A multiscale biophysical model delivers quantized metachronous waves in a lattice of beating cilia



Understanding the origin and properties of metachronous waves (MWs) in cilia arrays is a multiscale problem central to developmental biology, transport phenomena, and non-equilibrium physics, with potential biomedical applications. For 1D lattices of cilia, we report key mechanisms leading to the robust formation of MWs. Our modeling framework contains all the microscopic details of beating cilia. This helps us understand how ciliary bed morphology, beat patterns, and steric and hydrodynamic interactions work together to shape emergence dynamics on 1D lattices. Due to the novelty of our modeling and computation, we have deciphered the spatiotemporal self-assembly of nanometer-scale motor proteins in the coordination of collective dynamics that span millimeters: bridging length scales over six orders of magnitude.


Motile cilia are slender, hair-like cell appendages that spontaneously oscillate under the action of internal molecular motors and are typically found in dense arrays. These active filaments coordinate their beating to create metachronous waves that power fluid transport and long-distance locomotion. So far, our understanding of their collective behavior relies largely on the study of minimal models, which coarsen the relevant biophysics and hydrodynamics of slender structures. Here, we build on a detailed biophysical model to elucidate the emergence of millimeter-scale metachronous waves from nanometer-scale motor activity within single cilia. Our study of a one-dimensional ciliary lattice in the presence of hydrodynamic and steric interactions reveals how metachronous waves are formed and sustained. We find that these interactions result in multiple states of attraction in homogeneous cilia beds, all characterized by a conserved integer charge. This even allows us to design initial conditions that lead to predictable emerging states. Finally, and very importantly, we show that boundaries and inhomogeneities in patchy ciliary tissue provide a robust route to metachronous waves.


    • Accepted December 2, 2021.
  • Author Contributions: BC, SF and MJS drafted research papers, conducted research papers, contributed new reagents/analytical tools, analyzed data and wrote the paper.

  • The authors declare no competing interests.

  • This article is a PNAS Direct Submission.

  • This article has supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2113539119/-/DCSupplemental.

data availability

All study data are included in the article and/or supporting information. The simulation code is available from the authors upon request.


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