Distinct geometrical landscapes distinguish between modes of tristability in gene regulatory networks

Geometrical models have recently been used to construct landscapes for cell-fate decisions inferred directly from experimental data. However, such quantitative cell-fate data are available for only a few systems; instead, gene regulatory network dynamics have been studied across a broader set of biological decision-making scenarios. Thus, connecting the geometry of cell-fate decisions to their underlying regulatory networks remains an open question. Recently, two regulatory networks have been shown to exhibit tristability-a toggle switch with self-activation and a toggle triad. Here, we show that these two motifs are distinct from a geometrical point of view, and identify two bifurcations in their behaviour: the standard cusp and the elliptic umbilic. We study experimentally accessible signatures of the differences in tristability between the two motifs. We also show how the standard cusp can be used to quantitatively model cell-fate transition data for the epithelial to mesenchymal transition on TGF-β induction in the context of cancer cells. Our work uncovers geometrical signatures of gene regulatory motifs and demonstrates how different gene regulatory networks can encode tristability in dynamically distinct ways.