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SUMMARYPD-1 blockade unleashes the potent antitumor activity of CD8 cells but can also promote immunosuppressive T regulatory (Treg) cells, which may worsen response to immunotherapy. Tumor Treg inhibition is a promising strategy to overcome therapeutic resistance; however, the mechanisms supporting tumor Tregs during PD-1 immunotherapy are largely unexplored. Here, we report that PD-1 blockade increases tumor Tregs in mouse models of immunogenic tumors, including melanoma, and metastatic melanoma patients. Unexpectedly, Treg accumulation was not caused by Treg-intrinsic inhibition of PD-1 signaling but instead depended on an indirect effect of activated CD8 cells. CD8 cells colocalized with Tregs within tumors and produced IL-2, especially after PD-1 immunotherapy. IL-2 upregulated the anti-apoptotic protein ICOS on tumor Tregs, causing their accumulation. ICOS signaling inhibition before PD-1 immunotherapy resulted in increased control of immunogenic melanoma. Thus, interrupting the intratumor CD8:Treg crosstalk is a novel strategy that may enhance the efficacy of immunotherapy in patients.

The monocytic/macrophage system is essential for skeletal muscle homeostasis, but its dysregulation contributes to the pathogenesis of muscle degenerative disorders. Despite our increasing knowledge of the role of macrophages in degenerative disease, it still remains unclear how macrophages contribute to muscle fibrosis. Here, we used single-cell transcriptomics to determine the molecular attributes of dystrophic and healthy muscle macrophages. We identified six novel clusters. Unexpectedly, none corresponded to traditional definitions of M1 or M2 macrophage activation. Rather, the predominant macrophage signature in dystrophic muscle was characterized by high expression of fibrotic factors, galectin-3 and spp1. Spatial transcriptomics and computational inferences of intercellular communication indicated that spp1 regulates stromal progenitor and macrophage interactions during muscular dystrophy. Galectin-3+macrophages were chronically activated in dystrophic muscle and adoptive transfer assays showed that the galectin-3+phenotype was the dominant molecular program induced within the dystrophic milieu. Histological examination of human muscle biopsies revealed that galectin-3+macrophages were also elevated in multiple myopathies. These studies advance our understanding of macrophages in muscular dystrophy by defining the transcriptional programs induced in muscle macrophages, and reveal spp1 as a major regulator of macrophage and stromal progenitor interactions.

Diet can impact on gut health and disease by modulating intestinal stem cells (ISCs). However, it is largely unknown if and how the ISC niche responds to diet and influences ISC function. Here, we demonstrate that Lepr(+) mesenchymal cells (MCs) surrounding intestinal crypts sense diet change and provide a novel niche signal to maintain ISC and progenitor cell proliferation. The abundance of these MCs increases upon administration of a high-fat diet (HFD) but dramatically decreases upon fasting. Depletion of Lepr(+) MCs resulted in fewer intestinal stem/progenitor cells, compromised the architecture of crypt–villus axis and impaired intestinal regeneration. Furthermore, we showed that IGF1 secreted by Lepr(+) MCs is an important effector that promotes proliferation of ISCs and progenitor cells in the intestinal crypt. We conclude that Lepr(+) MCs sense diet alterations and, in turn, modulate intestinal stem/progenitor cell function via a stromal IGF1–epithelial IGF1R axis. These findings reveal that Lepr(+) MCs are important mediators linking systemic diet changes to local ISC function and might serve as a novel therapeutic target for gut diseases.

Cells make decisions through their communication with other cells and receiving signals from their environment. Using single-cell transcriptomics, computational tools have been developed to infer cell–cell communication through ligands and receptors. However, the existing methods only deal with signals sent by the measured cells in the data, the received signals from the external system are missing in the inference. Here, we present exFINDER, a method that identifies such external signals received by the cells in the single-cell transcriptomics datasets by utilizing the prior knowledge of signaling pathways. In particular, exFINDER can uncover external signals that activate the given target genes, infer the external signal-target signaling network (exSigNet), and perform quantitative analysis on exSigNets. The applications of exFINDER to scRNA-seq datasets from different species demonstrate the accuracy and robustness of identifying external signals, revealing critical transition-related signaling activities, inferring critical external signals and targets, clustering signal-target paths, and evaluating relevant biological events. Overall, exFINDER can be applied to scRNA-seq data to reveal the external signal-associated activities and maybe novel cells that send such signals.

Many approaches have been developed to overcome technical noise in single cell (and single nucleus) RNA-sequencing (scRNAseq). As researchers dig deeper into data— looking for rare cell types, subtleties of cell states, and details of gene regulatory networks—there is a growing need for algorithms with controllable accuracy and a minimum ofad hocparameters and thresholds. Impeding this goal is the fact that an appropriate null distribution for scRNAseq cannot simply be extracted from data in the event that ground truth about biological variation is unknown (i.e., most of the time). Here we approach this problem analytically, based on the assumption that scRNAseq data reflect only cell heterogeneity (what we seek to characterize), transcriptional noise (temporal fluctuations randomly distributed across cells), and sampling error (i.e., Poisson noise). We then analyze scRNAseq data without normalization—a step that can skew distributions, particular for sparse data—and calculatep-values associated with key statistics. We develop an improved method for the selection of features for cell clustering and the identification of gene-gene correlations, both positive and negative. Using simulated data, we show that this method, which we call BigSur (Basic Informatics andGeneStatistics fromUnnormalizedReads), accurately captures even weak yet significant correlation structures in scRNAseq data. Applying BigSur to data from a clonal human melanoma cell line, we identify tens of thousands of correlations that, when clustered without supervision into gene communities, both align with cellular components and biological processes, and point toward potentially novel cell biological relationships.

Neural communication networks form the fundamental basis for brain function. These communication networks are enabled by emitted ligands such as neurotransmitters, which activate receptor complexes to facilitate communication. Thus, neural communication is fundamentally dependent on the transcriptome. Here we develop NeuronChat, a method and package for the inference, visualization and analysis of neural-specific communication networks among pre-defined cell groups using single-cell expression data. We incorporate a manually curated molecular interaction database of neural signaling for both human and mouse, and benchmark NeuronChat on several published datasets to validate its ability in predicting neural connectivity. Then, we apply NeuronChat to three different neural tissue datasets to illustrate its functionalities in identifying interneural communication networks, revealing conserved or context-specific interactions across different biological contexts, and predicting communication pattern changes in diseased brains with autism spectrum disorder. Finally, we demonstrate NeuronChat can utilize spatial transcriptomics data to infer and visualize neural-specific cell-cell communication.

