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PubMed2004

A graph theory analysis of renal glomerular microvascular networks.

A graph theory model and its invariants are used to compare previously published renal glomerular networks of six adult rats, one adult uremic rat, and one newborn rat. Invariants calculated include order, size, cycle rank, eccentricity, root distance, planarity, and vertex degree distribution. These invariants enabled the differentiation of six normal adult glomerular microvascular networks from that of the uremic glomerulus and from that of the normal newborn glomerulus. These invariants might then be used to differentiate between normal and pathological vascular networks. Also proposed are graph theory invariants that might be used to develop a quantitative model for angiogenesis.

Animals↗

PubMed2022

A graph-based approach for the visualisation and analysis of bacterial pangenomes.

BACKGROUND: The advent of low cost, high throughput DNA sequencing has led to the availability of thousands of complete genome sequences for a wide variety of bacterial species. Examining and interpreting genetic variation on this scale represents a significant challenge to existing methods of data analysis and visualisation. RESULTS: Starting with the output of standard pangenome analysis tools, we describe the generation and analysis of interactive, 3D network graphs to explore the structure of bacterial populations, the distribution of genes across a population, and the syntenic order in which those genes occur, in the new open-source network analysis platform, Graphia. Both the analysis and the visualisation are scalable to datasets of thousands of genome sequences. CONCLUSIONS: We anticipate that the approaches presented here will be of great utility to the microbial research community, allowing faster, more intuitive, and flexible interaction with pangenome datasets, thereby enhancing interpretation of these complex data.

Bacteria↗

PubMed2004

Experimental simulation of quantum graphs by microwave networks.

We present the results of experimental and theoretical study of irregular, tetrahedral microwave networks consisting of coaxial cables (annular waveguides) connected by T joints. The spectra of the networks were measured in the frequency range 0.0001-16 GHz in order to obtain their statistical properties such as the integrated nearest neighbor spacing distribution and the spectral rigidity Delta(3) (L). The comparison of our experimental and theoretical results shows that microwave networks can simulate quantum graphs with time reversal symmetry. In particular, we use the spectra of the microwave networks to study the periodic orbits of the simulated quantum graphs. We also present experimental study of directional microwave networks consisting of coaxial cables and Faraday isolators for which the time reversal symmetry is broken. In this case our experimental results indicate that spectral statistics of directional microwave networks deviate from predictions of Gaussian orthogonal ensembles in random matrix theory approaching, especially for small eigenfrequency spacing s, results for Gaussian unitary ensembles. Experimental results are supported by the theoretical analysis of directional graphs.

Journal Article↗

PubMed2004

Spread of two linked social norms on complex interaction networks.

In this paper, we study the spread of social norms, such as rules and customs that are components of human cultures. We consider the spread of two social norms, which are linked through individual behaviors. Spreading social norms depend not only on the social network structure, but also on the learning system. We consider four social network structures: (1) complete mixing, in which each individual interacts with the others at random, (2) lattice, in which each individual interacts with its neighbors with some probability and with the others at random, (3) power-law network, in which a few influential people have more social contacts than the others, and (4) random graph network, in which the number of contacts follows a Poisson distribution. Using the lattice model, we also investigate the effect of the small-world phenomenon on the dynamics of social norms. In our models, each individual learns a social norm by trial and error (individual learning) and also imitates the other's social norm (social learning). We investigate how social network structure and learning systems affect the spread of two linked social norms. Our main results are: (1) Social learning does not lead to coexistence of social norms. Individual learning produces coexistence, and the dynamics of coexistence depend on which social norms are learned individually. (2) Social norms spread fastest in the power-law network model, followed by the random graph model, the complete mixing model, the two-dimensional lattice model and the one-dimensional lattice. (3) We see a "small world effect" in the one-dimensional model, but not in two dimensions.

Computer Simulation↗

PubMed2026

Network methods for diagonal integration of unpaired single-cell multiomics data: a review.

