Universal cell type identifier based on number theory.

Preface. Notation. Chapter 1 Fundamentals. 1.0 Introduction. 1.1 A Famous Sequence of Numbers. 1.2 The Euclidean ALgorithm. The Oldest Algorithm. Reversing the Euclidean Algorithm. The Extended GCD Algorithm. The Fundamental Theorem of Arithmetic. Two Applications. 1.3 Modular Arithmetic. 1.4 Fast Powers. A Fast Alforithm for ExponentiationPowers of Matrices, Big-O Notation. Chapter 2 Congruences, Equations, and Powers. 2.0 Introduction. 2.1 Solving Linear Congruences. Linear Diophantine Equations in Two Variables. The Conductor. An Importatnt Quadratic Congruence. 2.2 The Chinese Remainder Theorem. 2.3 PowerMod Patterns. Fermat's Little Theorem. More Patterns in Powers. 2.4 Pseudoprimes. Using the Pseudoprime Test. Chapter 3 Euler's Function. 3.0 Introduction. 3.1 Euler's Function. 3.2 Perfect Numbers and Their Relatives. The Sum of Divisors Function. Perfect Numbers. Amicalbe, Abundant, and Deficient Numbers. 3.3 Euler's Theorem. 3.4 Primitive Roots for Primes. The order of an Integer. Primes Have PRimitive roots. Repeating Decimals. 3.5 Primitive Roots for COmposites. 3.6 The Universal Exponent. Universal Exponents. Power Towers. The Form of Carmichael Numbers. Chapter 4 Prime Numbers. 4.0 Introduction. 4.1 The Number of Primes. We'll Never Run Out of Primes. The Sieve of Eratosthenes. Chebyshev's Theorem and Bertrand's Postulate. 4.2 Prime Testing and Certification. Strong Pseudoprimes. Industrial-Grade Primes. Prime Certification Via Primitive Roots. An Improvement. Pratt Certificates. 4.3 Refinements and Other Directions. Other PRimality Tests. Strong Liars are Scarce. Finding the nth Prime. 4.4 A Doszen Prime Mysteries. Chapter 5 Some Applications. 5.0 Introduction. 5.1 Coding Secrets. Tossing a Coin into a Well. The RSA Cryptosystem. Digital Signatures. 5.2 The Yao Millionaire Problem. 5.3 Check Digits. Basic Check Digit Schemes. A Perfect Check Digit Method. Beyond Perfection: Correcting Errors. 5.4 Factoring Algorithms. Trial Division. Fermat's Algorithm. Pollard Rho. Pollard p-1. The Current Scene. Chapter 6 Quadratic Residues. 6.0 Introduction. 6.1 Pepin's Test. Quadratic Residues. Pepin's Test. Primes Congruent to 1 (Mod. 6.2 Proof of Quadratic Reciprocity. Gauss's Lemma. Proof of Quadratic Recipocity. Jacobi's Extension. An Application to Factoring. 6.3 Quadratic Equations. Chapter 7 Continuec Faction. 7.0 Introduction. 7.1 FInite COntinued Fractions. 7.2 Infinite Continued Fractions. 7.3 Periodic Continued Fractions. 7.4 Pell's Equation. 7.5 Archimedes and the Sun God's Cattle. Wurm's Version: Using Rectangular Bulls. The Real Cattle Problem. 7.6 Factoring via Continued Fractions. Chapter 8 Prime Testing with Lucas Sequences. 8.0 Introduction. 8.1 Divisibility Properties of Lucas Sequencese. 8.2 Prime Tests Using Lucas Sequencesse. Lucas Certification. The Lucas-Lehmer Algorithm Explained. Luca Pseudoprimes. Strong Quadratic Pseudoprimes. Primality Testing's Holy Grail. Chapter 9 Prime Imaginaries and Imaginary Primes. 9.0 Introduction. 9.1 Sums of Two Squares. 9.2 The Gaussian Intergers. Complex Number Theory. Gaussian Primes. The Moat Problem. The Gaussian Zoo. 9.3 Higher Reciprocity 325. Appendix A. Maathematica Basics. 1.0 Introduction. A.1 Plotting. A.2 Typesetting. Sending Files By E-Mail. A.3 Types of Functions. A.4 Lists. A.5 Programs. A.6 Solving Equations. A.7 Symbolic Algebra. Appendix B Lucas Certificates Exist. References. Index of Mathematica Objects. Subject Index.

This chapter introduces basic concepts of elementary number theory such as divisibility, greatest common divisor, and prime and composite numbers. The principal results are Theorem 1.2, which establishes the existence of the greatest common divisor of any two integers, and Theorem 1.10 (the fundamental theorem of arithmetic), which shows that every integer greater than 1 can be represented as a product of prime factors in only one way (apart from the order of the factors). Many of the proofs make use of the following property of integers.

