Abstract
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 that relies on morphogens (diffusing proteins acting as chemical signals) and a more abstract version based on reliable message passing between cellular agents.
Much of developmental biology research focuses on concentration-based approaches, since morphogen gradients are an underlying mechanism in tissue patterning. We show that both model types 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 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 logarithmic or constant size state machines.
We hope that our insights may lay some groundwork for what kind of message passing abstractions or guarantees, if any, may be useful in analogy to cells communicating at long and short distances to solve patterning problems. We also hope our models and findings may be of interest in the design of nano-robots.
The authors were supported in part by NSF Award Numbers CCF-1461559 and CCF-0939370.
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Notes
- 1.
We note that a similar algorithm may use a single token rather than binary messages, at an additional constant-factor increase in round complexity.
References
Afek, Y., Alon, N., Barad, O., Hornstein, E., Barkai, N., Bar-Joseph, Z.: A biological solution to a fundamental distributed computing problem. Science 331(6014), 183–185 (2011)
Alberts, B.: Molecular Biology of the Cell. CRC Press, Boca Raton (2017)
Ancona, A., Bajwa, A., Lynch, N., Mallmann-Trenn, F.: How to color a french flag-biologically inspired algorithms for scale-invariant patterning. arXiv e-prints p. 1905.00342 (2019)
Dessaud, E., McMahon, A.P., Briscoe, J.: Pattern formation in the vertebrate neural tube: a sonic hedgehog morphogen-regulated transcriptional network. Development 135(15), 2489–2503 (2008)
Driever, W., Nüsslein-Volhard, C.: A gradient of bicoid protein in drosophila embryos. Cell 54(1), 83–93 (1988)
Flajolet, P.: Approximate counting: a detailed analysis. BIT 25(1), 113–134 (1985)
Green, J.B.A., Sharpe, J.: Positional information and reaction-diffusion: two big ideas in developmental biology combine. Development 142(7), 1203–1211 (2015)
Gregor, T., Bialek, W., van Steveninck, R.R.D.R., Tank, D.W., Wieschaus, E.F.: Diffusion and scaling during early embryonic pattern formation. Proc. Natl. Acad. Sci. 102(51), 18403–18407 (2005)
Habermann, N.: Parallel neighbor-sort (or the glory of the induction principle). Carnegie-Mellon University, Technical report (1972)
Jaeger, J., Martinez-Arias, A.: Getting the measure of positional information. PLoS Biol. 7(3), e1000081 (2009)
Miller, J.F.: Evolving a self-repairing, self-regulating, french flag organism. In: Deb, K. (ed.) GECCO 2004. LNCS, vol. 3102, pp. 129–139. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24854-5_12
Morgan, T.H.: “Polarity” considered as a phenomenon of gradation of materials. J. Exp. Zool. 2, 495–506 (1905)
Morris, R.: Counting large numbers of events in small registers. Commun. ACM 21(10), 840–842 (1978)
Nüsslein-Volhard, C., Wieschaus, E.: Mutations affecting segment number and polarity in drosophila. Nature 287(5785), 795–801 (1980)
Patten, I., Placzek, M.: The role of sonic hedgehog in neural tube patterning. Cellular Molecular Life Sci. CMLS 57(12), 1695–1708 (2000)
Stumpf, H.F.: Mechanism by which cells estimate their location within the body. Nature 212(5060), 430–431 (1966)
Summerbell, D., Lewis, J.H., Wolpert, L.: Positional information in chick limb morphogenesis. Nature 244(5417), 492–496 (1973)
Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Biol. Sci. 237(641), 37–72 (1952)
Wolpert, L.: The french flag problem: a contribution to the discussion on pattern development and regulation. Towards Theoret. Biol. 1, 125–133 (1968)
Wolpert, L.: Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25(1), 1–47 (1969)
Zadorin, A.S., et al.: Synthesis and materialization of a reaction–diffusion french flag pattern. Nat. Chem. 9(10), 990–996 (2017)
Acknowledgements
We thank Ama Koranteng, Adam Sealfon, and Vipul Vachharajani for valuable discussions and contributions.
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Ancona, B., Bajwa, A., Lynch, N., Mallmann-Trenn, F. (2020). How to Color a French Flag. In: Kohayakawa, Y., Miyazawa, F.K. (eds) LATIN 2020: Theoretical Informatics. LATIN 2021. Lecture Notes in Computer Science(), vol 12118. Springer, Cham. https://doi.org/10.1007/978-3-030-61792-9_33
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