Covariance Matrix Estimation for High-Throughput Biomedical Data with Interconnected Communities

Estimating a covariance matrix is central to high-dimensional data analysis. Empirical analyses of high-dimensional biomedical data, including genomics, proteomics, microbiome, and neuroimaging, among others, consistently reveal strong modularity in the dependence patterns. In these analyses, intercorrelated high-dimensional biomedical features often form communities or modules that can be interconnected with others. While the interconnected community structure has been extensively studied in biomedical research (e.g., gene co-expression networks), its potential to assist in the estimation of covariance matrices remains largely unexplored. To address this gap, we propose a procedure that leverages the commonly observed interconnected community structure in high-dimensional biomedical data to estimate large covariance and precision matrices. We derive the uniformly minimum variance unbiased estimators for covariance and precision matrices in closed forms and provide theoretical results on their asymptotic properties. Our proposed method enhances the accuracy of covariance- and precision-matrix estimation and demonstrates superior performance compared to the competing methods in both simulations and real data analyses.</p

Integrated Scanning Electrochemical Probes Assisted by ECSTM for In Situ Imaging of the Pitting Corrosion Process of Carbon Steel in SCCP Solutions

Pitting corrosion of steel is a common phenomenon featuring a random and dynamic process. An understanding of the mechanism and origin of pitting initiation in the early stage remains ambiguous because it is difficult to sense the pitting activity, topographical variations, and local environment at or around pits of carbon steel in situ in a simulated carbonated concrete pore (SCCP) solution. In this work, integrated scanning electrochemical probes assisted by electrochemical scanning tunneling microscopy (ECSTM) were developed to concurrently image the local topography, local pH, and pitting activity in situ in the vicinity of the pits. Ex situ surface topographies and element contents were also determined by scanning electron microscopy (SEM) and energy-dispersive spectroscopy (EDS). It was found that the pitting initiated at a high rate when carbon steel was immersed in SCCP solution, and then the pitting merged to form beam-like localized corrosion and uniform corrosion finally. The local pH became lower in the vicinity of the pits during the pitting initiation. The results demonstrated that the nuclei of pitting corrosion occurred close to the MnS inclusion or ferrite phase at the steel surface due to selective phase dissolution. Integrated scanning electrochemical probes enabled us to obtain correlated information about the pitting activity, topography, and local environment around pits, which is powerful for the in situ imaging of pitting corrosion with higher resolution and for further understanding the mechanism of localized corrosion of carbon steel in various service environments

<i>Irx7</i> is specifically expressed in the prospective INL during zebrafish retinal development.

<p>Whole-mount <i>in situ</i> hybridization was conducted to elucidate the expression dynamics of <i>irx7</i> in embryonic retinas. (A–G) Dissected eyes obtained from embryos between 38 to 96 hpf. Anterior is to the left and dorsal is up. The black arrowheads indicate the <i>irx7+</i> cells (blue colour) in the retina, the dashed lines indicate the choroid fissure, while the red arrowheads in (D and E) indicate the posterior dorsal region of the retina, the last region to express <i>irx7</i>. (H) A schematic diagram of <i>irx7</i> expression dynamics in the retina from 38 to 52 hpf. The Roman numerals indicate the order of five retinal regions in which <i>irx7</i> appears sequentially. (I–L) Transverse retinal section of the corresponding whole-mount embryo at 38, 50, 72 and 96 hpf. Lateral is to the left and dorsal is up. The black arrowheads indicate the <i>irx7+</i> cells in the retina. Scale bars = 50 µm.</p

Irx7 gene regulatory network for zebrafish retinal development.

