Multisite phosphorylation networks are encountered in many intracellular processes like signal transduction, cell-cycle control, or nuclear signal integration. In this contribution, networks describing the phosphorylation and dephosphorylation of a protein at n sites in a sequential distributive mechanism are considered. Multistationarity (i.e., the existence of at least two positive steady state solutions of the associated polynomial dynamical system) has been analyzed and established in several contributions. It is, for example, known that there exist values for the rate constants where multistationarity occurs. However, nothing else is known about these rate constants. Here, we present a sign condition that is necessary and sufficient for multistationarity in n-site sequential, distributive phosphorylation. We express this sign condition in terms of linear systems, and show that solutions of these systems define rate constants where multistationarity is possible. We then present, for n≥2, a collection of feasible linear systems, and hence give a new and independent proof that multistationarity is possible for n≥2. Moreover, our results allow to explicitly obtain values for the rate constants where multistationarity is possible. Hence, we believe that, for the first time, a systematic exploration of the region in parameter space where multistationarity occurs has become possible. One consequence of our work is that, for any pair of steady states, the ratio of the steady state concentrations of kinase-substrate complexes equals that of phosphatase-substrate complexes.

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, 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., 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.

Dreyer and Duderstadt [J. Stat. Phys. 123, 1 (2006)] proposed a modification of the standard mass-conserving Becker-Döring model. In this paper we solve for steadystate solutions to two versions of the Becker-Döring model. One is the modified massconserving model introduced by Dreyer and Duderstadt. The second one, which is a new version, is a modification of the so called constant free molecule Becker-Döring model. For practical purposes, there is a maximum cluster of size ν allowed in the system. For each version we study the two known truncations to finite system size. One is given by a zero flux to larger cluster sizes out of the system. The second one is obtained by setting the number of clusters larger than ν to zero. For each model and each truncation we determine the unique steady states by studying the null space of the flux matrix. The zero flux truncation gives equilibrium steady-states whereas the zero particle number truncation leads to non-equilibrium steady-states. C

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