Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Entropic size control of self-assembled filaments

Maximilian C. Hübl* and Carl P. Goodrich†

  • *Contact author: maximilian.huebl@ist.ac.at
  • †Contact author: carl.goodrich@ist.ac.at

Phys. Rev. Research 8, L012054 – Published 5 March, 2026

DOI: https://doi.org/10.1103/68rs-3qgn

Abstract

Controlling the size and shape of assembled structures is a fundamental challenge in self-assembly and is highly relevant in material design and biology. Here, we show that specific but promiscuous short-range binding interactions make it possible to economically assemble linear filaments of user-defined length. Our approach leads to independent control over the mean and width of the filament size distribution and allows us to smoothly explore design trade-offs between assembly quality (spread in size) and cost (number of particle species). We employ a simple hierarchical assembly protocol to minimize assembly times and show that multiple stages of hierarchy make it possible to extend our approach to the assembly of higher-dimensional structures. Our work provides a conceptually simple solution to size control that is applicable to a broad range of systems, from DNA nanoparticles to supramolecular polymers and beyond.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (70)

  1. B. Alberts, A. Johnson, J. Lewis, D. Morgan, M. Raff, K. Roberts, and P. Walter, Molecular Biology of the Cell, 7th ed. (W. W. Norton & Company, New York, 2022).
  2. N. Rhind, Cell-size control, Curr. Biol. 31, R1414 (2021).
  3. M. B. Ginzberg, R. Kafri, and M. Kirschner, On being the right (cell) size, Science 348, 1245075 (2015).
  4. M. F. Hagan and G. M. Grason, Equilibrium mechanisms of self-limiting assembly, Rev. Mod. Phys. 93, 025008 (2021).
  5. Y. Kim, P. Dalhaimer, D. A. Christian, and D. E. Discher, Polymeric worm micelles as nano-carriers for drug delivery, Nanotechnology 16, S484 (2005).
  6. M. Rad-Malekshahi, L. Lempsink, M. Amidi, W. E. Hennink, and E. Mastrobattista, Biomedical applications of self-assembling peptides, Bioconjugate Chem. 27, 3 (2016).
  7. C. Sigl, E. M. Willner, W. Engelen, J. A. Kretzmann, K. Sachenbacher, A. Liedl, F. Kolbe, F. Wilsch, S. A. Aghvami, U. Protzer, M. F. Hagan, S. Fraden, and H. Dietz, Programmable icosahedral shell system for virus trapping, Nat. Mater. 20, 1281 (2021).
  8. P. Vukusic and J. R. Sambles, Photonic structures in biology, Nature (London) 424, 852 (2003).
  9. E. R. Dufresne, H. Noh, V. Saranathan, S. G. J. Mochrie, H. Cao, and R. O. Prum, Self-assembly of amorphous biophotonic nanostructures by phase separation, Soft Matter 5, 1792 (2009).
  10. D. Satyabola, A. Prasad, H. Yan, and X. Zhou, Bioinspired photonic systems directed by designer DNA nanostructures, ACS Appl. Opt. Mater. 3, 552 (2025).
  11. D. Hayakawa, T. E. Videbæk, G. M. Grason, and W. B. Rogers, Symmetry-guided inverse design of self-assembling multiscale DNA origami tilings, ACS Nano 18, 19169 (2024).
  12. A. Hensley, T. E. Videbæk, H. Seyforth, W. M. Jacobs, and W. B. Rogers, Macroscopic photonic single crystals via seeded growth of DNA-coated colloids, Nat. Commun. 14, 4237 (2023).
  13. F. A. Aldaye, A. L. Palmer, and H. F. Sleiman, Assembling materials with DNA as the guide, Science 321, 1795 (2008).
  14. A. K. Pearce, T. R. Wilks, M. C. Arno, and R. K. O'Reilly, Synthesis and applications of anisotropic nanoparticles with precisely defined dimensions, Nat. Rev. Chem. 5, 21 (2021).
  15. A. Michelson, A. Subramanian, K. Kisslinger, N. Tiwale, S. Xiang, E. Shen, J. S. Kahn, D. Nykypanchuk, H. Yan, C.-Y. Nam, and O. Gang, Three-dimensional nanoscale metal, metal oxide, and semiconductor frameworks through DNA-programmable assembly and templating, Sci. Adv. 10, eadl0604 (2024).
  16. A. Michelson, L. Shani, J. S. Kahn, D. C. Redeker, W.-I. Lee, K. R. DeOlivares, K. Kisslinger, N. Tiwale, H. Yan, A. Pattammattel, C.-Y. Nam, V. S. Pribiag, and O. Gang, Scalable fabrication of chip-integrated 3D-nanostructured electronic devices via DNA-programmable assembly, Sci. Adv. 11, eadt5620 (2025).
  17. D. Hayakawa, T. E. Videbaek, D. M. Hall, H. Fang, C. Sigl, E. Feigl, H. Dietz, S. Fraden, M. F. Hagan, G. M. Grason, and W. B. Rogers, Geometrically programmed self-limited assembly of tubules using DNA origami colloids, Proc. Natl. Acad. Sci. USA 119, e2207902119 (2022).
