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

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.