Function first: classifying cell types and circuits of the retina

Article Figures and data Abstract Editor's evaluation Introduction Results Discussion Materials and methods Data availability References Decision letter Author response Article and author information Metrics Abstract Oral inflammatory diseases such as apical periodontitis are common bacterial infectious diseases that may affect the periapical alveolar bone tissues. A protective process occurs simultaneously with the inflammatory tissue destruction, in which mesenchymal stem cells (MSCs) play a primary role. However, a systematic and precise description of the cellular and molecular composition of the microenvironment of bone affected by inflammation is lacking. In this study, we created a single-cell atlas of cell populations that compose alveolar bone in healthy and inflammatory disease states. We investigated changes in expression frequency and patterns related to apical periodontitis, as well as the interactions between MSCs and immunocytes. Our results highlight an enhanced self-supporting network and osteogenic potential within MSCs during apical periodontitis-associated inflammation. MSCs not only differentiated toward osteoblast lineage cells but also expressed higher levels of osteogenic-related markers, including Sparc and Col1a1. This was confirmed by lineage tracing in transgenic mouse models and human samples from oral inflammatory-related alveolar bone lesions. In summary, the current study provides an in-depth description of the microenvironment of MSCs and immunocytes in both healthy and disease states. We also identified key apical periodontitis-associated MSC subclusters and their biomarkers, which could further our understanding of the protective process and the underlying mechanisms of oral inflammatory-related bone disease. Taken together, these results enhance our understanding of heterogeneity and cellular interactions of alveolar bone cells under pathogenic and inflammatory conditions. We provide these data as a tool for investigators not only to better appreciate the repertoire of progenitors that are stress responsive but importantly to help design new therapeutic targets to restore bone lesions caused by apical periodontitis and other inflammatory-related bone diseases. Editor's evaluation Data from scRNA-Seq analysis demonstrated that acute inflammation stimulates periodontal stem cells to differentiate into osteoblast lineage cells to protect the alveolar bone. In murine models and patients with apical periodontitis, the genes and proteins associated with osteogenesis were enriched. The studies help us understand how MSCs respond to inflammation during apical periodontitis disease progression. https://doi.org/10.7554/eLife.82537.sa0 Decision letter Reviews on Sciety eLife's review process Introduction Oral diseases, particularly dental caries and periodontal diseases, affect 3.5 billion people worldwide (Disease et al., 2018). Untreated dental caries can directly lead to pulp necrosis and periapical lesions, resulting in apical periodontitis (AP). Individuals with at least one tooth affected by AP comprise up to 52% of cases, indicating that AP is a highly prevalent disease (Tibúrcio-Machado et al., 2021). Inflammation in the oral cavity can lead to destruction of surrounding periapical tissues and resorption of hard tissues, a consequence of the unbalanced interaction between infection and the immune response (Gazivoda et al., 2009; Márton and Kiss, 2014). Restoring and regenerating the destroyed periapical alveolar bone structures have always been a challenging task in clinical practice. Active inflammation, the tissue injury and the protective process all occur simultaneously in the setting of chronic AP (Márton and Kiss, 2014). Importantly, there is a complex assemblage of immune cell types involved in the pathogenesis, highlighting the importance of polymorphonuclear leukocytes, lymphocytes, and monocyte/macrophages in periapical defense (Braz-Silva et al., 2019; Nair, 2004). Notably, an increasing number of studies report the involvement of mesenchymal stem cells (MSCs) in the protective action that occurs during oral inflammatory diseases, whereby MSCs exert immunomodulatory effects and have regenerative potential (Li et al., 2014; Márton and Kiss, 2000; Nair, 2004). MSC markers such as CD44, CD73, CD90, CD106, and STRO-1 have been observed in human periapical inflammatory tissues (Estrela et al., 2019; Liao et al., 2011). Cells isolated from the inflamed periapical region were able to produce colony-forming unit-fibroblasts (CFU-Fs) with high-osteogenic capacity. It is also reported that interference with MSC mobilization toward the periapex region in an AP mouse model led to enlargement of lesions, accompanied by decreased wound healing markers and increased inflammatory cytokines (Araujo-Pires et al., 2014). These findings indicate the involvement of MSCs in the repair and regeneration of oral inflammatory-related bone lesions. They also suggest that MSCs present promising targets for treating bone lesions, with great potential for modulating inflammation and promoting tissue regeneration. However, most studies have surveyed whole tissues to understand the transcriptomic and cellular profile of these diseases. Specific cell populations and their regulatory molecules, as well as the interaction among different cell populations, remain far from clear. The advances in single-cell technologies offer an unbiased approach for identifying heterogeneous cell subsets, tracking the trajectories of distinct cell clusters and uncovering regulatory relationships between genes (Hwang et al., 2018; Tang et al., 2009). In this study, we collected mandibular alveolar bone samples from control and AP in mice and subjected them to single-cell RNA sequencing (scRNA-seq). The atlas of the mandibular alveolar bone explored the distinct cell subsets and their expression profiles relevant to AP. We also investigated the relationship between MSCs and immune cell subsets. The results reveal the role of a subset of MSCs in inflammation, which showed increased frequency and which formed a self-supporting network. Moreover, MSCs exhibited upregulated osteogenic potential, which was confirmed in transgenic mouse models and human patients with chronic AP. These results advance our understanding of heterogeneity and interactions of alveolar bone cells in the pathogenesis of inflammatory-related bone diseases. Defining key cellular subsets such as MSCs and their biomarkers in inflamed tissue will be important for identifying new therapeutic targets for oral inflammatory-related bone diseases. Results Single-cell transcriptional profiling identified 15 discrete populations in homeostasis and chronic AP samples Individual cells were isolated from alveolar bone of healthy mice and mice with AP. We modeled AP using a well-established AP mouse model in which the mandibular first molar pulp was exposed and subsequently developed chronic AP over a 3 wk period (Taira et al., 2019). Bar-coded cDNA libraries from individual cells were obtained using the 10× Genomics Chromium Controller platform (Zheng et al., 2017; Figure 1A). The combined libraries from healthy and AP alveolar bone contained 15,148 individual cells. The median value of feature_RNA was between 1000 and 2000 (Figure 1—figure supplement 1B). After quality control filtering and removal of the batch effect between batches, the t-stochastic neighbor embedding (t-SNE) method was applied to reduce the dimensionality. Seurat's unbiased cluster detection algorithm identified 15 distinct cell populations (Figure 1B and C). Cluster-specific transcripts were utilized to annotate cell types with classic markers as described in a previous study (Lin et al., 2021). These included B cell (Cd79a), hematopoietic stem cell (HSC) (Cd34), MSC (Col1a1), natural killer (NK) cell (Klrd1), T cell (Cd3g), dendritic cell (Siglech), epithelial cell (Epcam), erythrocyte (Hbb-bt), macrophage (Adgre1), mast cell (Fcer1a), megakaryocyte (Gp1bb), monocyte (Ly6c2), myeloid progenitor (Mpo), neutrophil (S100a8), and pre-B cell (Vpreb1; Figure 1E and F). The top 20 enriched genes in each defined cluster were identified and compared (Figure 1G). Figure 1 with 1 supplement see all Download asset Open asset Identification of the single-cell atlas of alveolar bone using scRNA-sequencing (scRNA-seq) and unbiased clustering. (A) Schematic diagram of the experimental design. (B–C) t-Stochastic neighbor embedding (t-SNE) representation of aligned gene expression data in single cells extracted from mandibles of control mice (n=8340) and apical periodontitis (AP) mice (n=6808) showing 15 distinct clusters and cellular origin. (D) Relative abundance of 15 cell populations composing alveolar bone under healthy and AP conditions. (E) Expression of gene markers in distinct cell types. (F) Gene expression patterns projected onto t-SNE plots of marker genes. Scale: log-transformed gene expression. (G) Heatmap showing the 20 most upregulated genes (ordered by decreasing Padj value) in each cluster defined in B. Scale: log2 fold change. AP led to significant changes in frequency and transcriptional expression of cell populations All the identified cell clusters were present in both AP and control samples, but there were significant differences in the cellular compositions of particular clusters. T cell, B cell, NK cell, macrophage, epithelial cell, and MSC had significantly increased frequency in AP samples. Neutrophil, myeloid progenitor, monocyte, megakaryocyte, mast cell, HSC, and dendritic cell were markedly decreased (Figures 2A and 1D). Figure 2 Download asset Open asset Changes in frequency and transcriptional expression pattern in each cell population from control and apical periodontitis (AP) groups. (A) Bar plot of cells per cluster (AP versus control). Normalization to overall number of inputs per condition. Fisher's exact test with Bonferroni correction was used. *p<0.05, **p<0.01, and ****p<0.0001. All data were exhibited as mean ± SEM. (B) Violin plots of cluster-specific expression of representative genes. (C–G) Violin plots showing genes that significantly changed in each cluster from control and AP. AP is a complex inflammatory process involving innate and adaptive immune responses (Cotti et al., 2014). A variety of inflammatory cells such as neutrophils, mast cells, monocytes, macrophages, and lymphocytes are involved in periapical lesions, highlighting the direct involvement of the immune response in the pathogenesis of AP (Nair, 2004). Neutrophils are important components in the acute phase of AP as a first line of defense. But they are also important in the progression of AP by interacting with microorganisms, leading to tissue damage and chemotaxis (Braz-Silva et al., 2019). Single-cell differential expression analysis revealed that the most significantly enriched genes in neutrophils were various proinflammatory chemokines and cytokines. These included C-X-C motif chemokine ligand 2 (Cxcl2), C-C motif chemokine ligand 6 (Ccl6), NLR family pyrin domain containing 3 (Nlrp3), and Interleukin-1β (Il1b). Notably, we found that C-C motif chemokine receptor like 2 (Ccrl2) was upregulated in neutrophils during AP (Figure 2D). It is responsible for the innate defense against pathogens and is also involved in the regulation of neutrophil migration (Del Prete et al., 2017; Kolaczkowska and Kubes, 2013; Mantovani et al., 2011). Mast cells, monocytes, and macrophages have critical roles in the inflammatory infiltrate during chronic AP (Braz-Silva et al., 2019). The production of Interleukin-6 (Il6) was present in these cell populations with the highest expression level in mast cells. The pro-inflammatory cytokine IL-1β is a key mediator of host response to microbial infection and is associated with the persistence of AP (Morsani et al., 2011; Ng et al., 2008). We found Il1b transcripts in a series of cell types, such as monocyte, macrophage, mast cell, and neutrophil. Of these, macrophages had the highest Il1b expression. Another major cytokine, tumor necrosis factor (Tnf; Cotti et al., 2014), was detected in immunoresponsive cell clusters, such as monocyte, macrophage, mast cell, myeloid progenitor, neutrophil, and HSC, with the highest expression observed in the monocyte population (Figure 2B). Furthermore, gene signatures from monocytes showed that the interferon-induced transmembranes (IFITMs) protein 1 and 2 (Ifitm1 and Ifitm2; Figure 2E) were upregulated the most during AP. These factors have been associated with signal transduction of anti-inflammation activity in the immune system (Yánez et al., 2020). We also detected upregulated expression levels of Ccl9 in the monocyte population from AP. Ccl9 is an important cytokine and is involved in the survival of osteoclasts during the destruction of the periapical bone (Silva et al., 2007). Also, genes coding for pro-inflammatory calcium-binding S100 family proteins such as S100a9 and S100a11 had increased expression in the AP monocyte cluster (Figure 2E). Previous reports demonstrated that macrophages are capable of secreting pro- and anti-inflammatory substances which act on the development and repair of the AP lesions (Italiani and Boraschi, 2014; Shapouri-Moghaddam et al., 2018). Indeed, several genes encoding pro-inflammatory mediators, including Cxcl2, Cxcl16, Il1a, and Ptgs2, were upregulated in macrophages from AP samples (Figure 2C). Expression of anti-inflammatory-associated genes such as Ifitm1 and Ifitm2 was significantly increased in the AP macrophage cluster compared to control cells. Furthermore, Fcγ receptor IIB (Fcgr2b) was markedly upregulated in macrophages from AP samples. Fcgr2b is expressed in most tissue-resident macrophages (Gautier et al., 2012) and functions to inhibit Fcγ-dependent phagocytosis. It also inhibits release of cytokines such as IL-6, TNF-α, IL-1α, as well as neutrophil chemotactants (Clatworthy and Smith, 2004; Espéli et al., 2016). In addition, expression of Apolipoprotein E (Apoe), which can suppress the pro-inflammatory response (Jofre-Monseny et al., 2007), was significantly increased in the macrophage population (Figure 2C). These data indicated the activation of anti-inflammatory factors by macrophages during local inflammation by AP. The major classes of lymphocytes are T lymphocytes, B lymphocytes, and the NK cells. T and B lymphocytes comprise the majority of the inflammatory infiltrate in AP (Graunaite et al., 2012). A significant increase in the expression of inflammatory-associated genes, such as Srgn, Emb, Ctla4, and Il7r could be observed in the AP T cell population (Figure 2F). In the AP B lymphocytes cluster, inflammation-responsive genes (Fth1, Ftl1, Ebf1, and mt-Nd5) were upregulated (Figure 2G). Interestingly, Lars2, the gene encoding a mitochondrial leucyl tRNA synthase (Carminho-Rodrigues et al., 2020; 't Hart et al., 2005), was significantly upregulated in T and B lymphocytes, indicating changes in mitochondrial metabolism in both clusters. Inflammation induces osteoclasts differentiation leading to periapical alveolar bone destruction Bone destruction is a major pathological factor in chronic inflammatory diseases such as AP. Various cytokines including TNF-α, IL-1α, and IL-6 were released by immunocytes to recruit the osteoclast precursors and induce the maturation of osteoclasts (Lyu et al., 2022). We have detected osteoclast markers including Ctsk, Acp5, Mmp9, and Nfatc1 by scRNA-seq. Moreover, Csfr1, Cx3cr1, Itgam, and Tnfrs11a were used to identify osteoclast precursors. Markers of osteoclast and osteoclast precursors were highly expressed in the clusters of monocyte and macrophage (Figure 3A and B). Gene Ontology (GO) analysis showed that inflammation related immune reactions and bone resorption activity were significantly enriched in macrophage cluster (Figure 3C). To further study the differential trajectory of osteoclasts, pseudotime analysis was performed for the clusters of macrophage and monocyte. Two independent branch points were determined, and five monocyte/macrophage subclusters were scattered at different branches in the developmental tree (Figure 3D and G). The results showed that the monocyte cluster differentiated into the macrophage cluster (Figure 3E). During this trajectory, the gene expression pattern across pseudotime showed that osteoclastic genes, such as Ctsk, Acp5, Mmp9, Atp6v0d2, and Dcstamp, were progressively elevated (Figure 3F). Of note, we have observed a branch which was highly positive for Ctsk and Acp5 (Figure 3H), indicating the mature osteoclasts were differentiated from monocyte/macrophage lineage and contributed to inflammatory bone resorption during AP. We have also analyzed the expression of osteoclast related genes using the bulk RNA-seq library built on mandibular samples extracted from mice with AP. Markers of osteoclast and osteoclast precursors were significantly upregulated, confirming the osteoclasts activity in the inflammatory-related bone lesion (Figure 3I). Figure 3 with 1 supplement see all Download asset Open asset Inflammatory-related bone resorption under apical periodontitis (AP) situation. (A) The expression levels of markers of osteoclasts and osteoclast precursors. (B) Violin plots of the expression of osteoclastogenesis genes. (C) Gene Ontology (GO) enrichment analysis of the biological functions of macrophage cluster. (D) Trajectory order of the monocyte/macrophage populations by pseudotime value. (E) The differentiation trajectory of monocyte and macrophage clusters presented on a t-stochastic neighbor embedding (t-SNE) visualization. (F) The expression patterns of osteoclast markers during the trajectory of monocyte/macrophage populations. (G) Distribution of monocytes/macrophages on the