Hexagonal patterns in finite domains

In many mathematical models for pattern formation, a regular hexagonal pattern is stable in an infinite region. However, laboratory and numerical experiments are carried out in finite domains, and this imposes certain constraints on the possible patterns. In finite rectangular domains, it is shown that a regular hexagonal pattern cannot occur if the aspect ratio is rational. In practice, it is found experimentally that in a rectangular region, patterns of irregular hexagons are often observed. This work analyses the geometry and dynamics of irregular hexagonal patterns. These patterns occur in two different symmetry types, either with a reflection symmetry, involving two wavenumbers, or without symmetry, involving three different wavenumbers. The relevant amplitude equations are studied to investigate the detailed bifurcation structure in each case. It is shown that hexagonal patterns can bifurcate subcritically either from the trivial solution or from a pattern of rolls. Numerical simulations of a model partial differential equation are also presented to illustrate the behaviour.

Nme Gene Family Evolutionary History Reveals Pre-Metazoan Origins and High Conservation between Humans and the Sea Anemone, Nematostella vectensis

Background: The Nme gene family is involved in multiple physiological and pathological processes such as cellular differentiation, development, metastatic dissemination, and cilia functions. Despite the known importance of Nme genes and their use as clinical markers of tumor aggressiveness, the associated cellular mechanisms remain poorly understood. Over the last 20 years, several non-vertebrate model species have been used to investigate Nme functions. However, the evolutionary history of the family remains poorly understood outside the vertebrate lineage. The aim of the study was thus to elucidate the evolutionary history of the Nme gene family in Metazoans. Methodology/Principal Findings: Using a total of 21 eukaryote species including 14 metazoans, the evolutionary history of Nme genes was reconstructed in the metazoan lineage. We demonstrated that the complexity of the Nme gene family, initially thought to be restricted to chordates, was also shared by the metazoan ancestor. We also provide evidence suggesting that the complexity of the family is mainly a eukaryotic innovation, with the exception of Nme8 that is likely to be a choanoflagellate/metazoan innovation. Highly conserved gene structure, genomic linkage, and protein domains were identified among metazoans, some features being also conserved in eukaryotes. When considering the entire Nme family, the starlet sea anemone is the studied metazoan species exhibiting the most conserved gene and protein sequence features with humans. In addition, we were able to show that most of the proteins known to interact with human NME proteins were also found in starlet sea anemone. Conclusion/Significance: Together, our observations further support the association of Nme genes with key cellular functions that have been conserved throughout metazoan evolution. Future investigations of evolutionarily conserved Nme gene functions using the starlet sea anemone could shed new light on a wide variety of key developmental and cellular processes.

Global slowing of network oscillations in mouse neocortex by diazepam

SCHEFFZUK, C. , KUKUSHKA, V. , VYSSOTSKI, A. L. , DRAGUHN, A. , TORT, A. B. L. , BRANKACK, J. . Global slowing of network oscillations in mouse neocortex by diazepam. Neuropharmacology , v. 65, p. 123-133, 2013.

Existence and weak-strong uniqueness for the Navier-Stokes-Smoluchowski system over moving domains

This dissertation concerns the well-posedness of the Navier-Stokes-Smoluchowski system. The system models a mixture of fluid and particles in the so-called bubbling regime. The compressible Navier-Stokes equations governing the evolution of the fluid are coupled to the Smoluchowski equation for the particle density at a continuum level. First, working on fixed domains, the existence of weak solutions is established using a three-level approximation scheme and based largely on the Lions-Feireisl theory of compressible fluids. The system is then posed over a moving domain. By utilizing a Brinkman-type penalization as well as penalization of the viscosity, the existence of weak solutions of the Navier-Stokes-Smoluchowski system is proved over moving domains. As a corollary the convergence of the Brinkman penalization is proved. Finally, a suitable relative entropy is defined. This relative entropy is used to establish a weak-strong uniqueness result for the Navier-Stokes-Smoluchowski system over moving domains, ensuring that strong solutions are unique in the class of weak solutions.

