  Multistationarity in molecular systems underlies switch-like responses in
cellular decision making. Determining whether and when a system displays
multistationarity is in general a difficult problem. In this work we completely
determine the set of kinetic parameters that enable multistationarity in a
ubiquitous motif involved in cell signaling, namely a dual phosphorylation
cycle. In addition we show that the regions of multistationarity and
monostationarity are both path connected.
  We model the dynamics of the concentrations of the proteins over time by
means of a parametrized polynomial ordinary differential equation (ODE) system
arising from the mass-action assumption. Since this system has three linear
first integrals defined by the total amounts of the substrate and the two
enzymes, we study for what parameter values the ODE system has at least two
positive steady states after suitably choosing the total amounts. We employ a
suite of techniques from (real) algebraic geometry, which in particular concern
the study of the signs of a multivariate polynomial over the positive orthant
and sums of nonnegative circuit polynomials.
