  We introduce jump processes in R^k, called density-profile process, to model
biological signaling networks. They describe the macroscopic evolution of
finite-size spin-flip models with k types of spins interacting through a
non-reversible Glauber dynamics. We focus on the the k-dimensional
empirical-magnetization vector in the thermodynamic limit, and prove that,
within arbitrary finite time-intervals, its path converges almost surely to a
deterministic trajectory determined by a first-order (non-linear) differential
equation. As parameters of the spin-flip dynamics change, the associated
dynamical system may go through bifurcations, associated to phase transitions
in the statistical mechanical setting. We present a simple example of spin-flip
stochastic model leading to a dynamical system with Hopf and pitchfork
bifurcations; depending on the parameter values, the magnetization random path
can either converge to a unique stable fixed point, converge to one of a pair
of stable fixed points, or asymptotically evolve close to a deterministic orbit
in R^k.
