How to Color a French Flag

Biologically Inspired Algorithms for Scale-Invariant Patterning
  • Bertie Ancona
  • Ayesha BajwaEmail author
  • Nancy Lynch
  • Frederik Mallmann-Trenn
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11639)


In the French flag problem, initially uncolored cells on a grid must differentiate to become blue, white or red. The goal is for the cells to color the grid as a French flag, i.e., a three-colored triband, in a distributed manner. To solve a generalized version of the problem in a distributed computational setting, we consider two models: a biologically-inspired version relying on morphogens (gradients of chemicals acting as signals) and a more abstract version based on reliable message passing between cellular agents. We show that both models easily achieve a French ribbon - a French flag in the 1D case. However, extending the ribbon to the 2D flag in the concentration model is somewhat difficult unless each agent has additional positional information. Assuming that cells are are identical, it is impossible to achieve a French flag or even a close approximation. In contrast, using a message-based approach in the 2D case only requires assuming that agents can be represented as constant size state machines. We hope our insights may lay some groundwork for what kind of message passing abstractions or guarantees are useful in analogy to cells communicating at long and short distances to solve patterning problems. In addition, we hope that our models and findings may be of interest in the design of nano-robots.


  1. 1.
    Ancona, A., Bajwa, A., Lynch, N., Mallmann-Trenn, F.: How to color a French flag-biologically inspired algorithms for scale-invariant patterning. arXiv preprint arXiv:1905.00342 (2019)
  2. 2.
    Flajolet, P.: Approximate counting: a detailed analysis. BIT 25(1), 113–134 (1985)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Nüsslein-Volhard, C., Wieschaus, E.: Mutations affecting segment number and polarity in drosophila. Nature 287(5785), 795–801 (1980)CrossRefGoogle Scholar
  4. 4.
    Wolpert, L.: Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25(1), 1–47 (1969)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Bertie Ancona
    • 1
  • Ayesha Bajwa
    • 1
    Email author
  • Nancy Lynch
    • 1
  • Frederik Mallmann-Trenn
    • 1
  1. 1.Massachusetts Institute of TechnologyCambridgeUSA

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