than their planktonic counter parts, but their flagella density is comparable. By varying the wall separation, we analyze the confinement effect on the flagella arrangement, on the distribution of cells in the gap between the walls, and on the cell dynamics. We find only a weak dependence of confinement on the bundle structure and dynamics. The distribution of cells in the gap changes from a geometry-dominated behavior for very narrow to fluid-dominated behavior for wider gaps, where cells are preferentially located in the gap center for narrower gaps and stay preferentially next to one of the walls for wider gaps. Dynamically, the cells exhibit a wide spectrum of migration behaviors, depending on their flagella bundle arrangement, and ranges from straight swimming to wall rolling.
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Affiliation(s)
- Thomas Eisenstecken
- Theoretical Soft Matter and Biophysics, Institute for Advanced Simulation and Institute of Complex Systems, Forschungszentrum Jülich, D-52425 Jülich, Germany.
| | - Jinglei Hu
- Theoretical Soft Matter and Biophysics, Institute for Advanced Simulation and Institute of Complex Systems, Forschungszentrum Jülich, D-52425 Jülich, Germany. and Kuang Yaming Honors School, Nanjing University, 210023 Nanjing, China.
| | - Roland G Winkler
- Theoretical Soft Matter and Biophysics, Institute for Advanced Simulation and Institute of Complex Systems, Forschungszentrum Jülich, D-52425 Jülich, Germany.
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13
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Adhyapak TC, Stark H. Dynamics of a bacterial flagellum under reverse rotation. SOFT MATTER 2016; 12:5621-5629. [PMID: 27265475 DOI: 10.1039/c6sm00443a] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
To initiate tumbling of an E. coli, one of the helical flagella reverses its sense of rotation. It then transforms from its normal form first to the transient semicoiled state and subsequently to the curly-I state. The dynamics of polymorphism is effectively modeled by describing flagellar elasticity through an extended Kirchhoff free energy. However, the complete landscape of the free energy remains undetermined because the ground state energies of the polymorphic forms are not known. We investigate how variations in these ground state energies affect the dynamics of a reversely rotated flagellum of a swimming bacterium. We find that the flagellum exhibits a number of distinct dynamical states and comprehensively summarize them in a state diagram. As a result, we conclude that tuning the landscape of the extended Kirchhoff free energy alone cannot generate the intermediate full-length semicoiled state. However, our model suggests an ad hoc method to realize the sequence of polymorphic states as observed for a real bacterium. Since the elastic properties of bacterial flagella are similar, our findings can easily be extended to other peritrichous bacteria.
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Affiliation(s)
- Tapan Chandra Adhyapak
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany.
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14
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Jawed MK, Reis PM. Deformation of a soft helical filament in an axial flow at low Reynolds number. SOFT MATTER 2016; 12:1898-1905. [PMID: 26738932 DOI: 10.1039/c5sm02625c] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We perform a numerical investigation of the deformation of a rotating helical filament subjected to an axial flow, under low Reynolds number conditions, motivated by the propulsion of bacteria using helical flagella. Given its slenderness, the helical rod is intrinsically soft and deforms due to the interplay between elastic forces and hydrodynamic loading. We make use of a previously developed and experimentally validated computational tool framework that models the elasticity of the filament using the discrete elastic rod method and the fluid forces are treated using Lighthill's slender body theory. Under axial flow, and in the absence of rotation, the initially helical rod is extended. Above a critical flow speed its configuration comprises a straight portion connected to a localized helix near the free end. When the rod is also rotated about its helical axis, propulsion is only possible in a finite range of angular velocity, with an upper bound that is limited by buckling of the soft helix arising due to viscous stresses. A systematic exploration of the parameter space allows us to quantify regimes for successful propulsion for a number of specific bacteria.
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Affiliation(s)
- Mohammad K Jawed
- Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
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15
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Hu J, Yang M, Gompper G, Winkler RG. Modelling the mechanics and hydrodynamics of swimming E. coli. SOFT MATTER 2015; 11:7867-7876. [PMID: 26256240 DOI: 10.1039/c5sm01678a] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
The swimming properties of an E. coli-type model bacterium are investigated by mesoscale hydrodynamic simulations, combining molecular dynamics simulations of the bacterium with the multiparticle particle collision dynamics method for the embedding fluid. The bacterium is composed of a spherocylindrical body with attached helical flagella, built up from discrete particles for an efficient coupling with the fluid. We measure the hydrodynamic friction coefficients of the bacterium and find quantitative agreement with experimental results of swimming E. coli. The flow field of the bacterium shows a force-dipole-like pattern in the swimming plane and two vortices perpendicular to its swimming direction arising from counterrotation of the cell body and the flagella. By comparison with the flow field of a force dipole and rotlet dipole, we extract the force-dipole and rotlet-dipole strengths for the bacterium and find that counterrotation of the cell body and the flagella is essential for describing the near-field hydrodynamics of the bacterium.
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Affiliation(s)
- Jinglei Hu
- Theoretical Soft Matter and Biophysics, Institute for Advanced Simulation and Institute of Complex Systems, Forschungszentrum Jülich, D-52425 Jülich, Germany.
