[Paper Review] N-site phosphorylation systems with 2N-1 steady states

Dietrich Flockerzi, Katharina Holstein+1 more|arXiv (Cornell University)|Dec 17, 2013

Advanced Proteomics Techniques and Applications22 references4 citations

TL;DR

This paper demonstrates that n-site sequential distributive phosphorylation systems can exhibit up to 2N−1 steady states, contradicting a prior conjecture by Wang and Sontag that for odd n, the maximum is n and for even n, it is n+1. Using a scalar determining equation for multistationarity, the authors construct explicit parameter values showing 5 steady states in a 3-site system and 7 in a 4-site system, establishing the first counterexamples to the conjecture and revealing deep geometric constraints on steady state ratios via enzyme and substrate ratios.

ABSTRACT

Multisite protein phosphorylation plays a prominent role in intracellular processes like signal transduction, cell-cycle control and nuclear signal integration. Many proteins are phosphorylated in a sequential and distributive way at more than one phosphorylation site. Mathematical models of $n$-site sequential distributive phosphorylation are therefore studied frequently. In particular, in {\em Wang and Sontag, 2008,} it is shown that models of $n$-site sequential distributive phosphorylation admit at most $2n-1$ steady states. Wang and Sontag furthermore conjecture that for odd $n$, there are at most $n$ and that, for even $n$, there are at most $n+1$ steady states. This, however, is not true: building on earlier work in {\em Holstein et.al., 2013}, we present a scalar determining equation for multistationarity which will lead to parameter values where a $3$-site system has $5$ steady states and parameter values where a $4$-site system has $7$ steady states. Our results therefore are counterexamples to the conjecture of Wang and Sontag. We furthermore study the inherent geometric properties of multistationarity in $n$-site sequential distributive phosphorylation: the complete vector of steady state ratios is determined by the steady state ratios of free enzymes and unphosphorylated protein and there exists a linear relationship between steady state ratios of phosphorylated protein.

Motivation & Objective

• To resolve a longstanding conjecture by Wang and Sontag regarding the maximum number of steady states in n-site sequential distributive phosphorylation systems.
• To investigate the geometric structure of multistationarity in these systems, particularly the relationships between steady state ratios of phosphorylated species.
• To develop a scalar determining equation that enables explicit construction of parameter values supporting high numbers of steady states.
• To provide a graphical test for multistationarity based on collinearity of relative steady state ratios.

Proposed method

• Derives a scalar determining equation for multistationarity in n-site phosphorylation systems under mass-action kinetics.
• Uses the rational parameterization of toric steady states to express steady state ratios in terms of free enzyme and unphosphorylated protein ratios.
• Establishes that the complete vector of steady state ratios is determined by the ratios of kinase, phosphatase, and unphosphorylated protein concentrations.
• Introduces a linear relationship between steady state ratios of phosphorylated proteins, parameterized by the ratio of kinase to phosphatase concentrations.
• Applies the coset condition Z(b−a)=0 to identify when two steady states belong to the same multistationary manifold.
• Develops a graphical test based on collinearity of relative ratios (αi, βi) to detect whether two measured steady states can coexist in a multistationary system.

Experimental results

Research questions

• RQ1 Can n-site sequential distributive phosphorylation systems support more than 2n−1 steady states, as previously conjectured?
• RQ2 What geometric constraints govern the ratios of steady state concentrations in multistationary systems?
• RQ3 Can a scalar determining equation be derived to explicitly construct parameter values supporting multiple steady states?
• RQ4 Is there a graphical method to test whether two measured steady states belong to the same multistationary manifold?
• RQ5 Do the ratios of phosphorylated species in multistationary systems follow a linear relationship parameterized by enzyme ratios?

Key findings

• The paper constructs explicit parameter values for a 3-site system that yield 5 steady states, exceeding the conjectured maximum of n=3 for odd n.
• For a 4-site system, the authors find parameter values supporting 7 steady states, contradicting the conjectured upper bound of n+1=5 for even n.
• The results disprove the conjecture by Wang and Sontag that the maximum number of steady states is bounded by n for odd n and n+1 for even n.
• The steady state ratios of all species are fully determined by the ratios of free kinase, free phosphatase, and unphosphorylated protein concentrations.
• A linear relationship exists between steady state ratios of phosphorylated proteins, with slope ξ = ΓE₁ / ΓE₂, linking ratios across all phosphorylation levels.
• A graphical test based on collinearity of (αi, βi) pairs on the line β = ξα can determine whether two measured steady states belong to the same multistationary coset.