Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Dynamics of topological defects and structural synchronization in a forming periodic tissue


Living organisms form a large variety of hierarchically structured extracellular functional tissues. Remarkably, these materials exhibit regularity and structural coherence across multiple length scales, far beyond the size of a single cell. Here, synchrotron-based nanotomographic imaging in combination with machine-learning-based segmentation is used to reveal the structural synchronization process of nacre forming in the shell of the mollusc Unio pictorum. We show that the emergence of this highly regular layered structure is driven by a disorder-to-order transition achieved through the motion and interaction of screw-like structural dislocations with an opposite topological sign. Using an analogy to similar processes observed in liquid-crystalline systems, we demonstrate that these microstructural faults act as dissipative topological defects coupled by an elastic distortion field surrounding their cores. Their mutual annihilation results in structural synchronization that is simulated using the classical Kuramoto model. The developed experimental, theoretical and numerical framework provides a comprehensive physical view of the formation of biogenic materials.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

Fig. 1: The shell of U. pictorum, characterized by a scanning electron microscope.
Fig. 2: 3D visualization of the shell of U. pictorum using holographic X-ray nanotomography.
Fig. 3: Dynamics of topological defects.
Fig. 4: Kuramoto simulations.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

Code for the Kuramoto simulation is publicly available at


  1. Fratzl, P. & Weinkamer, R. Nature’s hierarchical materials. Prog. Mater. Sci. 52, 1263–1334 (2007).

    Article  Google Scholar 

  2. Bøggild, O. B. The shell structure of the mollusks. Kongel. Danske Vidensk. Selsk. Skr.: Naturvidensk. Math. Afd. 9, 231–326 (1930).

  3. Carter, J. G. & Clark, G. R. Classification and phylogenetic significance of molluscan shell microstructure. Notes Short Course: Stud. Geol. 13, 50–71 (1985).

    Article  Google Scholar 

  4. Lowenstam, H. A. & Weiner, S. On Biomineralization (Oxford Univ. Press, 1989).

  5. Currey, J. D. & Taylor, J. D. The mechanical behaviour of some molluscan hard tissues. J. Zool. 173, 395–406 (1974).

    Article  Google Scholar 

  6. Barthelat, F., Yin, Z. & Buehler, M. J. Structure and mechanics of interfaces in biological materials. Nat. Rev. Mater. 1, 16007 (2016).

    Article  ADS  Google Scholar 

  7. Zlotnikov, I. & Schoeppler, V. Thermodynamic aspects of molluscan shell ultrastructural morphogenesis. Adv. Funct. Mater. 27, 1700506 (2017).

    Article  Google Scholar 

  8. Schoeppler, V. et al. Biomineralization as a paradigm of directional solidification: a physical model for molluscan shell ultrastructural morphogenesis. Adv. Mater. 30, 1803855 (2018).

    Article  Google Scholar 

  9. Cartwright, J. H. E. & Checa, A. G. The dynamics of nacre self-assembly. J. R. Soc. Interface 4, 491–504 (2007).

    Article  Google Scholar 

  10. Dauphin, Y., Luquet, G., Salome, M., Bellot-Gurlet, L. & Cuif, J. P. Structure and composition of Unio pictorum shell: arguments for the diversity of the nacroprismatic arrangement in molluscs. J. Microsc. 270, 156–169 (2018).

    Article  Google Scholar 

  11. Wegst, U. G. K., Bai, H., Saiz, E., Tomsia, A. P. & Ritchie, R. O. Bioinspired structural materials. Nat. Mater. 14, 23–36 (2015).

    Article  ADS  Google Scholar 

  12. Ritchie, R. O. The conflicts between strength and toughness. Nat. Mater. 10, 817–822 (2011).

    Article  ADS  Google Scholar 

  13. Barthelat, F. Nacre from mollusk shells: a model for high-performance structural materials. Bioinspir. Biomim. 5, 035001 (2010).

    Article  ADS  Google Scholar 

  14. Checa, A. G. Physical and biological determinants of the fabrication of molluscan shell microstructures. Front. Mar. Sci. 5, 1–21 (2018).

