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Related Experiment Videos

Labyrinthine versus straight-striped patterns generated by two-dimensional Turing systems.

Hiroto Shoji1, Yoh Iwasa

  • 1Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan. shoji@yukawa.kyoto-u.ac.jp

Journal of Theoretical Biology
|June 7, 2005
PubMed
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Reaction-diffusion models explain fish skin patterns. Straight stripes form with narrow unstable spatial periods, while labyrinthine patterns emerge from wide unstable spatial periods, a finding consistent across different models.

Area of Science:

  • Mathematical Biology
  • Pattern Formation

Background:

  • Striped patterns on fish skin are frequently observed and often explained using reaction-diffusion (RD) Turing-type models.
  • These models involve two substances interacting to create spatial patterns from a homogeneous field.
  • RD models can generate both straight-striped and labyrinthine patterns, with variations in their tendency to produce one over the other.

Purpose of the Study:

  • To investigate the specific conditions that lead to the emergence of either labyrinthine or straight-striped patterns.
  • To define a quantitative index for stripe clearness (Sh) to differentiate between pattern types.

Main Methods:

  • Definition of a stripe clearness index (Sh) based on the range of unstable spatial periods.
  • Analysis of reaction-diffusion models with nonlinear reaction terms, including activator-inhibitor and substrate-depletion models.

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Main Results:

  • Straight-striped patterns (high Sh) occur when only a narrow range of spatial periods corresponds to an unstable mode.
  • Labyrinthine patterns (low Sh) arise when a wide range of spatial periods is unstable.
  • Specifically, labyrinthine patterns form if the maximum unstable spatial period is more than twice the minimum unstable spatial period; otherwise, straight-striped patterns form.

Conclusions:

  • The range of unstable spatial periods is the key determinant for generating straight-striped versus labyrinthine patterns in RD models.
  • These findings hold true for different types of nonlinear reaction terms within the RD models.
  • The study provides a clear criterion for predicting pattern morphology in biological systems modeled by reaction-diffusion dynamics.