Modified mean curvature flow of entire locally Lipschitz radial graphs in hyperbolic space

The asymptotic Plateau problem asks for the existence of smooth complete hypersurfaces of constant mean curvature with prescribed asymptotic boundary at infinity in the hyperbolic space $\mathbb{H}^{n+1}$. The modified mean curvature flow (MMCF) was firstly introduced by Xiao and the second author a few years back, and it provides a tool using geometric flow to find such hypersurfaces with constant mean curvature in $\mathbb{H}^{n+1}$. Similar to the usual mean curvature flow, the MMCF is the natural negative $L^2$-gradient flow of the area-volume functional $\mathcal{I}(\Sigma)=A(\Sigma)+\sigma V(\Sigma)$ associated to a hypersurface $\Sigma$. In this paper, we prove that the MMCF starting from an entire locally Lipschitz continuous radial graph exists and stays radially graphic for all time. In general one cannot expect the convergence of the flow as it can be seen from the flow starting from a horosphere (whose asymptotic boundary is degenerate to a point).Comment: 22pages, 2 figure

A Modified Mean Curvature Flow of Entire Locally Lipschitz Star-Shaped Hypersurfaces in Hyperbolic Space

In \cite{GS00}, B. Guan and J. Spruck showed the existence of smooth radial graphs of constant mean curvature with prescribed $C^0$, star-shaped boundary at infinity using elliptic PDE methods and the maximum principle. Surfaces of constant mean curvature are critical points of an area functional with a volume constraint. In \cite{DS09}, D. De Silva and J. Spruck showed the same result mentioned above in \cite{GS00} using variational methods. It is a natural question then to ask whether we can approach this problem using the negative gradient flow of that area-volume functional. Such a flow, called \textbf{modified mean curvature flow}, was first introduced by L. Lin and L. Xiao in \cite{LX}. There they showed, starting with a star-shaped hypersurface with a global $C^1$ bound, the longtime existence of the modified mean curvature flow. Moreover, they recovered the previous results by showing the flow converges to a stationary solution. This work is inspired by these three works. Here, we show the longtime existence of a smooth modified mean curvature flow of hypersurfaces in hyperbolic space if the initial hypersurface is locally Lipschitz and star-shaped. This result can be considered as a generalization of the main theorem of \cite{U03} by P. Unterberger, in which they show a longtime existence result of \textbf{mean curvature flow} in the same ambient and initial setting. It's also a hyperbolic version of the nonparametric mean curvature flow in Euclidean space studied by K. Ecker and G. Huisken in \cite{EH91}. There they found a locally Lipschitz vertical graph moving by its mean curvature becomes a smooth vertical graph for all time.\

A Modified Mean Curvature Flow of Entire Locally Lipschitz Star-Shaped Hypersurfaces in Hyperbolic Space

In \cite{GS00}, B. Guan and J. Spruck showed the existence of smooth radial graphs of constant mean curvature with prescribed $C^0$, star-shaped boundary at infinity using elliptic PDE methods and the maximum principle. Surfaces of constant mean curvature are critical points of an area functional with a volume constraint. In \cite{DS09}, D. De Silva and J. Spruck showed the same result mentioned above in \cite{GS00} using variational methods. It is a natural question then to ask whether we can approach this problem using the negative gradient flow of that area-volume functional. Such a flow, called \textbf{modified mean curvature flow}, was first introduced by L. Lin and L. Xiao in \cite{LX}. There they showed, starting with a star-shaped hypersurface with a global $C^1$ bound, the longtime existence of the modified mean curvature flow. Moreover, they recovered the previous results by showing the flow converges to a stationary solution. This work is inspired by these three works. Here, we show the longtime existence of a smooth modified mean curvature flow of hypersurfaces in hyperbolic space if the initial hypersurface is locally Lipschitz and star-shaped. This result can be considered as a generalization of the main theorem of \cite{U03} by P. Unterberger, in which they show a longtime existence result of \textbf{mean curvature flow} in the same ambient and initial setting. It's also a hyperbolic version of the nonparametric mean curvature flow in Euclidean space studied by K. Ecker and G. Huisken in \cite{EH91}. There they found a locally Lipschitz vertical graph moving by its mean curvature becomes a smooth vertical graph for all time.\

Direct Homo- and Hetero-Interactions of MeCP2 and MBD2.

