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Unnikrishnan S, Scott R, Ogundele E, Azad M, Ishimoto K, Suarez S, Tung CK. Hybrid motility mechanism of sperm at viscoelastic-solid interface. RESEARCH SQUARE 2024:rs.3.rs-4284452. [PMID: 38746416 PMCID: PMC11092832 DOI: 10.21203/rs.3.rs-4284452/v1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2024]
Abstract
To fertilize eggs, sperm must pass through narrow, complex channels filled with viscoelastic fluids in the female reproductive tract. While it is known that the topography of the surfaces plays a role in guiding sperm movement, sperm have been thought of as swimmers, i.e., their motility comes solely from sperm interaction with the surrounding fluid, and therefore, the surfaces have no direct role in the motility mechanism itself. Here, we examined the role of solid surfaces in the movement of sperm in a highly viscoelastic medium. By visualizing the flagellum interaction with surfaces in a microfluidic device, we found that the flagellum stays close to the surface while the kinetic friction between the flagellum and the surface is in the direction of sperm movement, providing thrust. Additionally, the flow field generated by sperm suggests slippage between the viscoelastic fluid and the solid surface, deviating from the no-slip boundary typically used in standard fluid dynamics models. These observations point to hybrid motility mechanisms in sperm involving direct flagellum-surface interaction in addition to flagellum pushing the fluid. This finding signifies an evolutionary strategy of mammalian sperm crucial for their efficient migration through narrow, mucus-filled passages of the female reproductive tract.
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Affiliation(s)
| | - Robert Scott
- North Carolina Agricultural and Technical State University
| | | | - Mohammad Azad
- North Carolina Agricultural and Technical State University
| | | | | | - Chih Kuan Tung
- North Carolina Agricultural and Technical State University
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2
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Ishimoto K, Gaffney EA, Smith DJ. Squirmer hydrodynamics near a periodic surface topography. Front Cell Dev Biol 2023; 11:1123446. [PMID: 37123410 PMCID: PMC10133482 DOI: 10.3389/fcell.2023.1123446] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2022] [Accepted: 03/15/2023] [Indexed: 05/02/2023] Open
Abstract
The behaviour of microscopic swimmers has previously been explored near large-scale confining geometries and in the presence of very small-scale surface roughness. Here, we consider an intermediate case of how a simple microswimmer, the tangential spherical squirmer, behaves adjacent to singly and doubly periodic sinusoidal surface topographies that spatially oscillate with an amplitude that is an order of magnitude less than the swimmer size and wavelengths that are also within an order of magnitude of this scale. The nearest neighbour regularised Stokeslet method is used for numerical explorations after validating its accuracy for a spherical tangential squirmer that swims stably near a flat surface. The same squirmer is then introduced to different surface topographies. The key governing factor in the resulting swimming behaviour is the size of the squirmer relative to the surface topography wavelength. For instance, directional guidance is not observed when the squirmer is much larger, or much smaller, than the surface topography wavelength. In contrast, once the squirmer size is on the scale of the topography wavelength, limited guidance is possible, often with local capture in the topography troughs. However, complex dynamics can also emerge, especially when the initial configuration is not close to alignment along topography troughs or above topography crests. In contrast to sensitivity in alignment and topography wavelength, reductions in the amplitude of the surface topography or variations in the shape of the periodic surface topography do not have extensive impacts on the squirmer behaviour. Our findings more generally highlight that the numerical framework provides an essential basis to elucidate how swimmers may be guided by surface topography.
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Affiliation(s)
- Kenta Ishimoto
- Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
- *Correspondence: Kenta Ishimoto,
| | - Eamonn A. Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - David J. Smith
- School of Mathematics, University of Birmingham, Birmingham, United Kingdom
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RAZAVI SEYEDESMAIL, FARHANGMEHR VAHID, MARANDI ELIAS. MODELING OF FLAGELLUM BEHAVIOR AND TWO-DIMENSIONAL SPERM CELL MOTILITY WITHIN THE CONTEXT OF FLUID–SOLID INTERACTIONS. J MECH MED BIOL 2021. [DOI: 10.1142/s0219519421500524] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this study, the flagellar motility of a swimmer microorganism as a model of a human sperm cell, inside a two-dimensional channel as a model of the female reproductive tract containing a viscous fluid, is numerically investigated. The Navier–Stokes equations governing the fluid are coupled with the equations governing the models flagellum via applying a fluid–solid interaction approach and then solved using the finite element method. To stimulate the flagellum to move, a prescribed sinusoidal waveform is applied to it. The strain induced by this waveform along the flagellum initiates a continuous interaction between the flagellum and the fluid. The simulations are validated using data available in the literature. A very good agreement is seen between them. The results show that by decreasing the Young modulus of the flagellum as well as increasing the fluid viscosity, the swimming velocity of the model significantly decreases. It is found that for lower Young modulus of the flagellum, the effect of the fluid viscosity on the flagellar deformation is stronger. It is also found that for higher amplitude of the waveform applied to stimulate the flagellum, both the swimming velocity of the model and the average work rate are greater. Moreover, it is found that in a channel with a smaller height, the model swims at a higher speed and with a higher average work rate.
