A multiscale biophysical model gives time-transcendental waves in a network of pulsating cilia


Understanding the genesis and properties of temporal waves (MWs) in ciliated arrays is a central multiscale problem in evolutionary biology, transport phenomenology, non-equilibrium physics, and with potential biomedical applications. For one-dimensional cilia networks, we report the main mechanisms that lead to the robust emergence of MWs. Our modeling framework includes the full finer details of lashing. This helps us understand how ciliary bed morphology, beating patterns, steric and hydrodynamic interactions work together to shape the dynamics arising at 1D synapses. Due to the novelty of our modeling and computation, we have revealed the spatiotemporal self-organization of nanoscale kinetic proteins in the coordination of collective dynamics spanning millimeters: bridge length scales of more than six orders of magnitude.


Motile cilia are thin, hair-like cellular appendages that oscillate spontaneously under the influence of intrinsic molecular motors and are usually found in dense arrays. These energetic filaments coordinate their strokes to generate transcendent waves that drive long-range movement and transport of fluids. So far, our understanding of their collective behavior largely comes from studying coarse-grain minimal models and related biophysics and hydrodynamics of slender structures. Here we build on a detailed biophysical model to illustrate the emergence of transient waves on millimeter scales of nanometer-scale locomotor activity within individual cilia. Our study of a one-dimensional network of cilia in the presence of hydrodynamic and steric interactions reveals how choppy waves are formed and maintained. We find that, in families of homologous cilia, these interactions lead to multiple attractions, all characterized by an integer charge that is conserved. This also allows us to design the initial conditions that lead to predictable emergent states. Finally, and most importantly, we show that in irregular ciliated tissues, boundaries and heterogeneity provide a robust pathway for bypass waves.


    • accepted December 2, 2021.
  • Authors contributions: Research designed by BC, SF, and MJS, conducted research, contributed new reagents/analytical tools, analyzed data, and wrote the paper.

  • The book declares no competing interest.

  • This article is a direct PNAS submission.

  • This article contains 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. Simulation code is available upon request of the authors.

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