Exact protein distribution in the three-stage gene-expression model

Determine the exact stationary protein distribution for the canonical three-stage gene-expression model with gene-state switching, transcription, translation, and mRNA and protein degradation, for which the stationary mRNA distribution is known in closed form but the protein distribution has not been obtained exactly.

Background

The paper analyzes a canonical three-stage gene-expression network involving promoter switching, mRNA synthesis, translation, and degradation of mRNA and protein. Under the assumption of one gene copy, the process is exponentially ergodic within a closed irreducible class, and its stationary distribution exists.

Although the stationary mRNA distribution can be written in closed form, the authors state that the exact protein distribution remains unresolved. This is a specific unresolved distributional characterization problem rather than a question about ergodicity, which the paper establishes.

References

We emphasize that the stationary mRNA distribution for the three-stage gene expression model can be obtained in closed form , whereas the exact protein distribution remains an open problem.

— Long-time behavior of reducible stochastic linear reaction networks: a spectral and structural classification  (2610.11443 - Wu et al., 8 Oct 2026) in Example (Section: Long-time behavior in the critical case)