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Affiliation(s)
- SEYED ESMAIL RAZAVI
- Faculty of Mechanical Engineering, University of Tabriz, Tabriz, East Azarbaijan, Iran
| | - VAHID FARHANGMEHR
- Department of Mechanical Engineering, University of Bonab, Bonab, East Azarbaijan, Iran
| | - ELIAS MARANDI
- Faculty of Mechanical Engineering, University of Tabriz, Tabriz, East Azarbaijan, Iran
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4
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Gaffney EA, Ishimoto K, Walker BJ. Modelling Motility: The Mathematics of Spermatozoa. Front Cell Dev Biol 2021; 9:710825. [PMID: 34354994 PMCID: PMC8329702 DOI: 10.3389/fcell.2021.710825] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2021] [Accepted: 06/25/2021] [Indexed: 11/23/2022] Open
Abstract
In one of the first examples of how mechanics can inform axonemal mechanism, Machin's study in the 1950s highlighted that observations of sperm motility cannot be explained by molecular motors in the cell membrane, but would instead require motors distributed along the flagellum. Ever since, mechanics and hydrodynamics have been recognised as important in explaining the dynamics, regulation, and guidance of sperm. More recently, the digitisation of sperm videomicroscopy, coupled with numerous modelling and methodological advances, has been bringing forth a new era of scientific discovery in this field. In this review, we survey these advances before highlighting the opportunities that have been generated for both recent research and the development of further open questions, in terms of the detailed characterisation of the sperm flagellum beat and its mechanics, together with the associated impact on cell behaviour. In particular, diverse examples are explored within this theme, ranging from how collective behaviours emerge from individual cell responses, including how these responses are impacted by the local microenvironment, to the integration of separate advances in the fields of flagellar analysis and flagellar mechanics.
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Affiliation(s)
- Eamonn A. Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Kenta Ishimoto
- Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
| | - Benjamin J. Walker
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
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5
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Abstract
Due to rising human infertility, sperm motility has been an important subject. Among the hundreds of millions of sperms on the journey up the oviducts, only a few excellent travelers will reach the eggs. This journey is affected by many factors, some of which include sperm quality, sperm density, fluid rheology and chemotaxis. In addition, the sperm swimming through different body tracks and fluids involves complex sperm flagellar, complex fluid environment, and multi-sperm and sperm-wall interactions. Therefore, this topic has generated substantial research interest. In this paper, we present a review of computational studies on sperm swimming from an engineering perspective with focus on both simplified theoretical methods and fluid–structure interaction methods. Several open issues in this field are highlighted.
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6
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Hyakutake T, Sugita K, Ujifuku S, Sakurai R, Murakami R, Hayamizu Y. Experimental study on the effect of flow in microfluidic channel on bovine sperm navigation. J Biomech 2021; 118:110290. [PMID: 33581442 DOI: 10.1016/j.jbiomech.2021.110290] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2020] [Revised: 12/17/2020] [Accepted: 01/23/2021] [Indexed: 10/22/2022]
Abstract
The navigation mechanism of mammalian sperm in the female reproductive tract is unclear owing to its complex process. This study performed an in vitro experiment using the microfluidic channel with two reservoirs to investigate the effect of fluid flow on the swimming properties of the bovine sperm. The width and height of the manufactured channel were 200 and 20 μm, respectively. The flow in the microchannel occurs because of the hydraulic head difference between the two reservoirs. Sperm with positive rheotaxis proceed in the opposite direction of the flow in the channel after swimming up the downstream reservoir. This study focused on the effect of the flow in the microfluidic channel on sperm motility. It was observed that sperm mostly moved along the channel wall and accumulated near the wall away from the downstream reservoir. The existence of fluid flow in the channel brought about an increase in the ratio of the sperm with positive rheotaxis. Furthermore, the experimental results indicated that the motility of sperm swimming against the flow along the wall increased away from the downstream reservoir. These results will provide useful information to understand the mechanism of sperm navigation for in vivo fertilization.
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Affiliation(s)
- Toru Hyakutake
- Faculty of Engineering, Yokohama National University, 79-5, Hodogaya, Yokohama 240-8501, Japan.
| | - Kenta Sugita
- Graduate School of Engineering, Yokohama National University, 79-5, Hodogaya, Yokohama 240-8501, Japan
| | - Shota Ujifuku
- Graduate School of Engineering, Yokohama National University, 79-5, Hodogaya, Yokohama 240-8501, Japan
| | - Rintaro Sakurai
- Graduate School of Engineering, Yokohama National University, 79-5, Hodogaya, Yokohama 240-8501, Japan
| | - Renta Murakami
- Graduate School of Engineering, Yokohama National University, 79-5, Hodogaya, Yokohama 240-8501, Japan
| | - Yasutaka Hayamizu
- National Institute of Technology, Yonago College, 4448 Hikona-cho, Yonago 683-8502, Japan
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Sprenger AR, Shaik VA, Ardekani AM, Lisicki M, Mathijssen AJTM, Guzmán-Lastra F, Löwen H, Menzel AM, Daddi-Moussa-Ider A. Towards an analytical description of active microswimmers in clean and in surfactant-covered drops. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2020; 43:58. [PMID: 32920676 DOI: 10.1140/epje/i2020-11980-9] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2020] [Accepted: 08/10/2020] [Indexed: 05/24/2023]
Abstract
Geometric confinements are frequently encountered in the biological world and strongly affect the stability, topology, and transport properties of active suspensions in viscous flow. Based on a far-field analytical model, the low-Reynolds-number locomotion of a self-propelled microswimmer moving inside a clean viscous drop or a drop covered with a homogeneously distributed surfactant, is theoretically examined. The interfacial viscous stresses induced by the surfactant are described by the well-established Boussinesq-Scriven constitutive rheological model. Moreover, the active agent is represented by a force dipole and the resulting fluid-mediated hydrodynamic couplings between the swimmer and the confining drop are investigated. We find that the presence of the surfactant significantly alters the dynamics of the encapsulated swimmer by enhancing its reorientation. Exact solutions for the velocity images for the Stokeslet and dipolar flow singularities inside the drop are introduced and expressed in terms of infinite series of harmonic components. Our results offer useful insights into guiding principles for the control of confined active matter systems and support the objective of utilizing synthetic microswimmers to drive drops for targeted drug delivery applications.