MOTIVATION: Advances in single-cell sequencing have enabled multiomics profiling at unprecedented resolution; however, mass spectrometry-based single-cell proteomics (scMS) remains inherently destructive, precluding simultaneous transcriptomic capture. Unlike antibody-based methods such as CITE-seq, which permit paired profiling but are restricted to targeted protein panels, scMS provides unbiased, genome-scale coverage of the intracellular proteome yet necessitates post hoc integration of unpaired datasets. This diagonal integration challenge, where transcriptomes and proteomes are measured in separate cells lacking shared anchors, remains underserved by existing reviews, which focus predominantly on vertical integration strategies enabled by non-destructive assays. RESULTS: We survey the complete computational pipeline for constructing mechanistic proteogenomic networks from unpaired single-cell data, covering: (i) unimodal network inference such as knowledge-based approaches, probabilistic graphical models, temporal directionality inference, and generative and foundation model strategies that establish the transcriptomic scaffold; (ii) cross-modal integration architectures such as network propagation, graph neural networks (scMRDR, scmFormer, scCotag), and consensus frameworks designed explicitly for the unpaired proteomics setting; and (iii) benchmarking paradigms spanning network reconstruction (BEELINE, GRETA, CausalBench) and multi-task integration evaluation (scMultiBench, SCMMIB), with guidance on metric selection under network sparsity and class imbalance. We identify three principal axes of future development: generative proteomic translation from transcriptomic precursors, inductive prior embedding in next-generation architectures, and perturbation-based causal benchmarking. AVAILABILITY AND IMPLEMENTATION: This is a review article; no novel software is distributed. A curated benchmark resource table, methods starter guide, and per-method bottleneck annotations are provided in the Supplementary Material.

Multiomics↗

PubMed2004

Metabolic pathway analysis web service (Pathway Hunter Tool at CUBIC).

MOTIVATION: Pathway Hunter Tool (PHT), is a fast, robust and user-friendly tool to analyse the shortest paths in metabolic pathways. The user can perform shortest path analysis for one or more organisms or can build virtual organisms (networks) using enzymes. Using PHT, the user can also calculate the average shortest path (Jungnickel, 2002 Graphs, Network and Algorithm. Springer-Verlag, Berlin), average alternate path and the top 10 hubs in the metabolic network. The comparative study of metabolic connectivity and observing the cross talk between metabolic pathways among various sequenced genomes is possible. RESULTS: A new algorithm for finding the biochemically valid connectivity between metabolites in a metabolic network was developed and implemented. A predefined manual assignment of side metabolites (like ATP, ADP, water, CO(2) etc.) and main metabolites is not necessary as the new concept uses chemical structure information (global and local similarity) between metabolites for identification of the shortest path.

Algorithms↗

PubMed2003

Subnetwork hierarchies of biochemical pathways.

MOTIVATION: The vastness and complexity of the biochemical networks that have been mapped out by modern genomics calls for decomposition into subnetworks. Such networks can have inherent non-local features that require the global structure to be taken into account in the decomposition procedure. Furthermore, basic questions such as to what extent the network (graph theoretically) can be said to be built by distinct subnetworks are little studied. RESULTS: We present a method to decompose biochemical networks into subnetworks based on the global geometry of the network. This method enables us to analyze the full hierarchical organization of biochemical networks and is applied to 43 organisms from the WIT database. Two types of biochemical networks are considered: metabolic networks and whole-cellular networks (also including for example information processes). Conceptual and quantitative ways of describing the hierarchical ordering are discussed. The general picture of the metabolic networks arising from our study is that of a few core-clusters centred around the most highly connected substances enclosed by other substances in outer shells, and a few other well-defined subnetworks. AVAILABILITY: An implementation of our algorithm and other programs for analyzing the data is available from http://www.tp.umu.se/forskning/networks/meta/ SUPPLEMENTARY INFORMATION: Supplementary material is available at http://www.tp.umu.se/forskning/networks/meta/

Algorithms↗

PubMed2004

RAG: RNA-As-Graphs database--concepts, analysis, and features.

