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Externally driven condensates show translation-induced polarization, directed coalescence, and anomalous diffusion in viscoelastic media

Andriy Goychuk*

  • *Contact author: andriy@goychuk.me
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Phys. Rev. E 113, 065402 – Published 3 June, 2026

DOI: https://doi.org/10.1103/n662-39rs

Abstract

Phase separation into compositionally and physically distinct domains is ubiquitous in (non)living matter ranging from alloys and emulsions to biomolecular condensates in cells. The organization of these domains can be controlled, for example, by nonequilibrium chemical reactions, external fields, or mechanical stresses. In this context, stationary states can emerge from effective long-range interactions resembling the electrostatics of charges. Here, I develop a theory predicting that externally controlled dynamic states, such as condensate motion, lead to an effective polarization and dipolar force fields even for microscopically nonpolarizable matter. The dipole-dipole interactions resulting from this translation-induced polarization cause directed coalescence of domains. This coarsening mechanism complements Ostwald ripening and coalescence due to Brownian motion or Marangoni flows, and has implications for controlling domains by electric fields or concentration gradients. Interestingly, the chemical potential gradients around a domain that nucleates material are exactly opposite to the hydrodynamic pressure gradients around an impermeable colloid that pushes the fluid, suggesting a competition between phase separation and hydrodynamics. In addition to chemical control, the motion of domains can also be driven by mechanical stresses. An example is the cell interior, where mechanical stresses are actively generated by molecular motors and opposed by passive viscoelastic stresses in the cytoplasm and nucleoplasm. Taking into account such active fluctuations in the theory predicts that the resulting fluid flows lead to Brownian motion with a suppressed or enhanced size scaling that modifies collision-coalescence. For active stresses with a long correlation time, the domains show superdiffusion on intermediate timescales. Together, these findings shed new light on the dynamics of domains in viscoelastic media and conserved order parameters in general.

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References (97)

  1. M. L. Huggins, Solutions of long chain compounds, J. Chem. Phys. 9, 440 (1941).
  2. P. J. Flory, Thermodynamics of high polymer solutions, J. Chem. Phys. 9, 660 (1941).
  3. P. J. Flory, Thermodynamics of high polymer solutions, J. Chem. Phys. 10, 51 (1942).
  4. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. interfacial free energy, J. Chem. Phys. 28, 258 (1958).
  5. J. W. Cahn, On spinodal decomposition, Acta Metall. 9, 795 (1961).
  6. M. Z. Bazant, Theory of chemical kinetics and charge transfer based on nonequilibrium thermodynamics, Acc. Chem. Res. 46, 1144 (2013).
  7. J. Joshi and K. Nandakumar, Computational modeling of multiphase reactors, Annu. Rev. Chem. Biomol. Eng. 6, 347 (2015).
  8. C. P. Brangwynne, C. R. Eckmann, D. S. Courson, A. Rybarska, C. Hoege, J. Gharakhani, F. Jülicher, and A. A. Hyman, Germline P granules are liquid droplets that localize by controlled dissolution/condensation, Science 324, 1729 (2009).
  9. A. A. Hyman, C. A. Weber, and F. Jülicher, Liquid-liquid phase separation in biology, Annu. Rev. Cell Dev. Biol. 30, 39 (2014).
  10. S. F. Banani, H. O. Lee, A. A. Hyman, and M. K. Rosen, Biomolecular condensates: Organizers of cellular biochemistry, Nat. Rev. Mol. Cell Biol. 18, 285 (2017).
  11. A. S. Lyon, W. B. Peeples, and M. K. Rosen, A framework for understanding the functions of biomolecular condensates across scales, Nat. Rev. Mol. Cell Biol. 22, 215 (2021).
  12. S. Alberti, A. Gladfelter, and T. Mittag, Considerations and challenges in studying liquid-liquid phase separation and biomolecular condensates, Cell 176, 419 (2019).
  13. Y. Shin and C. P. Brangwynne, Liquid phase condensation in cell physiology and disease, Science 357, eaaf4382 (2017).
