Soliton approximation in continuum models of leader-follower behavior.

Area of Science:

• Computational Biology
• Mathematical Biology
• Developmental Biology

Background:

• Complex biological systems exhibit emergent collective behavior across multiple scales.
• Modeling approaches include discrete agent-based and continuum methods.
• Angiogenesis involves tip and stalk cell dynamics guiding new blood vessel formation.

Purpose of the Study:

• To analyze hybrid stochastic agent-based models of branching morphogenesis and angiogenesis.
• To investigate the relationship between discrete cell-level dynamics and continuum partial differential equations (PDEs).
• To identify common features and provide insights into biological mechanisms governing collective cell dynamics.

Main Methods:

• Development of hybrid stochastic agent-based models for angiogenesis.
• Coarse-graining hybrid models to derive continuum PDEs for endothelial cell densities.
• Mathematical analysis of the leading-order PDE system in a quasi-one-dimensional geometry.
• Numerical solution of the PDE and identification of emergent wave phenomena.

Main Results:

• Hybrid models can be coarse-grained into continuum PDEs.
• At leading order, particularly with dominant chemotaxis, discrete and continuum models yield similar equations.
• Numerical solutions of the leading-order PDE are well-described by a soliton wave.
• This soliton wave acts as an attractor, guiding cell density towards growth factor sources.

Conclusions:

• Hybrid modeling approaches offer a unified framework for studying complex biological processes like angiogenesis.
• The emergent soliton wave behavior provides insights into the collective dynamics of endothelial cells.
• Mathematical analysis reveals commonalities between discrete and continuum modeling strategies in biology.