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Embryonic development leads to the reproducible and ordered appearance of complexity from egg to adult. The successive differentiation of different cell types that elaborate this complexity results from the activity of gene networks and was likened by Waddington to a flow through a landscape in which valleys represent alternative fates. Geometric methods allow the formal representation of such landscapes and codify the types of behaviors that result from systems of differential equations. Results from Smale and coworkers imply that systems encompassing gene network models can be represented as potential gradients with a Riemann metric, justifying the Waddington metaphor. Here, we extend this representation to include parameter dependence and enumerate all three-way cellular decisions realizable by tuning at most two parameters, which can be generalized to include spatial coordinates in a tissue. All diagrams of cell states vs. model parameters are thereby enumerated. We unify a number of standard models for spatial pattern formation by expressing them in potential form (i.e., as topographic elevation). Turing systems appear nonpotential, yet in suitable variables the dynamics are low dimensional and potential. A time-independent embedding recovers the original variables. Lateral inhibition is described by a saddle point with many unstable directions. A model for the patterning of theDrosophilaeye appears as relaxation in a bistable potential. Geometric reasoning provides intuitive dynamic models for development that are well adapted to fit time-lapse data. 

At first, embryos are made up of identical cells. Then, as the embryo develops, these cells specialize into different types, such as heart and brain cells. Chemical signals sent and received by the cells are key to forming the right type of cell at the right time and place. The cellular machinery that produces and interprets these signals is exceedingly complex and difficult to understand. In the 1950s, Conrad Waddington presented an alternative way of thinking about how an unspecialized cell progresses to one of many different fates. He suggested visualizing the developing cell as a ball rolling along a hilly landscape. As the ball travels, obstacles in its way guide it along particular paths. Eventually the ball comes to rest in a valley, with each valley in the landscape representing a different cell fate. Although this “landscape model” is an appealing metaphor for how signaling events guide cell specialization, it was not clear whether it could be put to productive use. The egg-laying organ in the worm species Caenorhabditis elegans is called the vulva, and is often studied by researchers who want to learn more about how organs develop. The vulva develops from a small number of identical cells that adopt one of three possible cell fates. Two chemical signals, called epidermal growth factor (EGF) and Notch, control this specialization process. Corson and Siggia have now constructed a simple landscape model that can reproduce the normal arrangement of cell types in the vulva. When adjusted to describe the effect of genetic mutations that affect either EGF or Notch, the model could predict the outcome of mutations that affect both signals at once. The twists and turns of cell paths in the landscape could also account for several non-intuitive cell fate outcomes that had been assumed to result from subtle regulation of EGF and Notch signals. Landscape models should be easy to apply to other developing tissues and organs. By providing an intuitive picture of how signals shape cellular decisions, the models could help researchers to learn how to control cell and tissue development. This could lead to new treatments to repair or replace failing organs, making regenerative medicine a reality.