<p>A gene regulatory network was constructed using the expression patterns of <i>irx7</i> downstream targets as characterized in this study. The specification circuit of the network consists of TFs that specify different retinal cell types while the differentiation circuit consists of genes that carry out cell type specific functions. For example, <i>opsins</i> in the photoreceptors are responsible for visual signal transduction. Genes that have not been fully characterized yet are represented by a generic gene (<i>cell type</i>-<i>genes</i>) in the differentiation circuit. The activation of these “<i>cell type</i>-genes” by <i>irx7</i>, as well as by other TFs, symbolizes the differentiation of the corresponding cell types driven by the specification circuit. For GCs, an additional <i>“GC genes for dendritic outgrowth”</i> is created to distinguish the specific effects of <i>irx7</i> knockdown on their dendritic outgrowth (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0036145#pone-0036145-g005" target="_blank">Figure 5</a>). If the actual location of the interaction is not well defined, the domain/cell type in which the effector gene is expressed will be used. In addition, different retinal regions, including, GCL, INL and ONL can have cellular interaction (≪-≫) that can trigger signal transduction and in turn modulate gene expression. Note that the network topology is a static global view which consists of information obtained from different stages and studies, thus some genes may have both positive and negative inputs if the regulation is dynamical during development. See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0036145#pone.0036145.s004" target="_blank">File S2</a> for supporting evidence of the connections and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0036145#pone.0036145.s002" target="_blank">Figure S2</a> for a comprehensive network with additional connections between these genes as extracted from other literature references.</p

Irx7 regulates the expression of TFs that specify retinal cell types in ONL and GCL at 52 and 72 hpf.

<p>Whole-mount <i>in situ</i> hybridization of <i>neurod</i> (A–D), <i>crx</i> (E–H), <i>nr2e3</i> (I–L), <i>nrl</i> (M–P) and <i>atoh7</i> (Q–T) was conducted. The most common staining pattern is shown. Embryos were imaged from the ventral side and anterior is up. See text for further discussions. Scale bar = 50 µm.</p

<i>Irx7</i> expresses in non-proliferative cells that are likely to be undifferentiated precursors in mature retina.

<p>To detect for the co-localization of <i>irx7</i> and ACs, BCs, HCs, MCs and proliferative cells in the more differentiated part of the WT retina, <i>in situ</i> hybridization of <i>irx7</i> was conducted in conjunction with immunostaining of anti-Islet1 for ACs, BCs and HCs, anti-GS for MCs and anti-BrdU for proliferative cells. (A) <i>irx7 in situ</i> hybridization with anti-Islet1 immunostaining at 72 hpf. The blue arrowheads indicate the <i>irx7+</i> cells (red colour) in the retina, while the red, cyan and white arrowheads indicate ACs, BCs and HCs respectively (all in green colour). (A′) The magnified view of the white box in (A). (B and D) The retina of embryos treated with BrdU from 60 to 72 hpf and 72 to 80 hpf respectively. The blue arrowheads indicate <i>irx7+</i> cells (red colour), while the pink arrowheads indicate BrdU+ cells (green colour). (B′ and D′) The magnified view of the white box in (B and D) respectively. (C) <i>irx7 in situ</i> hybridization with anti-GS immunostaining at 80 hpf. The blue arrowheads indicate the <i>irx7+</i> cells (red colour), while the yellow arrowheads indicate MCs (green colour). (C′) The magnified view of the white box in (C). Lateral is to the left and dorsal is up for all sections. Note that the RPE layer also showed red fluorescence but that was not a real signal. It was an artifact of the pigmentation in RPE. Since the images of the <i>in situ</i> hybridization were inverted before combining with the fluorescent images obtained from the immunostaining, the darker pigment in RPE, as well as the intense <i>in situ</i> colour, would appear as signal in this transformation. Scale bars = 20 µm.</p

Irx7 knockdown reduces eye size and compromises retinal lamination, INL cells differentiation and dendritic projection of GCs into the INL.

