Skip to main content

Bistability in the synchronization of actuated microfilaments

  • Hanliang Guo (a1), Lisa Fauci (a2), Michael Shelley (a3) (a4) and Eva Kanso (a1) (a3)

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.

Corresponding author
Email address for correspondence:
Hide All
Ainley, J., Durkin, S., Embid, R., Boindala, P. & Cortez, R. 2008 The method of images for regularized stokeslets. J. Comput. Phys. 227 (9), 46004616.
Audoly, B. & Pomeau, Y. 2010 Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells. Oxford University Press.
Brennen, C. & Winet, H. 1977 Fluid mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9 (1), 339398.
Brokaw, C. J. 1971 Bend propagation by a sliding filament model for flagella. J. Expl Biol. 55 (2), 289304.
Brokaw, C. J. 2009 Thinking about flagellar oscillation. Cytoskel. 66 (8), 425436.
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.
Brumley, D. R., Wan, K. Y., Polin, M. & Goldstein, R. E. 2014 Flagellar synchronization through direct hydrodynamic interactions. eLife 3, e02750.
Bruot, N. & Cicuta, P. 2016 Realizing the physics of motile cilia synchronization with driven colloids. Annu. Rev. Condens. Matter Phys. 7 (1), 323348.
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.
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).
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.
Cortez, R. & Varela, D. 2015 A general system of images for regularized stokeslets and other elements near a plane wall. J. Comput. Phys. 285, 4154.
Elfring, G. J. & Lauga, E. 2011 Synchronization of flexible sheets. J. Fluid Mech. 674, 163173.
Eloy, C. & Lauga, E. 2012 Kinematics of the most efficient cilium. Phys. Rev. Lett. 109 (3), 038101.
Faubel, R., Westendorf, C., Bodenschatz, E. & Eichele, G. 2016 Cilia-based flow network in the brain ventricles. Science 353 (6295), 176178.
Friedrich, B. M. & Jülicher, F. 2012 Flagellar synchronization independent of hydrodynamic interactions. Phys. Rev. Lett. 109 (13), 138102.
Fulford, G. R. & Blake, J. R. 1986 Muco-ciliary transport in the lung. J. Theor. Biol. 121 (4), 381402.
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), 1805818063.
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), 10981103.
Goldstein, R. E., Lauga, E., Pesci, A. I. & Proctor, M. R. 2016 Elastohydrodynamic synchronization of adjacent beating flagella. Phys. Rev. Fluids 1 (7), 073201.
Goldstein, R. E., Polin, M. & Tuval, I. 2009 Noise and synchronization in pairs of beating eukaryotic flagella. Phys. Rev. Lett. 103 (16), 168103.
Goldstein, R. E., Polin, M. & Tuval, I. 2011 Emergence of synchronized beating during the regrowth of eukaryotic flagella. Phys. Rev. Lett. 107 (14), 148103.
Golestanian, R., Yeomans, J. M. & Uchida, N. 2011 Hydrodynamic synchronization at low Reynolds number. Soft Matt. 7 (7), 30743082.
Gray, J. 1928 Ciliary Movement. Cambridge University Press.
Gueron, S. & Levit-Gurevich, K. 1999 Energetic considerations of ciliary beating and the advantage of metachronal coordination. Proc. Natl Acad. Sci. USA 96 (22), 1224012245.
Guirao, B. & Joanny, J.-F. 2007 Spontaneous creation of macroscopic flow and metachronal waves in an array of cilia. Biophys. J. 92 (6), 19001917.
Guo, H. & Kanso, E. 2016 Evaluating efficiency and robustness in cilia design. Phys. Rev. E 93 (3), 033119.
Guo, H., Nawroth, J. C., Ding, Y. & Kanso, E. 2014 Cilia beating patterns are not hydrodynamically optimal. Phys. Fluids 26 (9), 091901.
Kim, Y. W. & Netz, R. R. 2006 Pumping fluids with periodically beating grafted elastic filaments. Phys. Rev. Lett. 96 (15), 158101.
Kotar, J., Leoni, M., Bassetti, B., Lagomarsino, M. C. & Cicuta, P. 2010 Hydrodynamic synchronization of colloidal oscillators. Proc. Natl Acad. Sci. USA 107 (17), 76697673.
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), 215224.
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (9), 096601.
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.
Lindemann, C. B. 1994 A ‘geometric clutch’ hypothesis to explain oscillations of the axoneme of cilia and flagella. J. Theor. Biol. 168 (2), 175189.
Man, Y., Koens, L. & Lauga, E. 2016 Hydrodynamic interactions between nearby slender filaments. Europhys. Lett. 116 (2), 24002.
Mettot, C. & Lauga, E. 2011 Energetics of synchronized states in three-dimensional beating flagella. Phys. Rev. E 84 (6), 061905.
Mitran, S. M. 2007 Metachronal wave formation in a model of pulmonary cilia. Comput. Struct. 85 (11), 763774.
Niedermayer, T., Eckhardt, B. & Lenz, P. 2008 Synchronization, phase locking, and metachronal wave formation in ciliary chains. Chaos Interdiscipl. J. Nonlinear Sci. 18, 0370128.
Olson, S. D. & Fauci, L. J. 2015 Hydrodynamic interactions of sheets versus filaments: synchronization, attraction, and alignment. Phys. Fluids 27 (12), 121901.
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, 169187.
Osterman, N. & Vilfan, A. 2011 Finding the ciliary beating pattern with optimal efficiency. Proc. Natl Acad. Sci. USA 108 (38), 1572715732.
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), 487490.
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.
Riedel-Kruse, I. H., Hilfinger, A., Howard, J. & Jülicher, F. 2007 How molecular motors shape the flagellar beat. HFSP J. 1 (3), 192208.
Rüffer, U. & Nultsch, W. 1985 High-speed cinematographic analysis of the movement of Chlamydomonas . Cytoskel. 5 (3), 251263.
Rüffer, U. & Nultsch, W. 1987 Comparison of the beating of cis- and trans-flagella of Chlamydomonas cells held on micropipettes. Cytoskel. 7 (1), 8793.
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.
Taylor, G. 1951 Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209 (1099), 447461.
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.
Uchida, N. & Golestanian, R. 2011 Generic conditions for hydrodynamic synchronization. Phys. Rev. Lett. 106 (5), 058104.
Uchida, N. & Golestanian, R. 2012 Hydrodynamic synchronization between objects with cyclic rigid trajectories. Eur. Phys. J. E Soft Matt. 35 (12), 98139813.
Vilfan, A. & Jülicher, F. 2006 Hydrodynamic flow patterns and synchronization of beating cilia. Phys. Rev. Lett. 96 (5), 058102.
Wan, K. Y. & Goldstein, R. E. 2016 Coordinated beating of algal flagella is mediated by basal coupling. Proc. Natl Acad. Sci. USA 113 (20), E2784E2793.
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.
Wiggins, C. H. & Goldstein, R. E. 1998 Flexive and propulsive dynamics of elastica at low Reynolds number. Phys. Rev. Lett. 80, 38793882.
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), 22152223.
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), 27592768.
Yang, X., Dillon, R. H. & Fauci, L. J. 2008 An integrative computational model of multiciliary beating. Bull. Math. Biol. 70 (4), 11921215.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification

Type Description Title

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)

 Video (6.1 MB)
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)

 Video (16.3 MB)
16.3 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)

 Video (8.2 MB)
8.2 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)

 Video (9.7 MB)
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)

 Video (9.8 MB)
9.8 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)

 Video (11.3 MB)
11.3 MB


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed