  Many cellular networks rely on the regulated transport of their components to
transduce extracellular information into precise intracellular signals. The
dynamics of these networks is typically described in terms of compartmentalized
chemical reactions. There are many important situations, however, in which the
properties of the compartments change continuously in a way that cannot
naturally be described by chemical reactions. Here, we develop an approach
based on transport along a trafficking coordinate to precisely describe these
processes and we apply it explicitly to the TGF-{\beta} signal transduction
network, which plays a fundamental role in many diseases and cellular
processes. The results of this newly introduced approach accurately capture for
the first time the distinct TGF-{\beta} signaling dynamics of cells with and
without cancerous backgrounds and provide an avenue to predict the effects of
chemical perturbations in a way that closely recapitulates the observed
cellular behavior.
