Geometry of Gene Regulatory Dynamics

TL;DR

This paper develops a geometric framework for understanding gene regulatory dynamics, extending landscape models to include parameter dependence and unifying various spatial pattern formation models.

Contribution

It introduces a geometric approach to gene network models, enabling enumeration of cellular decisions and unification of spatial patterning models in potential form.

Findings

All 3-way cell fate decisions are enumerated with parameter tuning.

Standard spatial pattern models are unified in potential form.

Biological variables can be embedded in low-dimensional potential dynamics.

Abstract

Embryonic development leads to the reproducible and ordered appearance of complexity from egg to adult. The successive differentiation of different cell types, that elaborates this complexity, result 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 3-way cellular decisions realisable 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. Turing systems appear non-potential yet in suitable variables the dynamics are low dimensional, potential, and a time independent embedding recovers the biological 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.