Post-rift evolution of the Gulf of Lion margin tested by stratigraphic modelling

The sedimentary architecture of basins and passive margins is determined by a complex interaction of parameters, including subsidence, eustasy, and sediment supply. A quantification of the post-rift (20 Ma-0 Ma) vertical movements of the Gulf of Lion (West Mediterranean) is proposed here based on the stratigraphic study of sedimentary paleomarkers using a large 3D grid of reflection seismic data, correlations with existing drillings, and refraction data. Post-rift subsidence was measured by the direct use of sedimentary geometries analysed in 3D and validated by numerical stratigraphic modelling. Three domains of subsidence were found: on the continental shelf and slope, subsidence corresponds to a seaward tilting with different amplitudes, whereas the deep basin subsides purely vertically. We show that these domains fit with the deeper crustal domains highlighted by previous geophysical data, and that post-break-up subsidence follows the initial hinge lines of the rifting phase. Subsidence rates are quantified on each domain for each stratigraphic interval. At a constant distance from the rotational hinge line, the Plio-Quaternary subsidence rate is constant on the shelf overall. Conversely, Miocene subsidence rates are very different on the eastern and western shelves. Stratigraphic simulations focused on the Messinian salinity crisis (MSC) were also performed. Their results are discussed together with our post-rift subsidence estimates in order to provide ideas and hypotheses for future detailed quantifications of Miocene subsidence, including isostatic readjustments linked to the MSC.

Fundamental domains for proper affine actions of Coxeter groups in three dimensions

We study proper actions of groups $G \cong \Z/2\Z \ast \Z/2\Z \ast \Z/2\Z$ on affine space of three real dimensions. Since $G$ is nonsolvable, work of Fried and Goldman implies that it preserves a Lorentzian metric. A subgroup $\Gamma < G$ of index two acts freely, and $\R^3/\Gamma$ is a Margulis spacetime associated to a hyperbolic surface $\Sigma$. When $\Sigma$ is convex cocompact, work of Danciger, Gu{\'e}ritaud, and Kassel shows that the action of $\Gamma$ admits a polyhedral fundamental domain bounded by crooked planes. We consider under what circumstances the action of $G$ also admits a crooked fundamental domain. We show that it is possible to construct actions of $G$ that fail to admit crooked fundamental domains exactly when the extended mapping class group of $\Sigma$ fails to act transitively on the top-dimensional simplices of the arc complex of $\Sigma$. We also provide explicit descriptions of the moduli space of $G$ actions that admit crooked fundamental domains.

Microfluidic Encapsulation of Pickering Oil Microdroplets into Alginate Microgels for Lipophilic Compound Delivery

Alginate microgels are widely used as delivery systems in food, cosmetics, and pharmaceutical industries for encapsulation and sustained release of hydrophilic compounds and cells. However, the encapsulation of lipophilic molecules inside these microgels remains a great challenge because of the complex oil-core matrix required. The present study describes an original two-step approach allowing the easy encapsulation of several oil microdroplets within alginate microgels. In the first step, stable oil microdroplets were formed by preparing an oil-in-water (O/W) Pickering emulsion. To stabilize this emulsion, we used two solid particles, namely the cotton cellulose nanocrystals (CNC) and calcium carbonate (CaCO3). It was observed that the surface of the oil microdroplets formed was totally covered by a CNC layer, whereas CaCO3 particles were adsorbed onto the cellulose layer. This solid CNC shell efficiently stabilized the oil microdroplets, preventing them from undesired coalescence. In the second step, oil microdroplets resulting from the Pickering emulsion were encapsulated within alginate microgels using microfluidics. Precisely, the outermost layer of oil microdroplets composed of CaCO3 particles was used to initiate alginate gelation inside the microfluidic device, following the internal gelation mode. The released Ca2+ ions induced the gel formation through physical cross-linking with alginate molecules. This innovative and easy to carry out two-step approach was successfully developed to fabricate monodisperse alginate microgels of 85 pm in diameter containing around 12 oil microdroplets of 15 mu m in diameter. These new oil-core alginate microgels represent an attractive system for encapsulation of lipophilic compounds such as vitamins, aroma compounds or anticancer drugs that could be applied in various domains including food, cosmetics, and medical applications.

