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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.