MOTIVATION: Understanding RNA's structural diversity is vital for identifying novel RNA structures and pursuing RNA genomics initiatives. By classifying RNA secondary motifs based on correlations between conserved RNA secondary structures and functional properties, we offer an avenue for predicting novel motifs. Although several RNA databases exist, no comprehensive schemes are available for cataloguing the range and diversity of RNA's structural repertoire. RESULTS: Our RNA-As-Graphs (RAG) database describes and ranks all mathematically possible (including existing and candidate) RNA secondary motifs on the basis of graphical enumeration techniques. We represent RNA secondary structures as two-dimensional graphs (networks), specifying the connectivity between RNA secondary structural elements, such as loops, bulges, stems and junctions. We archive RNA tree motifs as 'tree graphs' and other RNAs, including pseudoknots, as general 'dual graphs'. All RNA motifs are catalogued by graph vertex number (a measure of sequence length) and ranked by topological complexity. The RAG inventory immediately suggests candidates for novel RNA motifs, either naturally occurring or synthetic, and thereby might stimulate the prediction and design of novel RNA motifs. AVAILABILITY: The database is accessible on the web at http://monod.biomath.nyu.edu/rna

Algorithms↗

PubMed2004

Molecular analysis of the dengue virus type 1 and 2 in Brazil based on sequences of the genomic envelope-nonstructural protein 1 junction region.

The genomic sequences of the Envelope-Non-Structural protein 1 junction region (E/NS1) of 84 DEN-1 and 22 DEN-2 isolates from Brazil were determined. Most of these strains were isolated in the period from 1995 to 2001 in endemic and regions of recent dengue transmission in São Paulo State. Sequence data for DEN-1 and DEN-2 utilized in phylogenetic and split decomposition analyses also include sequences deposited in GenBank from different regions of Brazil and of the world. Phylogenetic analyses were done using both maximum likelihood and Bayesian approaches. Results for both DEN-1 and DEN-2 data are ambiguous, and support for most tree bipartitions are generally poor, suggesting that E/NS1 region does not contain enough information for recovering phylogenetic relationships among DEN-1 and DEN-2 sequences used in this study. The network graph generated in the split decomposition analysis of DEN-1 does not show evidence of grouping sequences according to country, region and clades. While the network for DEN-2 also shows ambiguities among DEN-2 sequences, it suggests that Brazilian sequences may belong to distinct subtypes of genotype III.

Amino Acid Sequence↗

PubMed2026

Graph neural network-based risk stratification of prostate cancer using gene expression and SHAP interpretability.

Accurate risk stratification is essential for guiding treatment decisions and preventing over treatment of prostate cancer, which remains one of the most prevalent cancers among adult men. While the Gleason score, obtained from prostate biopsies, is routinely used to assess tumor aggressiveness, the biopsy procedure carries risks such as pain, infection, and, in some cases, serious complications such as sepsis. In this study, we proposed an artificial intelligence-based framework that integrates mRNA expression profiles with functional interaction networks to classify prostate cancer patients into low-, medium-, and high-risk groups defined by Gleason scores. The pipeline comprised five steps: (1) data collection from The Cancer Genome Atlas (TCGA), (2) preprocessing of gene expression data, (3) two-stage feature selection to identify informative biomarkers, (4) risk classification using a dual-branch graph neural network (GNN) that combines gene-gene interaction graphs with sample-level expression features, and (5) model interpretation using SHAP to quantify feature contributions. Differentially expressed genes were identified in the High (ASPN, GMNN, PEBP4, C2, KNCK17), Medium (C2, IGSF1, ASPN, CDKN3, AMH), and Low (TNMD, VWA5B2, ST6GALNAC5, CYP3A5, PHGR1) risk groups, underscoring the molecular heterogeneity of disease progression. On an independent held-out test set, the model achieved AUCs of 0.86, 0.88, and 0.95 for the low-, medium-, and high-risk groups, respectively, with an overall accuracy of 80%. These results suggest that combining GNN-based modeling with explainable AI can capture both global and local molecular patterns relevant to tumor aggressiveness. However, as the model was developed and evaluated solely on the TCGA cohort, the findings should be regarded as exploratory, and external validation will be required to establish generalizability. Within these limitations, the proposed framework highlights the potential of molecular profiling and graph-based deep learning to support more precise, potentially less invasive, risk assessment and individualized treatment planning in prostate cancer.

Prostatic Neoplasms↗

PubMed2025

Deep learning-based cell-specific gene regulatory networks inferred from single-cell multiome data.

