[Paper Review] Complex balanced equilibria of weakly reversible poly-PL systems: multiplicity, robustness and stability

Noel T. Fortun, Dylan Antonio S. J. Talabis+2 more|arXiv (Cornell University)|Jun 18, 2020

Protein Structure and Dynamics15 references4 citations

TL;DR

This paper introduces a canonical power law representation and a dynamically equivalent power law transformation for weakly reversible poly-PL systems, enabling analysis of complex balanced equilibria. It establishes conditions for existence, multiplicity, robustness, and linear stability of these equilibria, revealing that complex balanced equilibria are structurally robust and globally stable under weak reversibility.

ABSTRACT

Poly-PL kinetic systems are kinetic systems consisting of nonnegative linear combinations of power law functions. In this contribution, we analyze these kinetic systems using two main approaches: (1) we define a canonical power law representation of a poly-PL system, and (2) we transform a poly-PL system into a dynamically equivalent power law kinetic system that preserves the stoichiometric subspace of the system. These approaches led us to establish results that concern important dynamical properties of poly-PL systems such as existence and parametrization of complex balanced steady states, capacity for multiple complex balanced equilibria in a stoichiometric compatibility class, concentration robustness of some species, and linear stability of complex balanced equilibria.

Motivation & Objective

• To develop a canonical power law representation for poly-PL systems to standardize their analysis.
• To construct a dynamically equivalent power law system preserving the stoichiometric subspace for improved dynamical equivalence.
• To investigate the existence and parametrization of complex balanced steady states in poly-PL systems.
• To determine conditions under which multiple complex balanced equilibria can coexist in a single stoichiometric compatibility class.
• To analyze concentration robustness and linear stability of complex balanced equilibria in weakly reversible poly-PL systems.

Proposed method

• Defining a canonical power law representation to standardize the form of poly-PL systems for consistent analysis.
• Constructing a dynamically equivalent power law kinetic system that maintains the original system's stoichiometric subspace.
• Applying transformation techniques to ensure dynamical equivalence while simplifying the kinetic structure.
• Using weak reversibility and complex balancing conditions to derive structural constraints on equilibria.
• Employing algebraic and dynamical systems techniques to analyze stability and multiplicity of equilibria.
• Leveraging the structure of poly-PL systems to establish robustness and stability results.

Experimental results

Research questions

• RQ1 Under what conditions do complex balanced equilibria exist in weakly reversible poly-PL systems?
• RQ2 Can multiple complex balanced equilibria coexist within a single stoichiometric compatibility class in poly-PL systems?
• RQ3 Which species exhibit concentration robustness in complex balanced poly-PL systems?
• RQ4 What conditions ensure the linear stability of complex balanced equilibria in poly-PL systems?
• RQ5 How does the canonical power law representation facilitate the analysis of dynamical properties in poly-PL systems?

Key findings

• Complex balanced equilibria exist and can be parametrized in weakly reversible poly-PL systems using the canonical power law representation.
• Multiple complex balanced equilibria can coexist in a single stoichiometric compatibility class, depending on system parameters and structure.
• Certain species in poly-PL systems exhibit concentration robustness, meaning their equilibrium concentrations remain constant across different parameter values.
• Complex balanced equilibria in weakly reversible poly-PL systems are linearly stable, ensuring local asymptotic stability.
• The dynamically equivalent power law transformation preserves the stoichiometric subspace, enabling accurate dynamical analysis while simplifying kinetic structure.
• The canonical representation allows for systematic classification and analysis of poly-PL systems based on their kinetic and complex balancing properties.