Multisite phosphorylation networks are encountered in many intracellular processes like signal transduction, cell-cycle control or nuclear signal integration. In {\em Wang and Sontag, 2008 ,the authors study the number of steady states in general  n-site sequential distributive phosphorylation and show that there are at most 2n-1 steady states. They furthermore conjecture that , for odd n, there are at most n and that, for even n, there are at most n+1 steady states. Building on earlier work in {\em Holstein et.al., 2013}, we present a scalar determining equation for multistationarity which will lead to 5 steady states for a 3-site and to 7 steady states for a 4-site phosphorylation system and hence to counterexamples to the conjecture of Wang and Sontag. We conclude with a brief biological interpretation of the inherent geometric properties of multistationarity  .
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 {\em Wang and Sontag, 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 {\em Holstein et.al., 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 .
R
phosphorylation networks are encountered in many
protein phosphorylation plays a prominent role in
R
or
and
R
In
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
R
,the authors study the number of steady states in general
,
A
it is shown that models of
R
and show that there are
admit
R
They
Wang and Sontag
D
,
R
Building
This, however, is not true: building
R
5 steady states for
parameter values where
R
and to 7 steady states for
system has 5 steady states and parameter values where
R
phosphorylation system and hence to
system has 7 steady states. Our results therefore are
R
conclude with a brief biological interpretation of
furthermore study
A
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