[Paper Review] The kinetic space of multistationarity in dual phosphorylation

Elisenda Feliu, Nidhi Kaihnsa+2 more|arXiv (Cornell University)|Jan 22, 2020

Gene Regulatory Network Analysis41 references4 citations

TL;DR

This paper completely characterizes the kinetic parameter space enabling multistationarity in a dual phosphorylation cycle—a fundamental motif in cell signaling—using advanced algebraic geometry techniques. It proves that the regions of multistationarity and monostationarity are both path-connected, resolving a long-standing open case in the parameter space where previous criteria were inconclusive.

ABSTRACT

Multistationarity in molecular systems underlies switch-like responses in cellular decision making. Determining whether and when a system displays multistationarity is in general a difficult problem. In this work we completely determine the set of kinetic parameters that enable multistationarity in a ubiquitous motif involved in cell signaling, namely a dual phosphorylation cycle. In addition we show that the regions of multistationarity and monostationarity are both path connected. We model the dynamics of the concentrations of the proteins over time by means of a parametrized polynomial ordinary differential equation (ODE) system arising from the mass-action assumption. Since this system has three linear first integrals defined by the total amounts of the substrate and the two enzymes, we study for what parameter values the ODE system has at least two positive steady states after suitably choosing the total amounts. We employ a suite of techniques from (real) algebraic geometry, which in particular concern the study of the signs of a multivariate polynomial over the positive orthant and sums of nonnegative circuit polynomials.

Motivation & Objective

• To fully determine the set of kinetic parameters that enable multistationarity in a dual phosphorylation cycle under mass-action kinetics.
• To resolve the open case in parameter space where previous rational functions $a(\kappa)$ and $b(\kappa)$ failed to classify multistationarity.
• To establish topological properties of the multistationary and monostationary parameter regions, specifically their path-connectedness.
• To apply advanced techniques from real algebraic geometry to analyze sign patterns of multivariate polynomials over the positive orthant.
• To provide a complete and mathematically rigorous characterization of parameter regions enabling multiple positive steady states in a canonical signaling motif.

Proposed method

• Model the system as a parametrized polynomial ODE system with 12 rate constants under mass-action kinetics.
• Exploit three linear first integrals (total substrate, kinase, and phosphatase) to reduce the system to invariant subspaces parameterized by total amounts.
• Use sums of nonnegative circuit polynomials (SONC) and sign analysis of multivariate polynomials over the positive orthant to study steady-state multiplicities.
• Apply cylindrical algebraic decomposition (CAD) and Newton polytope analysis to examine the sign of critical polynomials in parameter space.
• Construct explicit paths in the parameter space using scaling transformations to prove path-connectedness of multistationary and monostationary regions.
• Leverage the structure of the parameter map $\pi$ to lift path-connectedness from the reduced 8-dimensional parameter space to the full 12-dimensional kinetic space.

Experimental results

Research questions

• RQ1 For which values of the 12 kinetic rate constants does the dual phosphorylation cycle exhibit multistationarity?
• RQ2 What is the topological structure of the parameter region enabling multistationarity—specifically, is it path-connected?
• RQ3 How can algebraic geometry tools such as SONC and Newton polytopes be used to analyze sign patterns of steady-state equations in biochemical networks?
• RQ4 Can the previously open case $a(\kappa) \geq 0, b(\kappa) < 0$ be resolved using advanced real algebraic geometry?
• RQ5 Does the preimage of the multistationary region under the parameter map $\pi$ remain path-connected in the full 12-dimensional kinetic space?

Key findings

• The paper fully characterizes the set of kinetic parameters that enable multistationarity in the dual phosphorylation cycle, resolving a long-standing open problem.
• The region of parameters enabling multistationarity is path-connected in the 8-dimensional reduced parameter space $\mathbb{R}_{>0}^8$, and this property lifts to the full 12-dimensional kinetic space.
• The monostationary region is also proven to be path-connected, establishing that both multistationary and monostationary regions are topologically well-behaved.
• The analysis confirms that the open case $a(\kappa) \geq 0, b(\kappa) < 0$ does not yield multistationarity, completing the classification of parameter regimes.
• The study demonstrates that the preimage of the multistationary region under the parameter map $\pi$ is path-connected in $\mathbb{R}_{>0}^{12}$, due to the connected fibers of the map.
• The use of Newton polytopes and circuit polynomial techniques successfully identifies critical sign patterns in the steady-state equations, enabling a complete classification.