Control of reaction-diffusion equations on time-evolving manifolds

Among the main actors of organism development there are morphogens, which are signaling molecules diffusing in the developing organism and acting on cells to produce local responses. Growth is thus determined by the distribution of such signal. Meanwhile, the diffusion of the signal is itself affected by the changes in shape and size of the organism. In other words, there is a complete coupling between the diffusion of the signal and the change of the shapes. In this paper, we introduce a mathematical model to investigate such coupling. The shape is given by a manifold, that varies in time as the result of a deformation given by a transport equation. The signal is represented by a density, diffusing on the manifold via a diffusion equation. We show the non-commutativity of the transport and diffusion evolution by introducing a new concept of Lie bracket between the diffusion and the transport operator. We also provide numerical simulations showing this phenomenon

Pattern formation by a moving morphogen source

Abstract During Drosophila melanogaster oogenesis, the follicular epithelium that envelops the germline cyst gives rise to an elaborate eggshell, which houses the future embryo and mediates its interaction with the environment. A prominent feature of the eggshell is a pair of dorsal appendages, which are needed for embryo respiration. Morphogenesis of this structure depends on broad, a zinc-finger transcription factor, regulated by the EGFR pathway. While much has been learned about the mechanisms of broad regulation by EGFR, current understanding of processes that shape the spatial pattern of broad expression is incomplete. We propose that this pattern is defined by two different phases of EGFR activation: an early, posterior-to-anterior gradient of EGFR signaling sets the posterior boundary of broad expression, while the anterior boundary is set by a later phase of EGFR signaling, distributed in a dorsoventral gradient. This model can explain the wild-type pattern of broad in D. melanogaster, predicts how this pattern responds to genetic perturbations, and provides insight into the mechanisms driving diversification of eggshell patterning. The proposed model of the broad expression pattern can be used as a starting point for the quantitative analysis of a large number of gene expression patterns in Drosophila oogenesis

Quantitative analysis of the GAL4/UAS system in Drosophila oogenesis

The GAL4/UAS system is extensively used for targeted gene expression in Drosophila, but the strength of the GAL4 drivers and their effects on target genes are rarely quantified. Quantitative information about the strength of the perturbations introduced by the GAL4/UAS system would further expand the usefulness of the GAL4/UAS system in studying gene functions and developmental processes. We have developed an assay to determine the relative level of expression for target genes tagged with green fluorescent protein (GFP). Our assay enables the relative quantitation of fluorescent proteins within specific cell types and developmental time windows in living eggs/embryos, and permits the analysis of samples from a broad expression range. We illustrate the assay using a panel of four GAL4 drivers and three UAS responder lines in Drosophila oogenesis, discuss the issues associated with the interpretation of the quantitative data, and correlate our results with the analysis of the GAL4/UAS system at the transcript level. The imaging-based strategy described here can be used to quantify other GAL4 drivers in Drosophila and other organisms

Simple Expression Domains Are Regulated by Discrete CRMs During Drosophila Oogenesis

Eggshell patterning has been extensively studied in Drosophila melanogaster. However, the cis-regulatory modules (CRMs), which control spatiotemporal expression of these patterns, are vastly unexplored. The FlyLight collection contains >7000 intergenic and intronic DNA fragments that, if containing CRMs, can drive the transcription factor GAL4. We cross-listed the 84 genes known to be expressed during D. melanogaster oogenesis with the ∼1200 listed genes of the FlyLight collection, and found 22 common genes that are represented by 281 FlyLight fly lines. Of these lines, 54 show expression patterns during oogenesis when crossed to an UAS-GFP reporter. Of the 54 lines, 16 recapitulate the full or partial pattern of the associated gene pattern. Interestingly, while the average DNA fragment size is ∼3 kb in length, the vast majority of fragments show one type of spatiotemporal pattern in oogenesis. Mapping the distribution of all 54 lines, we found a significant enrichment of CRMs in the first intron of the associated genes’ model. In addition, we demonstrate the use of different anteriorly active FlyLight lines as tools to disrupt eggshell patterning in a targeted manner. Our screen provides further evidence that complex gene patterns are assembled combinatorially by different CRMs controlling the expression of genes in simple domains

Quantitative analyses of EGFR localization and trafficking dynamics in the follicular epithelium

To bridge the gap between qualitative and quantitative analyses of the epidermal growth factor receptor (EGFR) in tissues, we generated an sfGFP-tagged EGF receptor (EGFR-sfGFP) in Drosophila The homozygous fly appears similar to wild type with EGFR expression and activation patterns that are consistent with previous reports in the ovary, early embryo, and imaginal discs. Using ELISA, we quantified an average of 1100, 6200 and 2500 receptors per follicle cell (FC) at stages 8/9, 10 and ≥11 of oogenesis, respectively. Interestingly, the spatial localization of the EGFR to the apical side of the FCs at early stages depended on the TGFα-like ligand Gurken. At later stages, EGFR localized to basolateral positions of the FCs. Finally, we followed the endosomal localization of EGFR in the FCs. The EGFR colocalized with the late endosome, but no significant colocalization of the receptor was found with the early endosome. The EGFR-sfGFP fly is an exciting new resource for studying cellular localization and regulation of EGFR in tissues