Mammary and extramammary Paget's Diseases (PD) are a malignant skin cancer characterized by the appearance of Paget cells. Although easily diagnosed, its pathogenesis remains unknown. Here, single-cell RNA-sequencing identified distinct cellular states, novel biomarkers, and signaling pathways - including mTOR, associated with extramammary PD. Interestingly, we identified MSI1 ectopic overexpression in basal epithelial cells of human PD skin, and show that Msi1 overexpression in the epidermal basal layer of mice phenocopies human PD at histopathological, single-cell and molecular levels. Using this mouse model, we identified novel biomarkers of Paget-like cells that translated to human Paget cells. Furthermore, single-cell trajectory, RNA velocity and lineage-tracing analyses revealed a putative keratinocyte-to-Paget-like cell conversion, supporting the in situ transformation theory of disease pathogenesis. Mechanistically, the Msi1-mTOR pathway drives keratinocyte-Paget-like cell conversion, and suppression of mTOR signaling with Rapamycin significantly rescued the Paget-like phenotype in Msi1-overexpressing transgenic mice. Topical Rapamycin treatment improved extramammary PD-associated symptoms in humans, suggesting mTOR inhibition as a novel therapeutic treatment in PD.

Single-cell RNA sequencing (scRNA-seq) provides details for individual cells; however, crucial spatial information is often lost. We present SpaOTsc, a method relying on structured optimal transport to recover spatial properties of scRNA-seq data by utilizing spatial measurements of a relatively small number of genes. A spatial metric for individual cells in scRNA-seq data is first established based on a map connecting it with the spatial measurements. The cell–cell communications are then obtained by “optimally transporting” signal senders to target signal receivers in space. Using partial information decomposition, we next compute the intercellular gene–gene information flow to estimate the spatial regulations between genes across cells. Four datasets are employed for cross-validation of spatial gene expression prediction and comparison to known cell–cell communications. SpaOTsc has broader applications, both in integrating non-spatial single-cell measurements with spatial data, and directly in spatial single-cell transcriptomics data to reconstruct spatial cellular dynamics in tissues., Dissociation of tissues allows high-throughput expression profiling of single cells, but spatial information is lost. Here the authors apply an unbalanced and structured optimal transport method to infer spatial and signalling relationships between cells from scRNA-seq data by integrating it with spatial imaging data.

SUMMARY Our knowledge of transcriptional heterogeneities in epithelial stem and progenitor cell compartments is limited. Epidermal basal cells sustain cutaneous tissue maintenance and drive wound healing. Previous studies have probed basal cell heterogeneity in stem and progenitor potential, but a comprehensive dissection of basal cell dynamics during differentiation is lacking. Using single-cell RNA sequencing coupled with RNAScope and fluorescence lifetime imaging, we identify three non-proliferative and one proliferative basal cell state in homeostatic skin that differ in metabolic preference and become spatially partitioned during wound re-epithelialization. Pseudotemporal trajectory and RNA velocity analyses predict a quasi-linear differentiation hierarchy where basal cells progress from Col17a1Hi/Trp63Hi state to early-response state, proliferate at the juncture of these two states, or become growth arrested before differentiating into spinous cells. Wound healing induces plasticity manifested by dynamic basal-spinous interconversions at multiple basal transcriptional states. Our study provides a systematic view of epidermal cellular dynamics, supporting a revised “hierarchical-lineage” model of homeostasis., Graphical Abstract, In Brief Haensel et al. performed a comprehensive dissection of the cellular makeup of skin during homeostasis and wound healing and the molecular heterogeneity and cellular dynamics within its stem-cell-containing epidermal basal layer. Their work provides insights and stimulates further investigation into the mechanism of skin maintenance and repair.

SUMMARY Neural crest (NC) cells migrate throughout vertebrate embryos to give rise to a huge variety of cell types, but when and where lineages emerge and their regulation remain unclear. We have performed single-cell RNA sequencing (RNA-seq) of cranial NC cells from the first pharyngeal arch in zebrafish over several stages during migration. Computational analysis combining pseudotime and real-time data reveals that these NC cells first adopt a transitional state, becoming specified mid-migration, with the first lineage decisions being skeletal and pigment, followed by neural and glial progenitors. In addition, by computationally integrating these data with RNA-seq data from a transgenic Wnt reporter line, we identify gene cohorts with similar temporal responses to Wnts during migration and show that one, Atp6ap2, is required for melanocyte differentiation. Together, our results show that cranial NC cell lineages arise progressively and uncover a series of spatially restricted cell interactions likely to regulate such cell-fate decisions., In brief Tatarakis et al. provide a single-cell transcriptomic timeline of cranial neural crest (NC) development in zebrafish and address long-standing questions surrounding the integration of NC cell migration and lineage specification. They find that lineages are specified mid-migration. These fate decisions correspond to shifts in Wnt signaling, and lineages rapidly segregate., Graphical Abstract

Simple Summary Phenotypic plasticity is emerging as a crucial feature across multiple axes of cancer progression. Cells undergoing the epithelial–mesenchymal transition (EMT) can fall into intermediate or hybrid epithelial/mesenchymal (E/M) cell states. Moreover, cancer cells across the EMT spectrum can exhibit the traits of cancer stem cells (CSCs) and communicate through several cell-cell signaling pathways. By integrating multiple analytical tools into a single computational framework, we investigated several single-cell RNA-sequencing (scRNA-seq) datasets and identified the emerging relationship between EMT, acquisition of CSC traits, and cell–cell communication. Our integrated analysis shows that the increase of EM plasticity correlates with high expression of CSC markers as well as intensification of cell–cell signaling between cancer cells. Furthermore, these observations are consistent across different cancer types and anatomical locations. Overall, our results shine light onto the interconnected and multi-dimensional landscape of cancer progression. Abstract Intermediate cell states (ICSs) during the epithelial–mesenchymal transition (EMT) are emerging as a driving force of cancer invasion and metastasis. ICSs typically exhibit hybrid epithelial/mesenchymal characteristics as well as cancer stem cell (CSC) traits including proliferation and drug resistance. Here, we analyze several single-cell RNA-seq (scRNA-seq) datasets to investigate the relation between several axes of cancer progression including EMT, CSC traits, and cell–cell signaling. To accomplish this task, we integrate computational methods for clustering and trajectory inference with analysis of EMT gene signatures, CSC markers, and cell–cell signaling pathways, and highlight conserved and specific processes across the datasets. Our analysis reveals that “standard” measures of pluripotency often used in developmental contexts do not necessarily correlate with EMT progression and expression of CSC-related markers. Conversely, an EMT circuit energy that quantifies the co-expression of epithelial and mesenchymal genes consistently increases along EMT trajectories across different cancer types and anatomical locations. Moreover, despite the high context specificity of signal transduction across different cell types, cells undergoing EMT always increased their potential to send and receive signals from other cells.