developmental tree by clusters. (H) Heatmap of differential genes of three states. (I) Heatmap of genes associated with osteoclastogenesis in bulk RNA-seq analysis. AP leads to reduced transcriptionally inferred cellular interactions with an increased self-supporting network in MSCs We next sought to characterize the cell-cell communication related to the perturbation of signaling pathways detected in the AP samples by employing CellphoneDB (Efremova et al., 2020; Nagai et al., 2021). We identified a close interaction among MSC, macrophage, and dendritic cells under homeostasis conditions (Figure 4A). A similarly close communication was found among MSC, macrophage, and dendritic cells under inflammatory conditions (Figure 4B). Next, we compared the differential cell-cell interaction (CCI) network between AP and control samples using CrossTalkeR (Nagai et al., 2021). The results suggested that AP is associated with an overall decrease in cellular interactions. However, MSC intercellular communication with mast cells and monocytes is upregulated and accompanied by the highest number of interactions within MSCs themselves (Figure 4C). These results indicate that, although cell populations lose their normal physiological interactions, MSCs were able to establish a self-interacting network and coordinate with certain types of cells during chronic AP. Next, we ranked the individual ligands by the number of their interactions. Inflammatory-related proteins (Lgals9, Tnf, and Ccl4), extracellular matrix protein (Fn1), and protein involved in biomineralization (Spp1) were among the highest interactions. Also, Tgfb1, Vegfb, and Vegfa were in the top 10 most abundant ligands (Figure 4D). Bar plots were also generated to display the top 10 upregulated gene/cell pairs, showing that inflammation associated genes (Tnf/neutrophil, Ccl3/mast cell, Ccl3/monocyte, and Il1b/macrophage) and matrix related genes (Sele/MSC, Fn1/MSC, and Fn1/monocyte) were the most influential ligands during AP when compared to control state (Figure 4E). We used a Sankey plot to further focus on MSC-related interactions (Figure 4F). The results indicated that Sele was primarily directed by MSC toward the MSC cluster via multiple receptors including Glg1, Selplg, and Cd44. Moreover, Fn1 was secreted by MSC, monocyte, macrophage, and mast cells toward MSCs (Figure 4G). Of note, Sele and Fn1 are important cell adhesion molecules that mediate cell homing and migration (Frenette et al., 1998; To and Midwood, 2011). This was in accordance with the increased frequency of the MSC cluster (Figure 2A) as well as the upregulated cellular interaction among MSCs and other cell populations. Figure 4 Download asset Open asset Apical periodontitis (AP) suppressed transcriptionally inferred cellular interactions and increased a self-supporting network within the mesenchymal stem cell (MSC) cluster. (A–C) Network plot of ligand-receptor activity in control (A), AP (B), and AP versus control (C). (D) Bar plot of top 10 most abundant ligands in all inferred ligand-receptor interactions. (E) Ranking of ligand/source regarding communication gains in AP state. (F and G) Sankey plot listing all predicted source, receptor, and receiver interactions associated with Fn1 and Sele. scRNA-seq based identification of AP-associated MSC population MSCs represented the non-immune cell population in alveolar bone, constituting 1.76% of total identified cells. This cell population could be decomposed into four subclusters (Figure 5A). The most dominant subcluster was characterized by high expression of Prrx1, platelet-derived growth factor receptor β (Pdgfrb), and hematopoiesis supporting factors such as C-X-C motif chemokine 12 (Cxcl12) and angiopoietin (Angpt1). It was also characterized by osteogenic-related markers, such as Runt-related transcription factor 2 (Runx2), Sp7, and was thus classified as the MSC_osteolineage cells (OLCs). The other subclusters were identified as MSC_endothelial and Figure and B). The cell composition of the MSC cluster indicated an of subcluster during the subcluster was reduced (Figure Of note, we performed lineage tracing using and transgenic mouse models to identify stem cells in alveolar bone and the surrounding periodontal tissues et al., 2020; et al., et al., 2020; et al., 2020). The results suggested increased of and as well as upregulated and periodontal stem cells in AP lesions (Figure supplement 1A). We compared the MSC marker genes, identified among the four enrichment analysis confirmed that there are four MSC populations. were enriched for tissue system and osteoblast MSC_endothelial cells enrichment for and cells were enriched for of and suggested highly of cells, including regulation of protein from mitochondrial and (Figure supplement Figure with 3 see all Download asset Open asset Identification and of apical periodontitis mesenchymal stem cell (MSC) population and (A) t-Stochastic neighbor embedding (t-SNE) representation an of single cells within the MSC cluster. (B) Violin plots of MSC expression of representative genes. (C) The of four subclusters of MSC population were in control and AP groups. (D) Violin plots the changes in the expression of top upregulated genes in the MSC cluster. and of Sparc (E) and (F) in (G) of in periodontal stem cells (H) tracing analysis of and AP stimulates MSC differentiation toward osteoblast lineage cells We next compared the data from the MSC cluster between AP and control groups. the top upregulated genes, we observed a significant in the single cell expression levels of such as secreted protein that is and in and accompanied by a toward increased expression (Figure We their by and bulk RNA analysis. The results revealed that and were significantly upregulated, and to increase during AP (Figure supplement B). In confirmed the upregulated expression of osteogenic-related markers, such as Sparc and in the AP were increased of cells in the bone the protective of MSC to differentiate into and under AP conditions (Figure and F). tracing analysis further revealed that to AP lesions and differentiated into (Figure Moreover, using mouse we identified cells in the alveolar bone, confirming that inflammation could the progenitor cells in the bone and toward thus to the protective during AP (Figure We investigated the heterogeneity of MSCs by branch expression analysis and pseudotime analysis in 2 et al., The results revealed three distinct states. Of these, state 2 and state 3 represented differentiated cell populations Figure We the of the MSCs with to their states. We the pseudotime of each cluster cell in the state and the results indicate that with MSC_endothelial cells, MSCs were able to differentiate into and cells. cells may to an state (Figure Interestingly, in state as osteogenic exhibited the highest toward differentiation with upregulated of and (Figure Notably, this MSC population increased significantly in the AP (Figure A of the of the gene expression pattern across pseudotime revealed of osteogenic genes such as and during inflammation when compared to the state (Figure the MSC subcluster exhibited gene expression levels of osteogenic markers that increased during AP. These results were in accordance with our previous single-cell analysis of gene expression in the AP (Figure Figure 6 Download asset Open asset Apical periodontitis (AP) stimulates mesenchymal stem cell (MSC) differentiation toward osteoblast lineage cells. (A) analysis of the MSC (B) cell differentiation trajectory of MSC populations in control and AP groups. (C) Heatmap of differential genes of three states. (D) Bar plot of changes between AP and control in as identified in pseudotime analysis. Fisher's exact test with Bonferroni correction was used. ****p<0.0001. All data were exhibited as mean ± SEM. (E) between the pseudotime gene trajectories of MSC showed of and line and indicate marker levels were identified in alveolar bone from patients with AP We next osteogenesis within the MSC population was in human patients with AP. The of and were significantly enhanced in alveolar bone from patients with accompanied by in and (Figure Moreover, detected a of and in AP alveolar bone confirming higher osteogenic potential under AP conditions (Figure of markers in AP lesions from patients is with our previous an of osteogenic within the MSC subcluster in alveolar bone under AP conditions. Figure Download asset Open asset alveolar bone in apical periodontitis (AP) is associated with higher subcluster (A) Gene expression of and in samples of healthy and AP patients showed an upregulated expression in AP in healthy and in AP All data are as the mean ± SEM. (B–C) of and in bone of human alveolar bone exhibited high osteogenic protein levels in AP. 10 In summary, the current study the of MSCs and immunoresponsive cells under healthy and chronic AP including heterogeneity in their of We that this analysis a