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Affiliation(s)
- Alexander R Sprenger
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D-40225, Düsseldorf, Germany.
| | - Vaseem A Shaik
- School of Mechanical Engineering, Purdue University, 47907, West Lafayette, IN, USA
| | - Arezoo M Ardekani
- School of Mechanical Engineering, Purdue University, 47907, West Lafayette, IN, USA
| | - Maciej Lisicki
- Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093, Warsaw, Poland
| | - Arnold J T M Mathijssen
- Department of Bioengineering, Stanford University, 443 Via Ortega, 94305, Stanford, CA, USA
- Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, 19104, Philadelphia, PA, USA
| | - Francisca Guzmán-Lastra
- Centro de Investigación DAiTA Lab, Facultad de Estudios Interdisciplinarios, Universidad Mayor, Av. Manuel Montt 367, Providencia, Santiago de Chile, Chile
| | - Hartmut Löwen
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D-40225, Düsseldorf, Germany
| | - Andreas M Menzel
- Institut für Physik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany
| | - Abdallah Daddi-Moussa-Ider
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D-40225, Düsseldorf, Germany
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8
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Schoeller SF, Holt WV, Keaveny EE. Collective dynamics of sperm cells. Philos Trans R Soc Lond B Biol Sci 2020; 375:20190384. [PMID: 32713305 DOI: 10.1098/rstb.2019.0384] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022] Open
Abstract
While only a single sperm may fertilize the egg, getting to the egg can be facilitated, and possibly enhanced, by sperm group dynamics. Examples range from the trains formed by wood mouse sperm to the bundles exhibited by echidna sperm. In addition, observations of wave-like patterns exhibited by ram semen are used to score prospective sample fertility for artificial insemination in agriculture. In this review, we discuss these experimental observations of collective dynamics, as well as describe recent mechanistic models that link the motion of individual sperm cells and their flagella to observed collective dynamics. Establishing this link in models involves negotiating the disparate time- and length scales involved, typically separated by a factor of 1000, to capture the dynamics at the greatest length scales affected by mechanisms at the shortest time scales. Finally, we provide some outlook on the subject, in particular, the open questions regarding how collective dynamics impacts fertility. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.
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Affiliation(s)
- Simon F Schoeller
- Department of Mathematics, Imperial College London, London SW7 2AZ, UK
| | - William V Holt
- Zoological Society of London, Regent's Park, London NW14RY, UK
| | - Eric E Keaveny
- Department of Mathematics, Imperial College London, London SW7 2AZ, UK
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9
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El Alaoui-Faris Y, Pomet JB, Régnier S, Giraldi L. Optimal actuation of flagellar magnetic microswimmers. Phys Rev E 2020; 101:042604. [PMID: 32422737 DOI: 10.1103/physreve.101.042604] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/02/2019] [Accepted: 01/18/2020] [Indexed: 11/07/2022]
Abstract
We present an automated procedure for the design of optimal actuation for flagellar magnetic microswimmers based on numerical optimization. Using this method, a magnetic actuation method is provided which allows these devices to swim significantly faster compared to the usual sinusoidal actuation. This leads to a novel swimming strategy which makes the swimmer perform a three-dimensional figure-eight trajectory. This shows that a faster propulsion is obtained when the swimmer is allowed to go out of plane. This approach is experimentally validated on a scaled-up flexible swimmer.
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Affiliation(s)
- Yacine El Alaoui-Faris
- Université Côte d'Azur, Inria, CNRS, LJAD, McTAO team, 06902 Nice, France and Sorbonne Université, CNRS, ISIR, 75005 Paris, France
| | | | | | - Laetitia Giraldi
- Université Côte d'Azur, Inria, CNRS, LJAD, McTAO team, 06902 Nice, France
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10
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Abstract
Micro-swimmers such as spermatozoa are able to efficiently navigate through viscous fluids that contain a sparse network of fibers or other macromolecules. We utilize the Brinkman equation to capture the fluid dynamics of sparse and stationary obstacles that are represented via a single resistance parameter. The method of regularized Brinkmanlets is utilized to solve for the fluid flow and motion of the swimmer in 2-dimensions when assuming the flagellum (tail) propagates a curvature wave. Extending previous studies, we investigate the dynamics of swimming when varying the resistance parameter, head or cell body radius, and preferred beat form parameters. For a single swimmer, we determine that increased swimming speed occurs for a smaller cell body radius and smaller fluid resistance. Progression of swimmers exhibits complex dynamics when considering hydrodynamic interactions; attraction of two swimmers is a robust phenomenon for smaller beat amplitude of the tail and smaller fluid resistance. Wall attraction is also observed, with a longer time scale of wall attraction with a larger resistance parameter.
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11
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Daddi-Moussa-Ider A, Kurzthaler C, Hoell C, Zöttl A, Mirzakhanloo M, Alam MR, Menzel AM, Löwen H, Gekle S. Frequency-dependent higher-order Stokes singularities near a planar elastic boundary: Implications for the hydrodynamics of an active microswimmer near an elastic interface. Phys Rev E 2019; 100:032610. [PMID: 31639990 DOI: 10.1103/physreve.100.032610] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/16/2019] [Indexed: 06/10/2023]
Abstract
The emerging field of self-driven active particles in fluid environments has recently created significant interest in the biophysics and bioengineering communities owing to their promising future for biomedical and technological applications. These microswimmers move autonomously through aqueous media, where under realistic situations they encounter a plethora of external stimuli and confining surfaces with peculiar elastic properties. Based on a far-field hydrodynamic model, we present an analytical theory to describe the physical interaction and hydrodynamic couplings between a self-propelled active microswimmer and an elastic interface that features resistance toward shear and bending. We model the active agent as a superposition of higher-order Stokes singularities and elucidate the associated translational and rotational velocities induced by the nearby elastic boundary. Our results show that the velocities can be decomposed in shear and bending related contributions which approach the velocities of active agents close to a no-slip rigid wall in the steady limit. The transient dynamics predict that contributions to the velocities of the microswimmer due to bending resistance are generally more pronounced than those due to shear resistance. Bending can enhance (suppress) the velocities resulting from higher-order singularities whereas the shear related contribution decreases (increases) the velocities. Most prominently, we find that near an elastic interface of only energetic resistance toward shear deformation, such as that of an elastic capsule designed for drug delivery, a swimming bacterium undergoes rotation of the same sense as observed near a no-slip wall. In contrast to that, near an interface of only energetic resistance toward bending, such as that of a fluid vesicle or liposome, we find a reversed sense of rotation. Our results provide insight into the control and guidance of artificial and synthetic self-propelling active microswimmers near elastic confinements.
