Spectral theory of stochastic gene expression: a Hilbert space framework

Area of Science:

• Systems Biology
• Theoretical Biology
• Mathematical Biology

Background:

• Existing literature shows discrepancies in exact solutions for stochastic gene expression spectra.
• Previous studies on self-repressing gene circuits report different eigenvalues for the generator matrix.

Purpose of the Study:

• To propose a unified Hilbert space framework for the spectral theory of stochastic gene expression.
• To analytically derive spectra for constitutive, bursty, and autoregulated gene expression models.

Main Methods:

• Developed a unified Hilbert space framework for spectral theory.
• Analytically derived eigenvalues and eigenvectors for gene expression models.
• Constructed spectral representations of time-dependent gene product distributions.

Main Results:

• Derived exact spectra for constitutive, bursty, and autoregulated gene expression.
• Demonstrated deterministic models fail to capture relaxation rates with strong autoregulation.
• Highlighted limitations of linear algebra for infinite-dimensional operators in this context.

Conclusions:

• A unified Hilbert space framework resolves discrepancies in stochastic gene expression spectral theory.
• Functional analysis is crucial for understanding infinite-dimensional operators in gene expression models.
• Accurate modeling of relaxation rates requires stochastic approaches, especially with autoregulation.