1
|
Gadêlha H, Hernández-Herrera P, Montoya F, Darszon A, Corkidi G. Human sperm uses asymmetric and anisotropic flagellar controls to regulate swimming symmetry and cell steering. SCIENCE ADVANCES 2020; 6:eaba5168. [PMID: 32789171 PMCID: PMC7399739 DOI: 10.1126/sciadv.aba5168] [Citation(s) in RCA: 24] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2019] [Accepted: 06/18/2020] [Indexed: 05/21/2023]
Abstract
Flagellar beating drives sperm through the female reproductive tract and is vital for reproduction. Flagellar waves are generated by thousands of asymmetric molecular components; yet, paradoxically, forward swimming arises via symmetric side-to-side flagellar movement. This led to the preponderance of symmetric flagellar control hypotheses. However, molecular asymmetries must still dictate the flagellum and be manifested in the beat. Here, we reconcile molecular and microscopic observations, reconnecting structure to function, by showing that human sperm uses asymmetric and anisotropic controls to swim. High-speed three-dimensional (3D) microscopy revealed two coactive transversal controls: An asymmetric traveling wave creates a one-sided stroke, and a pulsating standing wave rotates the sperm to move equally on all sides. Symmetry is thus achieved through asymmetry, creating the optical illusion of bilateral symmetry in 2D microscopy. This shows that the sperm flagellum is asymmetrically controlled and anisotropically regularized by fast-signal transduction. This enables the sperm to swim forward.
Collapse
Affiliation(s)
- Hermes Gadêlha
- Department of Engineering Mathematics, University of Bristol, BS8 1UB Bristol, UK
| | - Paul Hernández-Herrera
- Laboratorio de Imágenes y Visión por Computadora, Departamento de Ingeniería Celular y Biocatálisis, Universidad Nacional Autónoma de México, Cuernavaca, Mexico
| | - Fernando Montoya
- Laboratorio de Imágenes y Visión por Computadora, Departamento de Ingeniería Celular y Biocatálisis, Universidad Nacional Autónoma de México, Cuernavaca, Mexico
| | - Alberto Darszon
- Departamento de Genética del Desarrollo y Fisiología Molecular, Instituto de Biotecnología, Universidad Nacional Autónoma de México, Cuernavaca, Mexico
| | - Gabriel Corkidi
- Laboratorio de Imágenes y Visión por Computadora, Departamento de Ingeniería Celular y Biocatálisis, Universidad Nacional Autónoma de México, Cuernavaca, Mexico
| |
Collapse
|
2
|
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.
Collapse
|
3
|
Martindale JD, Fu HC. Autonomously responsive pumping by a bacterial flagellar forest: A mean-field approach. Phys Rev E 2018; 96:033107. [PMID: 29346873 DOI: 10.1103/physreve.96.033107] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/19/2017] [Indexed: 11/07/2022]
Abstract
This study is motivated by a microfluidic device that imparts a magnetic torque on an array of bacterial flagella. Bacterial flagella can transform their helical geometry autonomously in response to properties of the background fluid, which provides an intriguing mechanism allowing their use as an engineered element for the regulation or transport of chemicals in microscale applications. The synchronization of flagellar phase has been widely studied in biological contexts, but here we examine the synchronization of flagellar tilt, which is necessary for effective pumping. We first examine the effects of helical geometry and tilt on the pumping flows generated by a single rotating flagellum. Next, we explore a mean-field model for an array of helical flagella to understand how collective tilt arises and influences pumping. The mean-field methodology allows us to take into account possible phase differences through a time-averaging procedure and to model an infinite array of flagella. We find array separation distances, magnetic field strengths, and rotation frequencies that produce nontrivial self-consistent pumping solutions. For individual flagella, pumping is reversed when helicity or rotation is reversed; in contrast, when collective effects are included, self-consistent tilted pumping solutions become untilted nonpumping solutions when helicity or rotation is reversed.
Collapse
Affiliation(s)
- James D Martindale
- Department of Mechanical Engineering, University of Utah, Salt Lake City, Utah, USA
| | - Henry C Fu
- Department of Mechanical Engineering, University of Utah, Salt Lake City, Utah, USA
| |
Collapse
|
4
|
Guo H, Kanso E. Evaluating efficiency and robustness in cilia design. Phys Rev E 2016; 93:033119. [PMID: 27078459 DOI: 10.1103/physreve.93.033119] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2015] [Indexed: 12/14/2022]
Abstract
Motile cilia are used by many eukaryotic cells to transport flow. Cilia-driven flows are important to many physiological functions, yet a deep understanding of the interplay between the mechanical structure of cilia and their physiological functions in healthy and diseased conditions remains elusive. To develop such an understanding, one needs a quantitative framework to assess cilia performance and robustness when subject to perturbations in the cilia apparatus. Here we link cilia design (beating patterns) to function (flow transport) in the context of experimentally and theoretically derived cilia models. We particularly examine the optimality and robustness of cilia design. Optimality refers to efficiency of flow transport, while robustness is defined as low sensitivity to variations in the design parameters. We find that suboptimal designs can be more robust than optimal ones. That is, designing for the most efficient cilium does not guarantee robustness. These findings have significant implications on the understanding of cilia design in artificial and biological systems.
