Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.

Intraplaque hemorrhage accelerates atherosclerosis via oxidant stress and contributes to lesion development and destabilization. Normally, macrophages scavenge hemoglobin-haptoglobin (HbHp) complexes via CD163, and this process provokes the secretion of the anti-inflammatory atheroprotective cytokine interleukin (IL)-10. We therefore tested the hypothesis that HbHp complexes may drive monocyte differentiation to an atheroprotective phenotype. Examination of the macrophage phenotype in hemorrhaged atherosclerotic plaques revealed a novel hemorrhage-associated macrophage population (HA-mac), defined by high levels of CD163, but low levels of human leukocyte antigen-DR. HA-mac contained more iron, a pro-oxidant catalyst, but paradoxically had less oxidative injury, measured by 8-oxo-guanosine content. Differentiating monocytes with HbHp complexes reproduced the CD163(high) human leukocyte antigen-DR(low) HA-mac phenotype in vitro. These in vitro HA-mac cells cleared Hb more quickly, and consistently showed less hydrogen peroxide release, highly reactive oxygen species and oxidant stress, and increased survival. Differentiation to HA-mac was prevented by neutralizing IL-10 antibodies, indicating that IL-10 mediates an autocrine feedback mechanism in this system. Nonlinear dynamic modeling showed that an IL-10/CD163-positive feedback loop drove a discrete HA-mac lineage. Simulations further indicated an all-or-none switch to HA-mac at threshold levels of HbHp, and this conversion was experimentally verified. These data demonstrate the creation of a novel atheroprotective (HA-mac) macrophage subpopulation in response to intraplaque hemorrhage and raise the possibility that therapeutically reproducing this macrophage phenotype may be cardio-protective in cases of atherosclerosis.

IntroductionMultiple myeloma (MM) is an incurable plasma cell (PC) malignancy of the bone marrow (BM). Although acquired genetic events and the tumor microenvironment are well-established regulators of myeloma cell survival and proliferation pathways, the identity and functional properties of the myeloma-propagating cells have been a matter of controversy. 1,2 The terminal differentiation of normal mature B lymphocytes to immunoglobulin (Ig)-secreting PCs entails conversion of antigennaive to antigen-experienced B cells in the germinal center of secondary lymphoid organs and their subsequent differentiation to either memory B cells or PCs. 3,4 Each stage of B-cell differentiation can be defined by surface markers with naive and memory B cells expressing CD19 and terminally differentiated normal and malignant PCs, but not B cells, expressing CD138 (Syndecan-1). 5,6 Given this linear B-cell lineage developmental process, it was suggested that myeloma cell growth is sustained by a minority of cells more immature than the PC. This hypothesis is supported by the presence of CD19 ϩ CD138 Ϫ clonotypic B cells (ie, cells sharing the same Ig heavy chain [IgH] complementarity region 3 [CDR3] sequence with the myeloma PCs) in peripheral blood (PB) and BM of patients with MM. 7-10 Indeed, because CD138 Ϫ but not CD138 ϩ PCs were found to lead to myeloma engraftment in NOD/SCID mice, it was proposed that CD138 Ϫ cells were the principal myeloma-propagating or "myeloma stem" cells [11][12][13][14] Earlier studies, For personal use only. on May 9, 2018. by guest www.bloodjournal.org From though, using a huSCID mouse model, had concluded that mature PCs (defined as CD38 hi CD45 Ϫ ), and not the CD19 ϩ B-cell fraction, contained the entire myeloma-propagating activity, 15 whereas more recently, CD19 Ϫ CD138 Ϫ as well as CD138 ϩ cells engrafted SCID-rab mice with myeloma. 16 Furthermore, whereas earlier studies reported that the myeloma side population is enriched in clonogenic activity and identifies with CD138 Ϫ but not CD138 ϩ myeloma cells, 13 recent evidence shows that both CD138 ϩ and CD138 Ϫ cells are included in the highly clonogenic myeloma side population. 17 Whether these discrepancies result from different animal models and phenotypic definitions of PC is not clear. Here, through a detailed phenotypic and genetic analysis of primary human myeloma cells and a prospective, dynamic ex vivo and in vivo study of the constituents of the myeloma cellular architecture, we show that a phenotypic and functional interconvertible state between CD138 ϩ and CD138 Ϫ cells underpins myelomapropagating activity and clinical drug resistance. Methods Patient and normal donor BM and PB samplesPatient BM and PB samples were obtained after written informed consent and appropriate institutional ethics committee approval. Patient characteristics are shown in supplemental Table 1 (available on the Blood Web site; see the Supplemental Materials link at the top of the online article). Diagnosis, remission, and relapse of MM were defined accordi...