<p>Irx7 knockdown reduced eye size and compromised retinal lamination at 72 hpf, as indicated by the DAPI (A and B) and phalloidin (C and D) stains that highlight nuclei and plexiform layers respectively. The red arrows indicate the IPL and OPL. The insets of (A) and (B) also show that the normal elongated morphology of the photoreceptors in control was compromised in the Irx7 morphant. The INL cells differentiation was analyzed by anti-GS for MCs (E and F), anti-PKC for BCs (G and H) and anti-Islet1 for ACs, BCs and HCs (I–L) at the same stage. Irx7 knockdown did not decrease the zn8+ GCs (N) compared with the controls (M), except for the elimination of a fuzzy domain on the apical side of the GCL (compare Q and R). This domain likely represents the dendritic projections of the GCs into the IPL, as it overlapped with the phalloidin staining of the IPL substantially (O and S). This overlap was completely absent in the morphants (P and T). Lateral is to the left and dorsal is up for all sections, except for (K and L), in which the apical side of retina is up. In addition, the retinal region in the samples with weak signal is highlighted by a dotted yellow line. The features indicated by the arrowheads are further discussed in the text. Scale bars = 20 µm.</p

Irx7 regulates the expression of TFs that specify retinal cell types in INL at 52 and 72 hpf.

<p>Whole-mount <i>in situ</i> hybridization of <i>vsx1</i> (A–D), <i>vsx2</i> (E–H), <i>ptf1a</i> (I–L) was conducted. The most common staining pattern is shown. Embryos were imaged from the ventral side and anterior is up. See text for further discussions. The results for <i>neurod</i>, a TF that specifies cells in both INL and ONL, will be presented in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0036145#pone-0036145-g009" target="_blank">Figure 9</a>. Scale bar = 50 µm.</p

The phenotypes of <i>irx7</i>SMO and <i>irx7</i>MO2 morphants at 72 hpf.

<p>Phenotypic analysis of the optimized <i>irx7</i>SMO (A–E) and <i>irx7</i>MO2 (F–J) injection experiments. (A) The lateral view of an uninjected control embryo for <i>irx7</i>SMO injection experiments. (B) An embryo injected with 10 ng of Control MO. (C–E) Three phenotypic categories of the morphants after injected with 10 ng of <i>irx7</i>SMO. The percentage of embryos that had mild (C), intermediate (D) and severe (E) phenotypes was 13.52% (N = 106), 69.81% (N = 444) and 16.67% (N = 86) respectively. (F) The lateral view of an uninjected control embryo for <i>irx7</i>MO2 injection experiments. (G) An embryo injected with 3 ng of <i>irx7</i>MO2-6 bms. (H–J) Three phenotypic categories of morphants after injected with 3 ng of <i>irx7</i>MO2. The percentage of embryos that had mild (H), intermediate (I), and severe (J) phenotypes were 13.28% (N = 38), 73.78% (N = 211) and 12.94% (N = 37) respectively. Scale bar = 200 µm.</p

Irx7 knockdown compromises photoreceptor differentiation at 72 hpf.

<p>Irx7 knockdown compromised the staining of anti-zpr1 for red-green double cones and anti-zpr3 for rods in the morphants (B and D) compared with the controls at 72 hpf (A and C). (A′, B′, C′ and D′) The corresponding magnified view of the positive signal area in the white box in A, B, C and D. The features indicated by the arrowheads are further discussed in the text. Lateral is to the left and dorsal is up for all sections. In addition, the retinal region in the samples with weak fluorescent signal is highlighted by a dotted yellow line. (E–L) Whole-mount <i>in situ</i> hybridization of three cone <i>opsins</i> (<i>uv</i>, <i>blue</i> and <i>red</i>) and one rod <i>opsin</i> (<i>rho</i>) also indicate the differentiation of these photoreceptors was compromised. The most common staining pattern is shown. The black arrowheads indicate the specific staining (blue colour) of the ventral patch, while the red arrowheads indicate the staining in the ONL. Embryos were imaged from the ventral side and anterior is up. Scale bars = 20 µm for (A)–(D) and 50 µm for (E)–(L).</p