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Control of reaction-diffusion equations on time-evolving manifolds.

Francesco Rossi1, Nastassia Pouradier Duteil2, Nir Yakoby3

  • 1Aix Marseille Université, CNRS, ENSAM, Université de Toulon, LSIS UMR 7296, 13397, Marseille, France.

Proceedings of the ... IEEE Conference on Decision & Control. IEEE Conference on Decision & Control
|October 14, 2017
PubMed
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This study models how morphogen diffusion and organism shape changes are coupled during development. We reveal a mathematical non-commutativity between transport and diffusion processes, impacting growth signals.

Area of Science:

  • Developmental biology
  • Mathematical modeling
  • Biophysics

Background:

  • Morphogens are key signaling molecules regulating organism development.
  • Morphogen distribution influences cell responses and growth.
  • Organism shape changes dynamically affect morphogen diffusion.

Purpose of the Study:

  • To introduce a mathematical model investigating the coupling between morphogen diffusion and organism shape dynamics.
  • To analyze the interplay between signal diffusion and evolving biological structures.

Main Methods:

  • Utilizing a mathematical framework where organism shape is a time-varying manifold.
  • Modeling signal diffusion using a diffusion equation on the manifold.
  • Employing a transport equation to describe shape deformation.

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Main Results:

  • Demonstrating the non-commutativity of transport (shape change) and diffusion (signal spread) evolutions.
  • Introducing a novel concept of Lie bracket to quantify this interaction.
  • Presenting numerical simulations that illustrate the non-commutative phenomenon.

Conclusions:

  • The interaction between morphogen diffusion and changing organism shape is complex and non-commutative.
  • This coupling significantly influences developmental signaling and growth patterns.
  • The developed mathematical model provides new insights into developmental dynamics.