The emergence of spatial patterns for compartmental reaction kinetics coupled by two bulk diffusing species with...

Area of Science:

• Chemical kinetics
• Mathematical biology
• Nonlinear dynamics

Background:

• Turing's theory predicts spatial patterns from uniform states via diffusion and nonlinear reactions.
• Stable patterns typically require a large diffusivity ratio between activator and inhibitor species.
• This large diffusivity ratio is often biologically unrealistic.

Purpose of the Study:

• Investigate pattern formation in biologically realistic scenarios with comparable diffusion rates.
• Explore pattern emergence in a one-dimensional bulk-compartment model.
• Address the challenge of pattern formation when diffusion timescales are similar.

Main Methods:

• Developed a coupled one-dimensional partial differential equation-ordinary differential equation (PDE-ODE) model.
• Modeled reactions occurring in localized compartments coupled to bulk diffusion.
• Utilized Robin boundary conditions to represent exchange between bulk and compartments.

Main Results:

• Demonstrated symmetry-breaking bifurcations leading to stable asymmetric patterns with equal diffusivities.
• Showed pattern formation is achievable even with comparable diffusion rates.
• Identified conditions for oscillatory instabilities based on nonlinear kinetics.

Conclusions:

• The model successfully generates spatial patterns under biologically realistic diffusion conditions.
• The findings challenge the necessity of large diffusivity ratios for pattern formation.
• The study provides a framework for understanding pattern dynamics in systems with comparable molecular diffusion rates.