Collapse
Affiliation(s)
- Hanliang Guo
- Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, California 90089, USA
| | - Eva Kanso
- Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, California 90089, USA
| |
Collapse
|
5
|
Kurbatova P, Bessonov N, Volpert V, Tiddens HAWM, Cornu C, Nony P, Caudri D. Model of mucociliary clearance in cystic fibrosis lungs. J Theor Biol 2015; 372:81-8. [PMID: 25746843 DOI: 10.1016/j.jtbi.2015.02.023] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/10/2014] [Revised: 02/06/2015] [Accepted: 02/25/2015] [Indexed: 11/19/2022]
Abstract
Mucus clearance is a primary innate defense mechanism in the human airways. Cystic fibrosis (CF) is a genetic disease caused by mutations in the gene encoding the cystic fibrosis transmembrane conductance regulator (CFTR) protein. CF is characterized by dehydration of airway surface liquid and impaired mucociliary clearance. As a result, microorganisms are not efficiently removed from the airways, and patients experience chronic pulmonary infections and inflammation. We propose a new physiologically based mathematical model of muco-ciliary transport consisting of the two major components of the mucociliary clearance system: (i) periciliary liquid layer (PCL) and (ii) mucus layer. We study mucus clearance under normal conditions and in CF patients. Restoring impaired clearance of airway secretions in one of the major goals of therapy in patients with CF. We consider the action of the aerosolized and inhaled medication dornase alfa, which reduces the viscosity of cystic fibrosis mucus, by selectively cleaving the long DNA strands it contains. The results of the model simulations stress the potential relevance of the location of the drug deposition in the central or peripheral airways. Mucus clearance was increased in case the drug was primarily deposited peripherally, i.e. in the small airways.
Collapse
Affiliation(s)
- P Kurbatova
- University of Lyon 1, UMR 5558, CRNS Lyon, 8 rue Guillaume Paradin, BP8071, 69376 cedex 08, Lyon, France; UMR 663, Inserm-University Paris Descartes-CEA, Necker Hospital, Paris, France.
| | - N Bessonov
- Institute of Mechanical Engineering Problems, 199178 Saint Petersburg, Russia
| | - V Volpert
- University of Lyon 1, CNRS UMR 5208, Institut Camille Jordan 43 blvd du 11 novembre 1918, F-69622 Villeurbanne-Cedex, France
| | - H A W M Tiddens
- Erasmus University Medical Centre-Sophia Children׳s Hospital, PO Box 2060, 3000 CB Rotterdam, Netherlands
| | - C Cornu
- University of Lyon 1, UMR 5558, CRNS Lyon, 8 rue Guillaume Paradin, BP8071, 69376 cedex 08, Lyon, France; CHU Lyon, Service de Pharmacologie Clinique, 8 rue Guillaume Paradin, BP8071, 69376 cedex 08, Lyon, France; Hôpital Louis Pradel, Centre d׳Investigation Clinique, INSERM CIC201/UMR5558, Bron, France
| | - P Nony
- University of Lyon 1, UMR 5558, CRNS Lyon, 8 rue Guillaume Paradin, BP8071, 69376 cedex 08, Lyon, France; CHU Lyon, Service de Pharmacologie Clinique, 8 rue Guillaume Paradin, BP8071, 69376 cedex 08, Lyon, France
| | - D Caudri
- Erasmus University Medical Centre-Sophia Children׳s Hospital, PO Box 2060, 3000 CB Rotterdam, Netherlands
| |
Collapse
|
6
|
Siddiqui AM, Farooq AA, Rana MA. A mathematical model for the flow of a Casson fluid due to metachronal beating of cilia in a tube. ScientificWorldJournal 2015; 2015:487819. [PMID: 25789334 PMCID: PMC4350871 DOI: 10.1155/2015/487819] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2014] [Accepted: 12/14/2014] [Indexed: 11/17/2022] Open
Abstract
A mathematical model is developed to study the transport mechanism of a Casson fluid flow inspired by the metachronal coordination between the beating cilia in a cylindrical tube. A two-dimensional system of nonlinear equations governing the flow problem is formulated by using axisymmetric cylindrical coordinates and then simplified by employing the long wavelength and low Reynolds number assumptions. Exact solutions are derived for the velocity components, the axial pressure gradient, and the stream function. However, the expressions for the pressure rise and the volume flow rate are evaluated numerically. The features of the flow characteristics such as pumping and trapping are illustrated and discussed with the help of graphs. It is observed that the volume flow rate is influenced significantly by the width of plug flow region H p as well as the cilia length parameter ε. The analysis is also applied and compared with the estimated value of the volume flow rate of epididymal fluid in the ductus efferentes of the human male reproductive tract.
