A geometric analysis of fast-slow models for stochastic gene expression

Area of Science:

• Systems Biology
• Mathematical Biology
• Computational Biology

Background:

• Gene expression stochastic models often display dynamics across multiple timescales.
• Differences in mRNA and protein lifetimes create a natural small parameter for perturbation analysis in chemical master equations.

Purpose of the Study:

• To develop a framework for analyzing fast-slow gene expression models using geometric singular perturbation theory.
• To provide a rigorous mathematical approach for approximating gene expression dynamics.
• To extend and rigorously prove previous results on gene expression modeling.

Main Methods:

• Application of geometric singular perturbation theory to gene expression models.
• Systematic expansion of the probability-generating function to approximate propagator probabilities.
• Development of a composite fast-slow expansion for uniform time validity.

Main Results:

• A complete characterization of a standard two-stage gene expression model.
• First-order corrections significantly improve steady-state probability distributions, especially in biologically relevant regimes.
• Inclusion of fast-scale asymptotics yields superior uniform approximations for time-dependent propagator probabilities.

Conclusions:

• The developed framework provides accurate approximations for gene expression dynamics, improving upon existing methods.
• The first-order correction offers substantial improvements for both steady-state and time-dependent analyses.
• The geometric framework is generalizable to more complex, regulated gene expression models.