Figure 1

Waddington’s epigenetic landscape and potential surfaces. (A) Two different views of Waddington’s epigenetic landscape taken from “The Strategy of the Genes” published in 1957 [3]. The top panel shows a top view of the landscape. The path that the ball will follow represents the developmental trajectory (chreode) of a given system. Valleys indicate alternative differentiation pathways, branch points imply developmental decisions. The bottom panel shows the view from below the landscape. It illustrates how genes remodel the surface by pulling on it through ropes. Waddington used this sketch to show how the landscape’s topography changes during development and evolution. (B) Panel 1: Diagrammatic representation of the toggle switch network used in the simulations. Activating interactions are indicated by arrows, repressing ones by T-bar connectors. See Model and methods section for detailed parameter descriptions. Panel 2: Mathematical formulation of the toggle switch model. x and y indicate concentrations of the protein products of genes X and Y. Ordinary differential equations define the rate of change in protein concentrations ($\frac{\mathit{\text{dx}}}{\mathit{\text{dt}}}$ and $\frac{\mathit{\text{dy}}}{\mathit{\text{dt}}}$). Sigmoid functions with fixed Hill coefficients of 4 are used to represent auto-activation and mutual repression. Decay and external activation are taken to be linear. Parameters as in Panel 1. Panel 3: Phase portrait for a constant set of parameter values of the toggle switch model in the bistable regime. X- and Y-axes represent protein concentrations x and y. We use this example to illustrate relevant features of phase space: arrows indicate flow, blue points mark the position of stable steady states (attractors), the red point shows an unstable steady state (saddle) lying on the separatrix that divides the two basins of attraction (grey line). See the main text for detailed descriptions of the highlighted features. Panel 4: Quasi-potential landscape associated to the the phase portrait shown in Panel 3. The steepness of the quasi-potential surface correlates with the flow at each corresponding point on the phase portrait. Attractors, saddle, and separatrix are indicated as in Panel 3. See main text for details.