Collapse
Affiliation(s)
- A. M. Siddiqui
- Department of Mathematics, Pennsylvania State University, York Campus, York, PA 17403, USA
| | - A. A. Farooq
- Department of Basic Sciences, Riphah International University, Islamabad 44000, Pakistan
- COMSATS Institute of Information Technology, Tobe Camp, Abbottabad 22010, Pakistan
| | - M. A. Rana
- Department of Basic Sciences, Riphah International University, Islamabad 44000, Pakistan
| |
Collapse
|
7
|
|
8
|
Levy R, Hill DB, Forest MG, Grotberg JB. Pulmonary fluid flow challenges for experimental and mathematical modeling. Integr Comp Biol 2014; 54:985-1000. [PMID: 25096289 PMCID: PMC4296202 DOI: 10.1093/icb/icu107] [Citation(s) in RCA: 32] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/13/2022] Open
Abstract
Modeling the flow of fluid in the lungs, even under baseline healthy conditions, presents many challenges. The complex rheology of the fluids, interaction between fluids and structures, and complicated multi-scale geometry all add to the complexity of the problem. We provide a brief overview of approaches used to model three aspects of pulmonary fluid and flow: the surfactant layer in the deep branches of the lung, the mucus layer in the upper airway branches, and closure/reopening of the airway. We discuss models of each aspect, the potential to capture biological and therapeutic information, and open questions worthy of further investigation. We hope to promote multi-disciplinary collaboration by providing insights into mathematical descriptions of fluid-mechanics in the lung and the kinds of predictions these models can make.
Collapse
Affiliation(s)
- Rachel Levy
- *Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA; The Marsico Lung Institute, Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA; Department of Mathematics, Department of Biomedical Engineering, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA; NASA Bioscience and Engineering Institute, The University of Michigan, Ann Arbor, MI 48109, USA
| | - David B Hill
- *Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA; The Marsico Lung Institute, Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA; Department of Mathematics, Department of Biomedical Engineering, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA; NASA Bioscience and Engineering Institute, The University of Michigan, Ann Arbor, MI 48109, USA
| | - M Gregory Forest
- *Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA; The Marsico Lung Institute, Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA; Department of Mathematics, Department of Biomedical Engineering, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA; NASA Bioscience and Engineering Institute, The University of Michigan, Ann Arbor, MI 48109, USA
| | - James B Grotberg
- *Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA; The Marsico Lung Institute, Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA; Department of Mathematics, Department of Biomedical Engineering, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA; NASA Bioscience and Engineering Institute, The University of Michigan, Ann Arbor, MI 48109, USA
| |
Collapse
|
9
|
Jung I, Powers TR, Valles JM. Evidence for two extremes of ciliary motor response in a single swimming microorganism. Biophys J 2014; 106:106-13. [PMID: 24411242 DOI: 10.1016/j.bpj.2013.11.3703] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/20/2013] [Revised: 10/11/2013] [Accepted: 11/15/2013] [Indexed: 11/24/2022] Open
Abstract
Because arrays of motile cilia drive fluids for a range of processes, the versatile mechano-chemical mechanism coordinating them has been under scrutiny. The protist Paramecium presents opportunities to compare how groups of cilia perform two distinct functions, swimming propulsion and nutrient uptake. We present how the body cilia responsible for propulsion and the oral-groove cilia responsible for nutrient uptake respond to changes in their mechanical environment accomplished by varying the fluid viscosity over a factor of 7. Analysis with a phenomenological model of trajectories of swimmers made neutrally buoyant with magnetic forces combined with high-speed imaging of ciliary beating reveal that the body cilia exert a nearly constant propulsive force primarily by reducing their beat frequency as viscosity increases. By contrast, the oral-groove cilia beat at a nearly constant frequency. The existence of two extremes of motor response in a unicellular organism prompts unique investigations of factors controlling ciliary beating.
