[Paper Review] Parameter regions that give rise to 2[n/2]+1 positive steady states in the n-site phosphorylation system

Magalí Giaroli, Rick Rischter+2 more|arXiv (Cornell University)|Apr 26, 2019

Protein Structure and DynamicsBiochemistry, Genetics and Molecular Biology17 references3 citations

TL;DR

This paper identifies open, explicit parameter regions in the rate constants and total conservation constants of the n-site distributive phosphorylation system that guarantee up to $2[n/2]+1$ positive steady states. Using a polyhedral approach based on algebraic geometry and computer algebra, it systematically constructs multistationarity regions for any $n$, providing both analytical conditions and algorithmic implementation for detecting high-order multistationarity in biochemical networks.

ABSTRACT

The distributive sequential n-site phosphorylation/dephosphorylation system is an important building block in networks of chemical reactions arising in molecular biology, which has been intensively studied. In the nice paper of Wang and Sontag (2008) it is shown that for certain choices of the reaction rate constants and total conservation constants, the system can have 2[n/2]+1 positive steady states (that is, n+1 positive steady states for n even and n positive steady states for n odd). In this paper we give open parameter regions in the space of reaction rate constants and total conservation constants that ensure these number of positive steady states, while assuming in the modeling that roughly only 1/4 of the intermediates occur in the reaction mechanism. This result is based on the general framework developed by Bihan, Dickenstein, and Giaroli (2018), which can be applied to other networks. We also describe how to implement these tools to search for multistationarity regions in a computer algebra system and present some computer aided results.

Motivation & Objective

• To identify explicit, open parameter regions in the space of rate constants and total conservation constants where the n-site phosphorylation system exhibits high multistationarity.
• To extend prior results on multistationarity in the n-site system by providing systematic, algorithmically implementable conditions rather than isolated parameter choices.
• To develop a general framework applicable to other biochemical networks, such as enzymatic cascades, based on polyhedral decomposition and algebraic techniques.
• To demonstrate that multistationarity with $2[n/2]+1$ positive steady states can be achieved under realistic assumptions, including only a quarter of intermediates in the kinase pathway.
• To provide a computational pipeline for detecting multistationarity regions using computer algebra systems, enabling verification and extension to small-to-moderate network sizes.

Proposed method

• Applies a polyhedral decomposition method derived from the framework of Bihan, Dickenstein, and Giaroli (2018) to analyze the steady-state equations of the n-site phosphorylation system.
• Uses the law of mass action to derive a system of polynomial ordinary differential equations from the reaction network, with parameters including rate constants and conservation constants.
• Imposes stoichiometric conservation laws: $S_{\text{tot}} = \sum s_i + \sum y_i + \sum u_i$, $E_{\text{tot}} = e + \sum y_i$, $F_{\text{tot}} = f + \sum u_i$, which constrain the positive orthant of the state space.
• Applies a rescaling technique to the association rate constants ($k_{\text{on}_i}$, $\ell_{\text{on}_i}$) to construct open parameter regions where multistationarity is guaranteed.
• Employs computational algebra systems to implement the method and verify multistationarity regions for small $n$ (e.g., $n=3,4,5$), using symbolic and numerical tools.
• Leverages symmetry between kinase and phosphatase pathways to extend results from one case (e.g., intermediates only in $E$) to the full system, including the symmetric case with intermediates in $F$.

Experimental results

Research questions

• RQ1 What open parameter regions in the rate constants and conservation constants lead to $2[n/2]+1$ positive steady states in the n-site phosphorylation system?
• RQ2 Can a systematic, algorithmic method be developed to identify multistationarity regions in biochemical networks, rather than relying on isolated parameter choices?
• RQ3 How can the polyhedral approach be applied to networks with only partial intermediate formation (e.g., only in the kinase arm) to still achieve high multistationarity?
• RQ4 What is the relationship between the rescaling of association rate constants and the emergence of multiple positive steady states?
• RQ5 Can the method be extended to other networks, such as enzymatic cascades, and implemented effectively in computer algebra systems?

Key findings

• The paper establishes open parameter regions—defined by rescaling the association rate constants $k_{\text{on}_i}$ and $\ell_{\text{on}_i}$—that guarantee $2[n/2]+1$ positive steady states in the n-site phosphorylation system.
• For $n$ even, the system can have $n+1$ positive steady states; for $n$ odd, it can have $n$ positive steady states, matching the upper bound conjectured in prior work.
• The method applies under the assumption that only about $1/4$ of the intermediates are present in the reaction mechanism (specifically, only in the kinase pathway), yet still yields high multistationarity.
• The results are robust under symmetry: identical multistationarity regions exist when intermediates are instead assumed to form only in the phosphatase pathway.
• The approach is algorithmically implementable in computer algebra systems, enabling verification and extension to small $n$, with explicit results provided for $n=3,4,5$.
• The findings improve upon prior works (e.g., [3], [5], [9]) by providing open, explicit regions rather than isolated parameter choices, and by offering a generalizable framework for other networks.