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Affiliation(s)
- Abdallah Daddi-Moussa-Ider
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany
| | - Christina Kurzthaler
- Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - Christian Hoell
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany
| | - Andreas Zöttl
- Institute for Theoretical Physics, Technische Universität Wien, Wiedner Hauptstraße 8-10, 1040 Wien, Austria
| | - Mehdi Mirzakhanloo
- Department of Mechanical Engineering, University of California, Berkeley, California 94720, USA
| | - Mohammad-Reza Alam
- Department of Mechanical Engineering, University of California, Berkeley, California 94720, USA
| | - Andreas M Menzel
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany
| | - Hartmut Löwen
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany
| | - Stephan Gekle
- Biofluid Simulation and Modeling, Theoretische Physik VI, Universität Bayreuth, Universitätsstraße 30, 95440 Bayreuth, Germany
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12
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Swimming of Spermatozoa in a Maxwell Fluid. MICROMACHINES 2019; 10:mi10020078. [PMID: 30678348 PMCID: PMC6412255 DOI: 10.3390/mi10020078] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/11/2018] [Revised: 01/18/2019] [Accepted: 01/21/2019] [Indexed: 12/18/2022]
Abstract
It has been suggested that the swimming mechanism used by spermatozoa could be adopted for self-propelled micro-robots in small environments and potentially applied to biomedical engineering. Mammalian sperm cells must swim through a viscoelastic mucus layer to find the egg cell. Thus, understanding how sperm cells swim through viscoelastic liquids is significant not only for physiology, but also for the design of micro-robots. In this paper, we developed a numerical model of a sperm cell in a linear Maxwell fluid based on the boundary element slender-body theory coupling method. The viscoelastic properties were characterized by the Deborah number (De), and we found that, under the prescribed waveform, the swimming speed decayed with the Deborah number in the small-De regime (De < 1.0). The swimming efficiency was independent of the Deborah number, and the decrease in the swimming speed was not significantly affected by the wave pattern.
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13
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Rorai C, Zaitsev M, Karabasov S. On the limitations of some popular numerical models of flagellated microswimmers: importance of long-range forces and flagellum waveform. ROYAL SOCIETY OPEN SCIENCE 2019; 6:180745. [PMID: 30800342 PMCID: PMC6366169 DOI: 10.1098/rsos.180745] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2018] [Accepted: 11/08/2018] [Indexed: 05/30/2023]
Abstract
For a sperm-cell-like flagellated swimmer in an unbounded domain, several numerical models of different fidelity are considered based on the Stokes flow approximation. The models include a regularized Stokeslet method and a three-dimensional finite-element method, which serve as the benchmark solutions for several approximate models considered. The latter include the resistive force theory versions of Lighthill, and Gray and Hancock, as well as a simplified approximation based on computing the hydrodynamic forces exerted on the head and the flagellum separately. It is shown how none of the simplified models is robust enough with regards to predicting the effect of the swimmer head shape change on the swimmer dynamics. For a range of swimmer motions considered, the resulting solutions for the swimmer force and velocities are analysed and the applicability of the Stokes model for the swimmers in question is probed.
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Affiliation(s)
- C. Rorai
- School of Engineering and Materials Science, Queen Mary University of London, Mile End Road, London E1 4NS, UK
| | - M. Zaitsev
- Nuclear Safety Institute, ul. Bolshaja Tulskaja, 52, 115191 Moscow, Russia
| | - S. Karabasov
- School of Engineering and Materials Science, Queen Mary University of London, Mile End Road, London E1 4NS, UK
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14
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Boundary behaviours of Leishmania mexicana: A hydrodynamic simulation study. J Theor Biol 2018; 462:311-320. [PMID: 30465777 PMCID: PMC6333917 DOI: 10.1016/j.jtbi.2018.11.016] [Citation(s) in RCA: 20] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2018] [Revised: 11/13/2018] [Accepted: 11/15/2018] [Indexed: 01/08/2023]
Abstract
It is well established that the parasites of the genus Leishmania exhibit complex surface interactions with the sandfly vector midgut epithelium, but no prior study has considered the details of their hydrodynamics. Here, the boundary behaviours of motile Leishmania mexicana promastigotes are explored in a computational study using the boundary element method, with a model flagellar beating pattern that has been identified from digital videomicroscopy. In particular a simple flagellar kinematics is observed and quantified using image processing and mode identification techniques, suggesting a simple mechanical driver for the Leishmania beat. Phase plane analysis and long-time simulation of a range of Leishmania swimming scenarios demonstrate an absence of stable boundary motility for an idealised model promastigote, with behaviours ranging from boundary capture to deflection into the bulk both with and without surface forces between the swimmer and the boundary. Indeed, the inclusion of a short-range repulsive surface force results in the deflection of all surface-bound promastigotes, suggesting that the documented surface detachment of infective metacyclic promastigotes may be the result of their particular morphology and simple hydrodynamics. Further, simulation elucidates a remarkable morphology-dependent hydrodynamic mechanism of boundary approach, hypothesised to be the cause of the well-established phenomenon of tip-first epithelial attachment of Leishmania promastigotes to the sandfly vector midgut.