  14. J.-M. Choi, A. S. Holehouse, and R. V. Pappu, Physical principles underlying the complex biology of intracellular phase transitions, Annu. Rev. Biophys. 49, 107 (2020).
  15. D. Hnisz, K. Shrinivas, R. A. Young, A. K. Chakraborty, and P. A. Sharp, A phase separation model for transcriptional control, Cell 169, 13 (2017).
  16. B. R. Sabari, A. Dall'Agnese, A. Ann Boija, I. A. Klein, E. L. Coffey, K. Shrinivas, B. J. Abraham, N. M. Hannett, A. V. Zamudio, J. C. Manteiga, C. H. Li, Y. E. Guo, D. S. Day, J. Schuijers, E. Vasile, S. Malik, D. Hnisz, T. I. Lee, I. I. Cisse, R. G. Roeder, et al., Coactivator condensation at super-enhancers links phase separation and gene control, Science 361, eaar3958 (2018).
  17. K. Shrinivas, B. R. Sabari, E. L. Coffey, I. A. Klein, A. Boija, A. V. Zamudio, J. Schuijers, N. M. Hannett, P. A. Sharp, R. A. Young, and A. K. Chakraborty, Enhancer features that drive formation of transcriptional condensates, Mol. Cell 75, 549 (2019).
  18. J. E. Henninger, O. Oksuz, K. Shrinivas, I. Sagi, G. LeRoy, M. M. Zheng, J. O. Andrews, A. V. Zamudio, C. Lazaris, N. M. Hannett, T. I. Lee, P. A. Sharp, I. I. Cissé, A. K. Chakraborty, and R. A. Young, RNA-mediated feedback control of transcriptional condensates, Cell 184, 207 (2021).
  19. T. Hirose, K. Ninomiya, S. Nakagawa, and T. Yamazaki, A guide to membraneless organelles and their various roles in gene regulation, Nat. Rev. Mol. Cell Biol. 24, 288 (2023).
  20. G. P. Faber, S. Nadav-Eliyahu, and Y. Shav-Tal, Nuclear speckles – a driving force in gene expression, J. Cell Sci. 135, jcs259594 (2022).
  21. D. L. J. Lafontaine, J. A. Riback, R. Bascetin, and C. P. Brangwynne, The nucleolus as a multiphase liquid condensate, Nat. Rev. Mol. Cell Biol. 22, 165 (2021).
  22. S. F. Banani, A. Goychuk, P. Natarajan, M. M. Zheng, G. Dall'Agnese, J. E. Henninger, M. Kardar, R. A. Young, and A. K. Chakraborty, Active RNA synthesis patterns nuclear condensates, bioRxiv (2024).
  23. P. W. Voorhees, Ostwald ripening of two-phase mixtures, Annu. Rev. Mater. Sci. 22, 197 (1992).
  24. E. D. Siggia, Late stages of spinodal decomposition in binary mixtures, Phys. Rev. A 20, 595 (1979).
  25. R. Shimizu and H. Tanaka, A novel coarsening mechanism of droplets in immiscible fluid mixtures, Nat. Commun. 6, 7407 (2015).
  26. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Mechanics of Fluids and Transport Processes (Springer, Dordrecht, 1983).
  27. M. Driscoll and B. Delmotte, Leveraging collective effects in externally driven colloidal suspensions: Experiments and simulations, Curr. Opin. Colloid Interface Sci. 40, 42 (2019).
  28. E. Frey and K. Kroy, Brownian motion: A paradigm of soft matter and biological physics, Ann. Phys. 517, 20 (2005).
  29. T. M. Squires and T. G. Mason, Fluid mechanics of microrheology, Annu. Rev. Fluid Mech. 42, 413 (2010).
  30. H. Tanaka, Viscoelastic phase separation, J. Phys.: Condens. Matter 12, R207 (2000).
  31. T. Wiegand and A. A. Hyman, Drops and fibers — how biomolecular condensates and cytoskeletal filaments influence each other, Emerg. Top. Life Sci. 4, 247 (2020).
