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Reaction-diffusion systems and external morphogen gradients: the two-dimensional case, with an application to

Tilmann Glimm¹, Jianying Zhang, Yun-Qiu Shen

¹Department of Mathematics, Western Washington University, Bellingham, WA 98225, USA. glimmt@wwu.edu

Bulletin of Mathematical Biology

|October 13, 2011

Summary

This study explores reaction-diffusion systems with morphogen gradients, revealing how these gradients influence pattern formation like spots and stripes in two dimensions. The findings offer new insights into skeletal development, particularly in mouse limbs.

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Area of Science:

Mathematical Biology
Developmental Biology
Pattern Formation

Background:

Reaction-diffusion systems with activator-inhibitor dynamics can generate spatial patterns like spots and stripes via Turing bifurcation.
External morphogen gradients introduce space-dependent terms into these reaction-diffusion equations.
Previous studies primarily focused on one-dimensional systems or lacked explicit morphogen gradients.

Purpose of the Study:

To investigate pattern formation in a two-dimensional reaction-diffusion system with an external morphogen gradient.
To derive analytical expressions for pattern formation under morphogen influence.
To apply these findings to understand skeletal pattern formation in vertebrate limbs.

Main Methods:

Mathematical modeling using reaction-diffusion equations with space-dependent terms.
Derivation of first-order expansions for pattern formation in the presence of a morphogen gradient.
Analysis of both quasi-one-dimensional (thin rectangular) and two-dimensional (square) domains.
Application to a specific biological model of limb development.

Main Results:

The presence of a morphogen gradient modifies the conditions for Turing instability, affecting both stripe and spot formation.
Analytical expansions predict how morphogen gradients alter the characteristics of emerging patterns.
The study successfully generalizes one-dimensional findings to two-dimensional systems.
The model provides a potential explanation for experimental observations in mouse limb skeletal patterning.

Conclusions:

Reaction-diffusion systems with morphogen gradients offer a robust framework for understanding complex pattern formation.
The derived analytical results provide a mechanistic explanation for observed patterns in biological systems, such as vertebrate limb development.
This approach simplifies explanations for experimental findings that are challenging for traditional positional-information models.