Solution of the chemical master equation by radial basis functions approximation with interface tracking

Area of Science:

• Computational Chemistry
• Stochastic Processes
• Chemical Kinetics

Background:

• The chemical master equation governs stochastic chemical kinetics, describing probability evolution for system states.
• High-dimensional systems pose computational challenges due to the vast number of possible states.
• Existing methods often approximate by considering only states above a probability threshold.

Purpose of the Study:

• To develop a computationally efficient algorithm for solving the chemical master equation.
• To address challenges posed by high-dimensional systems and multi-modal probability distributions.
• To enable accurate approximation of probability density functions in complex chemical systems.

Main Methods:

• Introduced an algorithm utilizing two key principles for time integration.
• Maintained track of essential support (states with significant probabilities).
• Parametrized the probability distribution using Gaussian radial basis functions for approximation on essential support.

Main Results:

• The algorithm significantly reduces computational resources, particularly for multi-modal distributions.
• Demonstrated effectiveness on challenging models including gene regulation, bistable switches, and cell differentiation.
• Successfully recovered full representations of multi-dimensional distributions, even with drastic temporal transformations.

Conclusions:

• The proposed method offers a new numerical approach for previously intractable problems in stochastic chemical kinetics.
• It is particularly effective for models exhibiting complex structures like multi-stability.
• Enables accurate solutions for systems with high-dimensional state spaces and complex probability dynamics.