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15
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Ishimoto K, Gaffney EA. Hydrodynamic Clustering of Human Sperm in Viscoelastic Fluids. Sci Rep 2018; 8:15600. [PMID: 30349142 PMCID: PMC6197292 DOI: 10.1038/s41598-018-33584-8] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/16/2018] [Accepted: 10/02/2018] [Indexed: 12/19/2022] Open
Abstract
We have numerically investigated sperm clustering behaviours, modelling cells as superpositions of regularised flow singularities, coarse-grained from experimentally obtained digital microscopy of human sperm, both in watery medium and a highly viscous-weakly elastic, methylcellulose medium. We find that the cell yaw and cell pulling dynamics inhibit clustering in low viscosity media. In contrast clustering is readily visible in simulations modelling sperm within a methylcellulose medium, in line with previous observations that bovine sperm clustering is much more prominent in a rheological polyacrylamide medium. Furthermore, the fine-scale details of sperm flagellar movement substantially impact large-scale collective behaviours, further motivating the need for the digital microscopy and characterization of sperm to understand their dynamics.
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Affiliation(s)
- Kenta Ishimoto
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK. .,Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, 153-8914, Japan.
| | - Eamonn A Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
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16
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Daddi-Moussa-Ider A, Lisicki M, Mathijssen AJTM, Hoell C, Goh S, Bławzdziewicz J, Menzel AM, Löwen H. State diagram of a three-sphere microswimmer in a channel. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2018; 30:254004. [PMID: 29757157 DOI: 10.1088/1361-648x/aac470] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Geometric confinements are frequently encountered in soft matter systems and in particular significantly alter the dynamics of swimming microorganisms in viscous media. Surface-related effects on the motility of microswimmers can lead to important consequences in a large number of biological systems, such as biofilm formation, bacterial adhesion and microbial activity. On the basis of low-Reynolds-number hydrodynamics, we explore the state diagram of a three-sphere microswimmer under channel confinement in a slit geometry and fully characterize the swimming behavior and trajectories for neutral swimmers, puller- and pusher-type swimmers. While pushers always end up trapped at the channel walls, neutral swimmers and pullers may further perform a gliding motion and maintain a stable navigation along the channel. We find that the resulting dynamical system exhibits a supercritical pitchfork bifurcation in which swimming in the mid-plane becomes unstable beyond a transition channel height while two new stable limit cycles or fixed points that are symmetrically disposed with respect to the channel mid-height emerge. Additionally, we show that an accurate description of the averaged swimming velocity and rotation rate in a channel can be captured analytically using the method of hydrodynamic images, provided that the swimmer size is much smaller than the channel height.
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Affiliation(s)
- Abdallah Daddi-Moussa-Ider
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany
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17
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Balboa Usabiaga F, Delmotte B, Donev A. Brownian dynamics of confined suspensions of active microrollers. J Chem Phys 2018; 146:134104. [PMID: 28390356 DOI: 10.1063/1.4979494] [Citation(s) in RCA: 32] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We develop efficient numerical methods for performing many-body Brownian dynamics simulations of a recently observed fingering instability in an active suspension of colloidal rollers sedimented above a wall [M. Driscoll, B. Delmotte, M. Youssef, S. Sacanna, A. Donev, and P. Chaikin, Nat. Phys. (2016), preprint arXiv:1609.08673. We present a stochastic Adams-Bashforth integrator for the equations of Brownian dynamics, which has the same cost but is more accurate than the widely used Euler-Maruyama scheme, and use a random finite difference to capture the stochastic drift proportional to the divergence of the configuration-dependent mobility matrix. We generate the Brownian increments using a Krylov method and show that for particles confined to remain in the vicinity of a no-slip wall by gravity or active flows, the number of iterations is independent of the number of particles. Our numerical experiments with active rollers show that the thermal fluctuations set the characteristic height of the colloids above the wall, both in the initial condition and the subsequent evolution dominated by active flows. The characteristic height in turn controls the time scale and wavelength for the development of the fingering instability.
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Affiliation(s)
| | - Blaise Delmotte
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
| | - Aleksandar Donev
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
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18
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Ishimoto K, Gadêlha H, Gaffney EA, Smith DJ, Kirkman-Brown J. Human sperm swimming in a high viscosity mucus analogue. J Theor Biol 2018; 446:1-10. [PMID: 29462624 DOI: 10.1016/j.jtbi.2018.02.013] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2017] [Revised: 02/12/2018] [Accepted: 02/13/2018] [Indexed: 12/16/2022]
Abstract
Remarkably, mammalian sperm maintain a substantive proportion of their progressive swimming speed within highly viscous fluids, including those of the female reproductive tract. Here, we analyse the digital microscopy of a human sperm swimming in a highly viscous, weakly elastic mucus analogue. We exploit principal component analysis to simplify its flagellar beat pattern, from which boundary element calculations are used to determine the time-dependent flow field around the sperm cell. The sperm flow field is further approximated in terms of regularised point forces, and estimates of the mechanical power consumption are determined, for comparison with analogous low viscosity media studies. This highlights extensive differences in the structure of the flows surrounding human sperm in different media, indicating how the cell-cell and cell-boundary hydrodynamic interactions significantly differ with the physical microenvironment. The regularised point force decomposition also provides cell-level information that may ultimately be incorporated into sperm population models. We further observe indications that the core feature in explaining the effectiveness of sperm swimming in high viscosity media is the loss of cell yawing, which is related with a greater density of regularised point force singularities along the axis of symmetry of the flagellar beat to represent the flow field. In turn this implicates a reduction of the wavelength of the distal beat pattern - and hence dynamical wavelength selection of the flagellar beat - as the dominant feature governing the effectiveness of sperm swimming in highly viscous media.
