Jointly analyzing biological pathway maps and experimental data is critical for understanding how biological processes work in different conditions and why different samples exhibit certain characteristics. This joint analysis, however, poses a significant challenge for visualization. Current techniques are either well suited to visualize large amounts of pathway node attributes, or to represent the topology of the pathway well, but do not accomplish both at the same time. To address this we introduce enRoute, a technique that enables analysts to specify a path of interest in a pathway, extract this path into a separate, linked view, and show detailed experimental data associated with the nodes of this extracted path right next to it. This juxtaposition of the extracted path and the experimental data allows analysts to simultaneously investigate large amounts of potentially heterogeneous data, thereby solving the problem of joint analysis of topology and node attributes. As this approach does not modify the layout of pathway maps, it is compatible with arbitrary graph layouts, including those of hand-crafted, image-based pathway maps. We demonstrate the technique in context of pathways from the KEGG and the Wikipathways databases. We apply experimental data from two public databases, the Cancer Cell Line Encyclopedia (CCLE) and The Cancer Genome Atlas (TCGA) that both contain a wide variety of genomic datasets for a large number of samples. In addition, we make use of a smaller dataset of hepatocellular carcinoma and common xenograft models. To verify the utility of enRoute, domain experts conducted two case studies where they explore data from the CCLE and the hepatocellular carcinoma datasets in the context of relevant pathways. PTEN is a part of the phosphoinositide 3-kinase signaling pathway, which regulates cell-growth [PTEN is mutated it does not fulfill its function and shuts down the pathway, which can lead to tumor growth. Jointly analyzing experimental data and pathways can help in reasoning about and predicting such effects for different conditions. Knowledge about how pathways are modulated by the genetic profile of groups or individual samples can help improving prognosis, treatment, and patient well-being.Biological networks, such as interactions between proteins, biochemical reactions, and signaling processes are commonly depicted in pathway maps. Pathway maps are often hand-crafted and only show the part of the whole known biological network that is immediately relevant for a particular natural process, such as the tyrosine metabolism, or for a particular disease, such as HIV or diabetes. The network described by these pathways is based on published research on the interactions and interdependencies between the various nodes. As a consequence, pathway maps are static and are only valid for the specific processes or disease states they are designed for and fail to adapt to the variation found in real-world data. It is not uncommon, for example, that a de-activation of a node in a cascade invalidates reactions further downstream. For example, the gene l-growth . If PTENCaleydo, an open-source biomolecular visualization framework , which features various other visualization techniques for analyzing tabular and network data.Current approaches for visualizing interdependencies between pathways and experimental data do not scale to the now common large and heterogeneous experimental datasets, which often contain hundreds of experiments and multiple data types. We designed enRoute to remedy this. enRoute consists of two views: the pathway view, which shows the whole pathway and hints at interesting paths, and the enRoute view, which visualizes experimental data for parts of the pathway. In the pathway view, shown in Figure At the beginning of this paper, we give a brief introduction of the biological background, followed by a detailed analysis of the challenges of visualizing graphs with very large numbers of node-attributes. We continue by reviewing the literature and evaluating how existing approaches address the described challenges. Based on this discussion of the state-of-the-art and its limitations, we present our visualization technique, followed by a validation of our approach in case studies, conducted with experts in molecular biology. In the course of these case studies, we demonstrate how enRoute can be used to analyze large datasets in the context of pathways.This paper is based on and extends previously published work . In addiGlycolysis/Gluconeogenesis pathway of the KEGG database the buildup or degradation of glucose is described in great detail with all biochemical conversions and the connections to the Pentosephosphate pathway and the Citrate cycle pathway. A prominent example of the signaling pathway group is the MAPK signaling pathway, which contains a well-studied signaling cascade leading from an activated cell surface receptor to a phosphorylation cascade of several kinases to the activation of DNA binding complexes, which regulate transcription of genes involved in the proliferation of cells thus enabling the cell to react to growth stimuli from its environment. It is common for pathways to include the spatial organization of cells like cell walls, the Golgi apparatus, or the cell nucleus, thus allowing to depict transport of molecules and signaling through these compartments.Life scientists have accumulated intricate knowledge about biochemical and signaling processes in living cells, which has been used to build detailed biological interaction networks called pathways. These pathways summarize the molecular interactions in the biochemical conversions of molecules from source material to complex biomolecules or the signaling from a cell surface receptor via a second messenger to the nucleic transcription machinery. Several initiatives are drawing pathway maps and make these maps available to the scientific community, such as KEGG  or WikipPathway maps have become a valuable resource for molecular biologists summarizing broad knowledge about molecular interactions and presenting this information in a condensed view highlighting the functional aspects most interesting to the researcher. In the pathway maps nodes represent biological entities like proteins, metabolites, or chemical compounds. Protein nodes are annotated using the gene names from which these proteins are transcribed. These nodes usually contain several isoforms of the same protein and often several proteins of the same family catalyzing the same reaction, leading to extensive multi-mapping of many gene names to a single node. The nodes in a pathway are connected by links that depict biochemical reactions, activating or inhibiting modifications of proteins, or enzymatic reactions. The nodes together with the links between them allow to capture the interaction network of a biological system and present them in a structured graph with node-link diagrams following specific drawing conventions. Most pathway databases contain manually curated pathway maps that are trying to represent the molecular interactions in a visually appealing way while maintaining the scientific content.