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Signal transduction within biological cells is governed by networks ofinteracting proteins. Communication between these proteins is mediated bysignaling molecules which bind to receptors and induce stochastic transitionsbetween different conformational states. Signaling is typically a cooperativeprocess which requires the occurrence of multiple binding events so thatreaction rates have a nonlinear dependence on the amount of signaling molecule.It is this nonlinearity that endows biological signaling networks with robustswitch-like properties which are critical to their biological function. In thisstudy, we investigate how the properties of these signaling systems depend onthe network architecture. Our main result is that these nonlinear networksexhibit bistability where the network activity can switch between states thatcorrespond to a low and high activity level. We show that this bistable regimeemerges at a critical coupling strength that is determined by the spectralstructure of the network. In particular, the set of nodes that correspond tolarge components of the leading eigenvector of the adjacency matrix determinesthe onset of bistability. Above this transition, the eigenvectors of theadjacency matrix determine a hierarchy of clusters, defined by its spectralproperties, which are activated sequentially with increasing network activity.We argue further that the onset of bistability occurs either continuously ordiscontinuously depending upon whether the leading eigenvector is localized ordelocalized. Finally, we show that at low network coupling stochastictransitions to the active branch are also driven by the set of nodes thatcontribute more strongly to the leading eigenvector.

Nonlinear signaling on biological networks: the role of stochasticity and spectral clustering