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Mixture distributions in a stochastic gene expression model with delayed feedback: a WKB approximation approach.

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Production delays significantly impact gene expression noise. This study models protein production with delays and feedback, revealing how large delays create distinct steady-state distributions for inactive and active proteins.

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Area of Science:

  • Systems Biology
  • Biophysics
  • Computational Biology

Background:

  • Gene expression exhibits inherent noise, which can be influenced by various factors.
  • Protein production often involves delays and feedback mechanisms that regulate expression levels.
  • Understanding noise is crucial for comprehending cellular function and variability.

Purpose of the Study:

  • To investigate the effect of production delay on gene expression noise in a mathematical model.
  • To analyze the steady-state behavior of protein production in the slow-activation (large-delay) regime.
  • To derive and validate analytical approximations for protein distributions.

Main Methods:

  • Development of a mathematical model incorporating bursty protein production, a one-step production delay, and feedback control.
  • Application of a formal asymptotic approach to derive an ordinary differential equation for inactive protein in the large-delay limit.
  • Comparison of asymptotic results with numerical solutions of the chemical master equation.

Main Results:

  • In the monostable regime, steady-state distributions for inactive and active proteins are approximated by single Gaussian and Poisson modes, respectively.
  • In the bistable regime (driven by positive feedback), distributions are approximated by mixtures of Gaussian and Poisson modes.
  • The derived analytical approximations accurately reflect numerical simulations of the chemical master equation.

Conclusions:

  • Production delays, particularly large ones, play a critical role in shaping gene expression noise.
  • The mathematical framework developed provides accurate predictions for protein steady-state distributions under different feedback conditions.
  • This work offers insights into the dynamics of gene expression noise and its dependence on kinetic parameters.