~omics technologies has generated large amounts of data. These datasets include genetic and expression data from large consortia like TCGA (http://cancergenome.nih.gov) and ENCODE (http://encodeproject.org/), but also high throughput metabolic screens via nuclear magnetic resonance (NMR) measurements or large scale proteomics by mass spectrometry. Data generated from these analysis includes dynamic data like gene expression levels, metabolite levels, and protein levels in various tissues, cells, and disease states, and also information about the genetic constitution of samples like copy number variation of genes, mutations in genes, or methylation patterns. All this data can only be interpreted in the context of the biological system present in cells, which are captured in the aforementioned pathway maps. The purpose of the techniques presented in this paper is to support the researcher in dynamically mapping biological data onto pathways. This allows researchers to compare, reason, and ultimately explain the complex biological systems and signaling cascades.In recent years, the advent of high throughput As a foundation for the design of our visualization technique and the evaluation of the related work, we have conducted a requirement analysis based on interviews with our collaborators from the Medical University of Graz. Our analysis resulted in five requirements that must be met by a visualization system to successfully enable the joint analysis of pathways and experimental data.R I: The Scale Requirement - While the scalability of the graph is addressed by the sub-division of the biological network into individual pathway maps, the experimental datasets we consider are quite large. Consequently, a joint pathway and experimental data visualization system must be able to scale to dozens of experimental conditions or groups and hundreds of samples.R II: The Heterogeneity Requirement - Modern biological studies often include a wide array of complementary but heterogeneous experimental datasets. While, for example, mRNA expression data measures the gene activity, copy number or mutation data can be used to reason about deviating expression values. These heterogeneous datasets need to be presented using different visualization techniques, as they differ in terms of data type. For example, mRNA expression data is numerical, copy number data is a hybrid categorical/numerical dataset, which is often binned into ordinal  data, and mutation status data is nominal . In order to analyze these different kinds of data in context of pathways, the visualization system needs to handle all of them simultaneously and also represent each of them using suitable visual encodings.R III: The Multi-Mapping Requirement - Pathway nodes can represent various different gene products, such as enzymes, proteins, or RNA. In some cases, pathways also summarize a whole gene family into a single node, where different genes produce functionally similar proteins. This is what we call a multi-mapping: One node in a pathway actually represents multiple entities and therefore multiple entries of an experimental dataset can be associated with this node. As understanding this complexity is essential for judging effects of experimental data on a pathway, it is critical to convey multi-mappings adequately.R IV: The Layout Constraint Requirement - Layouts for pathway maps can be either produced automatically or manually. To make pathway maps easier to understand, a number of drawing conventions have been established. Examples are cycles being drawn in circles or the predominance of orthogonal edges in many popular databases. Also, manually curated pathway maps typically contain rich meta-information indicating, for example, the cell compartments in which specific processes occur. Automatically drawn pathway maps can either try to respect these conventions , or optR V: The Topology-Attribute Coexistence Requirement - We distinguish between two main types of tasks conducted on a pathway: tasks that are based on the topology of the underlying graph, and tasks that are based on the node or edge attributes of the graph [Topology-based tasks are concerned with the connectivity of the graph, e.g., which nodes can be reached from a given node, what are the articulation points of a graph, etc. An example for a topology-based task in pathway analysis is to find all nodes that might be influenced by the inhibition of a node at the beginning of a pathway. Attribute-based tasks are concerned with analyzing the properties of node or edge attributes. Edge attributes in pathways commonly describe the type of a relationship between two nodes, such as biochemical conversion, while mapping experimental data represents the majority of node attributes. An example for an attribute-based task for pathway analysis is to find all nodes in a pathway that are mutated in a large number of the mapping samples.he graph . TopologVisualization techniques for graphs are usually optimized for either topology-based task or for attribute-based tasks, but are rarely suitable for both at the same time. Node-link diagrams, are, for example, well-suited for topology-based tasks, while matrix layouts, where nodes are shown on the sides of a matrix and the cells contain information on whether there is an edge connecting the nodes, are ideally suited for edge-attribute-based tasks . When anscale requirement (R I) with respect to the encoding of node or edge attributes. For a comprehensive review of systems biology visualization refer to the article by Gehlenborg et al. [While there is a wide body of literature on graph drawing and graph visualization, we focus on the discussion of techniques that are either directly relevant for pathways or that can address the g et al. . We iden• on-node mapping,• using multiple instances of the graph with different on-node mappings ,• using separate linked views for the graph and the attributes, and• adapting the graph layout.on-node mapping is that it makes it easy to address the layout constraint requirement (R IV). Consequently, on-node mapping has been widely used to augment pathways with multiple colored rectangles, each representing a single experiment or an aggregation of multiple experiments [R I) as the amount of distinguishable colored rectangles inside a node is severely limited.The benefit of using eriments -11, whereriments . The bigsmall multiples. Small multiples show the same configuration of a plot multiple times while changing one variable ([R I) is a limiting factor. Depending on the pathway about four to ten multiples are reasonable.An alternative strategy to using multiple colors within each node in one graph is to use multiple graphs where each of them uses a single experiment or a single aggregate of experiments to drive its color-coding. This approach is commonly referred to as ariable  and the heterogeneity requirement (R II) is using separate linked views for the experimental data and the pathways. This, of course, also preserves the topology (R IV) and can be used to address multi-mappings (R III). Separate linked views use synchronized highlighting (linking & brushing) between the multiple views to communicate relationships. If, for instance, a user selects a node in the pathway, the corresponding experimental values are highlighted in the views depicting the experimental data. Shannon et al. [R V, to simultaneously enable topology-and attribute-based tasks. As separate linked views require interaction to show relationships between a single node and its associated data, the joint analysis of the topology and attributes is severely hindered.A technique that can easily address the scale  and choose a node size that enables in-place encoding of experimental data with various visual encodings .Finally, there are methods that .g., bar ,19 and l.g., bar ). While GraphDice by Bezerianos et al. [R IV and R