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  • Open Access

Entropic, active, and frictional forces in cytoskeletal crosslinking

Cédrik Barutel and Sebastian Fürthauer*

  • Institute of Applied Physics, TU Wien, Lehargasse 6, Vienna 1060, Austria

  • *Contact author: fuerthauer@iap.tuwien.ac.at

Phys. Rev. Research 8, 033122 – Published 29 July, 2026

DOI: https://doi.org/10.1103/6vjh-ym1l

Abstract

The forces that mixtures of motorized and passive crosslinking proteins collectively generate between cytoskeletal filaments within our cells are the key drivers of active cellular mechanics. Despite their importance, a unified theory connecting measurable crosslinker properties to force generation between filament pairs has so far been missing. In this paper, we derive a theory that predicts the forces generated collectively by crosslinking proteins linking two biopolymer filaments from experimentally accessible filament and crosslinker properties, using out-of-equilibrium thermodynamics. Our framework allows us to decompose the forces generated by crosslinkers into three separate components: entropic, active, and frictional. In doing so, it offers a clear physical interpretation of the fundamental mechanisms by which crosslinking proteins self-organize and collectively generate forces. We demonstrate the robustness and utility of this framework by applying it to four different experiments that probe the combined roles of passive and motorized crosslinkers. For each experiment, our theoretical approach allows us to disentangle the relative contributions of entropic, active, and frictional forces, clarifying how different physical processes underpin collective force production. In turn, this makes it possible to quantitatively compare and predict how various crosslinker combinations influence force generation between filaments, pattern formation along filaments, and the dynamics of filament pairs.

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References (76)

