Geometry of gene regulatory dynamics

David A. Rand, Archishman Raju, Meritxell Sáez, Francis Corson, Eric D. Siggia

Research output: Indexed journal article Articlepeer-review

30 Citations (Scopus)

Abstract

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 the Drosophila eye appears as relaxation in a bistable potential. Geometric reasoning provides intuitive dynamic models for development that are well adapted to fit time-lapse data.

Original languageEnglish
Article numbere2109729118
JournalProceedings of the National Academy of Sciences of the United States of America
Volume118
Issue number38
DOIs
Publication statusPublished - 21 Sept 2021
Externally publishedYes

Keywords

  • Bifurcation
  • Gene network
  • Morse-Smale
  • Turing model
  • Waddington landscape

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