  32. S. Mohapatra and S. Wegmann, Biomolecular condensation involving the cytoskeleton, Brain Res. Bull. 194, 105 (2023).
  33. K. Graham, A. Chandrasekaran, L. Wang, N. Yang, E. M. Lafer, P. Rangamani, and J. C. Stachowiak, Liquid-like condensates mediate competition between actin branching and bundling, Proc. Natl. Acad. Sci. USA 121, e2309152121 (2024).
  34. T. Quail, S. Golfier, M. Elsner, K. Ishihara, V. Murugesan, R. Renger, F. Jülicher, and J. Brugués, Force generation by protein–dna co-condensation, Nat. Phys. 17, 1007 (2021).
  35. A. R. Strom, Y. Kim, H. Zhao, Y.-C. Chang, N. D. Orlovsky, A. Košmrlj, C. Storm, and C. P. Brangwynne, Condensate interfacial forces reposition DNA loci and probe chromatin viscoelasticity, Cell 187, 5282 (2024).
  36. D. A. Fletcher and R. D. Mullins, Cell mechanics and the cytoskeleton, Nature (London) 463, 485 (2010).
  37. A. Zidovska, D. A. Weitz, and T. J. Mitchison, Micron-scale coherence in interphase chromatin dynamics, Proc. Natl. Acad. Sci. USA 110, 15555 (2013).
  38. R. Bruinsma, A. Y. Grosberg, Y. Rabin, and A. Zidovska, Chromatin hydrodynamics, Biophys. J. 106, 1871 (2014).
  39. H. A. Shaban, R. Barth, and K. Bystricky, Formation of correlated chromatin domains at nanoscale dynamic resolution during transcription, Nucl. Acids Res. 46, e77 (2018).
  40. R. W. Style, T. Sai, N. Fanelli, M. Ijavi, K. Smith-Mannschott, Q. Xu, L. A. Wilen, and E. R. Dufresne, Liquid-liquid phase separation in an elastic network, Phys. Rev. X 8, 011028 (2018).
  41. K. A. Rosowski, T. Sai, E. Vidal-Henriquez, D. Zwicker, R. W. Style, and E. R. Dufresne, Elastic ripening and inhibition of liquid–liquid phase separation, Nat. Phys. 16, 422 (2020).
  42. F. Yuan, H. Alimohamadi, B. Bakka, A. N. Trementozzi, K. J. Day, N. L. Fawzi, P. Rangamani, and J. C. Stachowiak, Membrane bending by protein phase separation, Proc. Natl. Acad. Sci. USA 118, e2017435118 (2021).
  43. Y. Qiang, C. Luo, and D. Zwicker, Nonlocal elasticity yields equilibrium patterns in phase separating systems, Phys. Rev. X 14, 021009 (2024).
  44. Q. Yu and A. Košmrlj, Pattern formation of lipid domains in bilayer membranes, Soft Matter 21, 4288 (2025).
  45. A. Winter, Y. Liu, A. Ziepke, G. Dadunashvili, and E. Frey, Phase separation on deformable membranes: Interplay of mechanical coupling and dynamic surface geometry, Phys. Rev. E 111, 044405 (2025).
  46. G. I. Taylor, The formation of emulsions in definable fields of flow, Proc. R. Soc. London A 146, 501 (1934).
  47. H. A. Stone, Dynamics of drop deformation and breakup in viscous fluids, Annu. Rev. Fluid Mech. 26, 65 (1994).
  48. F. Caballero and M. C. Marchetti, Activity-suppressed phase separation, Phys. Rev. Lett. 129, 268002 (2022).
  49. R. Adkins, I. Kolvin, Z. You, S. Witthaus, M. C. Marchetti, and Z. Dogic, Dynamics of active liquid interfaces, Science 377, 768 (2022).
  50. C. P. Brangwynne, G. H. Koenderink, F. C. MacKintosh, and D. A. Weitz, Intracellular transport by active diffusion, Trends Cell Biol. 19, 423 (2009).