  1. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Molecular Biology of the Cell, 4th ed. (Garland Science, New York, 2002).
  2. D. Bray, Cell movements: From Molecules to Motility (Garland Science, New York, 2000).
  3. J. Howard, Mechanics of Motor Proteins and The Cytoskeleton (Sinauer Associates, Sunderland, MA, 2001).
  4. J. M. Hill, S. Cai, M. D. Carver, and D. G. Drubin, A role for cross-linking proteins in actin filament network organization and force generation, Proc. Natl. Acad. Sci. USA 121, e2407838121 (2024).
  5. R. Vale, T. Reese, and M. Sheetz, Identification of a novel force-generating protein, kinesin, involved in microtubule-based motility, Cell 42, 39 (1985).
  6. K. Svoboda and S. M. Block, Force and velocity measured for single kinesin molecules, Cell 77, 773 (1994).
  7. N. J. Carter and R. A. Cross, Mechanics of the kinesin step, Nature (London) 435, 308 (2005).
  8. A. Hunt, F. Gittes, and J. Howard, The force exerted by a single kinesin molecule against a viscous load, Biophys. J. 67, 766 (1994).
  9. J. T. Finer, R. M. Simmons, and J. A. Spudich, Single myosin molecule mechanics: Piconewton forces and nanometre steps, Nature (London) 368, 113 (1994).
  10. V. Bormuth, V. Varga, J. Howard, and E. Schäffer, Protein friction limits diffusive and directed movements of kinesin motors on microtubules, Science 325, 870 (2009).
  11. S. Forth, K.-C. Hsia, Y. Shimamoto, and T. M. Kapoor, Asymmetric friction of nonmotor MAPs can lead to their directional motion in active microtubule networks, Cell 157, 420 (2014).
  12. O. Kučera, V. Siahaan, D. Janda, S. H. Dijkstra, E. Pilátová, E. Zatecka, S. Diez, M. Braun, and Z. Lansky, Anillin propels myosin-independent constriction of actin rings, Nat. Commun. 12, 4595 (2021).
  13. V. F. Geyer and S. Diez, Horizontal magnetic tweezers to directly measure the force-velocity relationship for multiple Kinesin motors, Small 19, 2300558 (2023).
  14. A. Alfieri, I. Gaska, and S. Forth, Two modes of PRC1-mediated mechanical resistance to kinesin-driven microtubule network disruption, Curr. Biol. 31, 2495 (2021).
  15. Z. Lansky, M. Braun, A. Lüdecke, M. Schlierf, P. R. ten Wolde, M. E. Janson, and S. Diez, Diffusible crosslinkers generate directed forces in microtubule networks, Cell 160, 1159 (2015).
  16. Y. Shimamoto, S. Forth, and T. M. Kapoor, Measuring pushing and braking forces generated by ensembles of kinesin-5 crosslinking two microtubules, Dev. Cell 34, 669 (2015).
  17. F. Jülicher and J. Prost, Cooperative molecular motors, Phys. Rev. Lett. 75, 2618 (1995).
  18. F. Jülicher, A. Ajdari, and J. Prost, Modeling molecular motors, Rev. Mod. Phys. 69, 1269 (1997).
  19. S. Klumpp, T. M. Nieuwenhuizen, and R. Lipowsky, Self-organized density patterns of molecular motors in arrays of cytoskeletal filaments, Biophys. J. 88, 3118 (2005).
  20. T. Guérin, J. Prost, P. Martin, and J.-F. Joanny, Coordination and collective properties of molecular motors: Theory, Curr. Opin. Cell Biol. 22, 14 (2010).
  21. A. Parmeggiani, T. Franosch, and E. Frey, Totally asymmetric simple exclusion process with Langmuir kinetics, Phys. Rev. E 70, 046101 (2004).
  22. I. S. Aranson and L. S. Tsimring, Theory of self-assembly of microtubules and motors, Phys. Rev. E 74, 031915 (2006).
  23. D. Johann, C. Erlenkämper, and K. Kruse, Length regulation of active biopolymers by molecular motors, Phys. Rev. Lett. 108, 258103 (2012).
  24. C. Leduc, K. Padberg-Gehle, V. Varga, D. Helbing, S. Diez, and J. Howard, Molecular crowding creates traffic jams of kinesin motors on microtubules, Proc. Natl. Acad. Sci. USA 109, 6100 (2012).
  25. C. P. Descovich, D. B. Cortes, S. Ryan, J. Nash, L. Zhang, P. S. Maddox, F. Nedelec, and A. S. Maddox, Cross-linkers both drive and brake cytoskeletal remodeling and furrowing in cytokinesis, Mol. Biol. Cell 29, 622 (2018).
  26. M. Lera-Ramirez and F. J. Nédélec, Theory of antiparallel microtubule overlap stabilization by motors and diffusible crosslinkers, Cytoskeleton 76, 600 (2019).
  27. H. Wierenga and P. R. ten Wolde, Diffusible cross-linkers cause superexponential friction forces, Phys. Rev. Lett. 125, 078101 (2020).