The algebraic structure and distribution of prime numbers remain two of the most fundamental problems in mathematics. The Fundamental Theorem of Arithmetic, proved by Euclid, and Goldbach’s conjecture, while universal in scope with respect to how numbers can be represented multiplicatively or additively, do not provide insights into the structure of primes. Similarly, the definition of a prime −as a number divisible only by 1 and itself− or a sieve algorithm, commonly used to generate primes by successively eliminating multiples, offer no insight into the structure of primes. The powerful and persistent consideration of prime numbers as universal “arithmetic quanta” has not necessitated an equally powerful need for parallel research into a deeper and possibly more insightful explanation of primeness, that is, a better understanding of “why” a number is prime. In this paper, prime and coprime numbers are represented and generated by algebraic expressions. Specifically, given the first &amp;lt;i&amp;gt;n&amp;lt;/i&amp;gt; primes, &amp;lt;i&amp;gt;p&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;,&amp;lt;i&amp;gt; p&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;,…, &amp;lt;i&amp;gt;p&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;, sufficient conditions are given for expressing primes greater than &amp;lt;i&amp;gt;p&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;, and coprimes with prime factors greater than &amp;lt;i&amp;gt;p&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;, as algebraic functions of &amp;lt;i&amp;gt;p&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;,&amp;lt;i&amp;gt; p&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;,…, &amp;lt;i&amp;gt;p&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;&amp;lt;/i&amp;gt;. Thus, primality and co-primality are shown to be mathematical properties with inherently evolutionary algebraic characteristics, since larger primes and coprimes can be generated algebraically from smaller ones. The methodology described in the paper can be a useful tool in the study and analysis of the complexity, structure, interrelationships and distribution of primes and coprimes.

