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Bistability in the synchronization of actuated microfilaments

Published online by Cambridge University Press:  11 December 2017

Hanliang Guo ,
Lisa Fauci ,
Michael Shelley  and
Eva Kanso Open the ORCID record for Eva Kanso [Opens in a new window]
Hanliang Guo
Affiliation:
Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, USA
Lisa Fauci
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA
Michael Shelley
Affiliation:
Center for Computational Biology, Flatiron Institute, Simons Foundation, New York, NY 10010, USA Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Eva Kanso*
Affiliation:
Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, USA Center for Computational Biology, Flatiron Institute, Simons Foundation, New York, NY 10010, USA
*
†Email address for correspondence: [email protected]
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Abstract

Cilia and flagella are essential building blocks for biological fluid transport and locomotion at the micrometre scale. They often beat in synchrony and may transition between different synchronization modes in the same cell type. Here, we investigate the behaviour of elastic microfilaments, protruding from a surface and driven at their base by a configuration-dependent torque. We consider full hydrodynamic interactions among and within filaments and no slip at the surface. Isolated filaments exhibit periodic deformations, with increasing waviness and frequency as the magnitude of the driving torque increases. Two nearby but independently driven filaments synchronize their beating in-phase or anti-phase. This synchrony arises autonomously via the interplay between hydrodynamic coupling and filament elasticity. Importantly, in-phase and anti-phase synchronization modes are bistable and coexist for a range of driving torques and separation distances. These findings are consistent with experimental observations of in-phase and anti-phase synchronization in pairs of cilia and flagella and could have important implications on understanding the biophysical mechanisms underlying transitions between multiple synchronization modes.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Low-Reynolds-number flows: Low-Reynolds-number flows Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 836 , 10 February 2018 , pp. 304 - 323
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Copyright
© 2017 Cambridge University Press 

