  Cells sense their surrounding by employing intracellular signaling pathways
that transmit hormonal signals from the cell membrane to the nucleus.
TGF-$\beta$/SMAD signaling encodes various cell fates, controls tissue
homeostasis and is deregulated in diseases such as cancer. The pathway shows
strong heterogeneity at the single-cell level, but quantitative insights into
mechanisms underlying fluctuations at various time scales are still missing,
partly due to inefficiency in the calibration of stochastic models that
mechanistically describe signaling processes. In this work we analyze
single-cell TGF-$\beta$/SMAD signaling and show that it exhibits temporal
stochastic bursts which are dose-dependent and whose number and magnitude
correlate with cell migration. We propose a stochastic modeling approach to
mechanistically describe these pathway fluctuations with high computational
efficiency. Employing high-order numerical integration and fitting to burst
statistics we enable efficient quantitative parameter estimation and
discriminate models that assume noise in different reactions at the receptor
level. This modeling approach suggests that stochasticity in the
internalization of TGF-$\beta$ receptors into endosomes plays a key role in the
observed temporal bursting. Further, the model predicts the single-cell
dynamics of TGF-$\beta$/SMAD signaling in untested conditions, e.g.,
successfully reflects memory effects of signaling noise and cellular
sensitivity towards repeated stimulation. Taken together, our computational
framework based on burst analysis, noise modeling and path computation scheme
is a suitable tool for the data-based modeling of complex signaling pathways,
capable of identifying the source of temporal noise.
