Geometric models for robust encoding of dynamical information into embryonic patterns

Abstract

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, respectively arising through local or global bifurcations. 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.

Novel Generic Models for Differentiating Stem Cells Reveal Oscillatory Mechanisms

“…There is a wider context for the results presented here; the identification of dynamical scenarios typical for repressilator-type GRNs, particularly in view of their proposed role in clock circuits underlying global patterning processes in development [33]. Although the developmental context considered there is different, the formulation and behaviour of the mathematical models has some points of similarity with our analysis, such as the occurrence of Hopf and SNIC bifurcations, which are generic bifurcations involving time-periodic oscillations.…”

“…A key difference with our work is ordering of the different regimes. Model 1 of [33] shows a progression from oscillations to a single stable equilibrium, and then a saddle-node bifurcation that creates multiple equilibria. Model 2 of [33] has no Hopf bifurcation but instead a SNIC separating oscillatory from multistable regime; there is no single equilibrium.…”

“…Model 1 of [33] shows a progression from oscillations to a single stable equilibrium, and then a saddle-node bifurcation that creates multiple equilibria. Model 2 of [33] has no Hopf bifurcation but instead a SNIC separating oscillatory from multistable regime; there is no single equilibrium. In contrast, our model robustly indicates a progression from a single equilibrium to oscillations via a Hopf bifurcation, with the oscillations then terminating at a global bifurcation where multiple equilibria emerge.…”

“…In terms of model construction a key point of difference is that both Model 1 and Model 2 in [33] are constructed by interpolating, rather artificially, between two mechanisms potentially driven by distinct and unrelated morphogens. Our dynamical model is structurally more robust and parsimonious as our work has shown that the transition from equilibrium to oscillations whose frequency drops to zero, to 'differentiated' equilibria, is a natural result of a single GRN mechanism.…”

Precartilage condensation during limb skeletogenesis occurs by tissue phase separation controlled by a bistable cell-state switch with suppressed oscillatory dynamics

“…Tyson et al (2001Tyson et al ( , 2003) and more abstract geometric models motivated by Waddington's epigenetic landscape analogy (e.g. Corson and Siggia (2017); Ferrell (2012); Jutras-Dub et al (2020)). Our model is simple, arising from a reduction of a seven-species model to a three-species system via elimination of "fast" variables, while firmly based on experimentaly well understood interactions.…”