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Automated Classification of Homeostasis Structure in Input-Output Networks

Xinni Lin, Fernando Antoneli, Yangyang Wang

TL;DR

A Python-based algorithm is developed that automates the identification of homeostasis subnetworks and their associated homeostasis conditions directly from network topology, and the resulting computational framework provides a scalable and systematic approach to classifying homeostatic mechanisms in complex biological networks.

Abstract

Homeostasis is widely observed in biological systems and refers to their ability to maintain an output quantity approximately constant despite variations in external disturbances. Mathematically, homeostasis can be formulated through an input-output function mapping an external parameter to an output variable. Infinitesimal homeostasis occurs at isolated points where the derivative of this input-output function vanishes, allowing tools from singularity theory and combinatorial matrix theory to characterize homeostatic mechanisms in terms of network topology. However, the required combinatorial enumeration becomes increasingly intractable as network size grows, and the reliance on advanced graph-theoretic concepts limits accessibility and practical use in biological applications. To overcome these limitations, we develop a Python-based algorithm that automates the identification of homeostasis subnetworks and their associated homeostasis conditions directly from network topology. Given an input-output network specified solely by its connectivity structure and designated input and output nodes, the algorithm identifies the relevant graph-theoretical structures and enumerates all homeostatic mechanisms. We demonstrate its applicability across a range of biological examples, including small and large networks, networks with single or multiple input nodes or parameters, and cases where input and output coincide. This wide applicability stems from our extension of the theoretical framework from single-input-single-output networks to networks with multiple input nodes through an augmented single-input-node representation. The resulting computational framework provides a scalable and systematic approach to classifying homeostatic mechanisms in complex biological networks, facilitating the application of advanced mathematical theory to a broad range of biological systems.

Automated Classification of Homeostasis Structure in Input-Output Networks

TL;DR

A Python-based algorithm is developed that automates the identification of homeostasis subnetworks and their associated homeostasis conditions directly from network topology, and the resulting computational framework provides a scalable and systematic approach to classifying homeostatic mechanisms in complex biological networks.

Abstract

Homeostasis is widely observed in biological systems and refers to their ability to maintain an output quantity approximately constant despite variations in external disturbances. Mathematically, homeostasis can be formulated through an input-output function mapping an external parameter to an output variable. Infinitesimal homeostasis occurs at isolated points where the derivative of this input-output function vanishes, allowing tools from singularity theory and combinatorial matrix theory to characterize homeostatic mechanisms in terms of network topology. However, the required combinatorial enumeration becomes increasingly intractable as network size grows, and the reliance on advanced graph-theoretic concepts limits accessibility and practical use in biological applications. To overcome these limitations, we develop a Python-based algorithm that automates the identification of homeostasis subnetworks and their associated homeostasis conditions directly from network topology. Given an input-output network specified solely by its connectivity structure and designated input and output nodes, the algorithm identifies the relevant graph-theoretical structures and enumerates all homeostatic mechanisms. We demonstrate its applicability across a range of biological examples, including small and large networks, networks with single or multiple input nodes or parameters, and cases where input and output coincide. This wide applicability stems from our extension of the theoretical framework from single-input-single-output networks to networks with multiple input nodes through an augmented single-input-node representation. The resulting computational framework provides a scalable and systematic approach to classifying homeostatic mechanisms in complex biological networks, facilitating the application of advanced mathematical theory to a broad range of biological systems.
Paper Structure (30 sections, 10 theorems, 51 equations, 20 figures, 6 tables)

This paper contains 30 sections, 10 theorems, 51 equations, 20 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Infinitesimal Homeostasis in Input-Output Networks
  3. Single-Input Single-Output Networks
  4. Appendage Homeostasis
  5. Structural Homeostasis
  6. Networks with Input = Output
  7. Algorithm for the Classification of Homeostasis Subnetworks
  8. Step 1:
  9. Step 2:
  10. Generalization to Multiple-Input Single-Output Networks
  11. Single-Input Single-Output Networks with Multiple Input Nodes
  12. Multiple-Input Single-Output Networks
  13. Reduction to the Single Input Node Case
  14. Algorithm Overview
  15. 12-node Network Example
  16. ...and 15 more sections

Key Result

Lemma 2.3

Let $(X_0,{\mathcal{I}}_0)$ be an asymptotically stable equilibrium of eq: io. The input-output function $x_o({\mathcal{I}})$ satisfies Hence, ${\mathcal{I}}_0$ is a point of infinitesimal homeostasis if and only if at $(X_o, {\mathcal{I}}_0)$.

Figures (20)

  • Figure 1: Network Diagram of the 12-node example, adapted from WHAG21.
  • Figure 2: Input data file for the 12-node network example
  • Figure 3: Algorithm-generated core network for the 12-node example from Fig. \ref{['fig:12-network_diagram']}. Blue: super-simple nodes; pink: simple but non-super-simple nodes; yellow: appendage nodes.
  • Figure 4: Algorithm-generated tables of simple paths, complementary subnetworks, and node categories for the 12-node example. In the node categories table, super-simple nodes are arranged in downstream order: $0>1>3>4>11$.
  • Figure 5: Algorithm-generated appendage homeostasis subnetworks for the 12-node example.
  • ...and 15 more figures

Theorems & Definitions (35)

  • Remark 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 25 more