Clustering and cell type classification are a vital step of analyzing scRNA-seq data to reveal the complexity of the tissue (e.g. the number of cell types and the transcription characteristics of the respective cell type). Recently, deep learning-based single-cell clustering algorithms become popular since they integrate the dimensionality reduction with clustering. But these methods still have unstable clustering effects for the scRNA-seq datasets with high dropouts or noise. In this study, a novel single-cell RNA-seq deep embedding clustering via convolutional autoencoder embedding and soft K-means (scCAEs) is proposed by simultaneously learning the feature representation and clustering. It integrates the deep learning with convolutional autoencoder to characterize scRNA-seq data and proposes a regularized soft K-means algorithm to cluster cell populations in a learned latent space. Next, a novel constraint is introduced to the clustering objective function to iteratively optimize the clustering results, and more importantly, it is theoretically proved that this objective function optimization ensures the convergence. Moreover, it adds the reconstruction loss to the objective function combining the dimensionality reduction with clustering to find a more suitable embedding space for clustering. The proposed method is validated on a variety of datasets, in which the number of clusters in the mentioned datasets ranges from 4 to 46, and the number of cells ranges from 90 to 30302. The experimental results show that scCAEs is superior to other state-of-the-art methods on the mentioned datasets, and it also keeps the satisfying compatibility and robustness. In addition, for single-cell datasets with the batch effects, scCAEs can ensure the cell separation while removing batch effects.

