Infinitesimal homeostasis in three-node input-output networks

Area of Science:

• Systems Biology
• Mathematical Biology
• Control Theory

Background:

• Homeostasis is crucial for biological system regulation, maintaining constant output variables despite input variations.
• Existing methods define homeostasis via derivatives, with infinitesimal homeostasis using singularity theory.
• Previous work identified examples like feedforward excitation and substrate inhibition in three-node systems.

Purpose of the Study:

• To provide a complete classification of three-node input-output network configurations leading to infinitesimal homeostasis.
• To identify and define the fundamental mechanisms underlying infinitesimal homeostasis in these networks.
• To connect graph theory, singularity theory, and network structure to understand homeostasis.

Main Methods:

• Analyzed 13 distinct three-node network structures, resulting in 78 input-output configurations.
• Utilized general admissible systems framework for classification, independent of specific biochemical models.
• Related homeostasis types to graph-theoretic properties, specifically simple path existence.

Main Results:

• Identified three basic mechanisms for infinitesimal homeostasis: structural, Haldane, and null-degradation homeostasis.
• Structural homeostasis requires a feedforward loop with two distinct paths (one excitatory, one inhibitory).
• Haldane homeostasis occurs on a single path with a zero-strength coupling; null-degradation homeostasis requires a zero degradation constant.

Conclusions:

• Established a comprehensive classification of infinitesimal homeostasis in three-node networks.
• Demonstrated the link between network topology (feedforward loops, simple paths) and homeostasis mechanisms.
• Provided a foundation for applying singularity theory to higher-codimension homeostasis singularities.