    Article  Google Scholar 

  15. Checa, A. G., Macías-Sánchez, E. & Ramírez-Rico, J. Biological strategy for the fabrication of highly ordered aragonite helices: the microstructure of the cavolinioidean gastropods. Sci. Rep. 6, 25989 (2016).

    Article  ADS  Google Scholar 

  16. Willinger, M. G., Checa, A. G., Bonarski, J. T., Faryna, M. & Berent, K. Biogenic crystallographically continuous aragonite helices: the microstructure of the planktonic gastropod Cuvierina. Adv. Funct. Mater. 26, 553–561 (2016).

    Article  Google Scholar 

  17. Currey, J. D. Bones: Structure and Mechanics (Princeton Univ. Press, 2013).

  18. Fabritius, H. et al. in Chitin (ed. Gupta, N. S.) 35–60 (Springer, 2011).

  19. Schoeppler, V. et al. Crystal growth kinetics as an architectural constraint on the evolution of molluscan shells. Proc. Natl Acad. Sci. USA 116, 20388–20397 (2019).

    Article  Google Scholar 

  20. Gilbert, P. U. P. A. et al. Gradual ordering in red abalone nacre. J. Am. Chem. Soc. 130, 17519–17527 (2008).

    Article  Google Scholar 

  21. Hovden, R. et al. Nanoscale assembly processes revealed in the nacroprismatic transition zone of Pinna nobilis mollusc shells. Nat. Commun. 6, 10097 (2015).

    Article  ADS  Google Scholar 

  22. Wada, K. Spiral growth of nacre. Nature 211, 1427 (1966).

  23. Yao, N., Epstein, A. K., Liu, W. W., Sauer, F. & Yang, N. Organic–inorganic interfaces and spiral growth in nacre. J. R. Soc. Interface 6, 367–376 (2009).

    Article  Google Scholar 

  24. Gao, Y., Guo, Z., Song, Z. & Yao, H. Spiral interface: a reinforcing mechanism for laminated composite materials learned from nature. J. Mech. Phys. Solids 109, 252–263 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  25. Cartwright, J. H. E., Checa, A. G., Escribano, B. & Sainz-Díaz, C. I. Spiral and target patterns in bivalve nacre manifest a natural excitable medium from layer growth of a biological liquid crystal. Proc. Natl Acad. Sci. USA 106, 10499–10504 (2009).

    Article  ADS  Google Scholar 

  26. Wise, S. W. Jr & deVilliers, J. Scanning electron microscopy of molluscan shell ultrastructures: screw dislocations in pelecypod nacre. Trans. Am. Microsc. Soc. 90, 376–380 (1971).

  27. Qi, L., Huang, Y., Zhou, Z. & Zhou, Z. The growth of the screw dislocation of nacreous layer on Pteria penguin. Sci. China Earth Sci. 54, 951–958 (2011).

    Article  ADS  Google Scholar 

  28. Wang, X., Miller, D. S., Bukusoglu, E., de Pablo, J. J. & Abbott, N. L. Topological defects in liquid crystals as templates for molecular self-assembly. Nat. Mater. 15, 106–112 (2016).

    Article  ADS  Google Scholar 

  29. Darmon, A., Benzaquen, M., Čopar, S., Dauchot, O. & Lopez-Leon, T. Topological defects in cholesteric liquid crystal shells. Soft Matter 12, 9280–9288 (2016).

  30. Mesaros, A. et al. Topological defects coupling smectic modulations to intra-unit-cell nematicity in cuprates. Science 333, 426–430 (2011).

    Article  ADS  Google Scholar 

  31. Pacureanu, A., Langer, M., Boller, E., Tafforeau, P. & Peyrin, F. Nanoscale imaging of the bone cell network with synchrotron X‐ray tomography: optimization of acquisition setup. Med. Phys. 39, 2229–2238 (2012).

  32. Beliaev, M. et al. Quantification of sheet nacre morphogenesis using X-ray nanotomography and deep learning. J. Struct. Biol. 209, 107432 (2019).

