Topological descriptors of the parameter region of multistationarity: Deciding upon connectivity

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 address a basic topological property of the parameter region of multistationary, namely whether it is connected. 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. 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. Although the algorithm can only be used to certify connectivity, we illustrate that the ideas behind the algorithm can be adapted on a case-by-case basis to also decide that the region is not connected. In particular, 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.Author summary: This work addresses the challenging problem of studying the set of parameters for which a system of ordinary differential equations has more than one steady state, a property termed multistationarity. In particular, we are interested in systems arising from the study of biochemical networks. The shape of the multistationarity region is linked to different types of switches that the network can display. We provide an algorithm to decide whether this set is path connected, meaning that any two points in the set are joined by a path completely contained in the set. We illustrate the algorithm with numerous relevant networks, for which we can conclude that the parameter region is path connected.