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Physical Sensing of Surface Properties by Microswimmers--Directing Bacterial Motion via Wall Slip. Sci Rep 2015; 5:9586. [PMID: 25993019 PMCID: PMC4438609 DOI: 10.1038/srep09586] [Citation(s) in RCA: 67] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2014] [Accepted: 03/10/2015] [Indexed: 11/09/2022] Open
Abstract
Bacteria such as Escherichia coli swim along circular trajectories adjacent to surfaces. Thereby, the orientation (clockwise, counterclockwise) and the curvature depend on the surface properties. We employ mesoscale hydrodynamic simulations of a mechano-elastic model of E. coli, with a spherocylindrical body propelled by a bundle of rotating helical flagella, to study quantitatively the curvature of the appearing circular trajectories. We demonstrate that the cell is sensitive to nanoscale changes in the surface slip length. The results are employed to propose a novel approach to directing bacterial motion on striped surfaces with different slip lengths, which implies a transformation of the circular motion into a snaking motion along the stripe boundaries. The feasibility of this approach is demonstrated by a simulation of active Brownian rods, which also reveals a dependence of directional motion on the stripe width.
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Elgeti J, Winkler RG, Gompper G. Physics of microswimmers--single particle motion and collective behavior: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2015; 78:056601. [PMID: 25919479 DOI: 10.1088/0034-4885/78/5/056601] [Citation(s) in RCA: 647] [Impact Index Per Article: 71.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/13/2023]
Abstract
Locomotion and transport of microorganisms in fluids is an essential aspect of life. Search for food, orientation toward light, spreading of off-spring, and the formation of colonies are only possible due to locomotion. Swimming at the microscale occurs at low Reynolds numbers, where fluid friction and viscosity dominates over inertia. Here, evolution achieved propulsion mechanisms, which overcome and even exploit drag. Prominent propulsion mechanisms are rotating helical flagella, exploited by many bacteria, and snake-like or whip-like motion of eukaryotic flagella, utilized by sperm and algae. For artificial microswimmers, alternative concepts to convert chemical energy or heat into directed motion can be employed, which are potentially more efficient. The dynamics of microswimmers comprises many facets, which are all required to achieve locomotion. In this article, we review the physics of locomotion of biological and synthetic microswimmers, and the collective behavior of their assemblies. Starting from individual microswimmers, we describe the various propulsion mechanism of biological and synthetic systems and address the hydrodynamic aspects of swimming. This comprises synchronization and the concerted beating of flagella and cilia. In addition, the swimming behavior next to surfaces is examined. Finally, collective and cooperate phenomena of various types of isotropic and anisotropic swimmers with and without hydrodynamic interactions are discussed.
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Affiliation(s)
- J Elgeti
- Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany
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Vogel R, Stark H. Rotation-induced polymorphic transitions in bacterial flagella. PHYSICAL REVIEW LETTERS 2013; 110:158104. [PMID: 25167316 DOI: 10.1103/physrevlett.110.158104] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2012] [Indexed: 06/03/2023]
Abstract
Bacteria propel themselves with the help of rotating helical flagella. They change their swimming direction during tumbling events in order to increase, for example, their supply of nutrients (chemotaxis). During tumbling a bacterial flagellum assumes different polymorphic states. Based on a continuum model for the motor-flagellum system, we demonstrate that a changing motor torque can initiate these polymorphic transformations. In particular, we investigate the run-and-stop tumble strategy of Rhodobacter sphaeroides which uses a coiled-to-normal transition in its single flagellum. We also show that torque reversal in single-flagellated Escherichia coli generates a normal-to-curly I transition as observed for tumbling E. coli that swim with a bundle of several flagella.
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Affiliation(s)
- Reinhard Vogel
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Holger Stark
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
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Abstract
The swimming of a bacterium or a biomimetic nanobot driven by a rotating helical flagellum is often interpreted using the resistive force theory developed by Gray and Hancock and by Lighthill, but this theory has not been tested for a range of physically relevant parameters. We test resistive force theory in experiments on macroscopic swimmers in a fluid that is highly viscous so the Reynolds number is small compared to unity, just as for swimming microorganisms. The measurements are made for the range of helical wavelengths λ, radii R, and lengths L relevant to bacterial flagella. The experiments determine thrust, torque, and drag, thus providing a complete description of swimming driven by a rotating helix at low Reynolds number. Complementary numerical simulations are conducted using the resistive force theories, the slender body theories of Lighthill and Johnson, and the regularized Stokeslet method. The experimental results differ qualitatively and quantitatively from the predictions of resistive force theory. The difference is especially large for and/or , parameter ranges common for bacteria. In contrast, the predictions of Stokeslet and slender body analyses agree with the laboratory measurements within the experimental uncertainty (a few percent) for all λ, R, and L. We present code implementing the slender body, regularized Stokeslet, and resistive force theories; thus readers can readily compute force, torque, and drag for any bacterium or nanobot driven by a rotating helical flagellum.