Unfolding Operator Method for Thin Domains with a Locally Periodic Highly Oscillatory Boundary

We analyze the behavior of solutions of the Poisson equation with homogeneous Neumann boundary conditions in a two-dimensional thin domain which presents locally periodic oscillations at the boundary. The oscillations are such that both the amplitude and period of the oscillations may vary in space. We obtain the homogenized limit problem and a corrector result by extending the unfolding operator method to the case of locally periodic media.

Finite difference calculations of permeability in large domains in a wide porosity range.

Determining effective hydraulic, thermal, mechanical and electrical properties of porous materials by means of classical physical experiments is often time-consuming and expensive. Thus, accurate numerical calculations of material properties are of increasing interest in geophysical, manufacturing, bio-mechanical and environmental applications, among other fields. Characteristic material properties (e.g. intrinsic permeability, thermal conductivity and elastic moduli) depend on morphological details on the porescale such as shape and size of pores and pore throats or cracks. To obtain reliable predictions of these properties it is necessary to perform numerical analyses of sufficiently large unit cells. Such representative volume elements require optimized numerical simulation techniques. Current state-of-the-art simulation tools to calculate effective permeabilities of porous materials are based on various methods, e.g. lattice Boltzmann, finite volumes or explicit jump Stokes methods. All approaches still have limitations in the maximum size of the simulation domain. In response to these deficits of the well-established methods we propose an efficient and reliable numerical method which allows to calculate intrinsic permeabilities directly from voxel-based data obtained from 3D imaging techniques like X-ray microtomography. We present a modelling framework based on a parallel finite differences solver, allowing the calculation of large domains with relative low computing requirements (i.e. desktop computers). The presented method is validated in a diverse selection of materials, obtaining accurate results for a large range of porosities, wider than the ranges previously reported. Ongoing work includes the estimation of other effective properties of porous media.

DnaG interacts with a linker region that joins the N- and C-domains of DnaB and induces the formation of 3-fold symmetric rings

Loading of the replicative ring helicase onto the origin of replication (oriC) is the final outcome of a well coordinated series of events that collectively constitute a primosomal cascade. Once the ring helicase is loaded, it recruits the primase and signals the switch to the polymerization mode. The transient nature of the helicase-primase (DnaB-DnaG) interaction in the Escherichia coli system has hindered our efforts to elucidate its structure and function. Taking advantage of the stable DnaB-DnaG complex in Bacillus stearothermophilus, we have reviewed conflicting mutagenic data from other bacterial systems and shown that DnaG interacts with the flexible linker that connects the N- and C-terminal domains of DnaB. Furthermore, atomic force microscopy (AFM) imaging experiments show that binding of the primase to the helicase induces predominantly a 3-fold symmetric morphology to the hexameric ring. Overall, three DnaG molecules appear to interact with the hexameric ring helicase but a small number of complexes with two and even one DnaG molecule bound to DnaB were also detected. The structural/functional significance of these data is discussed and a speculative structural model for this complex is suggested.

Shared active site architecture between archaeal PolD and multi-subunit RNA polymerases revealed by X-ray crystallography

Archaeal replicative DNA polymerase D (PolD) constitute an atypical class of DNA polymerases made of a proofreading exonuclease subunit (DP1) and a larger polymerase catalytic subunit (DP2), both with unknown structures. We have determined the crystal structures of Pyrococcus abyssi DP1 and DP2 at 2.5 and 2.2 angstrom resolution, respectively, revealing a catalytic core strikingly different from all other known DNA polymerases (DNAPs). Rather, the PolD DP2 catalytic core has the same 'double-psi beta-barrel' architecture seen in the RNA polymerase (RNAP) superfamily, which includes multi-subunit transcriptases of all domains of life, homodimeric RNA-silencing pathway RNAPs and atypical viral RNAPs. This finding bridges together, in non-viral world, DNA transcription and DNA replication within the same protein superfamily. This study documents further the complex evolutionary history of the DNA replication apparatus in different domains of life and proposes a classification of all extant DNAPs.

