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Stripes, spots, or reversed spots in two-dimensional Turing systems.

Hiroto Shoji1, Yoh Iwasa, Shigeru Kondo

  • 1Mathematical Biology Laboratory, Department of Biology, Kyushu University, 812-8581 Fukuoka-shi, Japan. shoji@bio-math10.biology.kyushu-u.ac.jp

Journal of Theoretical Biology
|August 28, 2003
PubMed
Summary

Turing models explain animal body patterns. Constraining activator levels in these models dictates whether striped or spotted patterns emerge, offering insights into pattern formation.

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Area of Science:

  • Developmental Biology
  • Mathematical Biology
  • Theoretical Ecology

Background:

  • Turing models are fundamental to understanding animal body pattern formation, generating striped or spotted patterns.
  • The selection between these patterns is influenced by the reaction terms within the models.

Purpose of the Study:

  • To investigate how reaction terms affect pattern selection in two-dimensional Turing models.
  • To analyze the role of constraint terms in determining pattern outcomes.
  • To relate pattern selection in nonlinear models to findings from linear models with constraints.

Main Methods:

  • Studied a one-dimensional model with linear reaction terms and constraint terms.
  • Examined a two-dimensional model with linear reaction terms and constraint terms.

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  • Analyzed nonlinear reaction term models (activator-inhibitor, activator-depletion substrate).

    • Main Results:

    • In 1D, periodic patterns require both lower and upper activator level constraints.
    • In 2D, pattern selection depends on the activator equilibrium's position relative to constraints.
    • Striped patterns form when equilibrium is equidistant from constraints; spotted patterns form when closer to one constraint.

    Conclusions:

    • The position of the activator equilibrium relative to constraints is key to pattern selection in Turing models.
    • Skewness of the activator level distribution differentiates striped (near zero skew) from spotted (positive/negative skew) patterns.
    • Findings provide a heuristic explanation for pattern selection based on constraint interactions.