Bifurcation analysis of bistable and oscillatory dynamics in biological networks using the root-locus method

Multistability in Macrophage Activation Pathways and Metabolic Implications

Macrophages are innate immune cells with a dynamic range of reversible activation states including the classical pro-inflammatory (M1) and alternative anti-inflammatory (M2) states. Deciphering how macrophages regulate their transition from one state to the other is key for a deeper understanding of inflammatory diseases and relevant therapies. Common regulatory motifs reported for macrophage transitions, such as positive or double-negative feedback loops, exhibit a switchlike behavior, suggesting the bistability of the system. In this review, we explore the evidence for multistability (including bistability) in macrophage activation pathways at four molecular levels. First, a decision-making module in signal transduction includes mutual inhibitory interactions between M1 (STAT1, NF-KB/p50-p65) and M2 (STAT3, NF-KB/p50-p50) signaling pathways. Second, a switchlike behavior at the gene expression level includes complex network motifs of transcription factors and miRNAs. Third, these changes impact metabolic gene expression, leading to switches in energy production, NADPH and ROS production, TCA cycle functionality, biosynthesis, and nitrogen metabolism. Fourth, metabolic changes are monitored by metabolic sensors coupled to AMPK and mTOR activity to provide stability by maintaining signals promoting M1 or M2 activation. In conclusion, we identify bistability hubs as promising therapeutic targets for reverting or blocking macrophage transitions through modulation of the metabolic environment.

Root locus-based stability analysis for biological systems

Background: The first objective for realizing and handling biological systems is to choose a suitable model prototype and then perform structure and parameter identification. Afterwards, a theoretical analysis is needed to understand the characteristics, abilities, and limitations of the underlying systems. Generalized Michaelis-Menten kinetics (MM) and S-systems are two well-known biochemical system theory-based models. Research on steady-state estimation of generalized MM systems is difficult because of their complex structure. Further, theoretical analysis of S-systems is still difficult because of the power-law structure, and even the estimation of steady states can be easily achieved via algebraic equations. Aim: We focus on how to flexibly use control technologies to perform deeper biological system analysis. Methods: For generalized MM systems, the root locus method (proposed by Walter R. Evans) is used to predict the direction and rate (flux) limitations of the reaction and to estimate the steady states and stability margins (relative stability). Mode analysis is additionally introduced to discuss the transient behavior and the setting time. For S-systems, the concept of root locus, mode analysis, and the converse theorem are used to predict the dynamic behavior, to estimate the setting time and to analyze the relative stability of systems. Theoretical results were examined via simulation in a Simulink/MATLAB environment. Results: Four kinds of small functional modules (a system with reversible MM kinetics, a system with a singular or nearly singular system matrix and systems with cascade or branch pathways) are used to describe the proposed strategies clearly. For the reversible MM kinetics system, we successfully predict the direction and the rate (flux) limitations of reactions and obtain the values of steady state and net flux. We observe that theoretically derived results are consistent with simulation results. Good prediction is observed ([Formula: see text]% accuracy). For the system with a (nearly) singular matrix, we demonstrate that the system is neither globally exponentially stable nor globally asymptotically stable but globally semistable. The system possesses an infinite gain margin (GM denoting how much the gain can increase before the system becomes unstable) regardless of how large or how small the values of independent variables are, but the setting time decreases and then increases or always decreases as the values of independent variables increase. For S-systems, we first demonstrate that the stability of S-systems can be determined by linearized systems via root loci, mode analysis, and block diagram-based simulation. The relevant S-systems possess infinite GM for the values of independent variables varying from zero to infinity, and the setting time increases as the values of independent variables increase. Furthermore, the branch pathway maintains oscillation until a steady state is reached, but the oscillation phenomenon does not exist in the cascade pathway because in this system, all of the root loci are located on real lines. The theoretical predictions of dynamic behavior for these two systems are consistent with the simulation results. This study provides a guideline describing how to choose suitable independent variables such that systems possess satisfactory performance for stability margins, setting time and dynamic behavior. Conclusion: The proposed root locus-based analysis can be applied to any kind of differential equation-based biological system. This research initiates a method to examine system dynamic behavior and to discuss operating principles.