Biochemical studies of postsynaptic density signaling proteins with a focus on synaptic GTPase activating protein and PDZ domains

Memory storage in the brain involves adjustment of the strength of existing synapses and formation of new neural networks. A key process underlying memory formation is synaptic plasticity, the ability of excitatory synapses to strengthen or weaken their connections in response to patterns of activity between their connected neurons. Synaptic plasticity is governed by the precise pattern of Ca²⁺ influx through postsynaptic N-methyl-D-aspartate-type glutamate receptors (NMDARs), which can lead to the activation of the small GTPases Ras and Rap. Differential activation of Ras and Rap acts to modulate synaptic strength by promoting the insertion or removal of 2-amino-3-(3-hydroxy-5-methyl-isoxazol-4-yl)propanoic acid receptors (AMPARs) from the synapse. Synaptic GTPase activating protein (synGAP) regulates AMPAR levels by catalyzing the inactivation of GTP-bound (active) Ras or Rap. synGAP is positioned in close proximity to the cytoplasmic tail regions of the NMDAR through its association with the PDZ domains of PSD-95. SynGAP’s activity is regulated by the prominent postsynaptic protein kinase, Ca²⁺/calmodulin-dependent protein kinase II (CaMKII) and cyclin-dependent kinase 5 (CDK5), a known binding partner of CaMKII. Modulation of synGAP’s activity by phosphorylation may alter the ratio of active Ras to Rap in spines, thus pushing the spine towards the insertion or removal of AMPARs, subsequently strengthening or weakening the synapse. To date, all biochemical studies of the regulation of synGAP activity by protein kinases have utilized impure preparations of membrane bound synGAP. Here we have clarified the effects of phosphorylation of synGAP on its Ras and Rap GAP activities by preparing and utilizing purified, soluble recombinant synGAP, Ras, Rap, CaMKII, CDK5, PLK2, and CaM. Using mass spectrometry, we have confirmed the presence of previously identified CaMKII and CDK5 sites in synGAP, and have identified novel sites of phosphorylation by CaMKII, CDK5, and PLK2. We have shown that the net effect of phosphorylation of synGAP by CaMKII, CDK5, and PLK2 is an increase in its GAP activity toward HRas and Rap1. In contrast, there is no effect on its GAP activity toward Rap2. Additionally, by assaying the GAP activity of phosphomimetic synGAP mutants, we have been able to hypothesize the effects of CDK5 phosphorylation at specific sites in synGAP. In the course of this work, we also found, unexpectedly, that synGAP is itself a Ca²⁺/CaM binding protein. While Ca²⁺/CaM binding does not directly affect synGAP activity, it causes a conformational change in synGAP that increases the rate of its phosphorylation and exposes additional phosphorylation sites that are inaccessible in the absence of Ca²⁺/CaM.

The postsynaptic density (PSD) is an electron-dense region in excitatory postsynaptic neurons that contains a high concentration of glutamate receptors, cytoskeletal proteins, and associated signaling enzymes. Within the PSD, three major classes of scaffolding molecules function to organize signaling enzymes and glutamate receptors. PDZ domains present in the Shank and PSD-95 scaffolds families serve to physically link AMPARs and NMDARs to signaling molecules in the PSD. Because of the specificity and high affinity of PDZ domains for their ligands, I reasoned that these interacting pairs could provide the core components of an affinity chromatography system, including affinity resins, affinity tags, and elution agents. I show that affinity columns containing the PDZ domains of PSD-95 can be used to purify active PDZ domain-binding proteins to very high purity in a single step. Five heterologously expressed neuronal proteins containing endogenous PDZ domain ligands (NMDAR GluN2B subunit Tail, synGAP, neuronal nitric oxide synthase PDZ domain, cysteine rich interactor of PDZ three and cypin) were purified using PDZ domain resin, with synthetic peptides having the sequences of cognate PDZ domain ligands used as elution agents. I also show that conjugation of PDZ domain-related affinity tags to Proteins Of Interest (POIs) that do not contain endogenous PDZ domains or ligands does not alter protein activity and enables purification of the POIs on PDZ domain-related affinity resins.

Holomorphic functions on non-Runge domains and related problems.

Let U be a domain in CN that is not a Runge domain. We study the topological and algebraic properties of the family of holomorphic functions on U which cannot be approximated by polynomials.