A graph-based approach for the approximate solution of the chemical master equation

Area of Science:

• Chemical Kinetics
• Stochastic Processes
• Computational Chemistry

Background:

• The chemical master equation (CME) is the standard for mesoscopic chemical kinetics.
• Exact CME solutions are limited to simple systems, necessitating approximation methods.

Purpose of the Study:

• To develop a novel perturbative approximation technique for the CME.
• To enhance the accuracy of stochastic kinetic modeling in mesoscopic systems.

Main Methods:

• A three-step perturbative approach based on graph theory.
• Expansion of transition state matrix eigenvalues and eigenvectors into series.
• Application to reversible dimerization and catalytic reactions.

Main Results:

• The proposed method provides an approximate solution to the CME.
• The approach demonstrates superior accuracy compared to the linear-noise approximation (LNA) for small system parameters.
• Effective for systems with low molecule numbers, common in biological cells.

Conclusions:

• The novel perturbative method offers a more accurate alternative for CME solutions.
• The technique is applicable to general reaction networks under specific conditions.
• This advancement is particularly valuable for simulating biological systems.