Free boundary morphogenesis in living matter.

Area of Science:

• * Developmental Biology
• * Biophysics
• * Mathematical Biology

Background:

• * Morphogenesis involves the creation and emergence of biological form.
• * Free boundary problems are crucial for understanding growth and development in biological systems.
• * Existing models often simplify complex mass transport dynamics during morphogenesis.

Purpose of the Study:

• * To propose a novel approach for studying free boundary problems in morphogenesis.
• * To investigate the role of mass fluxes and morphogen gradients in biological form.
• * To analyze the contour stability of growing materials using a simplified model.

Main Methods:

• * Development of a two-dimensional system model within a Hele-Shaw cell.
• * Coupling of mass fluxes with local morphogen gradients.
• * Analysis of contour stability through perturbation of a rectilinear free boundary.
• * Derivation of a dispersion relation to characterize system dynamics.

Main Results:

• * A model for mass transport during morphogenesis was established, linking traveling wave velocity to a dimensionless parameter.
• * The dispersion relation was derived, indicating complex stability behaviors.
• * Both moving front velocity and surface tension stabilize the system at small wavelengths.
• * The rectilinear border was found to be unstable at large wavelengths.

Conclusions:

• * The proposed model provides a framework for analyzing free boundary problems in biological systems.
• * Understanding mass transport dynamics is key to predicting the stability of biological forms.
• * The findings highlight the inherent instability of rectilinear borders in growing biological materials at larger scales.
• * This work offers potential insights into diverse morphogenetic processes and free boundary phenomena.