  51. A. Al Jord, G. Letort, S. Chanet, F.-C. Tsai, C. Antoniewski, A. Eichmuller, C. Da Silva, J.-R. Huynh, N. S. Gov, R. Voituriez, M.-E. Terret, and M.-H. Verlhac, Cytoplasmic forces functionally reorganize nuclear condensates in oocytes, Nat. Commun. 13, 5070 (2022).
  52. A. Bray, Theory of phase-ordering kinetics, Adv. Phys. 43, 357 (1994).
  53. C. A. Weber, C. F. Lee, and F. Jülicher, Droplet ripening in concentration gradients, New J. Phys. 19, 053021 (2017).
  54. P. C. Bressloff, Two-dimensional droplet ripening in a concentration gradient, J. Phys. A 53, 365002 (2020).
  55. E. Jambon-Puillet, A. Testa, C. Lorenz, R. W. Style, A. A. Rebane, and E. R. Dufresne, Phase-separated droplets swim to their dissolution, Nat. Commun. 15, 3919 (2024).
  56. V. S. Doan, I. Alshareedah, A. Singh, P. R. Banerjee, and S. Shin, Diffusiophoresis promotes phase separation and transport of biomolecular condensates, Nat. Commun. 15, 7686 (2024).
  57. A. Einstein, Über die von der molekularkinetischen theorie der wärme geforderte bewegung von in ruhenden flüssigkeiten suspendierten Teilchen, Ann. Phys. 322, 549 (1905).
  58. M. Schmitt and H. Stark, Active Brownian motion of emulsion droplets: Coarsening dynamics at the interface and rotational diffusion, Eur. Phys. J. E 39, 80 (2016).
  59. M. E. Cates and E. Tjhung, Theories of binary fluid mixtures: From phase-separation kinetics to active emulsions, J. Fluid Mech. 836, P1 (2018).
  60. S. Wilken, A. Chaderjian, and O. A. Saleh, Spatial organization of phase-separated DNA droplets, Phys. Rev. X 13, 031014 (2023).
  61. D. S. W. Lee, C.-H. Choi, D. W. Sanders, L. Beckers, J. A. Riback, C. P. Brangwynne, and N. S. Wingreen, Size distributions of intracellular condensates reflect competition between coalescence and nucleation, Nat. Phys. 19, 586 (2023).
  62. H. Zhang, F. Wang, L. Ratke, and B. Nestler, Brownian motion of droplets induced by thermal noise, Phys. Rev. E 109, 024208 (2024).
  63. R. Balian, From Microphysics to Macrophysics (Springer, Berlin, 2007).
  64. S. R. De Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover Books on Physics (Dover, 2013).
  65. D. Zwicker, The intertwined physics of active chemical reactions and phase separation, Curr. Opin. Colloid Interface Sci. 61, 101606 (2022).
  66. D. Zwicker, O. W. Paulin, and C. t. Burg, Physics of droplet regulation in biological cells, Rep. Prog. Phys. 88, 116601 (2025).
  67. A. Goychuk, L. Demarchi, I. Maryshev, and E. Frey, Self-consistent sharp interface theory of active condensate dynamics, Phys. Rev. Res. 6, 033082 (2024).
  68. J. Romano, R. Golestanian, and B. Mahault, Dynamics of phase-separated interfaces in inhomogeneous and driven mixtures, Soft Matter 21, 9245 (2025).
  69. E. Hückel and P. Debye, Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen, Physikal. Zeitschr. 24, 185 (1923).
  70. A. Kumar and S. A. Safran, Fluctuations and shape dependence of microphase separation in systems with long-range interactions, Phys. Rev. Lett. 131, 258401 (2023).
  71. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Butterworth-Heinemann, Oxford, 1987).
  72. G. K. Batchelor, Brownian diffusion of particles with hydrodynamic interaction, J. Fluid Mech. 74, 1 (1976).
  73. D. Goh, D. Kannan, P. Natarajan, A. Goychuk, and A. K. Chakraborty, RNA gradients can guide condensates toward promoters: Implications for enhancer–promoter contacts and condensate-promoter kissing, J. Chem. Phys. 163, 104905 (2025).
