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. (BullMath 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.