  Switch-like responses arising from bistability have been linked to cell
signaling processes and memory. Revealing the shape and properties of the set
of parameters that lead to bistability is necessary to understand the
underlying biological mechanisms, but is a complex mathematical problem. We
present an efficient approach to determine a basic topological property of the
parameter region of multistationary, namely whether it is connected or not. The
connectivity of this region can be interpreted in terms of the biological
mechanisms underlying bistability and the switch-like patterns that the system
can create.
  We provide an algorithm to assert that the parameter region of
multistationarity is connected, targeting reaction networks with mass-action
kinetics. We show that this is the case for numerous relevant cell signaling
motifs, previously described to exhibit bistability. However, we show that for
a motif displaying a phosphorylation cycle with allosteric enzyme regulation,
the region of multistationarity has two distinct connected components,
corresponding to two different, but symmetric, biological mechanisms. The
method relies on linear programming and bypasses the expensive computational
cost of direct and generic approaches to study parametric polynomial systems.
This characteristic makes it suitable for mass-screening of reaction networks.
