  Tumor metastasis is one of the main factors responsible for the high fatality
rate of cancer. Metastasis can occur after malignant cells transition from the
epithelial phenotype to the mesenchymal phenotype. This transformation allows
cells to migrate via the circulatory system and subsequently settle in distant
organs after undergoing the reverse transition from the mesenchymal to the
epithelial phenotypes. The core gene regulatory network controlling these
transitions consists of a system made up of coupled SNAIL/miRNA-34 and
ZEB1/miRNA-200 subsystems. In this work, we formulate a mathematical model of
the core regulatory motif and analyze its long-term behavior. We start by
developing a detailed reaction network with 24 state variables. Assuming fast
promoter and mRNA kinetics, we then show how to reduce our model to a monotone
four-dimensional system. For the reduced system, monotone dynamical systems
theory can be used to prove generic convergence to the set of equilibria for
all bounded trajectories. The theory does not apply to the full model, which is
not monotone, but we briefly discuss results for singularly-perturbed monotone
systems that provide a tool to extend convergence results from reduced to full
systems, under appropriate time separation assumptions.
