The leopard never changes its spots: realistic pigmentation pattern formation by coupling tissue growth with reaction-diffusion

Previous research in pattern formation using reaction-diffusion mostly focused on static domains, either for computational simplicity or mathematical tractability. In this work, we have explored the expressiveness of combining simple mechanisms as a possible explanation for pigmentation pattern formation, where tissue growth plays a crucial role.

Our motivation is not only to realistically reproduce natural patterns but also to get insights into the underlying biological processes. Therefore, we present a novel approach to generate realistic animal skin patterns. First, we describe the approximation of tissue growth by a series of discrete matrix expansion operations. Then, we combine it with an adaptation of Turing’s non-linear reaction-diffusion model, which enforces upper and lower bounds to the concentrations of the involved chemical reagents. We also propose the addition of a single-reagent continuous autocatalytic reaction, called reinforcement, to provide a mechanism to maintain an already established pattern during growth.

By careful adjustment of the parameters and the sequencing of operations, we closely match the appearance of a few real species. In particular, we reproduce in detail the distinctive features of the leopard skin, also providing a hypothesis for the simultaneous productions of the most common melanin types, eumelanin and pheomelanin.

@article {Malheiros2020,
  title   = {The leopard never changes its spots: realistic pigmentation pattern formation
             by coupling tissue growth with reaction-diffusion},
  author  = {Malheiros, Marcelo de G. and Fensterseifer, Henrique and Walter, Marcelo},
  year = {2020},
  issue_date = {July 2020},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA},
  volume = {39},
  number = {4},
  issn = {0730-0301},
  url = {https://doi.org/10.1145/3386569.3392478},
  doi = {10.1145/3386569.3392478},
  journal = {ACM Trans. Graph.},
  month = jul,
  articleno = {63},
  numpages = {13},
  keywords = {natural phenomena, texturing, pattern formation, reaction-diffusion, Turing model}
}