Multistationarity in the activation of a MAPK: parametrizing the relevant region in parameter space

Mathematical models of biochemical reaction networks in the form of ordinary differential equations can exhibit all sorts of complex dynamical behaviour. It is for example known, that even a single layer of a MAPK cascade can exhibit bistability (i.e. there exist multiple (positive) steady state solutions). It is almost a common-place that bistability or some other form of multistationarity are observed in many biochemical reaction networks, especially if the focus is on signal transduction or cell cycle regulation. However, multistationarity is only exhibited if the parameter vector is located in an appropriate region of parameter space. To find these regions, for example by using numerical tools like bifurcation analysis, is a non-trivial task as it amounts to searching the whole parameter space. In this paper we show that for a model of a single layer of a MAPK cascade it is possible to derive analytical descriptions of these regions, if mass action kinetics are used. Moreover, our results give rise to a straightforward explanation for the 'robust yet fragile' behaviour in the activation of a MAPK.