Geometric models for robust encoding of dynamical information into embryonic patterns.

During development, cells gradually assume specialized fates via changes of transcriptional dynamics, sometimes even within the same developmental stage. For anterior-posterior (AP) patterning in metazoans, it has been suggested that the gradual transition from a dynamic genetic regime to a static one is encoded by different transcriptional modules. In that case, the static regime has an essential role in pattern formation in addition to its maintenance function. In this work, we introduce a geometric approach to study such transition. We exhibit two types of genetic regime transitions arising through local or global bifurcations, respectively. We find that the global bifurcation type is more generic, more robust, and better preserves dynamical information. This could parsimoniously explain common features of metazoan segmentation, such as changes of periods leading to waves of gene expressions, 'speed/frequency-gradient' dynamics, and changes of wave patterns. Geometric approaches appear as possible alternatives to gene regulatory networks to understand development.

Modelling Time-Dependent Acquisition of Positional Information.

Theoretical and computational modelling are crucial to understand dynamics of embryonic development. In this tutorial chapter, we describe two models of gene networks performing time-dependent acquisition of positional information under control of a dynamic morphogen: a toy-model of a bistable gene under control of a morphogen, allowing for the numerical computation of a simple Waddington's epigenetic landscape, and a recently published model of gap genes in Tribolium under control of multiple enhancers. We present detailed commented implementations of the models using python and jupyter notebooks.