How a cell changes from one stable phenotype to another one is a fundamental problem in developmental and cell biology. Mathematically a stable phenotype corresponds to a stable attractor in a generally multi-dimensional state space, which needs to be destabilized so the cell relaxes to a new attractor. Two basic mechanisms for destabilizing a stable fixed point, pitchfork and saddle-node bifurcations, have been extensively studied theoretically, however direct experimental investigation at the single cell level remains scarce. Here we performed live cell imaging studies and analyses in the framework of dynamical systems theories on epithelial-to-mesenchymal transition (EMT). While some mechanistic details remain controversial, EMT is a cell phenotypic transition (CPT) process central to development and pathology. Through time-lapse imaging we recorded single cell trajectories of human A549/Vim-RFP cells undergoing EMT induced by different concentrations of exogenous TGF-β in a multi-dimensional cell feature space. The trajectories clustered into two distinct groups, indicating that the transition dynamics proceeds through parallel paths. We then reconstructed the reaction coordinates and the corresponding quasi-potentials from the trajectories. The potentials revealed a plausible mechanism for the emergence of the two paths where the original stable epithelial attractor collides with two saddle points sequentially with increased TGF-β concentration, and relaxes to a new one. Functionally the directional saddle-node bifurcation ensures a CPT proceeds towards a specific cell type, as a mechanistic realization of the canalization idea proposed by Waddington.

A problem ubiquitous in almost all scientific areas is escape from a metastable state, or relaxation from one stationary distribution to a new one 1 . More than a century of studies lead to celebrated theoretical and computational developments such as the transition state theory and reactive flux formulation. Modern transition path sampling and transition path theory focus on an ensemble of trajectories that connect the initial and final states in a state space 2, 3 . However, it is generally unfeasible to experimentally observe these trajectories in multiple dimensions and compare to theoretical results. Here we report and analyze single cell trajectories of human A549 cells undergoing TGF-β induced epithelial-to-mesenchymal transition (EMT) in a combined morphology and protein texture space obtained through time lapse imaging. From the trajectories we identify parallel reaction paths with corresponding reaction coordinates and quasi-potentials. Studying cell phenotypic transition dynamics will provide testing grounds for nonequilibrium reaction rate theories. AUTHOR CONTRIBUTIONSJ. X. designed the research; W. W. and J. X. performed the research and wrote the paper. SUPPLEMENTAL FIGURE CAPTIONFigure S1 Different views of the trajectory shown in Fig. 2E. (A) Representation in the Morphology PC1-Vimentin Haralick PC1 space. (B) Representation in the Vimentin Haralick PC3-Vimentin Haralick PC4 space.Figure S2 Iterative procedure of the finite temperature string method. (A) Flow chart of the procedure. (B) Example RC curves obtained at different iteration cycle.

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