V.There are also more radical adaption approaches for the graph layout. Schulz et al. , for exas et al.  probablyPathline tool [Curvemap view, which displays experimental data for both genes and metabolites recorded in time series. While Pathline was the main inspiration for our approach, it suffers from the unconventional pathway layout, which can hinder understanding the graph topology (R V). Also, it currently requires manual creation of the linearized pathways, thereby making it difficult to integrate the large existing databases of pathways.A different approach on adapting the graph layout was taken by Meyer et al. with their ine tool . PathlinR V) as the most critical requirement to address, as current techniques usually either support only topology-based or attribute-based tasks. Only small-multiples and direct on-node mapping are able to address requirement R V, however, both neither scale to many experiments (R I), nor do they allow to simultaneously present heterogeneous data (R II). Our solution to this problem makes use of an observation we made in discussions with our collaborators: they usually reason about and analyze experimental data associated with a single path at any given time in detail, while the rest of the network merely informs them about the context of this path. They of course continuously change the path of interest, but do not need to see detailed data for multiple paths at the same time. This temporal separation of high-level topology-based tasks and low-level attribute-based tasks allowed us to create a solution that meets all five requirements. The enRoute visualization technique, as depicted in Figure pathway view, showing the pathway map in its original graph layout (meeting R IV), and the enRoute view where a user-selected path is shown in a linear fashion together with a potentially large number of experimental data from multiple sources (R I and R II). Due to the linear arrangement of the nodes from top to bottom, it is possible to encode multi-mappings (R III) by giving them more vertical space. enRoute thus makes use of the temporal separation of analysis focus by presenting an overview in one and the details of a selected path in another view. In the following, we discuss the components of our approach and their interplay in more detail.The goal of the enRoute visualization technique is to jointly visualize experimental data and pathways in a way that addresses all five requirements discussed. We identify the topology-attribute coexistence requirement  or a single sample in its high-level topological context. This allows analysts to consider experimental data associated with nodes that are not in the currently extracted path. For this purpose, the pathway view can be configured to show only the mapping of selected samples.Glioma pathway with on-node mappings of mRNA data, while Figure To address the overview task where analysts want to get a rough indicator of the mapped experimental values, we calculate the average of all experimental sample values and multi-mappings, if applicable, and color-code the nodes accordingly. If multiple data types are available, the analyst can choose which of them should be mapped. Figure mTOR node in the lower right part of Figure For numerical and ordinal data we use a blue-white-red color map. We decided to use white as a neutral base of the color map to be able to intuitively represent data that has a neutral base, as, for example is the case with copy number data, which has a "normal" status. In addition, the blue-white-red color map avoids the drawbacks of the common red-black-green color map for red-green color blind users. A two-color gray-red color map is used for nominal data with two categories, such as mutation status data. To indicate cases where experimental data is missing, we show a small rectangle in the lower left corner of the node, as can be seen, for example, in the PDGFR gene in Figure Since the aggregation of all samples and possible multi-mappings into an average value hides all variation, we additionally provide the standard deviation encoded as a green bar below each node, as shown in Figure data rows) right next to the nodes they are associated with. As a node can have multiple mapped data rows, we adapt the spacing between nodes of the path so that all rows can be shown with a uniform height. Such multi-mappings or the occurrence of complex nodes (nodes that consist of multiple subnodes) in the path make it very hard, if not impossible, to determine which data row belongs to which node using their position alone. Therefore, we connect each node with corresponding data rows using ribbons, as shown in Figure Once a path has been selected in the pathway view, it can be analyzed in detail in context of experimental data in the enRoute view. The path is displayed in a linear, top-down layout, which is ideally suited to show rows of experimental data (heterogeneity requirement (R II) by allowing the individual groups to originate from different datasets. However, all experiments within a group must be from a single dataset.Following the divide-and-conquer visualization strategy , we grouIn addition to showing the extracted path top-down in the enRoute view, we also display branches that join or leave the path in order to preserve some of the topological information present in the pathway maps. We indicate a branch by showing its first node relative to the node where the branching occurs in the extracted path. In order to maintain a compact path representation, multiple branches that join or leave a single node of the path are abstracted into expandable nodes, one for all joining and one for all leaving branches, as shown in Figure As illustrated in Figure R I and R II). enRoute supports the visualization of quantitative, ordinal, and binary categorical data. As previously mentioned, we organize experimental data in rows and columns. Each row shows data that maps to a certain node in the path and columns group the data by a homogeneity criterion. Different groups may also have overlapping experiments. The captions of the individual groups are displayed at the top and at the bottom of the corresponding columns. Their background color indicates the dataset they belong to. For example, in Figure Being able to display large amounts of heterogeneous experimental data is an integral part of the enRoute visualization technique  [Up to this point, we have assumed that decisions on which datasets and which groupings of the datasets to show are already made. However, given a large set of datasets and a variety of alternative groupings to choose from for every dataset, this presumably easy task is in fact not trivial. To support analysts in the task of selecting datasets and groupings and assigning them to views, Caleydo provides a dedicated view, the or (DVI) . As showCaleydo, which is an open source visualization framework for biomolecular data [http://www.caleydo.org. Both, our visualization technique and the visualization framework are implemented in Java and OpenGL using the freely available JOGL library (available at https://jogamp.org/jogl). In order to highlight selected pathways, we make use of a free implementation of the Bubble Sets technique [The approach presented in this paper is implemented as a module of lar data . Binary echnique  that we http://tcga.caleydo.org), which contain up to 900 samples (depending on the subtype stratification) and ~20,000 genes each. As demonstrated in Figure Caleydo allows analysts to import data sets