Social and biological contagions are influenced by the spatial embeddedness of networks. Historically, many epidemics spread as a wave across part of the Earth’s surface; however, in modern contagions long-range edges—for example, due to airline transportation or communication media—allow clusters of a contagion to appear in distant locations. Here we study the spread of contagions on networks through a methodology grounded in topological data analysis and nonlinear dimension reduction. We construct “contagion maps” that use multiple contagions on a network to map the nodes as a point cloud. By analyzing the topology, geometry, and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modeling, forecast, and control of spreading processes. Our approach highlights contagion maps also as a viable tool for inferring low-dimensional structure in networks.

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The transition between the proliferation and differentiation of progenitor cells is a key step in organogenesis, and alterations in this process can lead to developmental disorders. The extracellular signal-regulated kinase 1/2 (ERK) signaling pathway is one of the most intensively studied signaling mechanisms that regulates both proliferation and differentiation. How a single molecule (e.g. ERK) can regulate two opposing cellular outcomes is still a mystery. Using both chick and mouse models, we shed light on the mechanism responsible for the switch from proliferation to differentiation of head muscle progenitors and implicate ERK subcellular localization. Manipulation of the fibroblast growth factor (FGF)-ERK signaling pathway in chick embryos in vitro and in vivo demonstrated that blockage of this pathway accelerated myogenic differentiation, whereas its activation diminished it. We next examined whether the spatial subcellular localization of ERK could act as a switch between proliferation (nuclear ERK) and differentiation (cytoplasmic ERK) of muscle progenitors. A myristoylated peptide that blocks importin 7-mediated ERK nuclear translocation induced robust myogenic differentiation of muscle progenitor/stem cells in both head and trunk. In the mouse, analysis of Sprouty mutant embryos revealed that increased ERK signaling suppressed both head and trunk myogenesis. Our findings, corroborated by mathematical modeling, suggest that ERK shuttling between the nucleus and the cytoplasm provides a switch-like transition between proliferation and differentiation of muscle progenitors.

We use topological data analysis to study "functional networks" that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into these functional networks, and we use persistence landscapes to interpret our results. Our first example uses time-series output from networks of coupled Kuramoto oscillators. Our second example consists of biological data in the form of functional magnetic resonance imaging data that were acquired from human subjects during a simple motor-learning task in which subjects were monitored for three days during a five-day period. With these examples, we demonstrate that (1) using persistent homology to study functional networks provides fascinating insights into their properties and (2) the position of the features in a filtration can sometimes play a more vital role than persistence in the interpretation of topological features, even though conventionally the latter is used to distinguish between signal and noise. We find that persistent homology can detect differences in synchronization patterns in our data sets over time, giving insight both on changes in community structure in the networks and on increased synchronization between brain regions that form loops in a functional network during motor learning. For the motor-learning data, persistence landscapes also reveal that on average the majority of changes in the network loops take place on the second of the three days of the learning process.

Modeling complex systems and data using the language of graphs and networks has become an essential topic across a range of different disciplines. Arguably, this network-based perspective derives is success from the relative simplicity of graphs: A graph consists of nothing more than a set of vertices and a set of edges, describing relationships between pairs of such vertices. This simple combinatorial structure makes graphs interpretable and flexible modeling tools. The simplicity of graphs as system models, however, has been scrutinized in the literature recently. Specifically, it has been argued from a variety of different angles that there is a need for higher-order networks, which go beyond the paradigm of modeling pairwise relationships, as encapsulated by graphs. In this survey article we take stock of these recent developments. Our goals are to clarify (i) what higher-order networks are, (ii) why these are interesting objects of study, and (iii) how they can be used in applications.