Rate and memory effects in bifurcation-induced tipping

A variation in the environment of a system, such as the temperature, the concentration of a chemical solution or the appearance of a magnetic field, may lead to a drift in one of the parameters. If the parameter crosses a bifurcation point, the system can tip from one attractor to another (bifurcation-induced tipping). Typically, this stability exchange occurs at a parameter value beyond the bifurcation value. This is what we call here the stability exchange shift. We study systematically how the shift is affected by the initial parameter value and its change rate. To that end, we present numerical and analytical results for different types of bifurcations and different paradigmatic systems. Finally, we deduce the scaling laws governing this phenomenon. We show that increasing the change rate and starting the drift further from the bifurcation can delay the tipping process. Furthermore, if the change rate is sufficiently small, the shift becomes independent of the initial condition (no memory) and the shift tends to zero as the square root of the change rate. Thus, the bifurcation diagram for the system with fixed parameters is recovered

A Toggle-Switch and a Feed-Forward Loop Engage in the Control of the Drosophila Retinal Determination Gene Network

Dipterans show a striking range of eye sizes, shapes, and functional specializations. Their eye is of the compound type, the most frequent eye architecture in nature. The development of this compound eye has been most studied in Drosophila melanogaster. The early development of the Drosophila eye is under the control of a gene regulatory network of transcription factors and signaling molecules called the retinal determination gene network (RDGN). Nodes in this network have been found to be involved not only in the development of different eye types in invertebrates and vertebrates, but also of other organs. Here we have analyzed the network properties in detail. First, we have generated quantitative expression profiles for a number of the key RDGN transcription factors, at a single-cell resolution. With these profiles, and applying a correlation analysis, we revisited several of the links in the RDGN. Our study uncovers a new link, that we confirm experimentally, between the transcription factors Hth/Meis1 and Optix/Six3 and indicates that, at least during the period of eye differentiation, positive feedback regulation from Eya and Dac on the Pax6 gene Ey is not operating. From this revised RDGN we derive a simplified gene network that we model mathematically. This network integrates three basic motifs: a coherent feedforward loop, a toggle-switch and a positive autoregulation which, together with the input from the Dpp/BMP2 signaling molecule, recapitulate the gene expression profiles obtained experimentally, while ensuring a robust transition from progenitor cells into retinal precursors.MINECOFEDER (BFU2012-34324, BFU2015-66040-P)Agencia Estatal de Investigacion (AEI) of Spai

A toggle-switch and a feed-forward loop engage in the control of the Drosophila retinal determination gene network

© 2019 Sánchez-Aragón, Cantisán-Gómez, Luque, Brás-Pereira, Lopes, Lemos and CasaresDipterans show a striking range of eye sizes, shapes, and functional specializations. Their eye is of the compound type, the most frequent eye architecture in nature. The development of this compound eye has been most studied in Drosophila melanogaster. The early development of the Drosophila eye is under the control of a gene regulatory network of transcription factors and signaling molecules called the retinal determination gene network (RDGN). Nodes in this network have been found to be involved not only in the development of different eye types in invertebrates and vertebrates, but also of other organs. Here we have analyzed the network properties in detail. First, we have generated quantitative expression profiles for a number of the key RDGN transcription factors, at a single-cell resolution. With these profiles, and applying a correlation analysis, we revisited several of the links in the RDGN. Our study uncovers a new link, that we confirm experimentally, between the transcription factors Hth/Meis1 and Optix/Six3 and indicates that, at least during the period of eye differentiation, positive feedback regulation from Eya and Dac on the Pax6 gene Ey is not operating. From this revised RDGN we derive a simplified gene network that we model mathematically. This network integrates three basic motifs: a coherent feedforward loop, a toggle-switch and a positive autoregulation which, together with the input from the Dpp/BMP2 signaling molecule, recapitulate the gene expression profiles obtained experimentally, while ensuring a robust transition from progenitor cells into retinal precursors.This work was funded by MINECO and the Agencia Estatal de Investigacion (AEI) of Spain, co-financed by FEDER funds (EU) through grants BFU2012-34324 and BFU2015-66040-P to FC, MDM-2016-0687 in which FC is participant researcher, and TIN2017-89842 P in which MCL is participant researcher