Single-cell RNA sequencing (scRNA-seq) has unveiled extensive cellular heterogeneity, yet precise cell type annotation and the identification of novel cell populations remain significant challenges. scHeteroNet, a novel graph neural network framework specifically designed to leverage heterophily in scRNA-seq data, is presented. Unlike traditional methods that assume homophily, scHeteroNet captures complex cell-cell interactions by integrating information from both immediate and extended cellular neighborhoods, resulting in highly accurate cell representations. Additionally, scHeteroNet incorporates an innovative novelty propagation mechanism that robustly detects previously uncharacterized cell types. Comprehensive evaluations across diverse scRNA-seq datasets demonstrate that scHeteroNet consistently outperforms state-of-the-art approaches in both cell type classification and novel cell detection. This heterophily-aware approach enhances the ability to uncover cellular diversity, providing deeper insights into complex biological systems and advancing the field of single-cellanalysis.

BrainCellR: A precise cell type nomenclature pipeline for comparative analysis across brain single-cell datasets

The original three cell type classification (eosinophilic, basophilic, chromophobe) of the pituitary has been invalidated by modern staining techniques and more recently by immunohistologic procedures. The resultant observations led to the establishment of the “one cell, one hormone” theory indicating that each pituitary hormone (somatotrophic, thyrotrophic, adrenocorticotrophic, melanotrophic, lactogenic) with the exeption of the two gonadotrophic hormones is produced by a particular cell type characterized by distinct staining properties. The two gonadotrophic hormones (follicle stimulating and luteinizing hormone) are probably produced by the same cell.—The ultrastructural examination of the normal human anterior pituitary led to the same conclusion. Differences in cell size and shape, number, distribution and morphology of granules in the cytoplasm and amount of rough-surfaced endoplasmic reticulum are all used to distinguish the individual cell types. The exact correlation of each hormone with its production site has been established only in experimental animals and no unanimity has been reached for all cell types in man. Several publications agree only in the description of three cells (somatotrophic, prolactin producing, thyrotrophic).—In addition to the six granulated hypophysial cells involved in hormone production there are nongranulated cells, which are involved in the delineation of the colloid containing follicles. These follicular cells have star-like processes which extend between the secretory cells. There is no evidence to suggest hormone production in the follicular cells. They seem to represent the only true chromophobe cell since the staining characteristics observed with modern staining methods is dependent upon the stored hormone granules.

With the advent of single cell/nucleus RNA sequencing (sc/snRNA-seq), the field of cell phenotyping is now a data-driven exercise providing statistical evidence to support cell type/state categorization. However, the task of classifying cells into specific, well-defined categories with the empirical data provided by sc/snRNA-seq remains nontrivial due to the difficulty in determining specific differences between related cell types with close transcriptional similarities, resulting in challenges with matching cell types identified in separate experiments. To investigate possible approaches to overcome these obstacles, we explored the use of supervised machine learning methods-logistic regression, support vector machines, random forests, neural networks, and light gradient boosting machine (LightGBM)-as approaches to classify cell types using snRNA-seq datasets from human brain middle temporal gyrus (MTG) and human kidney. Classification accuracy was evaluated using an F-beta score weighted in favor of precision to account for technical artifacts of gene expression dropout. We examined the impact of hyperparameter optimization and feature selection methods on F-beta score performance. We found that the best performing model for granular cell type classification in both datasets is a multinomial logistic regression classifier and that an effective feature selection step was the most influential factor in optimizing the performance of the machine learning pipelines.

BackgroundMultiplex immunofluorescence (mIF) can provide invaluable insights on spatial biology and the complexities of the immune tumor microenvironment (iTME), however, current analysis methods are laborious and user dependent. We applied...

Single cell transcriptomics measures gene expression data of large number of genes, concurrently, from tens of thousands of cells present in a studied biological sample. It is difficult to obtain good classification results due to high data dimensionality and variability of biological states. We performed a preliminary study to assess the feasibility of using supervised machine learning methods to classify peripheral blood mononuclear cell (PBMC) types from single cell gene expression data. We analyzed a large PBMC data set $(\sim 120,000$ PBMC cells), selected 47 genes (from 30698 features) suitable as SML classification features, and performed classification using 20 machine learning algorithms. Data sets represented three sample processing strategies: PBMC separation (two data sets), and experimental cell sorting by (two data sets). The accuracy in 5-class classification among 20 methods was 91-97% (PBMC separation), 97-100% (magnetic-activated cell sorting), and 82-99% (fluorescence-activated cell sorting). Our results indicate the feasibility of supervised machine learning for classification of cells into major PBMC cell types using a small number of classification features from single cell gene expression data.