with large amounts of experiments, such as mRNA, copy number, and mutation status datasets. We provide pre-packaged and preprocessed TCGA datasets , which contains the genetic and pharmacologic characterization of a large panel of human cancer cell lines. The second is a microarray dataset from a model of hepatocellular carcinoma collected at the Medical University of Graz, which can be found through the corresponding original article [We developed the enRoute visualization technique in close collaboration with a biologist from the Medical University of Graz, who is also an author of this paper. To evaluate the utility of the enRoute visualization technique, we conducted two case studies using different datasets together with this biologist. The first dataset is taken from the Broad-Novartis Cancer Cell Line Encyclopedia  for cell lines originating from ovary, lung, or breast tissue among others. By selecting all samples with an increased copy number of TNFSF10 in the histogram, the system also highlights these cell lines in the mRNA expression plots, as indicated by the arrows in Figure TNFSF10 in the mRNA expression plots. Based on this data, one could interpret that these cell lines should be prone to apoptosis, as they have high amounts of the cytokine triggering the cell death. However, given the nature of these cell lines which are derived from human tumors, which are by definition resistant to apoptosis, this conclusion would be misleading. An explanation for this seemingly strange behavior can be found by looking at the copy numbers of the genes TNFRSF10A to TNFRSF10D, which are receptors for TNFSF10 cytokine, further downstream the path. Here, it quickly becomes evident that these genes exhibit a frequent loss of copies, which is easy to see when looking at the many blue bars pointing downwards for these genes. This explains how these cells can still form a tumor, as the loss of the receptor makes the cells immune to the stimulus by the TNFSF10 cytokine, thus preventing apoptosis initiation.As first step of the analysis, the In summary, enRoute enabled the analyst to study copy number variation, gene expression, and mutation status of a large number of samples in parallel in a clearly visualized and linearized sub-path of the complex network initiating apoptosis. The researcher stated that the linearization of the signaling cascade and the clear association of the other data entities to the individual steps of this cascade are very intuitive and greatly facilitated the interpretation of the data and the deduction of a biological interpretation from the dataset.all-trans-Retinoate to all-trans-18-Hydroxy retinoic acid, which is a reaction contained in the Retinol (Vitamin A) metabolism pathway shown in Figure cytochrome p450 enzymes, which metabolize drugs and mediate oxidative stress. When studying the aforementioned conversion of retinoate to retinoic acid by cytochrome p450 enzymes, it can now be seen that the node CYP2A6 actually contains members of the whole CYP2 family of genes and that the expression of these genes is uniform in normal liver and cirrhosis, but very variable in HCC, dividing these cases into a low expressing and a high expressing group. It is immediately visible that the xenograft models are only representative of the low CYP2 expressing group of HCCs. Decomposition of the complex nodes allows the identification and investigation of the individual expression patterns of genes contained in the complex node. It becomes obvious that the expression of CYP2C18 (highlighted in yellow), contained in the node labeled CYP2A6, is higher in the Hep3B model than in the other cell lines.The second use case demonstrates the ability of the enRoute path extraction feature to aid in a common problem when interpreting pathway maps. In biological reaction systems enzymatic reactions are often carried out not by a single gene or protein but by a family of proteins encoded by several genes. Additionally, enzymatic reactions are often not specific to a single protein family, but can be substituted by other enzymes, which, however, often work with different efficiency. This biological diversity leads to functional nodes in pathways that contain a multitude of genes with historically designated gene names often not indicative of the real function of these genes. It is extremely difficult for the researcher to keep in mind which genes are behind a single node of a pathway and thus understand the mapped biological data. Cross referencing of expression levels to genes and functional nodes can thus only be achieved by resolving the multi-mapping in a node to gene names which can then be mapped to the individual gene expression levels. In enRoute, the researcher can select nodes upstream and downstream of the enzymatic reaction, which is then resolved in a linearized representation of all genes involved. An example is the conversion of CYP1A1 is highly expressed in HUH7 tissue culture cells. All this information was not visible to the researcher using conventional on-node mapping approaches and was successfully visualized using enRoute. Node decomposition is an integral feature of the enRoute path extraction and thus makes the association of many mapped nodes and their corresponding experimental data readily available to the researcher.Additionally, it can be detected that Enabling the joint analysis of biological pathways and large amount of experimental data by means of visualization is a challenging problem. In this paper, we provide a list of requirements that have to be met to support such analysis. We introduce the enRoute visualization technique, which addresses these requirements using a tightly-coupled dual-view approach. Our visualization technique allows experts to select a single path from a pathway map, which is then extracted and shown in linear form in a second view, the enRoute view. This view allows experts to investigate associated experimental data in detail by displaying it in a tabular layout right next to the extracted path. The conducted case studies using two different datasets confirmed the utility of our visualization technique.Although we currently already support pathways from both the Wikipathways and the KEGG database, we plan to integrate further resources for biological networks. For instance, an integration of the EBI IntAct database  will makenRoute allows to select nodes independent of the structure of the pathway graph. However, the boundaries of a single pathway graph also limit the potential of enRoute. Consequently, we plan to extend enRoute to also support the analysis of paths spanning multiple pathways, which introduces new challenges for scalability concerning the topology of the graph. Solving these problems is subject of future research. Another interesting area of research is the selection of pathways, which we have only briefly touched in this paper. We intend to implement common measures for identifying interesting pathways in the context of the available experimental data and integrate them in a unified data and pathway visual analysis workflow.The authors declare that they have no competing interests.CP, AL, MS, DK, and DS developed the concept of the enRoute visualization technique. CP, AL, MS, and DK implemented the Pathway view and the enRoute view for the Caleydo visualization framework. KK consulted on biological relevance and evaluated the visualization technique with regard to its applicability for biological data analysis. All authors contributed to this manuscript.