  18. T. E. Videbæk, D. Hayakawa, G. M. Grason, M. F. Hagan, S. Fraden, and W. B. Rogers, Economical routes to size-specific assembly of self-closing structures, Sci. Adv. 10, eado5979 (2024).
  19. C. M. Duque, D. M. Hall, B. Tyukodi, M. F. Hagan, C. D. Santangelo, and G. M. Grason, Limits of economy and fidelity for programmable assembly of size-controlled triply periodic polyhedra, Proc. Natl. Acad. Sci. USA 121, e2315648121 (2024).
  20. R. Saha, D. Hayakawa, T. E. Videbaek, M. Price, W.-S. Wei, J. Pombo, D. Duke, G. Arya, G. M. Grason, W. B. Rogers, and S. Fraden, Modular programming of interaction and geometric specificity enables assembly of complex DNA origami nanostructures, Nat. Commun. 16, 11392 (2025).
  21. F. Sciortino, S. Mossa, E. Zaccarelli, and P. Tartaglia, Equilibrium cluster phases and low-density arrested disordered states: The role of short-range attraction and long-range repulsion, Phys. Rev. Lett. 93, 055701 (2004).
  22. T. D. Nguyen, B. A. Schultz, N. A. Kotov, and S. C. Glotzer, Generic, phenomenological, on-the-fly renormalized repulsion model for self-limited organization of terminal supraparticle assemblies, Proc. Natl. Acad. Sci. USA 112, E3161 (2015).
  23. G. M. Grason, Perspective: Geometrically frustrated assemblies J. Chem. Phys. 145, 110901 (2016).
  24. M. Lenz and T. A. Witten, Geometrical frustration yields fibre formation in self-assembly, Nat. Phys. 13, 1100 (2017).
  25. H. Le Roy, M. M. Terzi, and M. Lenz, Collective deformation modes promote fibrous self-assembly in deformable particles, Phys. Rev. X 15, 011022 (2025).
  26. L. Koehler, M. Eder, C. Karfusehr, V. Ouazan-Reboul, P. Ronceray, F. C. Simmel, and M. Lenz, Topological defect engineering enables size and shape control in self-assembly, arXiv:2504.13073.
  27. D. Johann, C. Erlenkämper, and K. Kruse, Length regulation of active biopolymers by molecular motors, Phys. Rev. Lett. 108, 258103 (2012).
  28. A. Melbinger, L. Reese, and E. Frey, Microtubule length regulation by molecular motors, Phys. Rev. Lett. 108, 258104 (2012).
  29. M. Striebel, F. Brauns, and E. Frey, Length regulation drives self-organization in filament-motor mixtures, Phys. Rev. Lett. 129, 238102 (2022).
  30. H.-S. Kuan and M. D. Betterton, Biophysics of filament length regulation by molecular motors, Phys. Biol. 10, 036004 (2013).
  31. S. Datta, M. L. Saha, and P. J. Stang, Hierarchical assemblies of supramolecular coordination complexes, Acc. Chem. Res. 51, 2047 (2018).
  32. A. Murugan, J. Zou, and M. P. Brenner, Undesired usage and the robust self-assembly of heterogeneous structures, Nat. Commun. 6, 6203 (2015).
  33. M. He, J. P. Gales, E. Ducrot, Z. Gong, G.-R. Yi, S. Sacanna, and D. J. Pine, Colloidal diamond, Nature (London) 585, 524 (2020).
  34. C. A. Mirkin, R. L. Letsinger, R. C. Mucic, and J. J. Storhoff, A DNA-based method for rationally assembling nanoparticles into macroscopic materials, Nature (London) 382, 607 (1996).
  35. W. B. Rogers and V. N. Manoharan, Programming colloidal phase transitions with DNA strand displacement, Science 347, 639 (2015).
  36. Y. Wang, Y. Wang, D. R. Breed, V. N. Manoharan, L. Feng, A. D. Hollingsworth, M. Weck, and D. J. Pine, Colloids with valence and specific directional bonding, Nature (London) 491, 51 (2012).
  37. Y. Wang, Y. Wang, X. Zheng, E. Ducrot, J. S. Yodh, M. Weck, and D. J. Pine, Crystallization of DNA-coated colloids, Nat. Commun. 6, 7253 (2015).
  38. M.-P. Valignat, O. Theodoly, J. C. Crocker, W. B. Russel, and P. M. Chaikin, Reversible self-assembly and directed assembly of DNA-linked micrometer-sized colloids, Proc. Natl. Acad. Sci. USA 102, 4225 (2005).
  39. W. M. Jacobs and W. B. Rogers, Assembly of complex colloidal systems using DNA, Annu. Rev. Condens. Matter Phys. 16, 443 (2025).
  40. S. Sacanna, W. T. M. Irvine, P. M. Chaikin, and D. J. Pine, Lock and key colloids, Nature (London) 464, 575 (2010).
  41. S. Sacanna, M. Korpics, K. Rodriguez, L. Colón-Meléndez, S.-H. Kim, D. J. Pine, and G.-R. Yi, Shaping colloids for self-assembly, Nat. Commun. 4, 1688 (2013).
  42. P.-S. Huang, S. E. Boyken, and D. Baker, The coming of age of de novo protein design, Nature (London) 537, 320 (2016).
