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A living mesoscopic cellular automaton made of skin scales

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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Figure 1: Colour pattern ontogeny in ocellated lizards.
Figure 2: Different combinations of chromatophores generate different scale colours.
Figure 3: Neighbourhood distributions of real and random patterns.
Figure 4: Scale colour changes follow a probabilistic CA.
Figure 5: Sensitivity analysis.
Figure 6: From RD to CA.

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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

Authors and Affiliations

Authors

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.

Corresponding author

Correspondence to Michel C. Milinkovitch.

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The authors declare no competing financial interests.

Additional information

Nature thanks L. Edelstein-Keshet, T. Miura, C. Tarnita and T. Woolley for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Figure 1 Scale detection, positioning and identification of neighbourhood

. a, From left to right: the 3D high-resolution scan allows the curvature of the animal surface to be measured; curvature thresholding yields domains whose centroids define the positions of scales; and the latter are used to build the Voronoi diagram and its corresponding Delaunay triangulation graph that identifies the neighbourhood of all scales. b, Each scale position is refined by moving it to the centroid of the corresponding polygon of smallest curvature (a 2D simplified representation is shown in the left panel; the refined position is at equal distances d from the two flanking points of smallest curvature).

Extended Data Figure 2 Derivation of the scale colour change probability function.

a, Four examples of exponential (red) and cubic (green) fits of the raw data (black plain line) for green scales; the mean square difference per point is generally lower for the cubic than the exponential fit. The graph labelled (I) is an example of a time point for which the value for 6 neighbours is not available, so this value is estimated with the fitted curve. b, On the left is a polynomial cubic fit of the 13 colour change probability distributions corresponding to three different individuals (blue, red and green curves) at different time points; the four curves labelled with Roman numbers correspond to the four graphs in a; all values are normalized with respect to the highest probability. In the middle is a normalized colour change probability distribution; each of the curves is normalized such that the sum of probabilities is 1. On the right the 13 normalized curves define a mean (±s.d.) colour change probability distribution.

Extended Data Figure 3 The 14 possible first-ring neighbourhood states for a green scale in an hexagonal lattice.

Assuming rotational isotropy (counting symmetrical cases as one case), the central green scale can have one state with one black neighbour (a), three states with two black neighbours (b), four states with three black neighbours (c), three states with four black neighbours (d), or one state with five or zero or six black neighbours (e, f and g).

Extended Data Figure 4 Interactions between melanophores (M) and xanthophores (X).

The figure represents the model developed in ref. 9. Melanophores and xanthophores interact negatively with each other at short range (green arrows). W is the long-range inhibitor that affects melanophores (blue dashed arrow) but not xanthophores (light blue dashed arrow and null parameter c5). W is modulated (red dotted arrows) positively by melanophores and negatively by xanthophores. c1, c2, c4, c5, c7 and c8 represent variables in the partial differential equations (see Methods).

Extended Data Figure 5 Switching of scale colour.

Density plots of skin scale colour (across all scales) for continuous (a) and discrete (b) RD simulations. For the continuous model, the colour of a hexagonal scale is computed as the mean among the (u − v)/max(u, v) values of all elements (pixels) in that hexagon. Both models generate a scale colour switching behaviour. Continuous RD simulations require smaller time steps than discrete RD, hence the higher number of iterations in the former.

Extended Data Figure 6 Skin colour pattern and scale size.

a, RD simulations with reduced diffusion at the scale borders show that a pattern appears within large-enough scales (magnification factors are indicated). b, This prediction is confirmed by the absence of a colour pattern within the ocellated lizard body scales (left panel) but the presence of a colour pattern within most of the large tail scales (right panel).

Extended Data Table 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

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. (MOV 2718 kb)

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). (MOV 376 kb)

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. (MOV 362 kb)

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. (MP4 1068 kb)

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. (MP4 847 kb)

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. (MP4 941 kb)

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. (MP4 989 kb)

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. (MP4 672 kb)

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Manukyan, L., Montandon, S., Fofonjka, A. et al. A living mesoscopic cellular automaton made of skin scales. Nature 544, 173–179 (2017). https://doi.org/10.1038/nature22031

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