The microdomain that orchestrates action potential initiation in neurons is the axon initial segment (AIS). It has long been considered to be a rather homogeneous domain at the very proximal axon hillock with relatively stable length, particularly in cortical pyramidal cells. However, studies in other brain regions paint a different picture. In hippocampal CA1, up to 50% of axons emerge from basal dendrites. Further, in about 30% of thick-tufted layer V pyramidal neurons in rat somatosensory cortex, axons have a dendritic origin. Consequently, the AIS is separated from the soma. Recent in vitro and in vivo studies have shown that cellular excitability is a function of AIS length/position and somatodendritic morphology, undermining a potentially significant impact of AIS heterogeneity for neuronal function. We therefore investigated neocortical axon morphology and AIS composition, hypothesizing that the initial observation of seemingly homogeneous AIS is inadequate and needs to take into account neuronal cell types. Here, we biolistically transfected cortical neurons in organotypic cultures to visualize the entire neuron and classify cell types in combination with immunolabeling against AIS markers. Using confocal microscopy and morphometric analysis, we investigated axon origin, AIS position, length, diameter as well as distance to the soma. We find a substantial AIS heterogeneity in visual cortical neurons, classified into three groups: (I) axons with somatic origin with proximal AIS at the axon hillock; (II) axons with somatic origin with distal AIS, with a discernible gap between the AIS and the soma; and (III) axons with dendritic origin (axon-carrying dendrite cell, AcD cell) and an AIS either starting directly at the axon origin or more distal to that point. Pyramidal cells have significantly longer AIS than interneurons. Interneurons with vertical columnar axonal projections have significantly more distal AIS locations than all other cells with their prevailing phenotype as an AcD cell. In contrast, neurons with perisomatic terminations display most often an axon originating from the soma. Our data contribute to the emerging understanding that AIS morphology is highly variable, and potentially a function of the cell type.

Most children are aware that some numbers are even and the rest are odd, but they know little else about our number system. The “board-stretcher” model provides a means to discovering the idea of factors and obtaining a feeling for prime and composite numbers. It portrays the inverse relation between multiplication and division and, as a finale, it can be used to introduce the concept of multiplication by a fraction. This model suggests the notion that the sum of the prime factors of a number (except for four) is always less than the product of the factors. The Fundamental Theorem of Arithmetic (or Unique Factorization Theorem) is understood by youngsters when they see that every composite number can be expressed as a product of prime factors that is unique except for the order of the factors.

Spermatogonia are the precursors of male germ cells. They are a valuable genetic material for the production of transgenic poultry. This technology includes isolation of the spermatogonia from male donor’s testes, transformation, and transplantation of donor cells into the sterilized recipient’s testes. The transplanted spermatogonia subsequently differentiate into male sex cells (sperm). The aim of this study was to optimize the individual stages of donor spermatogonia transplantation into the recipient’s testes to increase the effectiveness of spermatogenesis recovery. In the first stage, the spermatogenesis in male chicken was examined to determine the optimal age for isolation of spermatogonia from testes. Histological examinations of male chicken testes (n = 80 birds) were done for 8 age categories, from 1 week to 3 months. It was found that under the age of 4 weeks, the cell population in the seminiferous tubules of male chickens was represented mainly by Sertoli cells and spermatogonia. Maximum percentage of spermatogonia was 69 ± 3% at 4 weeks. At the next stage, a culture of spermatogonia was obtained. Testes of 3-week-old male chickens were used. Separation of the spermatogonia from other types of cells was based on a differential adhesive capacity. The maximum homogeneity of the cell population was established by transfer (3 times) of the supernatant containing unattached cells after 24 h of cultivation into a new culture dish for further cultivation. The cell population is represented mainly by the spermatogonia (89 ± 3%). The lentiviral transduction (pHAGE vector, ZsGreen under CMV promotor) was used to transform the resulting culture of the spermatogonia. The efficiency of spermatogonia infection with lentiviral particles (TU/mL = 2.5 × 108) was 65 ± 2%. After transformation, spermatogonia were introduced into the testes of busulfan-sterilized recipients. The optimal concentration of busulfan treatment after series of experiments from 40 to 100 mg/kg was determined. The effective dose for the removal of own spermatogenic cells was revealed at a concentration of 80 mg/kg of live weight. With complete elimination of other types of spermatogenic cells, the number of Sertoli cells and spermatogonia in the testicle tubules decreased by 39 ± 2% and 98 ± 1%, respectively, compared with the control group. The efficiency of spermatogenesis recovery was assessed based on sperm analysis that was obtained from male recipients (n = 5 birds) 4 months after the introduction of donor cells using PCR. The presence of recombinant DNA (ZsGreen) in recipients’ sperm was shown. Thus, our results indicate the prospect of using spermatogonia as a genetic material for the production of transgenic poultry. Study was supported by the Russian Science Foundation (Project no.16-16-10059).