Epigenetic marks like methylation of cytosines at CpG dinucleotides are essential for mammalian development and play a major role in the regulation of gene expression and chromatin architecture. The methyl-cytosine binding domain (MBD) protein family recognizes and translates this methylation mark. We have recently shown that the level of MeCP2 and MBD2, two members of the MBD family, increased during differentiation and their ectopic expression induced heterochromatin clustering in vivo. As oligomerization of these MBD proteins could constitute a factor contributing to the chromatin clustering effect, we addressed potential associations among the MBD family performing a series of different interaction assays in vitro as well as in vivo. Using recombinant purified MBDs we found that MeCP2 and MBD2 showed the stronger self and cross association as compared to the other family members. Besides demonstrating that these homo- and hetero-interactions occur in the absence of DNA, we could confirm them in mammalian cells using co-immunoprecipitation analysis. Employing a modified form of the fluorescent two-hybrid protein-protein interaction assay, we could clearly visualize these associations in single cells in vivo. Deletion analysis indicated that the region of MeCP2 comprising amino acids 163-309 as well the first 152 amino acids of MBD2 are the domains responsible for MeCP2 and MBD2 associations. Our results strengthen the possibility that MeCP2 and MBD2 direct interactions could crosslink chromatin fibers and therefore give novel insight into the molecular mechanism of MBD mediated global heterochromatin architecture

<i>In vivo</i> homo- and hetero-interactions between domains of MeCP2 and MBD2.

<p>(<b>A</b>) GFP- and RFP-tagged domains of MeCP2 and MBD2 as well as GFP control were co-expressed in HEK293-EBNA cells as indicated. After cell lysis using 200 mM NaCl buffer conditions, the extract was incubated with GBP-bound beads for co-immunoprecipitation analysis under the same buffer conditions. The immobilized protein complexes were washed afterwards with the same buffer as used for lysis and co-immunoprecipitation. The immobilized GFP-labeled proteins (B) used for the interaction assay were visualized by western blot using anti GFP antibody. The input (I) and the co-immunoprecipitated fraction (B) of the RFP-labeled proteins were visualized through western blot using anti RFP antibody. The input (I) represents 7% of the total reaction volume. (<b>B</b> and <b>C</b>) Schematic representation of the domains responsible for the homo-and hetero-interactions of MeCP2 and MBD2 (dark grey) illustrating the outcome of the <i>in vivo</i> and <i>in vitro</i> interaction analyses. Numbers stand for amino acid (aa) coordinates. (<b>C</b>) Full-length (fl) MeCP2 and MBD2 directly bind to themselves and each other (green). In case of the MeCP2 homo-interaction, the ID-TRD (aa 163–309) is the domain of MeCP2 that mediates strong direct binding to fl MeCP2 (light grey) and further recognizes the ID-TRD domain independently (dark grey). Regarding MeCP2 and MBD2 hetero-interaction, MeCP2 ID-TRD domain exhibits strong association to fl MBD2 in comparison to other MeCP2 domains (light grey) and further directly and independently interacts to the NH₂-terminal domain (NTD, aa 1–152) of MBD2 (dark grey). The NTD is also the only domain of MBD2 that shows strong binding to fl MeCP2 (light grey) and strongly binds to MeCP2 ID-TRD independently (dark grey). In the case of the MBD2 homo-interaction, the NTD is again the region of MBD2 exerting the strongest binding to fl MBD2 (light grey) and further recognizes MBD2 NTD and COOH-terminal domain (dark grey).</p

Interactions among MBD proteins.

