Geometry of gene regulatory dynamics.

Area of Science:

• Developmental biology
• Mathematical biology
• Systems biology

Background:

• Embryonic development involves ordered cell differentiation driven by gene networks.
• Waddington's landscape metaphor visualizes cell fate decisions as flows through valleys.
• Previous work represented gene networks as potential gradients, supporting Waddington's metaphor.

Purpose of the Study:

• Extend geometric landscape representations to include parameter dependence.
• Enumerate all possible three-way cellular decisions based on parameter tuning.
• Unify and represent various spatial pattern formation models in a potential framework.

Main Methods:

• Applying geometric methods to gene network models.
• Representing systems as potential gradients with Riemann metrics.
• Analyzing parameter dependence and spatial coordinates within the landscape model.

Main Results:

• Enumerated all three-way cellular decisions realizable by tuning up to two parameters.
• Expressed standard spatial pattern formation models, including Turing systems, in potential form.
• Described lateral inhibition as a saddle point and Drosophila eye patterning as bistable potential relaxation.

Conclusions:

• Geometric reasoning provides intuitive and adaptable dynamic models for embryonic development.
• The potential landscape framework unifies diverse developmental models.
• These models are well-suited for fitting time-lapse developmental data.