Arnold tongue entrainment reveals dynamical principles of the embryonic segmentation clock

Abstract

Living systems exhibit an unmatched complexity, due to countless, entangled interactions across scales. Here, we aim to understand a complex system, that is, segmentation timing in mouse embryos, without a reference to these detailed interactions. To this end, we develop a coarse-grained approach, in which theory guides the experimental identification of the segmentation clock entrainment responses. We demonstrate period- and phase-locking of the segmentation clock across a wide range of entrainment parameters, including higher-order coupling. These quantifications allow to derive the phase response curve (PRC) and Arnold tongues of the segmentation clock, revealing its essential dynamical properties. Our results indicate that the somite segmentation clock has characteristics reminiscent of a highly non-linear oscillator close to an infinite period bifurcation and suggests the presence of long-term feedbacks. Combined, this coarse-grained theoretical-experimental approach reveals how we can derive simple, essential features of a highly complex dynamical system, providing precise experimental control over the pace and rhythm of the somite segmentation clock.

Nonreciprocal synchronization in embryonic oscillator ensembles

“…The more general insight from studying PSM oscillators in vitro is hence that of a highly asymmetric coupling mechanism that results in a constant sign response. Interestingly, recent entrainment experiments of segmentation clock oscillators performed on embryonic PSM tissue explants revealed a similar, highly asymmetric response, in this case to periodic entrainment pulses [32]. In these experiments, the segmentation clock oscillator’s phase response curve (PRC) to pulses of a Notch inhibitor (DAPT) was found to be close to 0 for a significant part of the cycle, and mostly negative otherwise, meaning that DAPT mostly delays the segmentation clock.…”

Foci, waves, excitability : self-organization of phase waves in a model of asymmetrically coupled embryonic oscillators

“…1. Importantly, it was proposed that such coupling reflects the properties of an internal oscillator poised close to a Saddle-Node on Invariant Cycle (SNIC) (or Saddle-Node Infinite Period (SNIPER)) bifurcation [29, 38]. Such asymmetric nature of the clock response is also consistent with the experimentally observed unidirectional nature of the Notch-Delta coupling, termed as the “walkie-talkie” model [40, 41].…”

“…In Ref. [29], Sanchez et al . used a dynamical systems-inspired approach to study the entrainment dynamics and phase response of the segmentation clock, when subjected to weak external perturbations in the form of periodic pulses of DAPT [30].…”

“…To quantify the influence of DAPT, a phase oscillator model was used. Phase oscillator models, wherein PSM cells are idealized as limit cycle oscillators with a single phase variable provide an expedient approach to investigate systems-level phenomena [28, 29, 31–39], including local and global synchronization. This method circumvents the requirement to introduce molecular (genetic) complexity into the problem, allowing for a focused exploration of the desired properties.…”

“…Motivated by these insights, a model for the phase response, called the Elliptic Radial Isochron Cycle (ERIC) model was proposed in Ref. [29], whose infinitesimal PRC is precisely of the form given by Eq. 1.…”