A Kinetic Finite Volume Discretization of the Multidimensional PIDE Model for Gene Regulatory Networks.

Area of Science:

• Computational Biology
• Systems Biology
• Mathematical Biology

Background:

• Gene regulatory networks (GRNs) involve complex protein distribution dynamics.
• Partial integro-differential equations (PIDEs) are often used to model these dynamics.
• Analyzing the stability and equilibria of these models can be mathematically challenging.

Purpose of the Study:

• To propose a novel finite volume discretization scheme for PIDEs in GRNs.
• To enable qualitative analysis of protein distribution dynamics using established theories.
• To provide a computationally efficient method for determining stationary distributions.

Main Methods:

• Developed a finite volume discretization scheme for PIDEs.
• Represented the resulting ordinary differential equations (ODEs) as a compartmental kinetic system.
• Applied the theory of nonnegative and compartmental systems for analysis.

Main Results:

• The scheme allows for straightforward analysis of the existence, uniqueness, and stability of equilibria.
• Computation of stationary probability distributions is reduced to solving linear equations.
• Demonstrated the precision and accuracy of the method through computational examples.

Conclusions:

• The proposed finite volume scheme offers a robust framework for analyzing protein dynamics in GRNs.
• It bridges the gap between complex PIDE models and established theories of compartmental systems.
• The method provides accurate results and facilitates efficient computation of key system properties.