During progression from carcinoma in situ to an invasive tumor, the immune system is engaged in complex sets of interactions with various tumor cells. Tumor cell plasticity alters disease trajectories via epithelial-to-mesenchymal transition (EMT). Several of the same pathways that regulate EMT are involved in tumor-immune interactions, yet little is known about the mechanisms and consequences of crosstalk between these regulatory processes. Here we introduce a multiscale evolutionary model to describe tumor-immune-EMT interactions and their impact on epithelial cancer progression from in situ to invasive disease. Through simulation of patient cohorts in silico, the model predicts that a controllable region maximizes invasion-free survival. This controllable region depends on properties of the mesenchymal tumor cell phenotype: its growth rate and its immune-evasiveness. In light of the model predictions, we analyze EMT-inflammation-associated data from The Cancer Genome Atlas, and find that association with EMT worsens invasion-free survival probabilities. This result supports the predictions of the model, and leads to the identification of genes that influence outcomes in bladder and uterine cancer, including FGF pathway members. These results suggest new means to delay disease progression, and demonstrate the importance of studying cancer-immune interactions in light of EMT., Bergman, Karikomi et al. propose a multiscale evolutionary model to describe tumor-immune-EMT interactions and their impact on epithelial cancer progression, which highlights a controllable region that maximizes invasion-free survival. Using EMT-inflammation associated TCGA data, the authors find EMT association worsens invasion-free survival and identify genes that influence outcomes in bladder and uterine cancer.

The second-order implicit integration factor method (IIF2) is effective at solving stiff reaction–diffusion equations owing to its nice stability condition. IIF has previously been applied primarily to systems in which the reaction contained no explicitly time-dependent terms and the boundary conditions were homogeneous. If applied to a system with explicitly time-dependent reaction terms, we find that IIF2 requires prohibitively small time-steps, that are relative to the square of spatial grid sizes, to attain its theoretical second-order temporal accuracy. Although the second-order implicit exponential time differencing (iETD2) method can accurately handle explicitly time-dependent reactions, it is more computationally expensive than IIF2. In this paper, we develop a hybrid approach that combines the advantages of both methods, applying IIF2 to reaction terms that are not explicitly time-dependent and applying iETD2 to those which are. The second-order hybrid IIF-ETD method (hIFE2) inherits the lower complexity of IIF2 and the ability to remain second-order accurate in time for large time-steps from iETD2. Also, it inherits the unconditional stability from IIF2 and iETD2 methods for dealing with the stiffness in reaction–diffusion systems. Through a transformation, hIFE2 can handle nonhomogeneous boundary conditions accurately and efficiently. In addition, this approach can be naturally combined with the compact and array representations of IIF and ETD for systems in higher spatial dimensions. Various numerical simulations containing linear and nonlinear reactions are presented to demonstrate the superior stability, accuracy, and efficiency of the new hIFE method.

Cell-cell communication is a fundamental process that shapes biological tissue. Historically, studies of cell-cell communication have been feasible for one or two cell types and a few genes. With the emergence of single-cell transcriptomics, we are now able to examine the genetic profiles of individual cells at unprecedented scale and depth. The availability of such data presents an exciting opportunity to construct a more comprehensive description of cell-cell communication. This review discusses the recent explosion of methods that have been developed to infer cell-cell communication from non-spatial and spatial single-cell transcriptomics, two promising technologies which have complementary strengths and limitations. We propose several avenues to propel this rapidly expanding field forward in meaningful ways.

Advances in single-cell technologies allow scrutinizing of heterogeneous cell states, however, detecting cell-state transitions from snap-shot single-cell transcriptome data remains challenging. To investigate cells with transient properties or mixed identities, we present MuTrans, a method based on multiscale reduction technique to identify the underlying stochastic dynamics that prescribes cell-fate transitions. By iteratively unifying transition dynamics across multiple scales, MuTrans constructs the cell-fate dynamical manifold that depicts progression of cell-state transitions, and distinguishes stable and transition cells. In addition, MuTrans quantifies the likelihood of all possible transition trajectories between cell states using coarse-grained transition path theory. Downstream analysis identifies distinct genes that mark the transient states or drive the transitions. The method is consistent with the well-established Langevin equation and transition rate theory. Applying MuTrans to datasets collected from five different single-cell experimental platforms, we show its capability and scalability to robustly unravel complex cell fate dynamics induced by transition cells in systems such as tumor EMT, iPSC differentiation and blood cell differentiation. Overall, our method bridges data-driven and model-based approaches on cell-fate transitions at single-cell resolution., How to infer transient cells and cell-fate transitions from snap-shot single cell transcriptome dataset remains a major challenge. Here the authors present a multiscale approach to construct single-cell dynamical manifold, quantify cell stability, and compute transition trajectory and probability between cell states.

Understanding global communications among cells requires accurate representation of cell-cell signaling links and effective systems-level analyses of those links. We construct a database of interactions among ligands, receptors and their cofactors that accurately represent known heteromeric molecular complexes. We then develop CellChat, a tool that is able to quantitatively infer and analyze intercellular communication networks from single-cell RNA-sequencing (scRNA-seq) data. CellChat predicts major signaling inputs and outputs for cells and how those cells and signals coordinate for functions using network analysis and pattern recognition approaches. Through manifold learning and quantitative contrasts, CellChat classifies signaling pathways and delineates conserved and context-specific pathways across different datasets. Applying CellChat to mouse and human skin datasets shows its ability to extract complex signaling patterns. Our versatile and easy-to-use toolkit CellChat and a web-based Explorer (http://www.cellchat.org/) will help discover novel intercellular communications and build cell-cell communication atlases in diverse tissues., Single-cell methods record molecule expressions of cells in a given tissue, but understanding interactions between cells remains challenging. Here the authors show by applying systems biology and machine learning approaches that they can infer and analyze cell-cell communication networks in an easily interpretable way.

Unveiling the molecular mechanisms underlying tissue regeneration provides new opportunities to develop treatments for diabetic ulcers and other chronic skin lesions. Here, we show that Ccl2 secretion by epidermal keratinocytes is directly orchestrated by Nrf2, a prominent transcriptional regulator of tissue regeneration that is activated early after cutaneous injury. Through a unique feedback mechanism, we find that Ccl2 from epidermal keratinocytes not only drives chemotaxis of macrophages into the wound but also triggers macrophage expression of EGF, which in turn activates basal epidermal keratinocyte proliferation. Notably, we find dysfunctional activation of Nrf2 in epidermal keratinocytes of diabetic mice after wounding, which partly explains regenerative impairments associated with diabetes. These findings provide mechanistic insight into the critical relationship between keratinocyte and macrophage signaling during tissue repair, providing the basis for continued investigation of the therapeutic value of Nrf2.

Differential network analysis has become an important approach in identifying driver genes in development and disease. However, most studies capture only local features of the underlying gene-regulatory network topology. These approaches are vulnerable to noise and other changes which mask driver-gene activity. Therefore, methods are urgently needed which can separate the impact of true regulatory elements from stochastic changes and downstream effects. We propose the differential network flow (DNF) method to identify key regulators of progression in development or disease. Given the network representation of consecutive biological states, DNF quantifies the essentiality of each node by differences in the distribution of network flow, which are capable of capturing comprehensive topological differences from local to global feature domains. DNF achieves more accurate driver-gene identification than other state-of-the-art methods when applied to four human datasets from The Cancer Genome Atlas and three single-cell RNA-seq datasets of murine neural and hematopoietic differentiation. Furthermore, we predict key regulators of crosstalk between separate networks underlying both neuronal differentiation and the progression of neurodegenerative disease, among which APP is predicted as a driver gene of neural stem cell differentiation. Our method is a new approach for quantifying the essentiality of genes across networks of different biological states.

