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From geometric optics to plants: the eikonal equation for buckling
Sergei Nechaev1, Kirill Polovnikov2
1J.-V. Poncelet Laboratory, CNRS, UMI 2615, 119002 Moscow, Russia. sergei.nechaev@gmail.com and P.N. Lebedev Physical Institute, RAS, 119991 Moscow, Russia.
Soft Matter
|January 26, 2017
Summary
This study explores optimal tissue buckling in growing plant leaves using a conformal approach. The research reveals that tissue growth and boundary profiles are governed by the 2D eikonal equation, influenced by local cell division protocols.
Area of Science:
- * Mathematical modeling
- * Developmental biology
- * Biophysics
Background:
- * Growing tissues, like plant leaves, exhibit complex buckling phenomena.
- * Understanding the geometric principles governing tissue growth is crucial for developmental biology.
Purpose of the Study:
- * To investigate the optimal buckling of exponentially growing tissues.
- * To establish a mathematical framework describing tissue boundary formation.
Main Methods:
- * Application of a conformal approach to model tissue growth.
- * Utilizing the 2D eikonal equation to describe boundary profiles.
- * Relating surface geometry to local growth protocols.
Main Results:
- * The boundary profile of a growing tissue follows the 2D eikonal equation.
- * This equation acts as a geometric optic approximation for wave propagation.
- * The spatial variation of the refraction coefficient dictates the emergent 3D surface geometry.
Conclusions:
- * The conformal approach provides insights into optimal tissue buckling.
- * Local growth protocols directly influence the macroscopic shape of developing tissues.
- * The eikonal equation is a powerful tool for understanding growth-driven morphogenesis.
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