  Signal transduction within biological cells is governed by networks of
interacting proteins. Communication between these proteins is mediated by
signaling molecules which bind to receptors and induce stochastic transitions
between different conformational states. Signaling is typically a cooperative
process which requires the occurrence of multiple binding events so that
reaction rates have a nonlinear dependence on the amount of signaling molecule.
It is this nonlinearity that endows biological signaling networks with robust
switch-like properties which are critical to their biological function. In this
study, we investigate how the properties of these signaling systems depend on
the network architecture. Our main result is that these nonlinear networks
exhibit bistability where the network activity can switch between states that
correspond to a low and high activity level. We show that this bistable regime
emerges at a critical coupling strength that is determined by the spectral
structure of the network. In particular, the set of nodes that correspond to
large components of the leading eigenvector of the adjacency matrix determines
the onset of bistability. Above this transition, the eigenvectors of the
adjacency matrix determine a hierarchy of clusters, defined by its spectral
properties, which are activated sequentially with increasing network activity.
We argue further that the onset of bistability occurs either continuously or
discontinuously depending upon whether the leading eigenvector is localized or
delocalized. Finally, we show that at low network coupling stochastic
transitions to the active branch are also driven by the set of nodes that
contribute more strongly to the leading eigenvector.