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Affiliation(s)
- Kenta Ishimoto
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK; The Hakubi Center for Advanced Research, Kyoto University, Kyoto 606-8501, Japan; Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.
| | - Hermes Gadêlha
- Department of Mathematics, University of York, York YO10 5DD, UK; Centre for Human Reproductive Science, Birmingham Women's and Children's NHS Foundation Trust, Birmingham B15 2TG, UK
| | - Eamonn A Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
| | - David J Smith
- School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK; Institute for Metabolism and Systems Research, College of Medical and Dental Sciences, University of Birmingham, Birmingham B15 2TT, UK; Centre for Human Reproductive Science, Birmingham Women's and Children's NHS Foundation Trust, Birmingham B15 2TG, UK
| | - Jackson Kirkman-Brown
- Institute for Metabolism and Systems Research, College of Medical and Dental Sciences, University of Birmingham, Birmingham B15 2TT, UK; Centre for Human Reproductive Science, Birmingham Women's and Children's NHS Foundation Trust, Birmingham B15 2TG, UK
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19
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Ishimoto K, Gaffney EA. Mechanical tuning of mammalian sperm behaviour by hyperactivation, rheology and substrate adhesion: a numerical exploration. J R Soc Interface 2017; 13:rsif.2016.0633. [PMID: 27852807 PMCID: PMC5134015 DOI: 10.1098/rsif.2016.0633] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2016] [Accepted: 10/25/2016] [Indexed: 12/14/2022] Open
Abstract
The great mammalian sperm race encounters numerous microenvironments to which sperm must adapt and a fundamental sperm response is the change in its waveform owing to both fluid rheology and capacitation, with the latter associated with a hyperactivated beat pattern. Hence, in this modelling study, we examine the effect of different flagellar waveforms for sperm behaviour near adhesive substrates, which are representative of epithelia in female tract sperm reservoirs and the zona pellucida (ZP), which surrounds the mammalian egg. On contact with an adhesive surface, virtual sperm rotate to become nearly tangential with the surface, as generally observed. Hyperactivation also induces many effects: sperm exert greater forces on the substrate and periodically tug way from adhesions under circumstances reflecting binding at sperm reservoirs, but with extensive fluid elasticity, as found in the cumulus surrounding the ZP, sperm instead continually push into the substrate. Furthermore, with weak adhesion, rheological media increase the duration hyperactivated sperm remain in the proximity of a substrate. More generally, such results predict that changes owing to both hyperactivation of the flagellar waveform and the rheology of the surrounding medium provide a means of tuning sperm behaviour near, or attached to, adhesive substrates.
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Affiliation(s)
- Kenta Ishimoto
- The Hakubi Center for Advanced Research, Kyoto University, Kyoto 606-8501, Japan .,Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
| | - Eamonn A Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
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20
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Ishimoto K, Gadêlha H, Gaffney EA, Smith DJ, Kirkman-Brown J. Coarse-Graining the Fluid Flow around a Human Sperm. PHYSICAL REVIEW LETTERS 2017; 118:124501. [PMID: 28388208 DOI: 10.1103/physrevlett.118.124501] [Citation(s) in RCA: 31] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/05/2016] [Indexed: 06/07/2023]
Abstract
The flagellar beat is extracted from human sperm digital imaging microscopy and used to determine the flow around the cell and its trajectory, via boundary element simulation. Comparison of the predicted cell trajectory with observation demonstrates that simulation can predict fine-scale sperm dynamics at the qualitative level. The flow field is also observed to reduce to a time-dependent summation of regularized Stokes flow singularities, approximated at leading order by a blinking force triplet. Such regularized singularity decompositions may be used to upscale cell level detail into population models of human sperm motility.
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Affiliation(s)
- Kenta Ishimoto
- The Hakubi Center for Advanced Research, Kyoto University, Kyoto 606-8501, Japan
- Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
| | - Hermes Gadêlha
- Department of Mathematics, University of York, York YO10 5DD, United Kingdom
- Institute for Metabolism and Systems Research, College of Medical and Dental Sciences, University of Birmingham, Birmingham B15 2TT, United Kingdom
- Centre for Human Reproductive Science, Birmingham Women's NHS Foundation Trust, Birmingham B15 2TG, United Kingdom
| | - Eamonn A Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - David J Smith
- School of Mathematics, University of Birmingham, Birmingham B15 2TT, United Kingdom
- Institute for Metabolism and Systems Research, College of Medical and Dental Sciences, University of Birmingham, Birmingham B15 2TT, United Kingdom
- Centre for Human Reproductive Science, Birmingham Women's NHS Foundation Trust, Birmingham B15 2TG, United Kingdom
| | - Jackson Kirkman-Brown
- Institute for Metabolism and Systems Research, College of Medical and Dental Sciences, University of Birmingham, Birmingham B15 2TT, United Kingdom
- Centre for Human Reproductive Science, Birmingham Women's NHS Foundation Trust, Birmingham B15 2TG, United Kingdom
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21
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Ishimoto K, Ikawa M, Okabe M. The mechanics clarifying counterclockwise rotation in most IVF eggs in mice. Sci Rep 2017; 7:43456. [PMID: 28256541 PMCID: PMC5335560 DOI: 10.1038/srep43456] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/16/2016] [Accepted: 01/18/2017] [Indexed: 11/16/2022] Open
Abstract
In mammalian fertilization, a small spermatozoon interacts with an egg that is a few thousand times larger in volume. In spite of the big difference in size and mass, when spermatozoa are bound to eggs, they begin rotating the eggs in in vitro observation. This was dubbed the ‘fertilization dance’. Interestingly, some papers reported that the rotation was counterclockwise, although the reason for this skewed rotation was not clarified. We focused on a chirality of helical beating of spermatozoa and found that eggs rotate counterclockwise in simulations under a certain geometrical condition where the eggs were situated. This theory of egg rotation was validated by demonstrating egg rotation in a clockwise direction by floating eggs to the upper surface of the IVF medium. The enigma of skewed rotation of IVF eggs was clarified.