    Article  Google Scholar 

  33. Shen, Y. & Dierking, I. Annihilation dynamics of topological defects induced by microparticles in nematic liquid crystals. Soft Matter 15, 8749–8757 (2019).

    Article  ADS  Google Scholar 

  34. Bogi, A., Martinot-Lagarde, P., Dozov, I. & Nobili, M. Anchoring screening of defects interaction in a nematic liquid crystal. Phys. Rev. Lett. 89, 225501 (2002).

  35. Dierking, I. et al. Anisotropy in the annihilation dynamics of umbilic defects in nematic liquid crystals. Phys. Rev. E 85, 021703 (2012).

  36. Rahman, A. Correlations in the motion of atoms in liquid argon. Phys. Rev. 136, A405–A411 (1964).

    Article  ADS  Google Scholar 

  37. Peach, M. & Koehler, J. S. The forces exerted on dislocations and the stress fields produced by them. Phys. Rev. 80, 436–439 (1950).

    Article  ADS  MathSciNet  Google Scholar 

  38. Nudelman, F. Nacre biomineralisation: a review on the mechanisms of crystal nucleation. Semin. Cell Dev. Biol. 46, 2–10 (2015).

    Article  Google Scholar 

  39. Mahler, S. et al. Dynamics of dissipative topological defects in coupled phase oscillators. J. Phys. B 52, 205401 (2019).

    Article  ADS  Google Scholar 

  40. Kuramoto, Y. in International Symposium on Mathematical Problems in Theoretical Physics Vol. 39 (ed. Araki, H.) 420–422 (Springer, 1975).

  41. Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence (Springer, 1983).

  42. Pargellis, A. N. et al. Defect dynamics and coarsening dynamics in smectic-C films. Phys. Rev. A 46, 7765–7776 (1992).

    Article  ADS  Google Scholar 

  43. Bouligand, Y. in Liquid Crystalline Order in Polymers (ed. Blumstein, A.) Ch. 8 (Academic Press, 1978).

  44. Bouligand, Y. in Liquid Crystals (ed. Liebert, L.) 259–294 (Academic Press, 1978).

  45. Hubert, M. et al. Efficient correction of wavefront inhomogeneities in X-ray holographic nanotomography by random sample displacement. Appl. Phys. Lett. 112, 203704 (2018).

    Article  ADS  Google Scholar 

Download references


I.Z. acknowledges financial support provided by Bundesministerium für Bildung und Forschung through grant 03Z22EN11. We acknowledge the ESRF for providing beam time on ID16A for proposals SC4155 and IHLS2846. Finally, we thank L. Bertinetti (B CUBE, Technische Universität Dresden) for critically assessing the manuscript.

Author information

Authors and Affiliations



M.B. and I.Z. designed the study. M.B. and D.Z. performed data analysis. A.P., P.Z. and I.Z. performed the X-ray experiments. M.B. and I.Z. wrote the manuscript, with input from all authors.

Corresponding author

Correspondence to Igor Zlotnikov.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Julyan Cartwright and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Video 1

Motion and annihilation of a dislocation pair. Each frame represents a 2D slice through the tomography data perpendicular to the direction of growth. Time in the video represents the growth of the nacreous ultrastructure. White and black represent the organic and mineral components, respectively. The right-handed and left-handed defects are marked in blue and red, respectively.

Supplementary Video 2

Simulated evolution of topography at the front of the growing structure, obtained using the Kuramoto model. Time in the video represents the growth of the structure.

Supplementary Video 3

Simulated evolution of the phase field at the front of the growing structure, obtained using the Kuramoto model. Time in the video represents the growth of the structure.

Supplementary Video 4

Simulated evolution of the elastic energy field at the front of the growing structure, obtained using the Kuramoto model. Time in the video represents the growth of the structure.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beliaev, M., Zöllner, D., Pacureanu, A. et al. Dynamics of topological defects and structural synchronization in a forming periodic tissue. Nat. Phys. 17, 410–415 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing