Euclidean random matrix theory: low-frequency non-analyticities and Rayleigh scattering

By calculating all terms of the high-density expansion of the euclidean random matrix theory (up to second-order in the inverse density) for the vibrational spectrum of a topologically disordered system we show that the low-frequency behavior of the self energy is given by $\Sigma(k,z)\propto k^2z^{d/2}$ and not $\Sigma(k,z)\propto k^2z^{(d-2)/2}$, as claimed previously. This implies the presence of Rayleigh scattering and long-time tails of the velocity autocorrelation function of the analogous diffusion problem of the form $Z(t)\propto t^{(d+2)/2}$.Comment: 27 page

DAnkrd49 and Bdbt act via casein kinase Iε to regulate planar polarity in Drosophila

The core planar polarity proteins are essential mediators of tissue morphogenesis, controlling both the polarised production of cellular structures and polarised tissue movements. During development the core proteins promote planar polarisation by becoming asymmetrically localised to opposite cell edges within epithelial tissues, forming intercellular protein complexes that coordinate polarity between adjacent cells. Here we describe a novel protein complex that regulates the asymmetric localisation of the core proteins in the Drosophila pupal wing. DAnkrd49 (an ankyrin repeat protein) and Bride of Doubletime (Bdbt, a non-canonical FK506 binding protein family member) physically interact, and regulate each other's levels in vivo. Loss of either protein results in a reduction in core protein asymmetry and disruption of the placement of trichomes at the distal edge of pupal wing cells. Post-translational modifications are thought to be important for the regulation of core protein behaviour and their sorting to opposite cell edges. Consistent with this, we find that loss of DAnkrd49 or Bdbt leads to reduced phosphorylation of the core protein Dishevelled and to decreased Dishevelled levels both at cell junctions and in the cytoplasm. Bdbt has previously been shown to regulate activity of the kinase Discs Overgrown (Dco, also known as Doubletime or Casein Kinase Iε), and Dco itself has been implicated in regulating planar polarity by phosphorylating Dsh as well as the core protein Strabismus. We demonstrate that DAnkrd49 and Bdbt act as dominant suppressors of Dco activity. These findings support a model whereby Bdbt and DAnkrd49 act together to modulate the activity of Dco during planar polarity establishment

Molecular mechanisms mediating asymmetric subcellular localisation of the core planar polarity pathway proteins

Planar polarity refers to cellular polarity in an orthogonal plane to apicobasal polarity, and is seen across scales from molecular distributions of proteins to tissue patterning. In many contexts it is regulated by the evolutionarily conserved ‘core' planar polarity pathway that is essential for normal organismal development. Core planar polarity pathway components form asymmetric intercellular complexes that communicate polarity between neighbouring cells and direct polarised cell behaviours and the formation of polarised structures. The core planar polarity pathway consists of six structurally different proteins. In the fruitfly Drosophila melanogaster, where the pathway is best characterised, an intercellular homodimer of the seven-pass transmembrane protein Flamingo interacts on one side of the cell junction with the seven-pass transmembrane protein Frizzled, and on the other side with the four-pass transmembrane protein Strabismus. The cytoplasmic proteins Diego and Dishevelled are co-localised with Frizzled, and Prickle co-localises with Strabismus. Between these six components there are myriad possible molecular interactions, which could stabilise or destabilise the intercellular complexes and lead to their sorting into polarised distributions within cells. Post-translational modifications are key regulators of molecular interactions between proteins. Several post-translational modifications of core proteins have been reported to be of functional significance, in particular phosphorylation and ubiquitination. In this review, we discuss the molecular control of planar polarity and the molecular ecology of the core planar polarity intercellular complexes. Furthermore, we highlight the importance of understanding the spatial control of post-translational modifications in the establishment of planar polarity

Molecular symmetry breaking in the Frizzled-dependent planar polarity pathway

The core planar polarity pathway consists of six proteins that form asymmetric intercellular complexes that segregate to opposite cell ends in developing tissues and specify polarized cell structures or behaviors. Within these complexes, the atypical cadherin Flamingo localizes on both sides of intercellular junctions, where it interacts homophilically in trans via its cadherin repeats, whereas the transmembrane proteins Frizzled and Strabismus localize to the opposite sides of apposing junctions. However, the molecular mechanisms underlying the formation of such asymmetric complexes are poorly understood. Using a novel tissue culture system, we determine the minimum requirements for asymmetric complex assembly in the absence of confounding feedback mechanisms. We show that complexes are intrinsically asymmetric and that an interaction of Frizzled and Flamingo in one cell with Flamingo in the neighboring cell is the key symmetry-breaking step. In contrast, Strabismus is unable to promote homophilic Flamingo trans binding and is only recruited into complexes once Frizzled has entered on the opposite side. This interaction with Strabismus requires intact intracellular loops of the seven-pass transmembrane domain of Flamingo. Once recruited, Strabismus stabilizes the intercellular complexes together with the three cytoplasmic core proteins. We propose a model whereby Flamingo exists in a closed conformation and binding of Frizzled in one cell results in a conformational change that allows its cadherin repeats to interact with a Flamingo molecule in the neighboring cell. Flamingo in the adjacent cell then undergoes a further change in the seven-pass transmembrane region that promotes the recruitment of Strabismus

How do the Fat–Dachsous and core planar polarity pathways act together and independently to coordinate polarized cell behaviours?

Planar polarity describes the coordinated polarization of cells within the plane of a tissue. This is controlled by two main pathways in Drosophila: the Frizzled-dependent core planar polarity pathway and the Fat–Dachsous pathway. Components of both of these pathways become asymmetrically localized within cells in response to long-range upstream cues, and form intercellular complexes that link polarity between neighbouring cells. This review examines if and when the two pathways are coupled, focusing on the Drosophila wing, eye and abdomen. There is strong evidence that the pathways are molecularly coupled in tissues that express a specific isoform of the core protein Prickle, namely Spiny-legs. However, in other contexts, the linkages between the pathways are indirect. We discuss how the two pathways act together and independently to mediate a diverse range of effects on polarization of cell structures and behaviours

A Dual Function for Prickle in Regulating Frizzled Stability during Feedback-Dependent Amplification of Planar Polarity

The core planar polarity pathway coordinates epithelial cell polarity during animal development, and loss of its activity gives rise to a range of defects, from aberrant morphogenetic cell movements to failure to correctly orient structures, such as hairs and cilia. The core pathway functions via a mechanism involving segregation of its protein components to opposite cells ends, where they form asymmetric intracellular complexes that couple cell-cell polarity. This segregation is a self-organizing process driven by feedback interactions between the core proteins themselves. Despite intense efforts, the molecular pathways underlying feedback have proven difficult to elucidate using conventional genetic approaches. Here we investigate core protein function during planar polarization of the Drosophila wing by combining quantitative measurements of protein dynamics with loss-of-function genetics, mosaic analysis, and temporal control of gene expression. Focusing on the key core protein Frizzled, we show that its stable junctional localization is promoted by the core proteins Strabismus, Dishevelled, Prickle, and Diego. In particular, we show that the stabilizing function of Prickle on Frizzled requires Prickle activity in neighboring cells. Conversely, Prickle in the same cell has a destabilizing effect on Frizzled. This destabilizing activity is dependent on the presence of Dishevelled and blocked in the absence of Dynamin and Rab5 activity, suggesting an endocytic mechanism. Overall, our approach reveals for the first time essential in vivo stabilizing and destabilizing interactions of the core proteins required for self-organization of planar polarity

Reciprocal action of casein kinase Iε on core planar polarity proteins regulates clustering and asymmetric localisation

The conserved core planar polarity pathway is essential for coordinating polarised cell behaviours and the formation of polarised structures such as cilia and hairs. Core planar polarity proteins localise asymmetrically to opposite cell ends and form intercellular complexes that link the polarity of neighbouring cells. This asymmetric segregation is regulated by phosphorylation through poorly understood mechanisms. We show that loss of phosphorylation of the core protein Strabismus in the Drosophila pupal wing increases its stability and promotes its clustering at intercellular junctions, and that Prickle negatively regulates Strabismus phosphorylation. Additionally, loss of phosphorylation of Dishevelled – which normally localises to opposite cell edges to Strabismus – reduces its stability at junctions. Moreover, both phosphorylation events are independently mediated by Casein Kinase Iε. We conclude that Casein Kinase Iε phosphorylation acts as a switch, promoting Strabismus mobility and Dishevelled immobility, thus enhancing sorting of these proteins to opposite cell edges

The distribution of Polycomb-group proteins during cell division and development in Drosophila embryos: impact on models for silencing

The subcellular three-dimensional distribution of three polycomb-group (PcG) proteins-polycomb, polyhomeotic and posterior sex combs-in fixed whole-mount Drosophila embryos was analyzed by multicolor confocal fluorescence microscopy. All three proteins are localized in complex patterns of 100 or more loci throughout most of the interphase nuclear volume. The rather narrow distribution of the protein intensities in the vast majority of loci argues against a PcG-mediated sequestration of repressed target genes by aggregation into subnuclear domains. In contrast to the case for PEV repression (Csink, A.K., and S. Henikoff. 1996. Nature. 381:529-531), there is a lack of correlation between the occurrence of PcG proteins and high concentrations of DNA, demonstrating that the silenced genes are not targeted to heterochromatic regions within the nucleus. There is a clear distinction between sites of transcription in the nucleus and sites of PcG binding, supporting the assumption that most PcG binding loci are sites of repressive complexes. Although the PcG proteins maintain tissue-specific repression for up to 14 cell generations, the proteins studied here visibly dissociate from the chromatin during mitosis, and disperse into the cytoplasm in a differential manner. Quantitation of the fluorescence intensities in the whole mount embryos demonstrate that the dissociated proteins are present in the cytoplasm. We determined that <2% of PH remains attached to late metaphase and anaphase chromosomes. Each of the three proteins that were studied has a different rate and extent of dissociation at prophase and reassociation at telophase. These observations have important implications for models of the mechanism and maintenance of PcG-mediated gene repression

The distribution of polycomb-group proteins during cell division and development in Drosophila embryos: Impact on models for silencing

The subcellular three-dimensional distribution of three polycomb-group (PcG) proteins—polycomb, polyhomeotic and posterior sex combs—in fixed whole-mount Drosophila embryos was analyzed by multicolor confocal fluorescence microscopy. All three proteins are localized in complex patterns of 100 or more loci throughout most of the interphase nuclear volume. The rather narrow distribution of the protein intensities in the vast majority of loci argues against a PcG-mediated sequestration of repressed target genes by aggregation into subnuclear domains. In contrast to the case for PEV repression (Csink, A.K., and S. Henikoff. 1996. Nature. 381:529–531), there is a lack of correlation between the occurrence of PcG proteins and high concentrations of DNA, demonstrating that the silenced genes are not targeted to heterochromatic regions within the nucleus. There is a clear distinction between sites of transcription in the nucleus and sites of PcG binding, supporting the assumption that most PcG binding loci are sites of repressive complexes. Although the PcG proteins maintain tissue-specific repression for up to 14 cell generations, the proteins studied here visibly dissociate from the chromatin during mitosis, and disperse into the cytoplasm in a differential manner. Quantitation of the fluorescence intensities in the whole mount embryos demonstrate that the dissociated proteins are present in the cytoplasm. We determined that <2% of PH remains attached to late metaphase and anaphase chromosomes. Each of the three proteins that were studied has a different rate and extent of dissociation at prophase and reassociation at telophase. These observations have important implications for models of the mechanism and maintenance of PcG- mediated gene repression

Molecular symmetry breaking in the Frizzled-dependent planar polarity pathway

The core planar polarity pathway consists of six proteins that form asymmetric intercellular complexes that segregate to opposite cell ends in developing tissues and specify polarized cell structures or behaviors. Within these complexes, the atypical cadherin Flamingo localizes on both sides of intercellular junctions, where it interacts homophilically in trans via its cadherin repeats, whereas the transmembrane proteins Frizzled and Strabismus localize to the opposite sides of apposing junctions. However, the molecular mechanisms underlying the formation of such asymmetric complexes are poorly understood. Using a novel tissue culture system, we determine the minimum requirements for asymmetric complex assembly in the absence of confounding feedback mechanisms. We show that complexes are intrinsically asymmetric and that an interaction of Frizzled and Flamingo in one cell with Flamingo in the neighboring cell is the key symmetry-breaking step. In contrast, Strabismus is unable to promote homophilic Flamingo trans binding and is only recruited into complexes once Frizzled has entered on the opposite side. This interaction with Strabismus requires intact intracellular loops of the seven-pass transmembrane domain of Flamingo. Once recruited, Strabismus stabilizes the intercellular complexes together with the three cytoplasmic core proteins. We propose a model whereby Flamingo exists in a closed conformation and binding of Frizzled in one cell results in a conformational change that allows its cadherin repeats to interact with a Flamingo molecule in the neighboring cell. Flamingo in the adjacent cell then undergoes a further change in the seven-pass transmembrane region that promotes the recruitment of Strabismus