How stem cells give rise to epidermis is unclear despite the crucial role the epidermis plays in barrier and appendage formation. Here we use single cell-RNA sequencing to interrogate basal stem cell heterogeneity of human interfollicular epidermis and find four spatially distinct stem cell populations at the top and bottom of rete ridges and transitional positions between the basal and suprabasal epidermal layers. Cell-cell communication modeling suggests that basal cell populations serve as crucial signaling hubs to maintain epidermal communication. Combining pseudotime, RNA velocity, and cellular entropy analyses point to a hierarchical differentiation lineage supporting multi-stem cell interfollicular epidermal homeostasis models and suggest that transitional basal stem cells are stable states essential for proper stratification. Finally, alterations in differentially expressed transitional basal stem cell genes result in severe thinning of human skin equivalents, validating their essential role in epidermal homeostasis and reinforcing the critical nature of basal stem cell heterogeneity., The mechanisms regulating stem cells to give rise to human interfollicular epidermis are unclear. Here, the authors use single cell RNA sequencing to identify heterogeneity within the human neonatal interfollicular epidermis and distinct spatial positioning of at least four basal stem cell populations.

SUMMARY A certain number of epithelial cells in intestinal crypts are DNA damage resistant and contribute to regeneration. However, the cellular mechanism underlying intestinal regeneration remains unclear. Using lineage tracing, we show that cells marked by an Msi1 reporter (Msi1+) are right above Lgr5high cells in intestinal crypts and exhibit DNA damage resistance. Single-cell RNA sequencing reveals that the Msi1+ cells are heterogeneous with the majority being intestinal stem cells (ISCs). The DNA damage-resistant subpopulation of Msi1+ cells is characterized by low-to-negative Lgr5 expression and is more rapidly cycling than Lgr5high radiosensitive crypt base columnar stem cells (CBCs). This enables an efficient repopulation of the intestinal epithelium at early stage when Lgr5high cells are not emerging. Furthermore, relative to CBCs, Msi1+ cells preferentially produce Paneth cells during homeostasis and upon radiation repair. Together, we demonstrate that the DNA damage-resistant Msi1+ cells are cycling ISCs that maintain and regenerate the intestinal epithelium., Graphical Abstract, In Brief Quiescent reserve stem cells in the intestine are thought to activate following irradiation to restore the depleted Lgr5high CBCs. Now, Sheng et al. demonstrate that cycling Msi1+ cells represent DNA damage-resistant ISCs that support efficient repopulation of the intestinal epithelium at the early stage of post-radiation repair, ahead of Lgr5high CBCs.

Gradients of decapentaplegic (Dpp) pattern Drosophila wing imaginal discs, establishing gene expression boundaries at specific locations. As discs grow, Dpp gradients expand, keeping relative boundary positions approximately stationary. Such scaling fails in mutants for Pentagone (pent), a gene repressed by Dpp that encodes a diffusible protein that expands Dpp gradients. Although these properties fit a recent mathematical model of automatic gradient scaling, that model requires an expander that spreads with minimal loss throughout a morphogen field. Here, we show that Pent's actions are confined to within just a few cell diameters of its site of synthesis and can be phenocopied by manipulating non-diffusible Pent targets strictly within the Pent expression domain. Using genetics and mathematical modeling, we develop an alternative model of scaling driven by feedback downregulation of Dpp receptors and co-receptors. Among the model's predictions is a size beyond which scaling fails-something we observe directly in wing discs.

Single cell technologies provide an unprecedented opportunity to explore the heterogeneity in a biological process at the level of single cells. One major challenge in analyzing single cell data is to identify cell subpopulations, stable cell states, and cells in transition between states. To elucidate the transition mechanisms in cell fate dynamics, it is highly desirable to quantitatively characterize cellular states and intermediate states. Here, we present scRCMF, an unsupervised method that identifies stable cell states and transition cells by adopting a nonlinear optimization model that infers the latent substructures from a gene-cell matrix. We incorporate a random coefficient matrix-based regularization into the standard nonnegative matrix decomposition model to improve the reliability and stability of estimating latent substructures. To quantify the transition capability of each cell, we propose two new measures: single-cell transition entropy (scEntropy) and transition probability (scTP). When applied to two simulated and three published scRNA-seq datasets, scRCMF not only successfully captures multiple subpopulations and transition processes in large-scale data, but also identifies transition states and some known marker genes associated with cell state transitions and subpopulations. Furthermore, the quantity scEntropy is found to be significantly higher for transition cells than other cellular states during the global differentiation, and the scTP predicts the “fate decisions” of transition cells within the transition. The present study provides new insights into transition events during differentiation and development.

Glioblastoma (GBM) is a fast-growing type of malignant primary brain tumor. To explore the mechanisms in GBM, complex biological networks are used to reveal crucial changes among different biological states, which reflect on the development of living organisms. It is critical to discover the kernel differential subgraph (KDS) that leads to drastic changes. However, identifying the KDS is similar to the Steiner Tree problem that is an NP-hard problem. In this paper, we developed a criterion to explore the KDS (CKDS), which considered the connectivity and scale of KDS, the topological difference of nodes and function relevance between genes in the KDS. The CKDS algorithm was applied to simulated datasets and three single-cell RNA sequencing (scRNA-seq) datasets including GBM, fetal human cortical neurons (FHCN) and neural differentiation. Then we performed the network topology and functional enrichment analyses on the extracted KDSs. Compared with the state-of-art methods, the CKDS algorithm outperformed on simulated datasets to discover the KDSs. In the GBM and FHCN, seventeen genes (one biomarker, nine regulatory genes, one driver genes, six therapeutic targets) and KEGG pathways in KDSs were strongly supported by literature mining that they were highly interrelated with GBM. Moreover, focused on GBM, there were fifteen genes (including ten regulatory genes, three driver genes, one biomarkers, one therapeutic target) and KEGG pathways found in the KDS of neural differentiation process from activated neural stem cells (aNSC) to neural progenitor cells (NPC), while few genes and no pathway were found in the period from NPC to astrocytes (Ast). These experiments indicated that the process from aNSC to NPC is a key differentiation period affecting the development of GBM. Therefore, the CKDS algorithm provides a unique perspective in identifying cell-type-specific genes and KDSs.

Summary The production of germ cells in vitro would open important new avenues for stem biology and human medicine, but the mechanisms of germ cell differentiation are not well understood. The chicken, as a great model for embryology and development, was used in this study to help us explore its regulatory mechanisms. In this study, we reported a comprehensive genome-wide DNA methylation landscape in chicken germ cells, and transcriptomic dynamics was also presented. By uncovering DNA methylation patterns on individual genes, some genes accurately modulated by DNA methylation were found to be associated with cancers and virus infection, e.g., AKT1 and CTNNB1. Chicken-unique markers were also discovered for identifying male germ cells. Importantly, integrated epigenetic mechanisms were explored during male germ cell differentiation, which provides deep insight into the epigenetic processes associated with male germ cell differentiation and possibly improves treatment options to male infertility in animals and humans., Graphical Abstract, Highlights • The mechanisms of stem cell differentiation were explored using the chick embryo model • The orchestrated stem cell differentiation involves multiple epigenetic events • The unique markers in chick embryo were discovered for identifying male germ cells, In this article, Song, Li, and their colleagues reported a comprehensive genome-wide DNA methylation landscape in chicken germ cells. The integrated epigenetic mechanisms were explored, specifically, germ cell differentiation experiences through a highly orchestrated process that involves multiple epigenetic events, which would give a clue for curing male infertility in animals and humans.

The initial rostrocaudal patterning of the neural tube leads to differential expression of Hox genes that contribute to the specification of motor neuron (MN) subtype identity. Although several 3′ Hox mRNAs are expressed in progenitors in a noisy manner, these Hox proteins are not expressed in the progenitors and only become detectable in postmitotic MNs. MicroRNA biogenesis impairment leads to precocious expression and propagates the noise of Hoxa5 at the protein level, resulting in an imprecise Hoxa5-Hoxc8 boundary. Here we uncover, using in silico simulation, two feed-forward Hox-miRNA loops accounting for the precocious and noisy Hoxa5 expression, as well as an ill-defined boundary phenotype in Dicer mutants. Finally, we identify mir-27 as a major regulator coordinating the temporal delay and spatial boundary of Hox protein expression. Our results provide a novel trans Hox-miRNA circuit filtering transcription noise and controlling the timing of protein expression to confer robust individual MN identity., In the spinal cord, some Hox genes are transcribed in progenitors while their proteins are only detected in differentiating postmitotic motor neurons. Here, the authors show that miRNAs (specifically mir-27) regulate post-transcriptional Hoxa5 expression in motor neurons.

Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for stochastic variants. One challenge in developing adaptive methods for stochastic differential equations (SDEs) is the construction of embedded schemes with direct error estimates. We present a new class of embedded stochastic Runge-Kutta (SRK) methods with strong order 1.5 which have a natural embedding of strong order 1.0 methods. This allows for the derivation of an error estimate which requires no additional function evaluations. Next we derive a general method to reject the time steps without losing information about the future Brownian path termed Rejection Sampling with Memory (RSwM). This method utilizes a stack data structure to do rejection sampling, costing only a few floating point calculations. We show numerically that the methods generate statistically-correct and tolerance-controlled solutions. Lastly, we show that this form of adaptivity can be applied to systems of equations, and demonstrate that it solves a stiff biological model 12.28x faster than common fixed timestep algorithms. Our approach only requires the solution to a bridging problem and thus lends itself to natural generalizations beyond SDEs.

Many signaling systems execute adaptation under circumstances that require noise attenuation. Here, we identify an intrinsic trade-off existing between sensitivity and noise attenuation in the three-node networks. We demonstrate that although fine-tuning timescales in three-node adaptive networks can partially mediate this trade-off in this context, it prolongs adaptation time and imposes unrealistic parameter constraints. By contrast, four-node networks can effectively decouple adaptation and noise attenuation to achieve dual function without a trade-off, provided that these functions are executed sequentially. We illustrate ideas in seven biological examples, including Dictyostelium discoideum chemotaxis and the p53 signaling network and find that adaptive networks are often associated with a noise attenuation module. Our approach may be applicable to finding network design principles for other dual and multiple functions.

Tissue renewal requires proliferative progenitors with long-lasting potential. Designated stem cells within specialized niches are considered to be the primary mechanism for this requirement. Recent studies show that dispersed equipotent progenitors are sufficient to account for fast-paced cellular dynamics in skin oil glands and foetal gut epithelium.

The epithelial-mesenchymal transition (EMT), as a crucial process in embryonic development and cancer metastasis, has been investigated extensively. However, how to quantify the global stability and transition dynamics for EMT under fluctuations remains to be elucidated. Starting from a core EMT genetic circuit composed of three key proteins or microRNAs (microRNA-200, ZEB and SNAIL), we uncovered the potential landscape for the EMT process. Three attractors emerge from the landscape, which correspond to epithelial, mesenchymal and partial EMT states respectively. Based on the landscape, we analyzed two important quantities of the EMT system: the barrier heights between different basins of attraction that describe the degree of difficulty for EMT or backward transition, and the mean first passage time (MFPT) that characterizes the kinetic transition rate. These quantities can be harnessed as measurements for the stability of cell types and the degree of difficulty of transitions between different cell types. We also calculated the minimum action paths (MAPs) by path integral approaches. The MAP delineates the transition processes between different cell types quantitatively. We propose two different EMT processes: a direct EMT from E to P, and a step-wise EMT going through an intermediate state, depending on different extracellular environments. The landscape and kinetic paths we acquired offer new physical and quantitative way for understanding the mechanisms of EMT processes, and indicate the possible roles for the intermediate states.

Simple Summary Despite the successes of immune checkpoint therapy in treating metastatic skin cancers, most patients either fail to respond or become unresponsive to immunotherapy. We found that the cell–cell communication of two different immune cell types, macrophages and memory B cells, correlate very strongly with the response to immunotherapy. We built a mathematical model based on these results that predict different skin cancers would have different response rates and predict that a high ratio of memory B cells to macrophages is optimal for a response to immunotherapy. Abstract The advent of immune checkpoint therapy for metastatic skin cancer has greatly improved patient survival. However, most skin cancer patients are refractory to checkpoint therapy, and furthermore, the intra-immune cell signaling driving response to checkpoint therapy remains uncharacterized. When comparing the immune transcriptome in the tumor microenvironment of melanoma and basal cell carcinoma (BCC), we found that the presence of memory B cells and macrophages negatively correlate in both cancers when stratifying patients by their response, with memory B cells more present in responders. Moreover, inhibitory immune signaling mostly decreases in melanoma responders and increases in BCC responders. We further explored the relationships between macrophages, B cells and response to checkpoint therapy by developing a stochastic differential equation model which qualitatively agrees with the data analysis. Our model predicts BCC to be more refractory to checkpoint therapy than melanoma and predicts the best qualitative ratio of memory B cells and macrophages for successful treatment.

Highlights • Our results emphasize the importance of using phylogenetic and phylodynamic analyses to provide insights into the roles of various interventions to limit the spread of SARS-CoV-2 in China and beyond. • Understanding epidemic dynamics of SARS-CoV-2 in real time is increasingly important for guiding prevention efforts., To investigate the evolutionary and epidemiological dynamics of the current COVID-19 outbreak, a total of 112 genomes of SARS-CoV-2 strains sampled from China and 12 other countries with sampling dates between 24 December 2019 and 9 February 2020 were analyzed. We performed phylogenetic, split network, likelihood-mapping, model comparison, and phylodynamic analyses of the genomes. Based on Bayesian time-scaled phylogenetic analysis with the best-fitting combination models, we estimated the time to the most recent common ancestor (TMRCA) and evolutionary rate of SARS-CoV-2 to be 12 November 2019 (95% BCI: 11 October 2019 and 09 December 2019) and 9.90 × 10-4 substitutions per site per year (95% BCI: 6.29 × 10-4–1.35 × 10-3), respectively. Notably, the very low Re estimates of SARS-CoV-2 during the recent sampling period may be the result of the successful control of the pandemic in China due to extreme societal lockdown efforts. Our results emphasize the importance of using phylodynamic analyses to provide insights into the roles of various interventions to limit the spread of SARS-CoV-2 in China and beyond.

The fusion of digits or toes, syndactyly, can be part of complex syndromes, including Van der Woude syndrome. A subset of Van der Woude cases is caused by dominant negative mutations in the epithelial transcription factor Grainyhead like-3 (GRHL3) and Grhl3(−/−) mice have soft tissue syndactyly. Although impaired interdigital cell death of mesenchymal cells causes syndactyly in multiple genetic mutants, Grhl3(−/−) embryos had normal interdigital cell death, suggesting alternative mechanisms for syndactyly. We found that in digit separation, the overlying epidermis forms a migrating interdigital epithelial tongue (IET) as the epithelium invaginates to separate the digits. Normally, the non-adhesive surface periderm allows the IET to bifurcate as the digits separate. In contrast, in Grhl3(−/−) embryos, the IET moves normally between the digits but fails to bifurcate because of abnormal adhesion of the periderm. Our study identifies epidermal developmental processes required for digit separation.

The mandibular or first pharyngeal arch forms the upper and lower jaws in all gnathostomes. A gene regulatory network that defines ventral, intermediate, and dorsal domains along the dorsal-ventral (D-V) axis of the arch has emerged from studies in zebrafish and mice, but the temporal dynamics of this process remain unclear. To define cell fate trajectories in the arches we have performed quantitative gene expression analyses of D-V patterning genes in pharyngeal arch primordia in zebrafish and mice. Using NanoString technology to measure transcript numbers per cell directly we show that, in many cases, genes expressed in similar D-V domains and induced by similar signals vary dramatically in their temporal profiles. This suggests that cellular responses to D-V patterning signals are likely shaped by the baseline kinetics of target gene expression. Furthermore, similarities in the temporal dynamics of genes that occupy distinct pathways suggest novel shared modes of regulation. Incorporating these gene expression kinetics into our computational models for the mandibular arch improves the accuracy of patterning, and facilitates temporal comparisons between species. These data suggest that the magnitude and timing of target gene expression help diversify responses to patterning signals during craniofacial development.

The transition of epithelial cells into a mesenchymal state (epithelial-to-mesenchymal transition or EMT) is a highly dynamic process implicated in various biological processes. During EMT, cells do not necessarily exist in 'pure' epithelial or mesenchymal states. There are cells with mixed (or hybrid) features of the two, which are termed as the intermediate cell states (ICSs). While the exact functions of ICS remain elusive, together with EMT it appears to play important roles in embryogenesis, tissue development, and pathological processes such as cancer metastasis. Recent single cell experiments and advanced mathematical modeling have improved our capability in identifying ICS and provided a better understanding of ICS in development and disease. Here, we review the recent findings related to the ICS in/or EMT and highlight the challenges in the identification and functional characterization of ICS.

Following injury, skin activates a complex wound healing programme. While cellular and signalling mechanisms of wound repair have been extensively studied, the principles of epidermal-dermal interactions and their effects on wound healing outcomes are only partially understood. To gain new insight into the effects of epidermal-dermal interactions, we developed a multiscale, hybrid mathematical model of skin wound healing. The model takes into consideration interactions between epidermis and dermis across the basement membrane via diffusible signals, defined as activator and inhibitor. Simulations revealed that epidermal-dermal interactions are critical for proper extracellular matrix deposition in the dermis, suggesting these signals may influence how wound scars form. Our model makes several theoretical predictions. First, basal levels of epidermal activator and inhibitor help to maintain dermis in a steady state, whereas their absence results in a raised, scar-like dermal phenotype. Second, wound-triggered increase in activator and inhibitor production by basal epidermal cells, coupled with fast re-epithelialization kinetics, reduces dermal scar size. Third, high-density fibrin clot leads to a raised, hypertrophic scar phenotype, whereas low-density fibrin clot leads to a hypotrophic phenotype. Fourth, shallow wounds, compared to deep wounds, result in overall reduced scarring. Taken together, our model predicts the important role of signalling across dermal-epidermal interface and the effect of fibrin clot density and wound geometry on scar formation. This hybrid modelling approach may be also applicable to other complex tissue systems, enabling the simulation of dynamic processes, otherwise computationally prohibitive with fully discrete models due to a large number of variables.

Recent anatomical evidence suggests a functionally significant back-projection pathway from the subiculum to the CA1. Here we show that the afferent circuitry of CA1-projecting subicular neurons is biased by inputs from CA1 inhibitory neurons and the visual cortex, but lacks input from the entorhinal cortex. Efferents of the CA1-projecting subiculum neurons also target the perirhinal cortex, an area strongly implicated in object-place learning. We identify a critical role for CA1-projecting subicular neurons in object-location learning and memory, and show that this projection modulates place-specific activity of CA1 neurons and their responses to displaced objects. Together, these experiments reveal a novel pathway by which cortical inputs, particularly those from the visual cortex, reach the hippocampal output region CA1. Our findings also implicate this circuitry in the formation of complex spatial representations and learning of object-place associations.

As our catalog of cell states expands, appropriate characterization of these states and the transitions between them is crucial. Here we discuss the roles of intermediate cell states (ICSs) in this growing collection. We begin with definitions and discuss evidence for the existence of ICSs and their relevance in various tissues. We then provide a list of possible functions for ICSs with examples. Finally, we describe means by which ICSs and their functional roles can be identified from single-cell data or predicted from models.

© 2015 Elsevier Inc. Numerical methods for partial differential equations in high-dimensional spaces are often limited by the curse of dimensionality. Though the sparse grid technique, based on a one-dimensional hierarchical basis through tensor products, is popular for handling challenges such as those associated with spatial discretization, the stability conditions on time step size due to temporal discretization, such as those associated with high-order derivatives in space and stiff reactions, remain. Here, we incorporate the sparse grids with the implicit integration factor method (IIF) that is advantageous in terms of stability conditions for systems containing stiff reactions and diffusions. We combine IIF, in which the reaction is treated implicitly and the diffusion is treated explicitly and exactly, with various sparse grid techniques based on the finite element and finite difference methods and a multi-level combination approach. The overall method is found to be efficient in terms of both storage and computational time for solving a wide range of PDEs in high dimensions. In particular, the IIF with the sparse grid combination technique is flexible and effective in solving systems that may include cross-derivatives and non-constant diffusion coefficients. Extensive numerical simulations in both linear and nonlinear systems in high dimensions, along with applications of diffusive logistic equations and Fokker-Planck equations, demonstrate the accuracy, efficiency, and robustness of the new methods, indicating potential broad applications of the sparse grid-based integration factor method.

Yeast cells form a single mating projection when exposed to mating pheromone, a classic example of cell polarity. Prolonged treatment with pheromone or specific mutations results in alternative cell polarity behaviours. The authors performed mathematical modelling to investigate these unusual cell morphologies from the perspective of balancing spatial amplification (i.e. positive feedback that localises components) with spatial tracking (i.e. negative feedback that allows sensing of gradient). First, they used generic models of cell polarity to explore different cell polarity behaviours that arose from changes in the model spatial dynamics. By exploring the positive and negative feedback loops in each stage of a two-stage model, they simulated a variety of cell morphologies including single bending projections, single straight projections, periodic multiple projections and simultaneous double projections. In the second half of the study, they used a two-stage mechanistic model of yeast cell polarity focusing on G-protein signalling to integrate the modelling results more closely with the authors' previously published experimental observations. In summary, the combination of modelling and experiments describes how yeast cells exhibit a diversity of cell morphologies arising from two-stage G-protein signalling dynamics modulated by positive and negative feedbacks.

Adaptation of feathered dinosaurs and Mesozoic birds to new ecological niches was potentiated by rapid diversification of feather vane shapes. The molecular mechanism driving this spectacular process remains unclear. Here, through morphology analysis, transcriptome profiling, functional perturbations and mathematical simulations, we find that mesenchyme-derived GDF10 and GREM1 are major controllers for the topologies of rachidial and barb generative zones (setting vane boundaries), respectively, by tuning the periodic-branching programme of epithelial progenitors. Their interactions with the anterior–posterior WNT gradient establish the bilateral-symmetric vane configuration. Additionally, combinatory effects of CYP26B1, CRABP1 and RALDH3 establish dynamic retinoic acid (RA) landscapes in feather mesenchyme, which modulate GREM1 expression and epithelial cell shapes. Incremental changes of RA gradient slopes establish a continuum of asymmetric flight feathers along the wing, while switch-like modulation of RA signalling confers distinct vane shapes between feather tracts. Therefore, the co-option of anisotropic signalling modules introduced new dimensions of feather shape diversification., Asymmetric feather vane shape was a critical innovation in feather evolution and adaptation for flight. Here, Li and colleagues characterize the multi-module regulatory network that controls feather vane shape and underlies feather diversification.

Summary During epithelial tissue morphogenesis, developmental progenitor cells undergo dynamic adhesive and cytoskeletal remodeling to trigger proliferation and migration. Transcriptional mechanisms that restrict such a mild form of epithelial plasticity to maintain lineage-restricted differentiation in committed epithelial tissues are poorly understood. Here, we report that simultaneous ablation of transcriptional repressor-encoding Ovol1 and Ovol2 results in expansion and blocked terminal differentiation of embryonic epidermal progenitor cells. Conversely, mice overexpressing Ovol2 in their skin epithelia exhibit precocious differentiation accompanied by smaller progenitor cell compartments. We show that Ovol1/Ovol2 -deficient epidermal cells fail to undertake α-catenin-driven actin cytoskeletal reorganization and adhesive maturation and exhibit changes that resemble epithelial-to-mesenchymal transition (EMT). Remarkably, these alterations and defective terminal differentiation are reversed upon depletion of EMT-promoting transcriptional factor Zeb1. Collectively, our findings reveal Ovol-Zeb1-α-catenin sequential repression and highlight Ovol1 and Ovol2 as gatekeepers of epithelial adhesion and differentiation by inhibiting progenitor-like traits and epithelial plasticity.

Robust multiple-fate morphogen gradients are essential for embryo development. Here, we analyze mathematically a model of morphogen gradient (such as Dpp in Drosophila wing imaginal disc) formation in the presence of non-receptors with both diffusion of free morphogens and the movement of morphogens bound to non-receptors. Under the assumption of rapid degradation of unbound morphogen, we introduce a method of functional boundary value problem and prove the existence, uniqueness and linear stability of a biologically acceptable steady-state solution. Next, we investigate the robustness of this steady-state solution with respect to significant changes in the morphogen synthesis rate. We prove that the model is able to produce robust biological morphogen gradients when production and degradation rates of morphogens are large enough and non-receptors are abundant. Our results provide mathematical and biological insight to a mechanism of achieving stable robust long distance morphogen gradients. Key elements of this mechanism are rapid turnover of morphogen to non-receptors of neighoring cells resulting in significant degradation and transport of non-receptor-morphogen complexes, the latter moving downstream through a "bucket brigade" process.

Epithelial-mesenchymal transition (EMT) is an instance of cellular plasticity that plays critical roles in development, regeneration and cancer progression. Recent studies indicate that the transition between epithelial and mesenchymal states is a multi-step and reversible process in which several intermediate phenotypes might coexist. These intermediate states correspond to various forms of stem-like cells in the EMT system, but the function of the multi-step transition or the multiple stem cell phenotypes is unclear. Here, we use mathematical models to show that multiple intermediate phenotypes in the EMT system help to attenuate the overall fluctuations of the cell population in terms of phenotypic compositions, thereby stabilizing a heterogeneous cell population in the EMT spectrum. We found that the ability of the system to attenuate noise on the intermediate states depends on the number of intermediate states, indicating the stem-cell population is more stable when it has more sub-states. Our study reveals a novel advantage of multiple intermediate EMT phenotypes in terms of systems design, and it sheds light on the general design principle of heterogeneous stem cell population.

X-chromosome inactivation (XCI) is the mammalian dosage compensation strategy for balancing sex chromosome content between females and males. While works exist on initiation of symmetric breaking, the underlying allelic choice mechanisms and dynamic regulation responsible for the asymmetric fate determination of XCI remain elusive. Here we combine mathematical modeling and experimental data to examine the mechanism of XCI fate decision by analyzing the signaling regulatory circuit associated with long noncoding RNAs (lncRNAs) involved in XCI. We describe three plausible gene network models that incorporate features of lncRNAs in their localized actions and rapid transcriptional turnovers. In particular, we show experimentally that Jpx (a lncRNA) is transcribed biallelically, escapes XCI and is asymmetrically dispersed between two X’s. Subjecting Jpx to our test of model predictions against previous experimental observations, we identify that a self-enhanced transport feedback mechanism is critical to XCI fate decision. In addition, the analysis indicates that an ultrasensitive response of Jpx signal on CTCF is important in this mechanism. Overall, our combined modeling and experimental data suggest that the self-enhanced transport regulation based on allele-specific nature of lncRNAs and their temporal dynamics provides a robust and novel mechanism for bi-directional fate decisions in critical developmental processes.

It is challenging to obtain reliable and optimal mapping between networks for alignment algorithms when both nodal and topological structures are taken into consideration due to the underlying NP-hard problem. Here, we introduce an adaptive hybrid algorithm that combines the classical Hungarian algorithm and the Greedy algorithm (HGA) for the global alignment of biomolecular networks. With this hybrid algorithm, every pair of nodes with one in each network is first aligned based on node information (e.g., their sequence attributes) and then followed by an adaptive and convergent iteration procedure for aligning the topological connections in the networks. For four well-studied protein interaction networks, i.e., C.elegans , yeast , D.melanogaster , and human , applications of HGA lead to improved alignments in acceptable running time. The mapping between yeast and human PINs obtained by the new algorithm has the largest value of common gene ontology (GO) terms compared to those obtained by other existing algorithms, while it still has lower Mean normalized entropy (MNE) and good performances on several other measures. Overall, the adaptive HGA is effective and capable of providing good mappings between aligned networks in which the biological properties of both the nodes and the connections are important.

Retinoic acid (RA) regulates many cellular behaviors during embryonic development and adult homeostasis. Like other morphogens, RA forms gradients through the use of localized sources and sinks, feedback, and interactions with other signals; this has been particularly well studied in the context of hindbrain segmentation in vertebrate embryos. Yet, as a small lipophilic molecule derived from a dietary source. -. vitamin A. -. RA differs markedly from better-studied polypeptide morphogens in its mechanisms of transport, signaling, and removal. Computational models suggest that the distinctive features of RA gradients make them particularly robust to large perturbations. Such features include combined positive and negative feedback effects via intracellular fatty acid binding proteins and RA-degrading enzymes. Here, we discuss how these features, together with feedback interactions among RA target genes, help enable RA to specify multiple, accurate pattern elements in the developing hindbrain, despite operating in an environment of high cellular and biochemical uncertainty and noise. © 2012 .

An inhomogeneous steady state pattern of nonlinear reaction-diffusion equations with no-flux boundary conditions is usually computed by solving the corresponding time-dependent reaction-diffusion equations using temporal schemes. Nonlinear solvers (e.g., Newton's method) take less CPU time in direct computation for the steady state; however, their convergence is sensitive to the initial guess, often leading to divergence or convergence to spatially homogeneous solution. Systematically numerical exploration of spatial patterns of reaction-diffusion equations under different parameter regimes requires that the numerical method be efficient and robust to initial condition or initial guess, with better likelihood of convergence to an inhomogeneous pattern. Here, a new approach that combines the advantages of temporal schemes in robustness and Newton's method in fast convergence in solving steady states of reaction-diffusion equations is proposed. In particular, an adaptive implicit Euler with inexact solver (AIIE) method is found to be much more efficient than temporal schemes and more robust in convergence than typical nonlinear solvers (e.g., Newton's method) in finding the inhomogeneous pattern. Application of this new approach to two reaction-diffusion equations in one, two, and three spatial dimensions, along with direct comparisons to several other existing methods, demonstrates that AIIE is a more desirable method for searching inhomogeneous spatial patterns of reaction-diffusion equations in a large parameter space.

Background: How morphogen gradients form has long been a subject of controversy. The strongest support for the view that morphogens do not simply spread by free diffusion has come from a variety of studies of the Decapentaplegic (Dpp) gradient of the Drosophila larval wing disc. Results: In the present study, we initially show how the failure, in such studies, to consider the coupling of transport to receptor-mediated uptake and degradation has led to estimates of transport rates that are orders of magnitude too low, lending unwarranted support to a variety of hypothetical mechanisms, such as "planar transcytosis" and "restricted extracellular diffusion." Using several independent dynamic methods, we obtain data that are inconsistent with such models and show directly that Dpp transport occurs by simple, rapid diffusion in the extracellular space. We discuss the implications of these findings for other morphogen systems in which complex transport mechanisms have been proposed. Conclusions: We believe that these findings resolve a major, longstanding question about morphogen gradient formation and provide a solid framework for interpreting experimental observations of morphogen gradient dynamics. © 2012 Elsevier Ltd. All rights reserved.

For reaction-diffusion-advection equations, the stiffness from the reaction and diffusion terms often requires very restricted time step size, while the nonlinear advection term may lead to a sharp gradient in localized spatial regions. It is challenging to design numerical methods that can efficiently handle both difficulties. For reaction-diffusion systems with both stiff reaction and diffusion terms, implicit integration factor (IIF) method and its higher dimensional analog compact IIF (cIIF) serve as an efficient class of time-stepping methods, and their second order version is linearly unconditionally stable. For nonlinear hyperbolic equations, weighted essentially non-oscillatory (WENO) methods are a class of schemes with a uniformly high-order of accuracy in smooth regions of the solution, which can also resolve the sharp gradient in an accurate and essentially non-oscillatory fashion. In this paper, we couple IIF/cIIF with WENO methods using the operator splitting approach to solve reaction-diffusion-advection equations. In particular, we apply the IIF/cIIF method to the stiff reaction and diffusion terms and the WENO method to the advection term in two different splitting sequences. Calculation of local truncation error and direct numerical simulations for both splitting approaches show the second order accuracy of the splitting method, and linear stability analysis and direct comparison with other approaches reveals excellent efficiency and stability properties. Applications of the splitting approach to two biological systems demonstrate that the overall method is accurate and efficient, and the splitting sequence consisting of two reaction-diffusion steps is more desirable than the one consisting of two advection steps, because CWC exhibits better accuracy and stability.

Implicit integration factor (IIF) method, a class of efficient semi-implicit temporal scheme, was introduced recently for stiff reaction-diffusion equations. To reduce cost of IIF, compact implicit integration factor (cIIF) method was later developed for efficient storage and calculation of exponential matrices associated with the diffusion operators in two and three spatial dimensions for Cartesian coordinates with regular meshes. Unlike IIF, cIIF cannot be directly extended to other curvilinear coordinates, such as polar and spherical coordinates, due to the compact representation for the diffusion terms in cIIF. In this paper, we present a method to generalize cIIF for other curvilinear coordinates through examples of polar and spherical coordinates. The new cIIF method in polar and spherical coordinates has similar computational efficiency and stability properties as the cIIF in Cartesian coordinate. In addition, we present a method for integrating cIIF with adaptive mesh refinement (AMR) to take advantage of the excellent stability condition for cIIF. Because the second order cIIF is unconditionally stable, it allows large time steps for AMR, unlike a typical explicit temporal scheme whose time step is severely restricted by the smallest mesh size in the entire spatial domain. Finally, we apply those methods to simulating a cell signaling system described by a system of stiff reaction-diffusion equations in both two and three spatial dimensions using AMR, curvilinear and Cartesian coordinates. Excellent performance of the new methods is observed.