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References

Ainley, J., Durkin, S., Embid, R., Boindala, P. & Cortez, R. 2008 The method of images for regularized stokeslets. J. Comput. Phys. 227 (9), 4600–4616.CrossRefGoogle Scholar
Audoly, B. & Pomeau, Y. 2010 Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells. Oxford University Press.Google Scholar
Brennen, C. & Winet, H. 1977 Fluid mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9 (1), 339–398.CrossRefGoogle Scholar
Brokaw, C. J. 1971 Bend propagation by a sliding filament model for flagella. J. Expl Biol. 55 (2), 289–304.CrossRefGoogle ScholarPubMed
Brokaw, C. J. 2009 Thinking about flagellar oscillation. Cytoskel. 66 (8), 425–436.CrossRefGoogle ScholarPubMed
Brumley, D. R., Polin, M., Pedley, T. J. & Goldstein, R. E. 2012 Hydrodynamic synchronization and metachronal waves on the surface of the colonial alga Volvox carteri . Phys. Rev. Lett. 109 (26), 268102.CrossRefGoogle ScholarPubMed
Brumley, D. R., Wan, K. Y., Polin, M. & Goldstein, R. E. 2014 Flagellar synchronization through direct hydrodynamic interactions. eLife 3, e02750.CrossRefGoogle ScholarPubMed
Bruot, N. & Cicuta, P. 2016 Realizing the physics of motile cilia synchronization with driven colloids. Annu. Rev. Condens. Matter Phys. 7 (1), 323–348.CrossRefGoogle Scholar
Bruot, N., Kotar, J., de Lillo, F., Lagomarsino, M. & Cosentino, C. P. 2012 Driving potential and noise level determine the synchronization state of hydrodynamically coupled oscillators. Phys. Rev. Lett. 109 (16), 164103.CrossRefGoogle ScholarPubMed
Buchmann, A., Cortez, R. & Fauci, L.2017  A sliding-control switch alters the stability of synchronized states in an elasto-hydrodynamic model of actuated cilia. (submitted).Google Scholar
Chrispell, J. C., Fauci, L. J. & Shelley, M. 2013 An actuated elastic sheet interacting with passive and active structures in a viscoelastic fluid. Phys. Fluids 25 (1), 013103.CrossRefGoogle Scholar
Cortez, R. & Varela, D. 2015 A general system of images for regularized stokeslets and other elements near a plane wall. J. Comput. Phys. 285, 41–54.CrossRefGoogle Scholar
Elfring, G. J. & Lauga, E. 2011 Synchronization of flexible sheets. J. Fluid Mech. 674, 163–173.CrossRefGoogle Scholar
Eloy, C. & Lauga, E. 2012 Kinematics of the most efficient cilium. Phys. Rev. Lett. 109 (3), 038101.CrossRefGoogle ScholarPubMed
Faubel, R., Westendorf, C., Bodenschatz, E. & Eichele, G. 2016 Cilia-based flow network in the brain ventricles. Science 353 (6295), 176–178.CrossRefGoogle ScholarPubMed
Friedrich, B. M. & Jülicher, F. 2012 Flagellar synchronization independent of hydrodynamic interactions. Phys. Rev. Lett. 109 (13), 138102.CrossRefGoogle ScholarPubMed
Fulford, G. R. & Blake, J. R. 1986 Muco-ciliary transport in the lung. J. Theor. Biol. 121 (4), 381–402.CrossRefGoogle ScholarPubMed
Geyer, V. F., Jülicher, F., Howard, J. & Friedrich, B. M. 2013 Cell-body rocking is a dominant mechanism for flagellar synchronization in a swimming alga. Proc. Natl Acad. Sci. USA 110 (45), 18058–18063.CrossRefGoogle Scholar
Geyer, V. F., Sartori, P., Friedrich, B. M., Jülicher, F. & Howard, J. 2016 Independent control of the static and dynamic components of the Chlamydomonas flagellar beat. Current Biol. 26 (8), 1098–1103.CrossRefGoogle ScholarPubMed
Goldstein, R. E., Lauga, E., Pesci, A. I. & Proctor, M. R. 2016 Elastohydrodynamic synchronization of adjacent beating flagella. Phys. Rev. Fluids 1 (7), 073201.CrossRefGoogle ScholarPubMed
Goldstein, R. E., Polin, M. & Tuval, I. 2009 Noise and synchronization in pairs of beating eukaryotic flagella. Phys. Rev. Lett. 103 (16), 168103.CrossRefGoogle ScholarPubMed
Goldstein, R. E., Polin, M. & Tuval, I. 2011 Emergence of synchronized beating during the regrowth of eukaryotic flagella. Phys. Rev. Lett. 107 (14), 148103.CrossRefGoogle ScholarPubMed
Golestanian, R., Yeomans, J. M. & Uchida, N. 2011 Hydrodynamic synchronization at low Reynolds number. Soft Matt. 7 (7), 3074–3082.CrossRefGoogle Scholar
Gray, J. 1928 Ciliary Movement. Cambridge University Press.Google Scholar
Gueron, S. & Levit-Gurevich, K. 1999 Energetic considerations of ciliary beating and the advantage of metachronal coordination. Proc. Natl Acad. Sci. USA 96 (22), 12240–12245.CrossRefGoogle ScholarPubMed
Guirao, B. & Joanny, J.-F. 2007 Spontaneous creation of macroscopic flow and metachronal waves in an array of cilia. Biophys. J. 92 (6), 1900–1917.CrossRefGoogle Scholar
Guo, H. & Kanso, E. 2016 Evaluating efficiency and robustness in cilia design. Phys. Rev. E 93 (3), 033119.Google ScholarPubMed
Guo, H., Nawroth, J. C., Ding, Y. & Kanso, E. 2014 Cilia beating patterns are not hydrodynamically optimal. Phys. Fluids 26 (9), 091901.CrossRefGoogle Scholar
Kim, Y. W. & Netz, R. R. 2006 Pumping fluids with periodically beating grafted elastic filaments. Phys. Rev. Lett. 96 (15), 158101.CrossRefGoogle ScholarPubMed
Kotar, J., Leoni, M., Bassetti, B., Lagomarsino, M. C. & Cicuta, P. 2010 Hydrodynamic synchronization of colloidal oscillators. Proc. Natl Acad. Sci. USA 107 (17), 7669–7673.CrossRefGoogle ScholarPubMed
