Phenotype structuring in collective cell migration: a tutorial of mathematical models and methods.

Area of Science:

• Mathematical Biology
• Computational Biology
• Population Dynamics

Background:

• Phenotypic diversity, the range of traits within a population, critically influences population-level dynamics.
• Cellular heterogeneity in traits like signaling, movement, and growth affects cancer progression and treatment response.

Purpose of the Study:

• To review and extend classical partial differential equation (PDE) models to incorporate phenotypic structuring.
• To introduce phenotype-structured partial differential equations (PS-PDEs) for modeling complex population dynamics.

Main Methods:

• Reviewing extensions of classic models (Fisher-KPP, Keller-Segel) into PS-PDEs.
• Presenting methods for deriving PS-PDEs from agent-based models.
• Analyzing traveling waves and concentration phenomena in PS-PDEs.
• Discussing numerical simulation techniques for PS-PDEs.

Main Results:

• PS-PDEs offer a sophisticated framework for modeling populations with phenotypic diversity.
• Phenotypic structuring can be deduced across traveling waves in PS-PDE models.
• Numerical methods, like the method of lines, can simulate PS-PDEs.

Conclusions:

• PS-PDEs provide a powerful tool for understanding how phenotypic diversity shapes population dynamics.
• Future research directions and mathematical challenges in PS-PDE modeling are highlighted.