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On structural contraction of biological interaction networks

M. Ali Al-Radhawi, David Angeli, Eduardo Sontag

TL;DR

This work develops a structural contraction framework for Biological Interaction Networks (BINs) by leveraging graphical Lyapunov functions (GLFs) defined on reaction rates and extended to concentration coordinates. The core idea is that the existence of an $\ell_\infty$ GLF implies nonexpansivity of BIN trajectories within stoichiometric classes, and, under verifiable conditions (siphons, weak contraction), strict contraction on positive compact sets. This contraction framework yields powerful consequences: trajectories converge within stoichiometric classes, persist away from the boundary, and entrain to periodic inputs (CSOST). The authors illustrate the approach with canonical biochemical motifs (PTM, three-body binding, phosphorelays, T-cell kinetic proofreading) and provide computational tools (LEARN) to verify GLFs and contraction properties on real networks.

Abstract

Biological networks are customarily described as structurally robust. This means that they often function extremely well under large forms of perturbations affecting both the concentrations and the kinetic parameters. In order to explain this property, various mathematical notions have been proposed in the literature. In this paper, we propose the notion of structural contractivity, building on the previous work of the authors. That previous work characterized the long-term dynamics of classes of Biological Interaction Networks (BINs), based on "rate-dependent Lyapunov functions". Here, we show that stronger notions of convergence can be established by proving structural contractivity with respect to non-standard polyhedral $\ell_\infty$-norms. In particular, we show that such networks are nonexpansive. With additional verifiable conditions, we show that they are strictly contractive over arbitrary positive compact sets. In addition, we show that such networks entrain to periodic inputs. We illustrate our theory with examples drawn from the modeling of intracellular signaling pathways.

On structural contraction of biological interaction networks

TL;DR

This work develops a structural contraction framework for Biological Interaction Networks (BINs) by leveraging graphical Lyapunov functions (GLFs) defined on reaction rates and extended to concentration coordinates. The core idea is that the existence of an GLF implies nonexpansivity of BIN trajectories within stoichiometric classes, and, under verifiable conditions (siphons, weak contraction), strict contraction on positive compact sets. This contraction framework yields powerful consequences: trajectories converge within stoichiometric classes, persist away from the boundary, and entrain to periodic inputs (CSOST). The authors illustrate the approach with canonical biochemical motifs (PTM, three-body binding, phosphorelays, T-cell kinetic proofreading) and provide computational tools (LEARN) to verify GLFs and contraction properties on real networks.

Abstract

Biological networks are customarily described as structurally robust. This means that they often function extremely well under large forms of perturbations affecting both the concentrations and the kinetic parameters. In order to explain this property, various mathematical notions have been proposed in the literature. In this paper, we propose the notion of structural contractivity, building on the previous work of the authors. That previous work characterized the long-term dynamics of classes of Biological Interaction Networks (BINs), based on "rate-dependent Lyapunov functions". Here, we show that stronger notions of convergence can be established by proving structural contractivity with respect to non-standard polyhedral -norms. In particular, we show that such networks are nonexpansive. With additional verifiable conditions, we show that they are strictly contractive over arbitrary positive compact sets. In addition, we show that such networks entrain to periodic inputs. We illustrate our theory with examples drawn from the modeling of intracellular signaling pathways.
Paper Structure (68 sections, 27 theorems, 106 equations, 3 figures)

This paper contains 68 sections, 27 theorems, 106 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A useful formalism
  3. Notions of robustness
  4. Contraction
  5. This paper
  6. Overview of Results
  7. Review of PWL Lyapunov functions in rate
  8. Nonexpansivity and global convergence
  9. Strict contraction over positive compact sets
  10. Entrainment
  11. Background and preliminaries
  12. Notations
  13. Biological Interaction Networks
  14. Definition.
  15. Kinetics.
  16. ...and 53 more sections

Key Result

Theorem 3

\newlabelt.comLF0 A function ${\widetilde{V}}$ is an GLF for a given network iff ${\widetilde{V}}$ is a common Lyapunov function for the set of linear systems $\dot r = Q_1 r, ..., \dot r = Q_s r$.

Figures (3)

  • Figure 1: The same BIN formalism $S+K \leftrightharpoons C \to P + K$ can represent processes as varied as: (a) an interaction between a catalyst (in the figure, an enzyme kinase) and its substrate, in which the two species can temporarily interact in a reversible fashion, sometimes resulting in the catalyst being released and the substrate being modified ("product"); (b) an interaction between an infective individual and a susceptible individual, in which the two individuals meet and either go their way without any new infection, or result in the susceptible individual becoming exposed to the disease; or (c) a predator/prey interaction in which the prey might not be harmed or the pray may end up injured or dead.
  • Figure 1: Nonexpansivity for two examples (a) The $B$-distance between 500 pairs of randomly generated trajectories for the PTM system with $R(x)=[se, c_1,pd , c_2]^T$, and its numerically-calculated time-derivative using MATLAB's diff subroutine. (b) An unbounded nonexpansive system. The time-derivative of the $B$-distance between 500 pairs of randomly generated trajectories for the system \ref{['e.unstable']} with $R(x)=[c,1,ab]^T$, and a plot of the unbounded trajectories.
  • Figure 1: Examples of nonlinear biochemical motifs analyzed in section 6.

Theorems & Definitions (52)

  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 3
  • Proposition 4: MA_cdc14MA_LEARN
  • Definition 5
  • Theorem 6
  • Definition 7
  • Definition 8
  • Proposition 9
  • ...and 42 more