Gene regulatory networks (GRNs) provide a global representation of how genetic/genomic information is transferred in living systems and are a key component in understanding genome regulation. Single-cell multiome data provide unprecedented opportunities to reconstruct GRNs at fine-grained resolution. However, the inference of GRNs is hindered by insufficient single omic profiles due to the characteristic high loss rate of single-cell sequencing data. In this study, we developed scMultiomeGRN, a deep learning framework to infer transcription factor (TF) regulatory networks via unique integration of single-cell genomic (single-cell RNA sequencing) and epigenomic (single-cell ATAC sequencing) data. We create scMultiomeGRN to elucidate these networks by conceptualizing TF network graph structures. Specifically, we build modality-specific neighbor aggregators and cross-modal attention modules to learn latent representations of TFs from single-cell multi-omics. We demonstrate that scMultiomeGRN outperforms state-of-the-art models on multiple benchmark datasets involved in diseases and health. Via scMultiomeGRN, we identified Alzheimer's disease-relevant regulatory network of SPI1 and RUNX1 for microglia. In summary, scMultiomeGRN offers a deep learning framework to identify cell type-specific gene regulatory network from single-cell multiome data.

Deep Learning↗

PubMed2004

Architectural analysis of oral cancer, dysplastic, and normal epithelia.

BACKGROUND: We present a novel, automated, and quantitative approach to evaluate local epithelial tissue architecture based on mathematical graph theory. METHODS: Four hundred forty-one images of three diagnostic classes of oral epithelium (normal, dysplastic, and neoplastic) were analysed. The epithelial compartment was partitioned into exclusive areas associated with each nucleus to approach the theoretical cell extents. The spatial arrangement of cells in neighbourhoods of two sizes was characterised by constructing graph networks based on the cell centroids and recording 29 statistical properties. We analysed 104,627 and 67,590 neighbourhoods of diameters 37.5 and 75 microm, respectively. RESULTS: The discrimination power of the architectural descriptors was evaluated by using discriminant analysis. The best neighbourhood discrimination rate was 75% for normal versus carcinoma. For the pooled data, discrimination into three classes based on largest number of neighbourhoods associated with each class was 100% correct. Case-wise, discrimination rates were 67%, 100%, and 80% correct for normal, premalignant, and malignant. When considering two classes, discrimination rates was 89% (normal) and 100% (malignant) correct, with 71% of premalignant cases assigned to the malignant class. CONCLUSIONS: The results indicate that unbiased and reproducible quantification of tissue architectural features is possible and may provide valuable morphological information for diagnostic purposes.

Cytodiagnosis↗

PubMed2005

The impact of environmental fluctuations on structured discrete time population models: resonance, synchrony and threshold behaviour.

External forcing of a discrete time ecological system does not just add variation to existing dynamics but can change the dynamics. We study the mechanisms that can bring this about, focusing on the key concepts of excitation and suppression which emerge when analysing the power spectra of the system in linear approximation. Excitation, through resonance between the system dynamics and the external forcing, is the greater the closer the system is to the boundary of the stability region. This amplification means that the extinction of populations becomes possible sooner than expected and, conversely, invasion can be significantly delayed. Suppression and the consequent redistribution of power within the spectrum proves to be a function both of the connectivity of the network graph of the system and the way that external forcing is applied to the system. It is also established that colour in stochastic forcing can have a major impact, by enhancing resonance and by greater redistribution of power. This can mean a higher risk of extinction through larger fluctuations in population numbers and a higher degree of synchrony between populations. The implications of external forcing for stage-structured species, for populations in competition and for trophic web systems are studied using the tools and concepts developed in the paper.

Ecosystem↗

PubMed2026

Dissecting spatial patterning and signaling with directional diffusion in spatial multi-omics.

Spatial multi-omics sequencing enables the simultaneous profiling of transcriptomics, proteomics, and epigenomics at a spatial resolution, offering insights into complex tissue organization and molecular regulation. However, the effective integration of multiple omics modalities in a spatial context remains a major challenge. Here, we present SpaDDM, a spatial multi-omics integration framework based on directional diffusion models (DDMs), which supports spatial pattern identification, cross-omics alignment, and inter-and intracellular signaling flow analysis. SpaDDM employs DDM-based graph networks to learn omics-specific representations by jointly incorporating spatial coordinates and molecular measurements within each modality, followed by an attention mechanism to align features across modalities. We benchmarked SpaDDM on diverse spatial multi-omics datasets, including transcriptomics-epigenomics and transcriptomics-proteomics combinations across multiple tissues and species. SpaDDM consistently outperformed existing methods by more accurately deciphering spatial tissue patterns and effectively reducing the boundary noise between spatial regions. Moreover, the learned low-dimensional coembedded representations of individual cells serve as integral mediators for inferring the signaling flows that underlie spatial patterning. Finally, we demonstrated that SpaDDM alignment of complementary information across multi-omics layers facilitates cross-omics translation and significantly improves the prediction of cell state alignments.