Collapse
Affiliation(s)
- Ilyong Jung
- Department of Physics, Brown University, Providence, Rhode Island
| | - Thomas R Powers
- Department of Physics, Brown University, Providence, Rhode Island; School of Engineering, Brown University, Providence, Rhode Island
| | - James M Valles
- Department of Physics, Brown University, Providence, Rhode Island.
| |
Collapse
|
10
|
Curtis MP, Gaffney EA. Three-sphere swimmer in a nonlinear viscoelastic medium. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:043006. [PMID: 23679512 DOI: 10.1103/physreve.87.043006] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/26/2012] [Indexed: 06/02/2023]
Abstract
A simple model for a swimmer consisting of three colinearly linked spheres attached by rods and oscillating out of phase to break reciprocal motion is analyzed. With a prescribed forcing of the rods acting on the three spheres, the swimming dynamics are determined analytically in both a Newtonian Stokes fluid and a zero Reynolds number, nonlinear, Oldroyd-B viscoelastic fluid with Deborah numbers of order one (or less), highlighting the effects of viscoelasticity on the net displacement of swimmer. For instance, the model predicts that the three-sphere swimmer with a sinusoidal, but nonreciprocal, forcing cycle within an Oldroyd-B representation of a polymeric Boger fluid moves a greater distance with enhanced efficiency in comparison with its motility in a Newtonian fluid of the same viscosity. Furthermore, the nonlinear contributions to the viscoelastic constitutive relation, while dynamically nontrivial, are predicted a posteriori to have no effect on swimmer motility at leading order, given a prescribed forcing between spheres.
Collapse
Affiliation(s)
- Mark P Curtis
- Oxford Centre for Collaborative Applied Mathematics, Mathematical Institute, University of Oxford, Oxford OX1 3LB, United Kingdom.
| | | |
Collapse
|
11
|
Norton MM, Robinson RJ, Weinstein SJ. Model of ciliary clearance and the role of mucus rheology. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:011921. [PMID: 21405727 DOI: 10.1103/physreve.83.011921] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/12/2010] [Indexed: 05/30/2023]
Abstract
It has been observed that the transportability of mucus by cilial mats is dependent on the rheological properties of the mucus. Mucus is a non-Newtonian fluid that exhibits a plethora of phenomena such as stress relaxation, tensile stresses, shear thinning, and yielding behavior. These observations motivate the analysis in this paper that considers the first two attributes in order to construct a transport model. The model developed here assumes that the mucus is transported as a rigid body, the metachronal wave exhibits symplectic behavior, that the mucus is thin compared to the metachronal wavelength, and that the effects of individual cilia can be lumped together to impart an average strain to the mucus during contact. This strain invokes a stress in the mucus, whose non-Newtonian rheology creates tensile forces that persist into unsheared regions and allow the unsupported mucus to move as a rigid body whereas a Newtonian fluid would retrograde. This work focuses primarily on the Doi-Edwards model but results are generalized to the Jeffrey's and Maxwell fluids as well. The model predicts that there exists an optimal mucus rheology that maximizes the shear stress imparted to the mucus by the cilia for a given cilia motion. We propose that this is the rheology that the body strives for in order to minimize energy consumption. Predicted optimal rheologies are consistent with results from previous experimental studies when reasonable model parameters are chosen.
Collapse
Affiliation(s)
- Michael M Norton
- Department of Mechanical Engineering, Rochester Institute of Technology, Rochester, New York 14623, USA.
| | | | | |
Collapse
|
12
|
Smith DJ. A boundary element regularized Stokeslet method applied to cilia- and flagella-driven flow. Proc Math Phys Eng Sci 2009. [DOI: 10.1098/rspa.2009.0295] [Citation(s) in RCA: 72] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
A boundary element implementation of the regularized Stokeslet method of Cortez is applied to cilia and flagella-driven flows in biology. Previously published approaches implicitly combine the force discretization and the numerical quadrature used to evaluate boundary integrals. By contrast, a boundary element method can be implemented by discretizing the force using basis functions, and calculating integrals using accurate numerical or analytic integration. This substantially weakens the coupling of the mesh size for the force and the regularization parameter, and greatly reduces the number of degrees of freedom required. When modelling a cilium or flagellum as a one-dimensional filament, the regularization parameter can be considered a proxy for the body radius, as opposed to being a parameter used to minimize numerical errors. Modelling a patch of cilia, it is found that: (i) for a fixed number of cilia, reducing cilia spacing reduces transport, (ii) for fixed patch dimension, increasing cilia number increases the transport, up to a plateau at 9×9 cilia. Modelling a choanoflagellate cell, it is found that the presence of a lorica structure significantly affects transport and flow outside the lorica, but does not significantly alter the force experienced by the flagellum.
Collapse
Affiliation(s)
- D. J. Smith
- School of Mathematics and School of Clinical and Experimental Medicine, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Centre for Human Reproductive Science, Birmingham Women’s NHS Foundation Trust, Metchley Park Road, Edgbaston, Birmingham B15 2TG, UK
| |
Collapse
|