N-Site phosphorylation systems with 2n-1 steady states

Area of Science:

• Biochemistry and Molecular Biology
• Mathematical Biology
• Systems Biology

Background:

• Multisite protein phosphorylation is essential for cellular signaling, cell-cycle control, and nuclear signal integration.
• Sequential and distributive phosphorylation at multiple sites is a common regulatory mechanism.
• Mathematical models of n-site sequential distributive phosphorylation are used to study these processes.

Purpose of the Study:

• To investigate the maximum number of steady states in n-site sequential distributive phosphorylation models.
• To challenge the conjecture by Wang and Sontag (2008) regarding the upper bound of steady states.
• To explore the geometric properties of multistationarity in these systems.

Main Methods:

• Developing a scalar determining equation for multistationarity.
• Identifying parameter values that lead to specific numbers of steady states.
• Analyzing the relationship between steady-state ratios of phosphorylated proteins and free enzymes.

Main Results:

• Demonstrated that a 3-site system can have 5 steady states, and a 4-site system can have 7 steady states.
• Provided counterexamples to the conjecture that odd n-site systems have at most n steady states and even n-site systems have at most n+1 steady states.
• Established that the complete vector of steady-state ratios is determined by free enzyme and unphosphorylated protein ratios.

Conclusions:

• The conjecture by Wang and Sontag on the maximum number of steady states in n-site sequential distributive phosphorylation is false.
• The number of steady states can exceed the previously proposed upper bounds.
• Geometric properties reveal linear relationships in steady-state ratios, offering insights into regulatory mechanisms.