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Shklarsh A, Finkelshtein A, Ariel G, Kalisman O, Ingham C, Ben-Jacob E. Collective navigation of cargo-carrying swarms. Interface Focus 2012; 2:786-98. [PMID: 24312731 DOI: 10.1098/rsfs.2012.0029] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2012] [Accepted: 08/06/2012] [Indexed: 11/12/2022] Open
Abstract
Much effort has been devoted to the study of swarming and collective navigation of micro-organisms, insects, fish, birds and other organisms, as well as multi-agent simulations and to the study of real robots. It is well known that insect swarms can carry cargo. The studies here are motivated by a less well-known phenomenon: cargo transport by bacteria swarms. We begin with a concise review of how bacteria swarms carry natural, micrometre-scale objects larger than the bacteria (e.g. fungal spores) as well as man-made beads and capsules (for drug delivery). A comparison of the trajectories of virtual beads in simulations (using different putative coupling between the virtual beads and the bacteria) with the observed trajectories of transported fungal spores implies the existence of adaptable coupling. Motivated by these observations, we devised new, multi-agent-based studies of cargo transport by agent swarms. As a first step, we extended previous modelling of collective navigation of simple bacteria-inspired agents in complex terrain, using three putative models of agent-cargo coupling. We found that cargo-carrying swarms can navigate efficiently in a complex landscape. We further investigated how the stability, elasticity and other features of agent-cargo bonds influence the collective motion and the transport of the cargo, and found sharp phase shifts and dual successful strategies for cargo delivery. Further understanding of such mechanisms may provide valuable clues to understand cargo-transport by smart swarms of other organisms as well as by man-made swarming robots.
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Affiliation(s)
- Adi Shklarsh
- School of Physics and Astronomy , Tel Aviv University , Tel Aviv 69978 , Israel ; The Blavatnik School of Computer Science , Tel Aviv University , Tel Aviv 69978 , Israel
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Vogel R, Stark H. Motor-driven bacterial flagella and buckling instabilities. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2012; 35:15. [PMID: 22395533 DOI: 10.1140/epje/i2012-12015-0] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2011] [Revised: 02/14/2012] [Accepted: 02/16/2012] [Indexed: 05/31/2023]
Abstract
Many types of bacteria swim by rotating a bundle of helical filaments also called flagella. Each filament is driven by a rotary motor and a very flexible hook transmits the motor torque to the filament. We model it by discretizing Kirchhoff's elastic-rod theory and develop a coarse-grained approach for driving the helical filament by a motor torque. A rotating flagellum generates a thrust force, which pushes the cell body forward and which increases with the motor torque. We fix the rotating flagellum in space and show that it buckles under the thrust force at a critical motor torque. Buckling becomes visible as a supercritical Hopf bifurcation in the thrust force. A second buckling transition occurs at an even higher motor torque. We attach the flagellum to a spherical cell body and also observe the first buckling transition during locomotion. By changing the size of the cell body, we vary the necessary thrust force and thereby obtain a characteristic relation between the critical thrust force and motor torque. We present a elaborate analytical model for the buckling transition based on a helical rod which quantitatively reproduces the critical force-torque relation. Real values for motor torque, cell body size, and the geometry of the helical filament suggest that buckling should occur in single bacterial flagella. We also find that the orientation of pulling flagella along the driving torque is not stable and comment on the biological relevance for marine bacteria.
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Affiliation(s)
- R Vogel
- Institute for Theoretical Physics, TU Berlin, Berlin, Germany.
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Speier C, Vogel R, Stark H. Modeling the bacterial flagellum by an elastic network of rigid bodies. Phys Biol 2011; 8:046009. [DOI: 10.1088/1478-3975/8/4/046009] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Janssen PJA, Graham MD. Coexistence of tight and loose bundled states in a model of bacterial flagellar dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:011910. [PMID: 21867216 DOI: 10.1103/physreve.84.011910] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2011] [Revised: 05/20/2011] [Indexed: 05/31/2023]
Abstract
Many microorganisms propel themselves through their fluid environment by means of multiple rotating flagella that self-organize to form bundles, a process that is complex and poorly understood. In the present work, the bundling behavior of a pair of flexible flagella, each driven by a constant torque motor, is investigated, using a mathematical model incorporating the fluid motion generated by each flagellum as well as the finite flexibility of the flagella. The initial stage of bundling is driven purely by hydrodynamics but the final state of the bundle is determined by a nontrivial balance between hydrodynamics and elasticity. As the flexibility of the flagella increases a regime is found where, depending on initial conditions, one finds bundles that are either tight, with the flagella in mechanical contact, or loose, with the flagella intertwined but not touching. That is, multiple coexisting states of bundling are found. The parameter regime (in terms of flexibility and distance between motors) at which this multiplicity occurs is comparable to the parameters for a number of bacteria.
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Affiliation(s)
- P J A Janssen
- Department of Chemical and Biological Engineering, University of Wisconsin - Madison, 53706 Madison, Wisconsin, USA
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