The pseudo-compartment method for coupling partial differential equation and compartment-based models of diffusion.

Area of Science:

• Computational Biology
• Mathematical Modeling
• Biophysics

Background:

• Spatial reaction-diffusion models are crucial for understanding biological phenomena.
• Traditional partial differential equation (PDE) models assume high particle densities, limiting their application in low-concentration regions.
• Individual-based models (IBMs) offer accuracy for stochastic effects but are computationally intensive.

Purpose of the Study:

• To develop and validate hybrid algorithms coupling PDE and compartment-based models for reaction-diffusion systems.
• To address the challenge of particle transport between regions with different modeling assumptions (continuum vs. stochastic).
• To improve the efficiency of simulating systems with both high and low particle concentrations.

Main Methods:

• Development of two hybrid algorithms integrating PDE and compartment-based models.
• Algorithm 1: Redefining continuous PDE concentration as a probability distribution for particle transport.
• Algorithm 2: A simplified, more efficient continuum-limit implementation of Algorithm 1.

Main Results:

• Hybrid algorithms accurately simulate particle transport between different model descriptions.
• Algorithm 1 demonstrates strong convergence to analytical PDE solutions.
• Algorithm 2 provides a more computationally efficient alternative with comparable accuracy.

Conclusions:

• The developed hybrid methods offer a robust and efficient approach for simulating reaction-diffusion systems with multi-scale concentrations.
• These algorithms provide a more fundamental understanding of particle movement across model scales.
• The findings advance computational strategies for complex biological modeling.