N-site Phosphorylation Systems with 2N-1 Steady States

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 Wang and Sontag (J Math Biol 57:29–52, 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 Holstein et al. (Bull Math Biol 75(11):2028–2058, 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.

Appendix: The Network Matrices for \(n\ge 2\)

The matrices ${\mathcal {Y}}$, $Z$, $E$ and $L$ can be obtained from Eqs. (7), (8), (9) and (10) of this manuscript. We recall the definition of the stoichiometric matrix $S$ from Sect. 3 of Holstein et al. (2013). With the following sub-matrices

$$\begin{aligned} n_{11}&= \left[ \begin{array}{rrrrrr} -1&{} 1 &{} 1 &{} 0 &{} 0 &{} 0 \\ -1&{} 1 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} -1&{} 1 &{} 1 \end{array} \right] ,\ \ n_{12}&= \left[ \begin{array}{rrrrrr} -1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1&{} 1 &{} 1 \end{array} \right] ,\\ n_{21}&= \left[ \begin{array}{rrrrrr} 1 &{} -1&{} -1&{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} -1&{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} -1&{} -1 \end{array} \right] ,\ \ n_{22}&= \left[ \begin{array}{rrrrrr} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ -1&{} 1 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \end{array} \right] . \end{aligned}$$

of dimension $3\times 6$, one has

For the convenience of the reader, we close this appendix with the data for $n=3$: