How the zebra got its stripes: Curvature-dependent diffusion orients Turing patterns on three-dimensional surfaces

Area of Science:

• Mathematical Biology
• Developmental Biology
• Computational Modeling

Background:

• Animals exhibit diverse patterns (stripes, spots) generated by biological processes.
• Reaction-diffusion systems, like Turing models, simulate pattern formation.
• Controlling pattern orientation in these models remains a challenge.

Purpose of the Study:

• To propose a novel mechanism for orienting patterns in reaction-diffusion systems.
• To investigate the role of surface geometry, specifically curvature, in pattern orientation.
• To simulate and validate the proposed mechanism on animal models.

Main Methods:

• Developed a reaction-diffusion model incorporating surface curvature.
• Modified anisotropic diffusion rates based on local surface curvature.
• Performed numerical simulations using models of zebra and cat patterns.

Main Results:

• The model successfully recaptured the correct orientation of stripes on simulated animals.
• Local surface curvature was shown to influence diffusion rates and global pattern formation.
• Demonstrated robust pattern generation controlled by local geometric information.

Conclusions:

• Surface curvature is a key factor in orienting reaction-diffusion patterns.
• This model provides a mechanism for global pattern control using local geometric cues.
• Offers an alternative to morphogen gradients for conveying shape and positioning information in development.