# The Interplay Between Non-Instantaneous Dynamics of mRNA and Bounded Extrinsic Stochastic Perturbations for a Self-Enhancing Transcription Factor

## TL;DR

The paper explores how mRNA delays and random perturbations affect the behavior of a self-enhancing transcription factor system.

## Contribution

The study reveals how mRNA dynamics and extrinsic noise influence phase transitions and energy spectrum patterns in gene regulatory motifs.

## Key findings

- Non-instantaneous mRNA dynamics significantly impact phase transitions in the system over long timescales.
- Extrinsic stochastic perturbations are amplified with more complex feedback and translation delays.
- Power-law behavior in energy spectra can arise from the filtering nature of the motif structure.

## Abstract

In this work, we consider a simple bistable motif constituted by a self-enhancing Transcription Factor (TF) and its mRNA with non-instantaneous dynamics. In particular, we mainly numerically investigated the impact of bounded stochastic perturbations of Sine–Wiener type affecting the degradation rate/binding rate constant of the TF on the phase-like transitions of the system. We show that the intrinsic exponential delay in the TF positive feedback, due to the presence of a mRNA with slow dynamics, deeply affects the above-mentioned transitions for long but finite times. We also show that, in the case of more complex delays in the feedback and/or in the translation process, the impact of the extrinsic stochasticity is further amplified. We also briefly investigate the power-law behavior (PLB) of the averaged energy spectrum of the TF by showing that, in some cases, the PLB is simply due to the filtering nature of the motif. A similar analysis can also be applied to biological models having a qualitatively similar structure, such as the well-known Capasso and Paveri–Fontana model of cholera spreading.