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by claude@2026-07, 2026-07-05
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This paper is a unified survey-and-theory work that addresses discrepancies in claimed “exact” spectra for stochastic gene expression models, focusing on self-repressing gene circuits and proposing a Hilbert space (functional analytic) framework to derive spectra consistently. Using this framework, the authors analytically obtain eigenvalues and eigenvectors for constitutive, bursty, and autoregulated (self-repressing) gene expression models and use them to build an exact spectral representation for the time-dependent distribution of gene product numbers. They compare the spectral gap (the real part of the first nonzero eigenvalue) to deterministic predictions and find deterministic models fail to capture relaxation rates when autoregulation is strong, attributing this to the failure of standard linear algebra intuitions for infinite-dimensional operators. The paper does not explicitly discuss endometriosis or adenomyosis; it was included in the corpus via a keyword match in the upstream search index.
Abstract
A survey of the literature reveals notable discrepancies among the purported exact results for the spectra of stochastic gene expression models. For self-repressing gene circuits, previous studies ([Phys. Rev. Lett. 99, 108103 (2007)], [Phys. Rev. E 83,062902 (2011)], [J. Chem. Phys. 160, 074105 (2024)], and [bioRxiv 2025.02.05.635946 (2025)]) have provided different exact solutions for the eigenvalues of the generator matrix. In this work, we propose a unified Hilbert space framework for the spectral theory of stochastic gene expression. Based on this framework, we analytically derive the spectra for models of constitutive, bursty, and autoregulated gene expression. The eigenvalues and eigenvectors obtained are then used to construct an exact spectral representation of the time-dependent distribution of gene product numbers. The spectral gap between the zero eigenvalue and the first nonzero eigenvalue, which reflects the relaxation rate of the system towards its steady state, is then compared with the prediction of the deterministic model, and we find that deterministic modeling fails to capture the relaxation rate when autoregulation is strong. In particular, our results demonstrate that for infinite-dimensional operators such as in stochastic gene expression models, many conclusions in linear algebra do not apply, and one must rely on the modern theory of functional analysis.
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Abstract
A survey of the literature reveals notable discrepancies among the purported exact results for the spectra of stochastic gene expression models. For self-repressing gene circuits, previous studies ([Phys. Rev. Lett. 99, 108103 (2007)], [Phys. Rev. E 83,062902 (2011)], [J. Chem. Phys. 160, 074105 (2024)], and [bioRxiv 2025.02.05.635946 (2025)]) have provided different exact solutions for the eigenvalues of the generator matrix. In this work, we propose a unified Hilbert space framework for the spectral theory of stochastic gene expression. Based on this framework, we analytically derive the spectra for models of constitutive, bursty, and autoregulated gene expression. The eigenvalues and eigenvectors obtained are then used to construct an exact spectral representation of the time-dependent distribution of gene product numbers. The spectral gap between the zero eigenvalue and the first nonzero eigenvalue, which reflects the relaxation rate of the system towards its steady state, is then compared with the prediction of the deterministic model, and we find that deterministic modeling fails to capture the relaxation rate when autoregulation is strong. In particular, our results demonstrate that for infinite-dimensional operators such as in stochastic gene expression models, many conclusions in linear algebra do not apply, and one must rely on the modern theory of functional analysis.
Competing Interest Statement
The authors have declared no competing interest.
Footnotes
In the revised manuscript, we have expanded the discussion on the application of spectral theory to stochastic gene expression. Notably, we use the eigenvalues and eigenvectors obtained to construct an exact spectral representation of the time-dependent distribution of gene product numbers for the three models. The magnitude of the real part of the first nonzero eigenvalue, which reflects the relaxation rate to the steady state, is also compared with the prediction of the deterministic model. We demonstrate that deterministic modeling accurately capture the relaxation rate for constitutive and bursty genes, while it fails to reproduce the relaxation rate for self-repressing genes.
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