Precipitation is one of the most important meteorological variables for defining the climate dynamics, but the spatial patterns of precipitation have not been fully investigated yet. The complex network theory, which provides a robust tool to investigate the statistical interdependence of many interacting elements, is used here to analyze the spatial dynamics of annual precipitation over seventy years (1941–2010). The precipitation network is built associating a node to a geographical region, which has a temporal distribution of precipitation, and identifying possible links among nodes through the correlation function.The precipitation network reveals significant spatial variability with barely connected regions, as Eastern China and Japan, and highly connected regions, such as the African Sahel, Eastern Australia and, to a lesser extent, Northern Europe. Sahel and Eastern Australia are remarkably dry regions, where low amounts of rainfall are uniformly distributed on continental scales and small-scale extreme events are rare. As a consequence, the precipitation gradient is low, making these regions well connected on a large spatial scale. On the contrary, the Asiatic South-East is often reached by extreme events such as monsoons, tropical cyclones and heat waves, which can all contribute to reduce the correlation to the short-range scale only.Some patterns emerging between mid-latitude and tropical regions suggest a possible impact of the propagation of planetary waves on precipitation at a global scale. Other links can be qualitatively associated to the atmospheric and oceanic circulation. To analyze the sensitivity of the network to the physical closeness of the nodes, short-term connections are broken. The African Sahel, Eastern Australia and Northern Europe regions again appear as the supernodes of the network, confirming furthermore their long-range connection structure. Almost all North-American and Asian nodes vanish, revealing that extreme events can enhance high precipitation gradients, leading to a systematic absence of long-range patterns. By combining graph theory and statistical physics, complex network theory provides a powerful tool to investigate the structure and function of complex systems with a large number of interacting elements. The development and characterization of complex networks climate networks, which usually rely on gridded time series of meteorological preprocessed variables. The nodes of the network are identified by geographical regions corresponding to single points of measurement on the spatial grid of the underlying climate database. Each node has a measured state variable which varies in time. A link between two nodes exists if there is a significant statistical interdependence between their time series. The linear cross-correlation function is typically used as the simplest possible measure of the statistical interdependence of the temporal series. However, the influence of the choice of an association measure on the topology of the climate network has been studied, by accounting how the temporal complexity of time series influences the absolute correlations The extension of complex network theory to climate sciences is a very recent area yielding Until now, attention has been mainly devoted to networks based on the global surface temperature field to understand the influence of El Niño and La Niña events in regions which are far from the El Niño-Southern Oscillation (ENSO) area. Although the temperatures in different zones of the world are not significantly affected by El Niño and La Niña, it was surprisingly found that the climate network during these events is sensitively influenced by showing a different structure and a consistent amount of broken links et al.Apart from some works where different meteorological variables such as equipotential heights The gap concerning global precipitation in climate networks is evident even though precipitation teleconnections, especially related to the ENSO occurrence in situ observations . The precipitation network is built identifying a node with a geographical region, which has a temporal distribution of measured precipitation, and using the linear correlation function to evaluate possible links between nodes. If the statistical interdependence between two nodes is above a suitably chosen threshold, a link between the two nodes is established. The precipitation network is described through classical tools of the complex network theory - such as the degree centrality, the betweenness centrality and the clustering coefficient - as well as measures introduced here for the first time: the weighted average topological distance, which generalizes the average topological distance definition, and the average physical distance of a node from the rest of the network. To analyze the sensitivity of the network to the physical closeness of the nodes, short-term connections are broken and edges between physically distant nodes only survive. In so doing, the unavoidable spatial correlation between physical neighbors is left aside in favor of highlighting the possible interdependence between not confining regions.The present work arises in this scenario and aims at being a first step towards filling this gap in climate network analysis; in fact, precipitation, together with surface temperature and wind, atmospheric pressure and humidity, is one of the most important meteorological variables in defining the climate dynamics. The global annual precipitation over seventy years (1941–2010) is analyzed by means of the complex network theory. We use the Global Precipitation Climatology Centre (GPCC) Database In this section the database used to define the precipitation network is described. Details on the pre-processing analysis of data are then offered. Afterwards, we summarize some fundamental concepts in complex network theory The present investigation uses the Global Precipitation Climatology Centre (GPCC) Full Data Reanalysis Version 6, which consists of monthly land-surface precipitation data from rain-gauges built on Global Telecommunication System (GTS) and historic data The monthly global precipitation is reported on a regular grid with a spatial resolution of in situ precipitation data sets. In fact, the wide spatial and temporal coverage makes this database suitable for verification of models The GPCC Full Database provides one of the most accurate and complete Although the GPCC Full Database is a very accurate and detailed dataset, there are two main concerns to use it, as it is, to define a complex network. In this section we discuss these two issues and propose our solutions to overcome them.et al.First, the regular grid based on the angular partition of the terrestrial surface  leads to the definition of gridcells with different geometric area. This heterogeneity, which becomes more evident by approaching regions far from the equator, may induce substantial bias and spurious correlation when building the precipitation network. One way to avoid this bias is to use a tessellation technique 2. The new graticule, which is here proposed, is built maintaining this cell dimension . However, in these cases, the global precipitation information are recovered through the interpolation of global and mean data offered by the GPCC Precipitation Climatology Database. In so doing, the spatio-temporal coverage is complete but some precipitation values can be fully based on interpolated data. This aspect becomes important when the oldest data are analyzed, since fewer measurements were available. In order to consider data mainly based on in situ observations, we define a grid cell as active for a fixed year if there is at least one measurement for every month of the year. Otherwise the grid cell is not active. We then restrict the analysis to a temporal window of 70 years starting from 1941 to 2010, and consider only cells which are active  for at least 50 years over the temporal window of 70 years. In this way, more than 90% of the stations will be included in the spatio-temporal precipitation distribution and, in the worst cases, less than 30% of the distribution (20 years over 70) will rely on interpolated data. The number of active cells is adjacency matrix, A network (or graph) is defined by a set degree centrality of a node is defined asdegree distribution, The weighted average topological distance of a node asWe here propose the local clustering coefficient of a node isglobal clustering coefficient is the mean value of The betweenness centrality of a node isThe et al.The number of active cells, as previously described, is average physical distance of an active cell (node) is defined here asThe edge densityThe To define the adjacency matrix, The properties of the precipitation network are here presented and discussed. Among the network measures introduced in the As mentioned, the precipitation network is made of The data analysis is completed by describing the properties of another precipitation network with an additional constraint: an edge, To focus on possible links between regions physically far from each other, we set supernodes. Beside the supernode areas, there exist regions with fairly high degree centrality values: Northern Europe, Central Asia, Southern Africa, Western US, and Northeastern Brazil. It should be noted the great difference in terms of degree centrality values, 2, equivalent to the total area of the countries in the European Union.We start considering the degree centrality for the basic network, see supernodes) and a great amount of barely connected cells. However, because of the finite size of the network, data can have a rather strong intrinsic noise. To smooth the fluctuations generally present in the tails of the distribution, it is often verified if the cumulative distribution function, Real networks are often scale-free, that is power-law degree distributions are displayed The weighted average topological distance, As a first comment, there is quite a notable correspondence between high degree centrality and low weighted average topological distance, and viceversa. This is especially evident, on one hand, for large At this stage, it is useful to compare the weighted average topological distance map, To carry out a sensitivity analysis of the network with respect to the physical neighborhood of the nodes, we here define that an edge between nodes The degree centrality and the weighted average topological distance are presented in the left and right panels of The evident absence of nodes in Asia and North-America (and the poor connection of the few remaining cells) can be thought in terms of the presence of strong precipitation variation on a relatively short spatial scale, thereby leading to the emergence of high precipitation gradients. A high precipitation gradient can be, for instance, enhanced by the occurrence of regional extreme events  which are usually localized in time and space. The Asian and North-American continents, due to their huge land mass extension, experience the most imponent extreme phenomena In the supernode regions  extreme events in general occur less frequently. Medium (Western Europe) or very low  rainfall spread out more uniformly on a continental scale, the precipitation gradient is weaker and nodes remain connected in the long-range.Coming back to the original network with From a qualitative point of view, the local clustering coefficient presents a strong heterogeneity in the central part of Africa, in Eastern Asia and South America. Some patterned zones with high The betweenness centrality unveils the importance of a node in the network. Bottom panel of The shortest path distribution for 4 significant nodes can be observed in Some of the patterns linking mid-latitude to tropical nodes  seem to suggest the impact of the propagation of planetary waves on precipitation at a global scale level Nevertheless, some links can be qualitatively related to the oceanic and atmospheric circulation The different patterns expressed by the four nodes in In , an animWe conclude this section offering a possible climatological interpretation of two measures, the weighted average topological distance, The recent development of complex network theory is offering new quantitative tools to disentangle the global climate dynamics. In spite of teleconnections having long been studied in climatology, the idea to read climatic correlations among different Earth regions as forming a complex network is relatively new. Starting from this point of view, we have focused on the global precipitation network. To this aim, we have used reliable datasets of measured land-surface precipitation that have only recently become available. Paying attention not to introduce spurious correlations due to uneven partitions of the Earth surface and to have a sufficient number of measured data in each cell, correlation analysis (with cut-off at 0.5) performed on a 70 year-window has yielded an undirected and symmetric network with 1674 nodes and 9481 links. We have investigated the structure of this precipitation network by some topological properties of nodes  and, in particular, we have introduced a weighted form of the average topological distance in order to prevent some misleading behaviors when the graph has disconnected components.Some key aspects of the precipitation network clearly emerge. Firstly, supernodes  occur in the Sahel region in Africa and in Eastern Australia, and a scaling-law behavior is revealed in the node degree distribution. Sahel and Eastern Australia regions are some of the most arid areas in the world. Very low rainfall is uniformly distributed on continental scales and huge extreme events are rare. As a consequence, the precipitation gradient tends to weaken, making these regions well connected on a large spatial scale. This long-range connection is confirmed by the fact that Sahel and Eastern Australia remain supernode regions also in the long-range network see .Strongly connected zones are evident also in Northern Europe, Central Asia, Western US, and Northeastern Brazil, even if they are not always characterized by a high betweenness centrality. Remarkably, strong connection differences occur between Northern and Mediterranean Europe as well as between Western and Eastern US. The European differences can be explained by the fact that Northern regions are mostly influenced by the oceanic circulation of the Gulf Stream and the atmospheric effects of the NAO Click here for additional data file.Figure S1Shortest path of the Australia node  represented by a pink square.(EPS)Click here for additional data file.Figure S2Shortest path of the Africa node  represented by a pink square.(EPS)Click here for additional data file.Figure S3Shortest path of the US node  represented by a pink square.(EPS)Click here for additional data file.Movie S1Animated representation of the shortest path of the Australia node  displayed in  The node is indicated by a pink square.(AVI)Click here for additional data file.Movie S2Animated representation of the shortest path of the Africa node  displayed in  The node is indicated by a pink square.(AVI)Click here for additional data file.Movie S3Animated representation of the shortest path of the US node  displayed in  The node is indicated by a pink square.(AVI)Click here for additional data file.Movie S4Animated representation of the shortest path of the Northern Europe node  displayed in  of the main text. The node is indicated by a pink square.(AVI)Click here for additional data file.Movie S5Animated representation of the shortest path of the Southern Europe node  displayed in  of the main text. The node is indicated by a pink square.(AVI)Click here for additional data file.Movie S6Animated representation of the shortest path of the South America node  displayed in  of the main text. The node is indicated by a pink square.(AVI)Click here for additional data file.Movie S7Animated representation of the shortest path of the Asia node  displayed in  of the main text. The node is indicated by a pink square.(AVI)Click here for additional data file.


Large-scale sequencing of genomes has enabled the inference of phylogenies based on the evolution of genomic architecture, under such events as rearrangements, duplications, and losses. Many evolutionary models and associated algorithms have been designed over the last few years and have found use in comparative genomics and phylogenetic inference. However, the assessment of phylogenies built from such data has not been properly addressed to date. The standard method used in sequence-based phylogenetic inference is the bootstrap, but it relies on a large number of homologous characters that can be resampled; yet in the case of rearrangements, the entire genome is a single character. Alternatives such as the jackknife suffer from the same problem, while likelihood tests cannot be applied in the absence of well established probabilistic models.We present a new approach to the assessment of distance-based phylogenetic inference from whole-genome data; our approach combines features of the jackknife and the bootstrap and remains nonparametric. For each feature of our method, we give an equivalent feature in the sequence-based framework; we also present the results of extensive experimental testing, in both sequence-based and genome-based frameworks. Through the feature-by-feature comparison and the experimental results, we show that our bootstrapping approach is on par with the classic phylogenetic bootstrap used in sequence-based reconstruction, and we establish the clear superiority of the classic bootstrap for sequence data and of our corresponding new approach for rearrangement data over proposed variants. Finally, we test our approach on a small dataset of mammalian genomes, verifying that the support values match current thinking about the respective branches.Our method is the first to provide a standard of assessment to match that of the classic phylogenetic bootstrap for aligned sequences. Its support values follow a similar scale and its receiver-operating characteristics are nearly identical, indicating that it provides similar levels of sensitivity and specificity. Thus our assessment method makes it possible to conduct phylogenetic analyses on whole genomes with the same degree of confidence as for analyses on aligned sequences. Extensions to search-based inference methods such as maximum parsimony and maximum likelihood are possible, but remain to be thoroughly tested. Large-scale sequencing of whole genomes has enabled the inference of phylogenies based on the evolution of genomic architecture, under such events as rearrangements, duplications, and losses. Many evolutionary models and associated algorithms have been designed over the last few years and have found use in comparative genomics and phylogenetic inference see-3). Howe. Howe3])We describe a novel approach to the assessment of distance-based phylogenetic inference from whole-genome data. Our approach restates the main characteristics of the jacknife and bootstrap in terms of noise modeling, itself a longstanding approach to robustness assessment in engineering.A number of new resampling methods are discussed along with extensive experimental testing, in both sequence-based and genome-based frameworks. We show empirically that our new approaches provide bootstrap support values that are as acceptable as those provided by the classic bootstrap for sequence data. . Finally, we test our approach on a small dataset of mammalian genomes, verifying that the bootstrap values match current thinking about support values of the respective values obtained from sequence-based studies. The focus on distance-based methods is due in part to simplicity and convenience: by reducing the input genomes to a distance matrix, these methods not only simplify the characteristics of the input data, but also make it straightforward to compare our method with methods for sequence-based inference. The focus is also due in part to two other characteristics of distance-based methods: they are very efficient compared to optimization searches such as maximum parsimony and maximum likelihood; and they remain the most commonly used in routine phylogenetic reconstruction—neighbor-joining and miniWe briefly review the bootstrap and jackknife approaches, as well as relevant characteristics of rearrangement data and of distance-based methods for phylogenetic inference.Quenouille introducn data points X = {x1,…,xn} and a statistical estimator E, a bootstrap replicate is a fictional datasetX. From each such fictional dataset a value of the estimator E can be obtained. The key idea of bootstrapping is that the distribution of values thus obtained closely matches the original distribution of E and can be used to estimate the confidence limits on the estimator. The advantage of the method lies in its applicability to arbitrary and complicated estimators that may be analytically intractable as thresholds.A Receiver-Operator-Characteristic (ROC) curve is drawn for every method we investigate. In this plot, a point is a particular bootstrapping test, defined by its e)%-jackknifing (37JK), and parsimony jackknifing (PJ)—are nearly indistinguishable and clearly dominate the others. The analogs of all the other methods developed for rearrangement data  are clearly worse than the above four, with BP and BPC being comparable and the most primitive noise-shaping method, BGN, doing the worst.FigureBC*, PJ*, 50JK*, and 37JK* are nearly indistinguishable and clearly dominate all others. They are followed by BP* and BPC*, which are comparable, while the Gaussian noise approach, BGN*, again does the worst. JG*, the marker-based jackknifing technique of Shi et al., is better than BGN*, but trails all other methods. The differences are particularly marked at very high levels of specificity; at 98% specificity, for instance, the top four methods retain nearly 90% sensitivity, but JG* drops to 80%. Very high specificity is the essential characteristic of a good bootstrap method.FigureBC*, PJ*, 50JK*, and 37JK* perform equally well and dominate BP* and BPC*. Since the reconstructed trees using FastME are more accurate, the sensitivity of the top four methods at high levels of specificity are even higher compared to the sensitivity attained when NJ was used , but fails to identify e1as the least supported edge, while JG* erroneously gives high support values to both (100% for e1 and 90% for e2).We also tested our bootstrapping methods on a real dataset: the genomes of 10 species from the Ensembl Mercator/Pecan alignments with 8,380 common markers. Four of these genomes  are not well assembled: their draft genomes have nearly twice as many contigs as there are chromosomes—but the effect on our adjacency-based distance estimator is minimal, given the large number of markers. FigureOur new approach for whole-genome data, based on the sampling of adjacencies, matches the classical bootstrap and parsimony jackknife approaches and thus provides the first reliable method for assessing the quality of phylogenetic reconstruction from such data.In the process of testing various methods, we also confirmed past findings about the superiority of the phylogenetic bootstrap and of the parsimony jackknife. Our results clearly indicate that duplicate samples play no role in the process—parsimony jackknifing works at least as well and occasionally slightly better. Indeed, the best sampling strategy appears to be a random sampling of half of the characters. Given the very high computational cost of the bootstrap, using half the number of characters in sequence-based analyses appears a worthwhile computational shortcut, especially as it delivers even better results.PJ*), each original genome is represented by a set of contiguous regions in the bootstrap; if the reconstruction method can handle such inputs, then this is the best method. Encoding rearrangement data into sequences was proposed many years ago  that are unable to handle such data include parsimony jackknifing and direct encoding of adjacencies into sequences. In parsimony jackknifing (ago (see) in two ago (see suggestsThe authors declare that they have no competing interests.YL and VR contributed equally to this work. YL and VR conceived the idea, implemented the methods, and conducted the experiments. BMEM directed the project. YL, VR and BMEM collaborated closely in the elaboration of the project, the analysis of the data, and the writing of the manuscript. All authors read and approved the final manuscript.