Lagomarsino, M. C., Capuani, F. & Lowe, C. P. 2003 A simulation study of the dynamics of a driven filament in an Aristotelian fluid. J. Theor. Biol. 224 (2), 215–224.CrossRefGoogle Scholar
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (9), 096601.CrossRefGoogle Scholar
Leptos, K. C., Wan, K. Y., Polin, M., Tuval, I., Pesci, A. I. & Goldstein, R. E. 2013 Antiphase synchronization in a flagellar-dominance mutant of Chlamydomonas . Phys. Rev. Lett. 111 (15), 158101.CrossRefGoogle Scholar
Lindemann, C. B. 1994 A ‘geometric clutch’ hypothesis to explain oscillations of the axoneme of cilia and flagella. J. Theor. Biol. 168 (2), 175–189.CrossRefGoogle Scholar
Man, Y., Koens, L. & Lauga, E. 2016 Hydrodynamic interactions between nearby slender filaments. Europhys. Lett. 116 (2), 24002.CrossRefGoogle Scholar
Mettot, C. & Lauga, E. 2011 Energetics of synchronized states in three-dimensional beating flagella. Phys. Rev. E 84 (6), 061905.Google ScholarPubMed
Mitran, S. M. 2007 Metachronal wave formation in a model of pulmonary cilia. Comput. Struct. 85 (11), 763–774.CrossRefGoogle Scholar
Niedermayer, T., Eckhardt, B. & Lenz, P. 2008 Synchronization, phase locking, and metachronal wave formation in ciliary chains. Chaos Interdiscipl. J. Nonlinear Sci. 18, 0370128.Google ScholarPubMed
Olson, S. D. & Fauci, L. J. 2015 Hydrodynamic interactions of sheets versus filaments: synchronization, attraction, and alignment. Phys. Fluids 27 (12), 121901.CrossRefGoogle Scholar
Olson, S. D., Lim, S. & Cortez, R. 2013 Modeling the dynamics of an elastic rod with intrinsic curvature and twist using a regularized stokes formulation. J. Comput. Phys. 238, 169–187.CrossRefGoogle Scholar
Osterman, N. & Vilfan, A. 2011 Finding the ciliary beating pattern with optimal efficiency. Proc. Natl Acad. Sci. USA 108 (38), 15727–15732.CrossRefGoogle ScholarPubMed
Polin, M., Tuval, I., Drescher, K., Gollub, J. P. & Goldstein, R. E. 2009 Chlamydomonas swims with two ‘gears’ in a eukaryotic version of run-and-tumble locomotion. Science 325 (5939), 487–490.CrossRefGoogle Scholar
Quaranta, G., Aubin-Tam, M.-E. & Tam, D. 2015 Hydrodynamics versus intracellular coupling in the synchronization of eukaryotic flagella. Phys. Rev. Lett. 115 (23), 238101.CrossRefGoogle ScholarPubMed
Riedel-Kruse, I. H., Hilfinger, A., Howard, J. & Jülicher, F. 2007 How molecular motors shape the flagellar beat. HFSP J. 1 (3), 192–208.CrossRefGoogle ScholarPubMed
Rüffer, U. & Nultsch, W. 1985 High-speed cinematographic analysis of the movement of Chlamydomonas . Cytoskel. 5 (3), 251–263.Google Scholar
Rüffer, U. & Nultsch, W. 1987 Comparison of the beating of cis- and trans-flagella of Chlamydomonas cells held on micropipettes. Cytoskel. 7 (1), 87–93.CrossRefGoogle Scholar
Sartori, P., Geyer, V. F., Scholich, A., Jülicher, F. & Howard, J. 2016 Dynamic curvature regulation accounts for the symmetric and asymmetric beats of chlamydomonas flagella. eLife 5, e13258.CrossRefGoogle ScholarPubMed
Taylor, G. 1951 Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209 (1099), 447–461.Google Scholar
Teran, J., Fauci, L. J. & Shelley, M. 2010 Viscoelastic fluid response can increase the speed and efficiency of a free swimmer. Phys. Rev. Lett. 104 (3), 038101.CrossRefGoogle ScholarPubMed
Uchida, N. & Golestanian, R. 2011 Generic conditions for hydrodynamic synchronization. Phys. Rev. Lett. 106 (5), 058104.CrossRefGoogle ScholarPubMed
Uchida, N. & Golestanian, R. 2012 Hydrodynamic synchronization between objects with cyclic rigid trajectories. Eur. Phys. J. E Soft Matt. 35 (12), 9813–9813.Google ScholarPubMed
Vilfan, A. & Jülicher, F. 2006 Hydrodynamic flow patterns and synchronization of beating cilia. Phys. Rev. Lett. 96 (5), 058102.CrossRefGoogle ScholarPubMed
Wan, K. Y. & Goldstein, R. E. 2016 Coordinated beating of algal flagella is mediated by basal coupling. Proc. Natl Acad. Sci. USA 113 (20), E2784–E2793.CrossRefGoogle ScholarPubMed
Wan, K. Y., Leptos, K. C. & Goldstein, R. E. 2014 Lag, lock, sync, slip: the many ‘phases’ of coupled flagella. J. R. Soc. Interface 11 (94), 20131160.CrossRefGoogle ScholarPubMed
Wiggins, C. H. & Goldstein, R. E. 1998 Flexive and propulsive dynamics of elastica at low Reynolds number. Phys. Rev. Lett. 80, 3879–3882.CrossRefGoogle Scholar
Woolley, D. M., Crockett, R. F., Groom, W. D. & Revell, S. G. 2009 A study of synchronisation between the flagella of bull spermatozoa, with related observations. J. Expl Biol. 212 (14), 2215–2223.CrossRefGoogle ScholarPubMed
Xu, G., Wilson, K. S., Okamoto, R. J., Shao, J.-Y., Dutcher, S. K. & Bayly, P. V. 2016 Flexural rigidity and shear stiffness of flagella estimated from induced bends and counterbends. Biophys. J. 110 (12), 2759–2768.CrossRefGoogle ScholarPubMed
Yang, X., Dillon, R. H. & Fauci, L. J. 2008 An integrative computational model of multiciliary beating. Bull. Math. Biol. 70 (4), 1192–1215.CrossRefGoogle ScholarPubMed