  43. N. P. King, W. Sheffler, M. R. Sawaya, B. S. Vollmar, J. P. Sumida, I. André, T. Gonen, T. O. Yeates, and D. Baker, Computational design of self-assembling protein nanomaterials with atomic level accuracy, Science 336, 1171 (2012).
  44. T. F. A. d. Greef and E. W. Meijer, Supramolecular polymers, Nature (London) 453, 171 (2008).
  45. T. Aida, E. W. Meijer, and S. I. Stupp, Functional supramolecular polymers, Science 335, 813 (2012).
  46. H. M. M. t. Eikelder, B. Adelizzi, A. R. A. Palmans, and A. J. Markvoort, Equilibrium model for supramolecular copolymerizations, J. Phys. Chem. B 123, 6627 (2019).
  47. E. D. Klein, R. W. Perry, and V. N. Manoharan, Physical interpretation of the partition function for colloidal clusters, Phys. Rev. E 98, 032608 (2018).
  48. A. I. Curatolo, O. Kimchi, C. P. Goodrich, R. K. Krueger, and M. P. Brenner, A computational toolbox for the assembly yield of complex and heterogeneous structures, Nat. Commun. 14, 8328 (2023).
  49. M. Holmes-Cerfon, S. J. Gortler, and M. P. Brenner, A geometrical approach to computing free-energy landscapes from short-ranged potentials, Proc. Natl. Acad. Sci. USA 110, E5 (2013).
  50. M. Holmes-Cerfon, Sticky-sphere clusters, Annu. Rev. Condens. Matter Phys. 8, 77 (2017).
  51. M. C. Hübl and C. P. Goodrich, Accessing semiaddressable self-assembly with efficient structure enumeration, Phys. Rev. Lett. 134, 058204 (2025).
  52. M. C. Hübl, T. E. Videbæk, D. Hayakawa, W. B. Rogers, and C. P. Goodrich, A polyhedral structure controls programmable self-assembly, Nat. Phys. (2026), doi: 10.1038/s41567-025-03120-3.
  53. Strictly speaking, the distribution is geometric, since n is an integer and 1≤n<∞.
  54. M. P. Brenner, Mathematics and Materials Ideas About Self-Assembly, IAS/Park City Mathematics Series (American Mathematical Society, Providence, 2017), Vol. 23.
  55. This remains true if the twofold symmetry of a filament is exploited, which cuts down the required particle species by half but leaves the prohibitive linear scaling unaffected.
  56. D. Nykypanchuk, M. M. Maye, D. v. d. Lelie, and O. Gang, DNA-guided crystallization of colloidal nanoparticles, Nature (London) 451, 549 (2008).
  57. See Supplemental Material at https://link.aps.org/supplemental/10.1103/68rs-3qgn for additional details of the kinetic calculations and supplementary data, which includes Ref. [70].
  58. The length of 〈n〉/(2m) is necessary since there is a roughly 50% chance that a single-species segment binds with another segment of the same species before encountering a filament of the subsequent species.
  59. O. G. Hayes, B. E. Partridge, and C. A. Mirkin, Encoding hierarchical assembly pathways of proteins with DNA, Proc. Natl. Acad. Sci. USA 118, e2106808118 (2021).
  60. E. W. Gehrels, W. B. Rogers, Z. Zeravcic, and V. N. Manoharan, Programming directed motion with DNA-grafted particles, ACS Nano 16, 9195 (2022).
  61. H. Xiong, D. van der Lelie, and O. Gang, Phase behavior of nanoparticles assembled by DNA linkers, Phys. Rev. Lett. 102, 015504 (2009).
  62. J. Lowensohn, B. Oyarzún, G. N. Paliza, B. M. Mognetti, and W. B. Rogers, Linker-mediated phase behavior of DNA-coated colloids, Phys. Rev. X 9, 041054 (2019).
  63. M. C. Hübl and C. P. Goodrich, Entropic size control of self-assembled filaments, Zenodo (2025), doi: 10.5281/zenodo.17898642.
  64. M. Kardar, Statistical Physics of Particles (Cambridge University Press, New York, 2007).
  65. D. T. Gillespie, Stochastic simulation of chemical kinetics, Annu. Rev. Phys. Chem. 58, 35 (2007).
  66. N. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier Science Publishers, Amsterdam, 1992).
  67. P. J. Blatz and A. V. Tobolsky, Note on the kinetics of systems manifesting simultaneous polymerization-depolymerization phenomena, J. Phys. Chem. 49, 77 (1945).
  68. W. H. Stockmayer, Theory of molecular size distribution and gel formation in branched-chain polymers, J. Chem. Phys. 11, 45 (1943).
  69. P. G. J. v. Dongen and M. H. Ernst, Kinetics of reversible polymerization, J. Stat. Phys. 37, 301 (1984).
  70. R. D. Vigil, On equilibrium solutions of aggregation–fragmentation problems, J. Colloid Interface Sci. 336, 642 (2009).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation