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A living mesoscopic cellular automaton made of skin scales

Nature 544, 173–179 (13 April 2017) | Download Citation

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Abstract

In vertebrates, skin colour patterns emerge from nonlinear dynamical microscopic systems of cell interactions. Here we show that in ocellated lizards a quasi-hexagonal lattice of skin scales, rather than individual chromatophore cells, establishes a green and black labyrinthine pattern of skin colour. We analysed time series of lizard scale colour dynamics over four years of their development and demonstrate that this pattern is produced by a cellular automaton (a grid of elements whose states are iterated according to a set of rules based on the states of neighbouring elements) that dynamically computes the colour states of individual mesoscopic skin scales to produce the corresponding macroscopic colour pattern. Using numerical simulations and mathematical derivation, we identify how a discrete von Neumann cellular automaton emerges from a continuous Turing reaction–diffusion system. Skin thickness variation generated by three-dimensional morphogenesis of skin scales causes the underlying reaction–diffusion dynamics to separate into microscopic and mesoscopic spatial scales, the latter generating a cellular automaton. Our study indicates that cellular automata are not merely abstract computational systems, but can directly correspond to processes generated by biological evolution.

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Acknowledgements

We thank A. Debry and F. Montange for technical assistance with animals and B. Chopard for comments on the manuscript. A. Martins advised on R2OBBIE scans. This work was supported by grants to M.C.M. from the University of Geneva (Switzerland), the Swiss National Science Foundation (FNSNF, grants 31003A_140785 and SINERGIA CRSII3_132430), and the SystemsX.ch initiative (project EpiPhysX). S.S. was supported by the ERC AG COMPASP, the FNSNF, the NCCR SwissMAP and the Russian Science Foundation.

Author information

    • Liana Manukyan
    • , Sophie A. Montandon
    •  & Anamarija Fofonjka

    These authors contributed equally to this work.

Affiliations

  1. Laboratory of Artificial and Natural Evolution (LANE), Department of Genetics and Evolution, University of Geneva, Geneva, Switzerland

    • Liana Manukyan
    • , Sophie A. Montandon
    • , Anamarija Fofonjka
    •  & Michel C. Milinkovitch

  2. SIB Swiss Institute of Bioinformatics, Geneva, Switzerland

    • Liana Manukyan
    • , Anamarija Fofonjka
    •  & Michel C. Milinkovitch

  3. Department of Mathematics, University of Geneva, Geneva, Switzerland

    • Stanislav Smirnov

  4. Skolkovo Institute of Science and Technology, Skolkovo, Russia

    • Stanislav Smirnov

  5. Chebyshev Laboratory, Department of Mathematics and Mechanics, St Petersburg State University, St Petersburg, Russia

    • Stanislav Smirnov

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Contributions

M.C.M. initiated the ocellated lizard breeding colony, identified the CA behaviour and proposed that the CA emerges from the superposition of skin geometry with a continuous RD system. S.A.M. performed 3D scanning and histology. S.A.M. and L.M. performed 3D geometry and colour texture reconstructions. L.M. and S.A.M. performed the alignments among 3D scans and the colour assignment of scales. L.M. and M.C.M. performed the statistical analyses and numerical modelling on real lizard lattices (all code written by L.M.). A.F., S.S. and M.C.M. performed the analyses and simulations (all code written by A.F.) on hexagonal lattices. S.S. proposed the discrete RD model, performed the mathematical derivation of discrete RD parameters from the continuous RD models and advised on numerical simulations. A.F. proposed the CA probability numerical derivation. M.C.M. supervised the whole study and wrote the manuscript. All authors agreed on the interpretation of data and approved the final version of the manuscript.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Michel C. Milinkovitch.

Nature thanks L. Edelstein-Keshet, T. Miura, C. Tarnita and T. Woolley 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.

Extended data

Extended data figures

  1. 1.

    Scale detection, positioning and identification of neighbourhood

  2. 2.

    Derivation of the scale colour change probability function.

  3. 3.

    The 14 possible first-ring neighbourhood states for a green scale in an hexagonal lattice.

  4. 4.

    Interactions between melanophores (M) and xanthophores (X).

  5. 5.

    Switching of scale colour.

  6. 6.

    Skin colour pattern and scale size.

Extended data tables

  1. 1.

    Eleven time points (TP0 to TP10), corresponding to age, expressed in weeks after hatching, at which three male ocellated lizards (LL09, LL10 and LL16) were scanned for colour texture and 3D reconstruction

Supplementary information

Videos

  1. 1.

    Time evolution of an ocellated lizard over three years of post-hatching development — real colours.

    The juvenile pattern is made of white ocelli on a brown background and develops into a labyrinthine pattern of green and black scales. Four examples of scales switching colour are shown: blue circles, from green to black; green circle, from black to green, light-blue circle, from green to black to green.

  2. 2.

    Time evolution of a male ocellated lizard over three years of post-hatching development — scale-colour switching.

    Time evolution for the same individual as in Supplementary Video 1 but after scale colour assignment (see Methods).

  3. 3.

    Cellular Automaton (CA)

    Time evolution simulation of an ocellated lizard skin pattern using a CA made of skin scales. The CA probability distribution has been inferred from actual time series of colour changes in real ocellated lizards.

  4. 4.

    Continuous reaction diffusion (cRD) simulation.

    Time evolution of the colour pattern on a hexagonal lattice using a cRD model. Discretisation is such that each hexagon (skin scale) contains about 300 elements. Diffusion is reduced specifically along the scale boundaries.

  5. 5.

    Continuous reaction diffusion (cRD) simulations generate a scale-colour switching behaviour

    The colour of each element (pixel) is a continuous variable that can take any value between ‘black’ and ‘green’. Right panel, colour evolution of three scales; left panel, the corresponding patch of skin. The scales marked by a red or yellow dot switch from green to black or black to green, respectively. The scale marked by a blue dot starts to switch to black but then reverts to green because most of its neighbours become black.

  6. 6.

    Discrete reaction diffusion (dRD) simulation

    Time evolution of the colour pattern on a hexagonal lattice using a dRD model. The entire reptile scales (hexagons) are used as discretisation units. Colour change is occurring according to RD equations.

  7. 7.

    Discrete reaction diffusion (dRD) simulations generate a scale-colour switching behaviour

    The colour of each hexagon is a continuous variable that can take any value between ‘black’ and ‘green’. Right panel, colour evolution of three scales; left panel, the corresponding patch of skin. The scales marked by a red or yellow dot switch from green to black or black to green, respectively. The scale marked by a blue dot starts to switch to black but then reverts to green because most of its neighbours become black.

  8. 8.

    Cellular Automaton (CA) simulation

    Time evolution of the colour pattern on a hexagonal lattice using a CA whose elements are the hexagons. The CA probability distribution has been inferred from the dynamic of colour changes in dRD simulations. Iterations are shown to indicate that the many colour changes occurring during the first iteration have been distributed across multiple frames of the video.

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https://doi.org/10.1038/nature22031

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