  28. S. A. Fiorenza, D. G. Steckhahn, and M. D. Betterton, Modeling spatiotemporally varying protein–protein interactions in CyLaKS, the cytoskeleton lattice-based kinetic simulator, Eur. Phys. J. E 44, 105 (2021).
  29. M. Striebel, F. Brauns, and E. Frey, Length regulation drives self-organization in filament-motor mixtures, Phys. Rev. Lett. 129, 238102 (2022).
  30. W. Yan, S. Ansari, A. Lamson, M. A. Glaser, R. Blackwell, M. D. Betterton, and M. Shelley, Toward the cellular-scale simulation of motor-driven cytoskeletal assemblies, eLife 11, e74160 (2022).
  31. A. Parmeggiani, T. Franosch, and E. Frey, Phase coexistence in driven one-dimensional transport, Phys. Rev. Lett. 90, 086601 (2003).
  32. K. Kruse and F. Jülicher, Actively contracting bundles of polar filaments, Phys. Rev. Lett. 85, 1778 (2000).
  33. K. Kruse, S. Camalet, and F. Jülicher, Self-propagating patterns in active filament bundles, Phys. Rev. Lett. 87, 138101 (2001).
  34. K. Kruse and F. Jülicher, Self-organization and mechanical properties of active filament bundles, Phys. Rev. E 67, 051913 (2003).
  35. K. Kruse, J. F. Joanny, F. Jülicher, J. Prost, and K. Sekimoto, Generic theory of active polar gels: A paradigm for cytoskeletal dynamics, Eur. Phys. J. E 16, 5 (2005).
  36. T. B. Liverpool and M. C. Marchetti, Bridging the microscopic and the hydrodynamic in active filament solutions, Europhys. Lett. 69, 846 (2005).
  37. D. Saintillan and M. J. Shelley, Instabilities, pattern formation, and mixing in active suspensions, Phys. Fluids 20, 123304 (2008).
  38. S. Fürthauer, B. Lemma, P. J. Foster, S. C. Ems-McClung, C.-H. Yu, C. E. Walczak, Z. Dogic, D. J. Needleman, and M. J. Shelley, Self-straining of actively crosslinked microtubule networks, Nat. Phys. 15, 1295 (2019).
  39. S. Fürthauer, D. J. Needleman, and M. J. Shelley, A design framework for actively crosslinked filament networks, New J. Phys. 23, 013012 (2021).
  40. F. De Luca, I. Maryshev, and E. Frey, Supramolecular assemblies in active motor-filament systems: Micelles, bilayers, and foams, Phys. Rev. X 14, 031031 (2024).
  41. I. Maryshev, D. Marenduzzo, A. B. Goryachev, and A. Morozov, Kinetic theory of pattern formation in mixtures of microtubules and molecular motors, Phys. Rev. E 97, 022412 (2018).
  42. I. Maryshev, A. B. Goryachev, D. Marenduzzo, and A. Morozov, Dry active turbulence in a model for microtubule–motor mixtures, Soft Matter 15, 6038 (2019).
  43. M. Striebel, I. R. Graf, and E. Frey, A mechanistic view of collective filament motion in active nematic networks, Biophys. J. 118, 313 (2020).
  44. S. R. De Groot and P. Mazur, Non-Equilibrium Thermodynamics (Dover Publications, New York, 1984), originally published in 1962 by North-Holland.
  45. D. Kondepudi and I. Prigogine, Modern Thermodynamics: From Heat Engines to Dissipative Structures (John Wiley & Sons, Chichester, 2015), pp. 403–420.
  46. L. D. Landau and E. M. Lifshitz, Statistical Physics: Part 1, 3rd ed., Course of Theoretical Physics, Vol. 5 (Pergamon Press, Oxford, 1980).
  47. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, 2nd ed. (Pergamon Press, Oxford, 1987), Vol. 6.
  48. S. Fürthauer, M. Neef, S. W. Grill, K. Kruse, and F. Jülicher, The Taylor–Couette motor: Spontaneous flows of active polar fluids between two coaxial cylinders, New J. Phys. 14, 023001 (2012).
  49. F. Jülicher, S. W. Grill, and G. Salbreux, Hydrodynamic theory of active matter, Rep. Prog. Phys. 81, 076601 (2018).
  50. M. Braun, Z. Lansky, F. Hilitski, Z. Dogic, and S. Diez, Entropic forces drive contraction of cytoskeletal networks, BioEssays 38, 474 (2016).
  51. T. L. Hill, Theoretical formalism for the sliding filament model of contraction of striated muscle Part I, Prog. Biophys. Mol. Biol. 28, 267 (1974).
  52. I. Gaska, M. E. Armstrong, A. Alfieri, and S. Forth, The mitotic crosslinking protein PRC1 acts like a mechanical dashpot to resist microtubule sliding, Dev. Cell 54, 367 (2020).
  53. M. M. Tirado and J. G. de la Torre, Translational friction coefficients of rigid, symmetric top macromolecules. Application to circular cylinders, J. Chem. Phys. 71, 2581 (1979).