<p>(<b>A</b>) Upper panel: schematic representation of methyl-cytosine binding domain (MBD) proteins. The numbers stand for amino acid coordinates. (MBD) methyl-cytosine binding domain, (TRD) transcriptional repression domain, (CxxC) cysteine rich domain. Lower panel: In vitro pull down experiments using purified strep (st) fused MeCP2 or MBD2 and GFP-labeled full-length (fl) MBD proteins immobilized to GFP-binding protein (GBP) bound sepharose beads. The proteins were extracted using 0.5 M NaCl containing lysis buffer. The interaction assays were performed either in PBS supplemented with 0.05% NP-40 (stMeCP2) or in PBS plus 150 mM NaCl and 0.05% NP-40 (MBD2st). Interacting st-taged fl MeCP2 and MBD2 were assessed by Western blot using st-HRP conjugate. Coomassie Brilliant Blue (CBB) staining of the SDS-gel after protein transfer shows GFP-labeled immobilized MBD proteins used for the pull-down assay. (<b>B, C, D</b>) Interactions between MeCP2 and MBD2 are not bridged by DNA. <i>In vitro</i> pull-down experiments were performed using immobilized RFP-fused MeCP2 (MeCP2R) or MBD2 (MBD2R) and GFP-labeled MeCP2 (MeCP2G) or MBD2 (MBD2G) either with or without addition of ethidium bromide (EtBr; 10 µg/ml). All proteins were extracted in 1 M NaCl containing lysis buffer. In the case of MeCP2 homo-interactions (<b>B</b>), the interaction was performed in PBS plus 0.05% NP-40 buffer. For the homo-interactions of MBD2 (<b>C</b>), PBS was additionally supplemented with 110 mM NaCl and 0.05% NP-40, and for the hetero-binding of MBD2 and MeCP2 (<b>D</b>), PBS plus 125 mM NaCl and 0.05% NP-40 was used. (<b>B, C, D</b>) For input control (I), ¼ of the protein amount used for the interaction assay of the immobilized RFP-tagged proteins was loaded on SDS-PAGE and stained with CBB. Also ¼ of the GFP-tagged proteins used for the pull-down were visualized by western blot using anti GFP (I). Interacting GFP-fused MeCP2 or MBD2 (B) were assessed by western blot using anti GFP antibody.</p

Mapping of domains responsible for MeCP2 and MBD2 homo- and hetero-interactions.

<p>(<b>A</b>) In vitro pull-down experiments with immobilized YFP- or GFP-fused MeCP2 constructs as illustrated and full-length (fl) RFP-labelled MeCP2 (MeCP2R) and MBD2 (MBD2R). The interactions were performed in PBS buffer supplemented with 125 mM NaCl and 0.05% NP-40. Interacting RFP-tagged proteins (B) were assessed by western blot with anti-RFP and Coomassie Brilliant Blue (CBB) staining of the gel after protein transfer was performed to visualize the immobilized YFP- or GFP-fused constructs (B). For input control (I), ¼ of the protein amounts used for the interaction assay was taken and stained by CBB or western blot using anti RFP. (<b>B</b>) Pull-down experiments using Cherry- (Ch) or RFP-fused MBD2 constructs as indicated, immobilized to RFP-binding protein (RBP) bound sepharose beads, and GFP-labelled fl MBD2 (MBD2G) and MeCP2 (MeCP2G). The assays were performed in PBS supplemented with 125 mM NaCl and 0.05% NP-40. The interacting proteins (B) were analyzed by western blot with anti GFP and CBB staining of the gel after protein transfer for the immobilized Cherry- or RFP-fused MBD2 constructs (B). As for (<b>A</b>), ¼ of the protein amounts used for the interaction assay were loaded as input control (I) and visualized either by western blotting with anti GFP or CBB respectively. (<b>C</b>) In vitro binding assays using YFP- or GFP-labelled MeCP2 or MBD2 constructs as indicated, immobilized to GFP-binding protein (GBP) bound beads, and RFP-fused MBD2 NH2-terminal domain (NTD) and Cherry-fused MeCP2 ID-TRD. The interaction was performed in PBS supplemented with 125 mM NaCl and 0.05% NP-40. Interacting Cherry- or RFP-tagged proteins (B) were assessed by western blot with anti RFP and Coomassie Brilliant Blue (CBB) staining of the gel after protein transfer was performed to visualize the immobilized YFP- or GFP-fused constructs (B). ¼ of the protein amounts used for the interaction assay were loaded as input control (I) and visualized either by western blotting with anti RFP or CBB. G, R and Ch stand for GFP, RFP and Cherry respectively.</p

Homo and hetero-interactions of MeCP2 and MBD2 <i>in vivo</i>.

