Free boundary problem for cell protrusion formations: theoretical and numerical aspects

Area of Science:

• Mathematical Biology
• Cell Biology
• Biophysics

Background:

• Cell protrusion formation is crucial for cell migration and invasion.
• Understanding the chemical signaling pathways driving these processes is essential.
• Existing models may not fully capture the complex interplay of forces and chemical signals.

Purpose of the Study:

• To develop and analyze a free boundary problem for modeling cell protrusion formation.
• To investigate the role of chemical interactions and actin polymerization in invadopodia and pseudopodia.
• To provide a theoretical and numerical framework for single-cell migration.

Main Methods:

• Utilized a level set function to describe the cell membrane.
• Modeled using coupled Laplace equations with specific boundary conditions.
• Developed a first-order Cartesian finite-difference method for numerical solutions.
• Proved well-posedness of the free boundary problem under a sign condition.

Main Results:

• Demonstrated the model's ability to reproduce distinct protrusion types (invadopodia, pseudopodia) based on source terms.
• Established the consistency and validity of the free boundary problem.
• The numerical method effectively solves the model.
• The model is applicable in both 2D and 3D.

Conclusions:

• The developed free boundary model provides a robust framework for studying cell protrusion dynamics.
• The model successfully links chemical signaling and actin polymerization to protrusion morphology.
• This work lays the foundation for future research into specific cell membrane chemical reactions and advanced cell migration modeling.