Stability and Bifurcation Analysis of a Diffusive miR-9/Hes1 Network With Time Delay

Area of Science:

• Systems biology
• Mathematical modeling
• Cellular dynamics

Background:

• The miR-9/Hes1 interaction network plays a crucial role in cellular differentiation and development.
• Understanding the dynamics of this network, including the effects of time delays and diffusion, is essential for comprehending biological processes.

Purpose of the Study:

• To investigate the stability and bifurcation phenomena in a mathematical model of the miR-9/Hes1 interaction network.
• To analyze the impact of time delay and diffusion on the network's dynamics, including the formation of spatial patterns.
• To develop an algorithm for characterizing bifurcating periodic solutions.

Main Methods:

• Analysis of the characteristic equation to determine stability and bifurcations (local Hopf and Turing-Hopf).
• Development of an algorithm to analyze the direction, stability, and period of bifurcating periodic solutions.
• Numerical simulations to verify theoretical findings and visualize spatio-temporal patterns.

Main Results:

• Time delay can induce oscillations in quiescent progenitors (high Hes1) but minimally affects differentiated cells (low Hes1).
• The combined effects of time delay and diffusion can lead to spatially inhomogeneous patterns.
• Simultaneous existence of homogeneous and inhomogeneous periodic solutions depends on diffusion coefficients; small coefficients allow both, while large ones favor homogeneous solutions.
• Time delay promotes a shift in Hes1 protein from high to low concentration (ON to OFF state), correlating with high miR-9 levels.

Conclusions:

• The miR-9/Hes1 network dynamics are sensitive to time delays and diffusion, impacting cellular states.
• Spatially inhomogeneous patterns can emerge due to the interplay of delay and diffusion.
• Time delay acts as a critical factor in switching Hes1 protein levels, potentially regulating cell fate decisions.