  54. M. M. Tirado, C. L. Martínez, and J. G. de la Torre, Comparison of theories for the translational and rotational diffusion coefficients of rod-like macromolecules. Application to short DNA fragments, J. Chem. Phys. 81, 2047 (1984).
  55. D. M. Heyes, Translational and rotational diffusion of rod shaped molecules by molecular dynamics simulations, J. Chem. Phys. 150, 184901 (2019).
  56. R. Milo and R. Phillips, Cell Biology by the Numbers (Garland Science, New York, 2015).
  57. E. J. Kramer, P. Green, and C. J. Palmstrøm, Interdiffusion and marker movements in concentrated polymer-polymer diffusion couples, Polymer 25, 473 (1984).
  58. J. M. Ortiz de Zárate, J. L. Hita, and J. V. Sengers, Fluctuating hydrodynamics and concentration fluctuations in ternary mixtures, Comptes Rendus. Mécanique 341, 483 (2013).
  59. H. van Beijeren, K. W. Kehr, and R. Kutner, Diffusion in concentrated lattice gases. III. Tracer diffusion on a one-dimensional lattice, Phys. Rev. B 28, 5711 (1983).
  60. M. Braun, Z. Lansky, A. Szuba, F. W. Schwarz, A. Mitra, M. Gao, A. Lüdecke, P. R. ten Wolde, and S. Diez, Changes in microtubule overlap length regulate kinesin-14-driven microtubule sliding, Nat. Chem. Biol. 13, 1245 (2017).
  61. M. Braun, Z. Lansky, G. Fink, F. Ruhnow, S. Diez, and M. E. Janson, Adaptive braking by Ase1 prevents overlapping microtubules from sliding completely apart, Nat. Cell Biol. 13, 1259 (2011).
  62. H. P. Erickson, Size and shape of protein molecules at the nanometer level determined by sedimentation, gel filtration, and electron microscopy, Biol. Proced. Online 11, 32 (2009).
  63. C. Barutel, Forces between transiently crosslinked biopolymers, GitHub, 2025, https://github.com/CedrikBarutel/ForcesBetweenTransientlyCrosslinkedBiopolymers.
  64. K. J. Burns, G. M. Vasil, J. S. Oishi, D. Lecoanet, and B. P. Brown, Dedalus: A flexible framework for numerical simulations with spectral methods, Phys. Rev. Res. 2, 023068 (2020).
  65. See Supplemental Material at https://link.aps.org/supplemental/10.1103/6vjh-ym1l for Supplementary movies.
  66. S. Fürthauer and M. J. Shelley, How cross-link numbers shape the large-scale physics of cytoskeletal materials, Annu. Rev. Condens. Matter Phys. 13, 365 (2022).
  67. R. A. Banks, V. Galstyan, H. J. Lee, S. Hirokawa, A. Ierokomos, T. D. Ross, Z. Bryant, M. Thomson, and R. Phillips, Motor processivity and speed determine structure and dynamics of microtubule-motor assemblies, eLife 12, e79402 (2023).
  68. R. Karmakar, S. Sastry, S. K. Kumar, and T. K. Patra, Rouse mode analysis of chain relaxation in reversibly crosslinked polymer melts, Phys. Chem. Chem. Phys. 27, 6777 (2025).
  69. R. W. Clarke, T. Sandmeier, K. A. Franklin, D. Reich, X. Zhang, N. Vengallur, T. K. Patra, R. J. Tannenbaum, S. Adhikari, S. K. Kumar, et al., Dynamic crosslinking compatibilizes immiscible mixed plastics, Nature (London) 616, 731 (2023).
  70. R. Karmakar, N. Venkatareddy, Himanshu, M. Valsecchi, P. K. Maiti, S. Sastry, S. K. Kumar, and T. K. Patra, Computer simulations of entropic cohesion in reversibly crosslinked polymers, Soft Matter 21, 348 (2025).
  71. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed., International Series of Monographs on Physics, Vol. 83 (Oxford University Press, Oxford, 1993).
  72. L. Onsager, Reciprocal relations in irreversible processes. I., Phys. Rev. 37, 405 (1931).
  73. P. J. Flory, Thermodynamics of high polymer solutions, J. Chem. Phys. 10, 51 (1942).
  74. A. W. C. Lau, D. Lacoste, and K. Mallick, Nonequilibrium fluctuations and mechanochemical couplings of a molecular motor, Phys. Rev. Lett. 99, 158102 (2007).
  75. S. M. J. L. van den Wildenberg, Single-protein motion on microtubules and in cell membranes, Ph.D. thesis, Vrije Universiteit Amsterdam, 2011.
  76. K. Furuta and Y. Y. Toyoshima, Minus-end-directed motor Ncd exhibits processive movement that is enhanced by microtubule bundling in vitro, Curr. Biol. 18, 152 (2008).

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