Statics and diffusive dynamics of surfaces driven by p-atic topological defects

Area of Science:

• Mathematical Biology
• Soft Matter Physics
• Surface Geometry

Background:

• Epithelial morphogenesis involves complex surface shaping processes.
• Topological defects play a crucial role in biological and physical systems.
• Understanding the interplay between intrinsic and extrinsic geometry is key.

Purpose of the Study:

• To develop a minimal model for surface shaping driven by p-atic topological defects.
• To investigate the dynamic generation and evolution of conical shapes from defects.
• To predict the final shape and geometric properties of membranes with embedded polar order.

Main Methods:

• Utilized a minimal mathematical model for surface dynamics.
• Analyzed the behavior of positive and negative topological defects.
• Exploited the coupling between extrinsic and intrinsic geometry for axisymmetric surfaces.

Main Results:

• Positive (negative) defects dynamically generate (hyperbolic) cones with diffusive shape evolution.
• A defect of charge +1/p predicts a final semi-cone angle β satisfying a specific inequality.
• Stationary membranes with negligible bending modulus and polar order form deformed lemon shapes with antipodal defects.

Conclusions:

• The study provides a theoretical framework for defect-driven surface morphogenesis.
• Results offer insights into the formation of conical and lemon-like surface geometries.
• Findings may extend to shape transitions in other closed spheroidal surfaces, like pollen grains.