  74. Y. Baek, A. P. Solon, X. Xu, N. Nikola, and Y. Kafri, Generic long-range interactions between passive bodies in an active fluid, Phys. Rev. Lett. 120, 058002 (2018).
  75. S. Ro, Y. Kafri, M. Kardar, and J. Tailleur, Disorder-induced long-ranged correlations in scalar active matter, Phys. Rev. Lett. 126, 048003 (2021).
  76. U. S. Schwarz and S. A. Safran, Elastic interactions of cells, Phys. Rev. Lett. 88, 048102 (2002).
  77. S. Bose, P. S. Noerr, A. Gopinathan, A. Gopinath, and K. Dasbiswas, Collective states of active particles with elastic dipolar interactions, Front. Phys. 10, 876126 (2022).
  78. F. Meng, D. Matsunaga, and R. Golestanian, Clustering of magnetic swimmers in a Poiseuille flow, Phys. Rev. Lett. 120, 188101 (2018).
  79. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435 (1977).
  80. J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. 195, 127 (1990).
  81. B. Barker, J. B. Bell, and A. L. Garcia, Fluctuating hydrodynamics and the Rayleigh–Plateau instability, Proc. Natl. Acad. Sci. USA 120, e2306088120 (2023).
  82. J. B. Bell, A. Nonaka, and A. L. Garcia, Comment on “Brownian motion of droplets induced by thermal noise,” Phys. Rev. E 111, 056201 (2025).
  83. H. C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res. 1, 27 (1949).
  84. R. W. Zwanzig, Hydrodynamic fluctuations and Stokes' law friction, J. Res. Natl. Bur. Stand. B 68B, 143 (1964).
  85. R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966).
  86. J. Laprade, L. B. Frechette, C. Amey, A. T. Cusi, A. Baskaran, W. B. Rogers, and G. Duclos, The coarsening of biomimetic condensates in an active fluid is non-self-similar, Nat. Phys. 22, 612 (2026).
  87. Y. I. Li and M. E. Cates, Non-equilibrium phase separation with reactions: A canonical model and its behaviour, J. Stat. Mech. (2020) 053206.
  88. J. Decayeux, V. Dahirel, M. Jardat, and P. Illien, Spontaneous propulsion of an isotropic colloid in a phase-separating environment, Phys. Rev. E 104, 034602 (2021).
  89. L. Demarchi, A. Goychuk, I. Maryshev, and E. Frey, Enzyme-enriched condensates show self-propulsion, positioning, and coexistence, Phys. Rev. Lett. 130, 128401 (2023).
  90. J. F. Robinson, T. Machon, and T. Speck, Universal limiting behaviour of reaction-diffusion systems with conservation laws, Phys. Rev. E 111, 065417 (2025).
  91. A. Goychuk, D. Kannan, A. K. Chakraborty, and M. Kardar, Polymer folding through active processes recreates features of genome organization, Proc. Natl. Acad. Sci. USA 120, e2221726120 (2023).
  92. A. Goychuk, D. Kannan, and M. Kardar, Delayed excitations induce polymer looping and coherent motion, Phys. Rev. Lett. 133, 078101 (2024).
  93. S. Banerjee, R. B. Griffiths, and M. Widom, Thermodynamic limit for dipolar media, J. Stat. Phys. 93, 109 (1998).
  94. W. R. Inc., Mathematica, version 14.3, Champaign, IL, 2025, https://www.wolfram.com/mathematica.
  95. A. Goychuk, Source code - Externally driven condensates show translation-induced polarization, directed coalescence, and anomalous diffusion in viscoelastic media, Zenodo, 2026, https://doi.org/10.5281/zenodo.20110341.
  96. J. Rotne and S. Prager, Variational treatment of hydrodynamic interaction in polymers, J. Chem. Phys. 50, 4831 (1969).
  97. H. Yamakawa, Transport properties of polymer chains in dilute solution: Hydrodynamic interaction, J. Chem. Phys. 53, 436 (1970).

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Phys. Rev. E 113, 065402– Published 3 June, 2026
  • Received 23 September 2025
  • Accepted 6 May 2026
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