<p>(<b>A, B, C</b>) GFP alone and GFP- and RFP-tagged MeCP2 and MBD2 were co-expressed in HEK293-EBNA cells as indicated. After extraction, co-immunoprecipitation assays were performed at 137 mM (<b>A</b>) or 274 mM (<b>B</b> and <b>C</b>) NaCl using GBP-bound beads. The bound fraction (B) of the immobilized proteins used for the interaction assay was visualized by western blot using anti GFP antibody. The input (I) and bound fraction (B) of the interacting RFP-labelled proteins were visualized using anti RFP antibody. The input (I) represents 3.75% (<b>A</b>) and 7.5% (<b>B</b> and <b>C</b>) of the total reaction volume. (<b>A</b>) Homo-interactions of MeCP2, (<b>B</b>) binding of MBD2 to itself and (<b>C</b>) hetero-interactions of MeCP2 and MBD2. (<b>D, E, F</b>) C2C12 mouse cells were transfected with plasmids coding for (<b>D</b>) RFP and GFP-fused MeCP2, (<b>E</b>) RFP-fused MeCP2 or MBD2, GFP control and a protein fusion of the GFP binding protein (GBP) and lamin B1 (GBP-laminB1), (<b>F</b>) two fluorescently labeled methyl-cytosine binding domain (MBD) proteins as indicated and GBP-laminB1. Shown are representative images of mouse cells expressing the proteins as indicated. Scale bar: 5 µm. The graphs represent % of cells with co-localization of the fluorescent signals. The experiment was repeated twice, analyzing 100 cells (n = 100) each time. Right side: Schematic illustrations of the interaction assay. <b>(D</b>, right side<b>)</b> Localization of RFP- and GFP-fused MeCP2 proteins at pericentric heterochromatin in mouse cells. <b>(E</b>, right side<b>)</b> Mouse cell expressing RFP-labelled MBD protein, GFP control and GBP-laminB1. Due to GBP-laminB1, GFP is recruited at the lamina. The RFP-MBD protein is localized to heterochromatin. <b>(F</b>, right side<b>)</b> Mouse cells expressing GFP and RFP-tagged MBD proteins and GBP-laminB1. In case of an interaction between both fluorescently labeled MBDs, the RFP and GFP signals co-localize. G and R stand for GFP and RFP respectively.</p

Phenotypic analysis of Δ<i>tfeα1</i>/Δ<i>tfeα1</i> cell.

<p>(A) growth curve of WT and Δ<i>tfeα1</i>/Δ<i>tfeα1</i> cell knock cells in glucose-rich (SDM79 with 10 mM glucose) or glucose-free (SDM79GluFree) conditions. (B) Global protein abundance in the partially purified glycosome fraction of WT (x-axis) and Δ<i>tfeα1</i>/Δ<i>tfeα1</i> cell knock cells (y-axis). Each protein identification is presented by a point at log₁₀ of normalized peptide count values taken from the proteome data in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0114628#pone.0114628.s004" target="_blank">S4 Figure</a>. Proteins on the dashed grey line have identical normalized peptide counts in both samples; the grey lines represent a 2-fold abundance in one condition.</p

TAG species analysis and uptake of labeled oleate.

<p>(A) Dominant TAG species in procyclic <i>T. brucei</i> cells identified by ESI/MS/MS after oleate feeding for three days (black columns) or in the control (white columns). For a complete list of TAG species detected see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0114628#pone.0114628.s001" target="_blank">S1 Figure</a>. The nomenclature 54:X indicates the total carbon number of all three acyl chains and the sum of all unsaturated double bonds within the acyl chains. (B) Uptake kinetics upon growth in the presence of radiolabeled oleate for up to 8 h. The incorporation of ¹⁴C oleate into lipid species was quantified by HPTLC and a Storm 860 phosphorimager. PPL, phospholipids; TAG, triacylglycerol; SE, Steryl-esters; DAG, diacylglycerol.</p