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Affiliation(s)
- Kenta Ishimoto
- The Hakubi Center for Advanced Research, Kyoto University, Kyoto, Japan.,Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
| | - Masahito Ikawa
- Research Institute for Microbial Diseases, Osaka University, Suita, Japan
| | - Masaru Okabe
- Research Institute for Microbial Diseases, Osaka University, Suita, Japan
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22
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Ishimoto K. Hydrodynamic evolution of sperm swimming: Optimal flagella by a genetic algorithm. J Theor Biol 2016; 399:166-74. [PMID: 27063642 DOI: 10.1016/j.jtbi.2016.03.041] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2015] [Revised: 03/29/2016] [Accepted: 03/30/2016] [Indexed: 01/12/2023]
Abstract
Swimming performance of spermatozoa is an important index for the success of fertilization. For many years, numerous studies have reported the optimal swimming of flagellar organisms. Nevertheless, there is still a question as to which is optimal among planar, circular helical and ellipsoidal helical beating. In this paper, we use a genetic algorithm to investigate the beat pattern with the best swimming efficiency based on hydrodynamic dissipation and internal torque exertion. For the parameters considered, our results show that the planar beat is optimal for small heads and the helical flagellum is optimum for a larger heads, while the ellipsoidal beat is never optimal. Also, the genetic optimization reveals that the wavenumber and shape of wave envelope are relevant parameters, whereas the wave shape and head geometry have relatively minor effects on efficiency. The optimal beat with respect to the efficiency based on the internal torque exertion of an active elastic flagellum is characterized by a small-wavenumber and large-amplitude wave in a lower-viscosity medium. The obtained results on the optimal waveform are consistent with observations for planar waveforms, but in many respects, the results suggest the necessity of a detailed flagellar structure-fluid interaction to address whether real spermatozoa exhibit hydrodynamically efficient swimming. The evolutional optimization approach used in this study has distinguished biologically important parameters, and the methodology can potentially be applicable to various swimmers.
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Affiliation(s)
- Kenta Ishimoto
- The Hakubi Center for Advanced Research, Kyoto University, Kyoto 606-8501, Japan; Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.
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23
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Ishimoto K, Gaffney EA. Fluid flow and sperm guidance: a simulation study of hydrodynamic sperm rheotaxis. J R Soc Interface 2016; 12:rsif.2015.0172. [PMID: 25878133 DOI: 10.1098/rsif.2015.0172] [Citation(s) in RCA: 33] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
How does a sperm find its way? The study of guidance cues has fascinated sperm biologists and in particular the prospect of rheotaxis, that is a fluid flow orienting the direction of sperm swimming, has been the subject of extensive recent study, as readily motivated by the prospect that such guidance may be active in the mammalian female reproductive tract. For instance, it has been hypothesized that helical sperm flagellar beating is necessary for such guidance, whereas there is an extensive diversity of flagellar beating patterns, with planar sperm beating readily observed in human cells for example. In particular, such cells will not be guided by fluid flow according to hypothesized mechanisms for rheotaxis presented thus far. Here, using simulation methods, we investigate rheotaxis for a wide range of flagellar beat patterns. Providing the virtual sperm firstly does not possess a tightly circling trajectory in the absence of a background flow and secondly, remains within a region of low shear to prevent being washed away by the background flow, rheotaxis is generally observed with the sperm swimming into the flow together with a possible transverse velocity. Tight circling sperm motility, as observed in select hyperactivated sperm and CatSper mutants, is predicted to disrupt the rheotactic response, whereas confinement to low shear regions generally requires boundary accumulation, thus introducing subtleties in the relationship between rheotactic behaviours and the flagellar waveform and sperm characteristics. Nonetheless, such predictions suggest such rheotactic guidance may be more common and robust than previously thought, and we document simple criteria for the presence of rheotaxis that are consistent with our simulations and understanding, as well as reported observations to date.
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Affiliation(s)
- Kenta Ishimoto
- The Hakubi Center for Advanced Research, Kyoto University, Kyoto, 606-8501, Japan Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan
| | - Eamonn A Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
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24
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Abstract
Mammalian sperm cells are required to swim over long distances, typically around 1000-fold their own length. They must orient themselves and maintain a swimming motion to reach the ovum, or egg cell. Although the mechanism of long-distance navigation is still unclear, one possible mechanism, rheotaxis, was reported recently. This work investigates the mechanism of the rheotaxis in detail by simulating the motions of a sperm cell in shear flow adjacent to a flat surface. A phase diagram was developed to show the sperm's swimming motion under different shear rates, and for varying flagellum waveform conditions. The results showed that, under shear flow, the sperm is able to hydrodynamically change its swimming direction, allowing it to swim upwards against the flow, which suggests that the upward swimming of sperm cells can be explained using fluid mechanics, and this can then be used to further understand physiology of sperm cell navigation.
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Affiliation(s)
- Toshihiro Omori
- Department of Bioengineering and Robotics, Tohoku University, Aoba 6-6-01 Sendai Miyagi, Japan
| | - Takuji Ishikawa
- Department of Bioengineering and Robotics, Tohoku University, Aoba 6-6-01 Sendai Miyagi, Japan
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25
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Yang J, Kupka I, Schuss Z, Holcman D. Search for a small egg by spermatozoa in restricted geometries. J Math Biol 2015; 73:423-46. [PMID: 26707857 PMCID: PMC4940446 DOI: 10.1007/s00285-015-0955-3] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/05/2015] [Revised: 09/19/2015] [Indexed: 10/26/2022]
Abstract
The search by swimmers for a small target in a bounded domain is ubiquitous in cellular biology, where a prominent case is that of the search by spermatozoa for an egg in the uterus. This is one of the severest selection processes in animal reproduction. We present here a mathematical model of the search, its analysis, and numerical simulations. In the proposed model the swimmers' trajectories are rectilinear and the speed is constant. When a trajectory hits an obstacle or the boundary, it is reflected at a random angle and continues the search with the same speed. Because hitting a small target by a trajectory is a rare event, asymptotic approximations and stochastic simulations are needed to estimate the mean search time in various geometries. We consider searches in a disk, in convex planar domains, and in domains with cusps. The exploration of the parameter space for spermatozoa motion in different uterus geometries leads to scaling laws for the search process.