Multiomics↗

PubMed2003

A segment alignment approach to protein comparison.

MOTIVATION: Local structure segments (LSSs) are small structural units shared by unrelated proteins. They are extensively used in protein structure comparison, and predicted LSSs (PLSSs) are used very successfully in ab initio folding simulations. However, predicted or real LSSs are rarely exploited by protein sequence comparison programs that are based on position-by-position alignments. RESULTS: We developed a SEgment Alignment algorithm (SEA) to compare proteins described as a collection of predicted local structure segments (PLSSs), which is equivalent to an unweighted graph (network). Any specific structure, real or predicted corresponds to a specific path in this network. SEA then uses a network matching approach to find two most similar paths in networks representing two proteins. SEA explores the uncertainty and diversity of predicted local structure information to search for a globally optimal solution. It simultaneously solves two related problems: the alignment of two proteins and the local structure prediction for each of them. On a benchmark of protein pairs with low sequence similarity, we show that application of the SEA algorithm improves alignment quality as compared to FFAS profile-profile alignment, and in some cases SEA alignments can match the structural alignments, a feat previously impossible for any sequence based alignment methods.

Algorithms↗

PubMed2005

Decomposing protein networks into domain-domain interactions.

UNLABELLED: The application of novel experimental techniques has generated large networks of protein-protein interactions. Frequently, important information on the structure and cellular function of protein-protein interactions can be gained from the domains of interacting proteins. We have designed a Cytoscape plugin that decomposes interacting proteins into their respective domains and computes a putative network of corresponding domain-domain interactions. To this end, the network graph of proteins has been extended by additional node and edge types for domain interactions, including different node and edge shapes and coloring schemes used for visualization. An additional plugin provides supplementary web links to Internet resources on domain function and structure. AVAILABILITY: Both Cytoscape plugins can be downloaded from http://www.cytoscape.org

Algorithms↗

PubMed2005

Integration of metabolic networks and gene expression in virtual reality.

MOTIVATION: Metabolic networks combine metabolism and regulation. These complex networks are difficult to understand and visualize due to the amount and diverse types of information that need to be represented. For example, pathway information gives indications of interactions. Experimental data, such as transcriptomics, proteomics and metabolomics data, give snapshots of the system state. Stereoscopic virtual environments provide a true three-dimensional representation of metabolic networks, which can be intuitively manipulated, and may help to manage the data complexity. RESULTS: MetNet3D, a 3D virtual reality system, allows a user to explore gene expression and metabolic pathway data simultaneously. Normalized gene expression data are processed in R and visualized as a 3D plot. Users can find a particular gene of interest or a cluster of genes that behave similarly and see how these genes function in metabolic networks from MetNetDB, a database of Arabidopsis metabolic networks, using animated network graphs. Interactive virtual reality, with its enhanced ability to display more information, makes such integration more effective by abstracting key relationships. AVAILABILITY: MetNet3D and some sample datasets are available at http://www.vrac.iastate.edu/research/sites/metnet/Download/Download.htm. SUPPLEMENTARY INFORMATION: Color snapshots and movies are available at http://www.vrac.iastate.edu/research/sites/metnet/Bioinformatics/SupplementaryInformation.htm.

Algorithms↗

PubMed2005

Quantum pumping: the charge transported due to a translation of a scatterer.

The amount of charge that is pushed by a moving scatterer is dQ= -GdX , where dX is the displacement of the scatterer. The question is: what is G ?. Does it depend on the transmission g(0) of the scatterer? Does the answer depend on whether the system is open (with leads attached to reservoirs) or closed? In the latter case what are the implications of having "quantum chaos" and/or coupling to the environment? The answers to these questions illuminate some fundamental aspects of the theory of quantum pumping. For the analysis we take a network (graph) as a model system, and use the Kubo formula approach.

Journal Article↗