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Inherent Dynamics Visualizer, an Interactive Application for Evaluating and Visualizing Outputs from a Gene Regulatory Network Inference Pipeline
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A model of regulatory dynamics with threshold-type state-dependent delay.

Qingwen Hu1

  • 1Department of Mathematical Sciences, The University of Texas at Dallas, 800 W. Campbell Road, FO. 35, Richardson, TX, 75080, USA.

Mathematical Biosciences and Engineering : MBE
|November 1, 2018
PubMed
Summary

State-dependent delays in intracellular regulatory dynamics can cause Hopf bifurcations, influencing oscillations. This research models these effects, revealing how diffusion time impacts stability and dynamics.

Keywords:
normal formDiffusion timeHopf bifurcationmultiple time scalesstate-dependent delay

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Area of Science:

  • Systems Biology
  • Mathematical Biology
  • Biophysics

Background:

  • Intracellular regulatory dynamics are often modeled using differential equations with time delays.
  • Classical models typically assume constant or zero time delays, which may not capture complex biological realities.
  • State-dependent delays and diffusion times are crucial factors in biological system regulation.

Purpose of the Study:

  • To develop a general model for intracellular regulatory dynamics incorporating threshold-type state-dependent delays and state-dependent diffusion time.
  • To analyze the stability of steady states and the occurrence/stability of periodic oscillations.
  • To investigate the impact of state-dependent diffusion time on regulatory dynamics and bifurcations.

Main Methods:

  • Development of a generalized differential equation model.
  • Application of the method of multiple time scales to compute the normal form.
  • Analysis of stability and bifurcation phenomena.
  • Numerical simulations of a Hes1 regulatory dynamics prototype model.

Main Results:

  • The generalized model extends classical models by including state-dependent delays and diffusion.
  • State-dependent diffusion time can induce both supercritical and subcritical Hopf bifurcations.
  • The study provides a framework for understanding complex oscillatory behaviors in biological systems.
  • Numerical simulations validate the theoretical findings for Hes1 dynamics.

Conclusions:

  • State-dependent diffusion time is a significant factor in intracellular regulatory dynamics.
  • Hopf bifurcations, leading to oscillations, are influenced by state-dependent diffusion.
  • The developed model offers new insights into the stability and dynamics of biological regulatory networks.