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Affiliation(s)
- J Yang
- Applied Mathematics and Computational Biology, Ecole Normale Supérieure, IBENS, 46 rue d'Ulm, 75005, Paris, France.,School of Life and Environmental Sciences, Guilin University of Electronic Technology, Guilin, 541004, Guangxi, China
| | - I Kupka
- Applied Mathematics and Computational Biology, Ecole Normale Supérieure, IBENS, 46 rue d'Ulm, 75005, Paris, France
| | - Z Schuss
- Department of Mathematics, Tel-Aviv University, 69978, Tel-Aviv, Israel
| | - D Holcman
- Applied Mathematics and Computational Biology, Ecole Normale Supérieure, IBENS, 46 rue d'Ulm, 75005, Paris, France. .,Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Rd, Oxford, OX2 6GG, UK.
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26
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Ishimoto K, Cosson J, Gaffney EA. A simulation study of sperm motility hydrodynamics near fish eggs and spheres. J Theor Biol 2015; 389:187-97. [PMID: 26542943 DOI: 10.1016/j.jtbi.2015.10.013] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2015] [Revised: 10/06/2015] [Accepted: 10/08/2015] [Indexed: 01/18/2023]
Abstract
For teleost fish fertilisation, sperm must proceed through a small opening on the egg surface, referred to as the micropyle. In this paper, we have used boundary element simulations to explore whether the hydrodynamic attraction between sperm and a fish egg can be a sperm guidance cue. Hydrodynamical egg-sperm interactions alone do not increase the chances of an egg encounter, nor do they induce surface swimming for virtual turbot fish sperm across smooth spheres with a diameter of 1mm, which is representative of a turbot fish egg. When a repulsive surface force between the virtual turbot sperm and the egg is introduced, as motivated by surface charge and van-der-Waals interactions for instance, we find that extended surface swimming of the virtual sperm across a model turbot egg occurs, but ultimately the sperm escapes from the egg. This is due to the small exit angle of the scattering associated with the initial sperm-egg interaction at the egg surface, leading to a weak drift away from the egg, in combination with a weak hydrodynamical attraction between both gametes, though the latter is not sufficient to prevent eventual escape. The resulting transience is not observed experimentally but is a detailed quantitative difference between theory and observation in that stable surface swimming is predicted for eggs with radii larger than about 1.8mm. Regardless, the extended sperm swimming trajectory across the egg constitutes a two-dimensional search for the micropyle and thus the egg is consistently predicted to provide a guidance cue for sperm once they are sufficiently close. In addition, the observation that the virtual turbot sperm swims stably next to a flat plane given repulsive surface interactions, but does not swim stably adjacent to a turbot-sized egg, which is extremely large by sperm-lengthscales, also highlights that the stability of sperm swimming near a boundary is very sensitive to geometry.
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Affiliation(s)
- Kenta Ishimoto
- The Hakubi Center for Advanced Research, Kyoto University, Kyoto 606-8501, Japan; Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.
| | - Jacky Cosson
- University of South Bohemia in Ceske Budejovice, Faculty of Fisheries and Protection of Waters, South Bohemian Research Center of Aquaculture and Biodiversity of Hydrocenoses, Research Institute of Fish Culture and Hydrobiology, Czech Republic.
| | - Eamonn A Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK.
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27
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Shum H, Gaffney EA. Hydrodynamic analysis of flagellated bacteria swimming near one and between two no-slip plane boundaries. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:033012. [PMID: 25871207 DOI: 10.1103/physreve.91.033012] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2014] [Indexed: 05/27/2023]
Abstract
The motility of swimming bacteria near solid surfaces has implications in a wide range of scenarios, including water treatment facilities, microfluidics, and biomedical implants. Using the boundary element method to numerically solve the equations of low Reynolds number fluid flow, we investigate the dynamics of a model swimmer propelled by rotating a single helical flagellum. Building on previous simulation results for swimmers near a single plane boundary, we introduce a second, parallel boundary and show that the bacterial trajectories change as the two plates are brought closer together. Analysis of this dynamical system shows that the configuration in the center of the channel and parallel to the walls is an unstable equilibrium state for large plate separations, but it becomes the only stable position for swimmers when the plate separation is reduced to three to four times the cell width. Our model also predicts that transient trajectories, i.e., those not at steady states, can exhibit curvature in the opposite sense to that expected from the well-known explanation for circular bacterial paths near a single wall.
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Affiliation(s)
- Henry Shum
- Department of Chemical & Petroleum Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, USA
| | - Eamonn A Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
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28
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Simons J, Fauci L, Cortez R. A fully three-dimensional model of the interaction of driven elastic filaments in a Stokes flow with applications to sperm motility. J Biomech 2015; 48:1639-51. [PMID: 25721767 DOI: 10.1016/j.jbiomech.2015.01.050] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/28/2015] [Accepted: 01/29/2015] [Indexed: 11/27/2022]
Abstract
In many animals, sperm flagella exhibit primarily planar waveforms. An isolated sperm with a planar flagellar beat in a three-dimensional unbounded fluid domain would remain in a plane. However, because sperm must navigate through complex, three-dimensional confined spaces along with other sperm, forces that bend or move the flagellum out of its current beat plane develop. Here we present an extension of previous models of an elastic sperm flagellar filament whose shape change is driven by the pursuit of a preferred curvature wave. In particular, we extend the energy of the generalized elastica to include a term that penalizes out-of-plane motion. We are now able to study the interaction of free-swimmers in a 3D Stokes flow that do not start out beating in the same plane. We demonstrate the three-dimensional nature of swimming behavior as neighboring sperm swim close to each other and affect each others' trajectories via fluid-structure coupling.
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Affiliation(s)
- Julie Simons
- Center for Computational Science and Mathematics Department, Tulane University, 6823 St. Charles Ave., New Orleans, LA 70118, USA.
| | - Lisa Fauci
- Center for Computational Science and Mathematics Department, Tulane University, 6823 St. Charles Ave., New Orleans, LA 70118, USA.
| | - Ricardo Cortez
- Center for Computational Science and Mathematics Department, Tulane University, 6823 St. Charles Ave., New Orleans, LA 70118, USA.
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