Guo et al. supplementary movie 1

Long term dynamics of single filament with M_b=1, vertical initial condition. (Figure 2a in the main text)

Download Guo et al. supplementary movie 1(Video)
Video 6.1 MB

Guo et al. supplementary movie 2

Long term dynamics of single filament with M_b=1, tilted initial condition. (Figure 4 in the main text)

Download Guo et al. supplementary movie 2(Video)
Video 16.3 MB

Guo et al. supplementary movie 3

Long term dynamics of single filament with M_b=3, vertical initial condition. (Figure 2b in the main text)

Download Guo et al. supplementary movie 3(Video)
Video 9.7 MB

Guo et al. supplementary movie 4

Long term dynamics of single filament with M_b=3, tilted initial condition. (Figure 4 in the main text)

Download Guo et al. supplementary movie 4(Video)
Video 9.8 MB

Guo et al. supplementary movie 5

Long term dynamics of a pair of filaments with M_b=1, d=0.7, initial phase difference dphi_0=0.49. (Figure 5a in the main text)

Download Guo et al. supplementary movie 5(Video)
Video 8.2 MB

Guo et al. supplementary movie 6

Long term dynamics of a pair of filaments with M_b=3, d=0.7, initial phase difference dphi_0=0.49. (Figure 5b in the main text)

Download Guo et al. supplementary movie 6(Video)
Video 11.3 MB