Going from microscopic to macroscopic on nonuniform growing domains

Area of Science:

• Mathematical Biology
• Developmental Biology
• Computational Biology

Background:

• Chemical cues (morphogens) guide tissue development and their distribution is influenced by tissue growth.
• Partial differential equations (PDEs) are commonly used to model morphogen dynamics.
• Previous work established a link between discrete stochastic and continuum models for particle migration on growing domains.

Purpose of the Study:

• To develop a more physically realistic model for domain growth in biological systems.
• To establish the equivalence between individual-based stochastic models and PDE models on nonuniformly partitioned domains.
• To derive and validate a PDE model for particle density on growing domains with incremental growth.

Main Methods:

• Developed an incremental domain growth model with element splitting upon reaching a threshold size.
• Utilized the master equation formalism to derive a PDE for particle density.
• Demonstrated the equivalence of stochastic and PDE models on nonuniform domains.
• Corroborated derived models using numerical simulations.

Main Results:

• An individual-based stochastic model on a nonuniform domain partition is equivalent to a PDE model on a static domain.
• Derived transition rates for particle migration on nonuniformly partitioned domains.
• Successfully derived a PDE for particle density on a growing domain with incremental growth.
• Numerical simulations confirmed the accuracy of the derived PDE model.

Conclusions:

• The developed incremental growth model provides a more physically accurate representation of tissue development.
• The study bridges discrete stochastic and continuum modeling approaches for particle dynamics on growing domains.
• The derived PDE offers a powerful tool for analyzing morphogen distribution and tissue functional specification during development.