We introduce an electromechanical model for human cardiac tissue which couples a biophysical model of cardiac excitation  and tension development  with a discrete elastic mass-lattice model. The equations for the excitation processes are solved with a finite difference approach, and the equations of the mass-lattice model are solved using Verlet integration. This allows the coupled problem to be solved with high numerical resolution. Passive mechanical properties of the mass-lattice model are described by a generalized Hooke's law for finite deformations . Active mechanical contraction is initiated by changes of the intracellular calcium concentration, which is a variable of the electrical model. Mechanical deformation feeds back on the electrophysiology via stretch-activated ion channels whose conductivity is controlled by the local stretch of the medium. We apply the model to study how stretch-activated currents affect the action potential shape, restitution properties, and dynamics of spiral waves, under constant stretch, and dynamic stretch caused by active mechanical contraction. We find that stretch conditions substantially affect these properties via stretch-activated currents. In constantly stretched medium, we observe a substantial decrease in conduction velocity, and an increase of action potential duration; whereas, with dynamic stretch, action potential duration is increased only slightly, and the conduction velocity restitution curve becomes biphasic. Moreover, in constantly stretched medium, we find an increase of the core size and period of a spiral wave, but no change in rotation dynamics; in contrast, in the dynamically stretching medium, we observe spiral drift. Our results may be important to understand how altered stretch conditions affect the heart's functioning. The heartbeat is governed by electrical waves of excitation that periodically propagate through the cardiac muscle and initiate its contraction. Abnormal electrical excitation of the heart may result in cardiac arrhythmias disturbing the heart's pumping function. Heart failure due to cardiac arrhythmias is a major cause of death in the industrialized world Electrical waves of excitation are affected by the deformation of the heart via the mechano-electrical feedback phenomenon. It has been shown that the rapid stretching of cardiac tissue  has a significant effect on the heart's functioning, for example, due to the initiation of electrical waves A valuable method to study mechano-electrical feedback is mathematical modeling allowing to study the coupled mechanical and electrical activity of the heart, which is a difficult problem in experimental research. Generic electromechanical models for heart tissue are successfully applied, for instance, to investigate the effect of mechano-electrical feedback on pacemaking and spiral wave activity To demonstrate the value of the discrete electromechanical model we apply it to study effects of deformation on basic properties of cardiac tissue, such as the action potential shape, restitution properties, and the dynamics of spiral waves. In this study we consider two mechanical conditions. The first one is a constantly stretched medium, a simple assumption which was widely used in initial studies on mechano-electric feedback We use the 2006 version of the Tusscher Noble Noble Panfilov model for human epicardial myocytes (TP06) In our model we add a stretch-activated depolarizing current We will now describe the coupling of the electrical excitation process of the cardiomyocytes to their tension development.We model myocyte excitation-contraction coupling in our model with a numerically improved version of the Niederer, Hunter, Smith (NHS) model The NHS model takes the dynamics of sarcomere length into account. We follow previous studies We followed changes on the original version of the NHS model The NHS model describes the tension It has been shown that in strongly coupled electromechanical models for cardiac tissue, in which the equations for excitation processes and mechanical processes are jointly solved, computational difficulties may occur due to the velocity and length-dependency of a sarcomere We will now describe the passive elastic properties of the medium.We use the mass-lattice framework introduced in ke's law . The equ1 to the mass points 2 and 3. Each mass point is connected to its In section “Model for Excitation-Contraction Coupling” we defined a pseudo normalized sarcomere length ings see . Parametel,  see .We will now describe how we model mechano-electrical feedback via stretch-activated currents.The deformation of a cardiomyocyte affects its excitation processes. It has been shown in studies of excised cardiac tissue and the whole heart that a direct electrophysiological influence of the deformation of cardiac tissue is a depolarizing stretch-activated current We solved the model applying an explicit Euler method for the TP06 and NHS models, and Verlet integration The acceleration of a mass point During the mechanical iteration algorithm the length and velocity dependent tension scaling functions eps  see , thus thThe numerical coupling and integration of the Euler and the Verlet scheme require the choice of several parameters. We will first discuss the integration parameters and then parameters for the coupling of the numerical grids to assure efficient and stable computations.It has been shown that the TP06 model together with the improved NHS model coupled with a whole heart continuum mechanics model can be stably integrated with the Euler method using integration parameters f  as in . We findThe system of coupled, damped, driven, mechanical oscillators described by Eqs.(8)-(10) has been shown in In The main determinant of cardiac contraction is the [CaWe applied our discrete electromechanical model to study the effects of stretch-activated currents and stretch conditions on action potential duration (APD) and conduction velocity (CV) restitution, and spiral wave dynamics. The results of these studies are shown in this section.We consider two mechanical conditions, a constantly stretched medium, and a contracting medium. The condition of sustained stretch in the medium has been assumed previously, for example, in a model to study how dilated cardiomyopathies may affect defibrillation efficacy We used a The effect of To measure restitution properties in the constantly stretched medium we applied the same setup as above, but varied the stimulation period from For the simulations in a contracting medium we applied the same system size, stimulation protocol, and parameter setting as for the constantly stretched medium; however, the medium is deforming due to excitation-contraction waves, and the boundaries are fixed space . In In  compare . FurtherLet us now compare the results for a constantly stretched and contracting medium. We found, that under both, dynamical and static stretch conditions, increasing  compare , and 8A. compare , and for compare , so thatWe studied effects of deformation on spiral wave dynamics in our model. We initiated a spiral wave with an S1–S2 protocol in the medium, and then simulated spiral rotation for We found that in the constantly stretched medium (assumed to be stretched to nditions .imal see  the incrOur results on spiral wave drift in contracting tissue are qualitatively similar to that observed in We introduced a discrete electromechanical model of the human heart which couples a biophysical model of cardiac excitation It has been shown that the dynamics of spiral waves in the heart manifests itself in the type of cardiac arrhythmia, for example, a drifting spiral wave can induce a polymorphic ventricular tachycardia which is a known precursor for ventricular fibrillation Our results on restitution properties suggest that in dynamic stretch-conditions The computation time of our mechanical model scales linearly against the number of mechanical nodes, which allows to solve the model with a higher mechanical node density and thus high spatial resolution of The passive elasticity of the heart is most commonly described by hyperelastic constitutive relations in finite element formulations of continuum mechanics, for example, the Guccione material relation in As another next step, the discrete electromechanical model can be extended to three-dimensional simulations to study the effect of mechano-electrical feedback on the dynamics of scroll waves.Our modeling framework can potentially be used to estimate effects of mechanical or electrical components in experimental studies of wave propagation in the heart. However, it needs to be adjusted to the specific tissue type and mechanical properties of the experimental system. It can be done by changing the parameters on our model based on direct measurements.We set up the new electromechanical model using a standard form for the stretch-activated currents The effect of deformation of the medium on the metric tensor are neglected in the model, as we assume that the main resistivity between cells is constituted by gap junctions. We used this assumption also in