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Kong J, Kim J, Kim D, Lee K, Lee J, Han SK, Kim I, Lim S, Park M, Shin S, Lee WY, Yun SH, Kim HC, Hong HK, Cho YB, Park D, Kim S. Information about immune cell proportions and tumor stage improves the prediction of recurrence in patients with colorectal cancer. PATTERNS (NEW YORK, N.Y.) 2023; 4:100736. [PMID: 37409049 PMCID: PMC10318368 DOI: 10.1016/j.patter.2023.100736] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/27/2022] [Revised: 11/21/2022] [Accepted: 03/28/2023] [Indexed: 07/07/2023]
Abstract
Predicting cancer recurrence is essential to improving the clinical outcomes of patients with colorectal cancer (CRC). Although tumor stage information has been used as a guideline to predict CRC recurrence, patients with the same stage show different clinical outcomes. Therefore, there is a need to develop a method to identify additional features for CRC recurrence prediction. Here, we developed a network-integrated multiomics (NIMO) approach to select appropriate transcriptome signatures for better CRC recurrence prediction by comparing the methylation signatures of immune cells. We validated the performance of the CRC recurrence prediction based on two independent retrospective cohorts of 114 and 110 patients. Moreover, to confirm that the prediction was improved, we used both NIMO-based immune cell proportions and TNM (tumor, node, metastasis) stage data. This work demonstrates the importance of (1) using both immune cell composition and TNM stage data and (2) identifying robust immune cell marker genes to improve CRC recurrence prediction.
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Affiliation(s)
- JungHo Kong
- Department of Life Sciences, Pohang University of Science and Technology, Pohang 37673, Korea
| | - Jinho Kim
- Precision Medicine Center, Future Innovation Research Division, Seoul National University Bundang Hospital, Seongnam 13620, Korea
| | - Donghyo Kim
- Department of Life Sciences, Pohang University of Science and Technology, Pohang 37673, Korea
| | - Kwanghwan Lee
- Department of Life Sciences, Pohang University of Science and Technology, Pohang 37673, Korea
| | - Juhun Lee
- Department of Life Sciences, Pohang University of Science and Technology, Pohang 37673, Korea
| | - Seong Kyu Han
- Department of Life Sciences, Pohang University of Science and Technology, Pohang 37673, Korea
| | - Inhae Kim
- Institute for Future Medicine, Samsung Medical Center, Seoul 06351, Korea
- Department of Health Sciences and Technology, SAIHST, Sungkyunkwan University, Seoul 06351, Korea
| | - Seongsu Lim
- School of Interdisciplinary Bioscience and Bioengineering, Pohang University of Science and Technology, Pohang 37673, Korea
| | - Minhyuk Park
- Department of Life Sciences, Pohang University of Science and Technology, Pohang 37673, Korea
| | | | - Woo Yong Lee
- Department of Surgery, Samsung Medical Center, Sungkyunkwan University School of Medicine, Seoul 06351, Korea
| | - Seong Hyeon Yun
- Department of Surgery, Samsung Medical Center, Sungkyunkwan University School of Medicine, Seoul 06351, Korea
| | - Hee Cheol Kim
- Department of Surgery, Samsung Medical Center, Sungkyunkwan University School of Medicine, Seoul 06351, Korea
| | - Hye Kyung Hong
- Institute for Future Medicine, Samsung Medical Center, Seoul 06351, Korea
| | - Yong Beom Cho
- Department of Surgery, Samsung Medical Center, Sungkyunkwan University School of Medicine, Seoul 06351, Korea
- Department of Health Sciences and Technology, SAIHST, Sungkyunkwan University, Seoul 06351, Korea
| | | | - Sanguk Kim
- Department of Life Sciences, Pohang University of Science and Technology, Pohang 37673, Korea
- Institute of Convergence Science, Yonsei University, Seoul 120-749, Korea
- School of Interdisciplinary Bioscience and Bioengineering, Pohang University of Science and Technology, Pohang 37673, Korea
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2
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Manipur I, Giordano M, Piccirillo M, Parashuraman S, Maddalena L. Community Detection in Protein-Protein Interaction Networks and Applications. IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 2023; 20:217-237. [PMID: 34951849 DOI: 10.1109/tcbb.2021.3138142] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/04/2023]
Abstract
The ability to identify and characterize not only the protein-protein interactions but also their internal modular organization through network analysis is fundamental for understanding the mechanisms of biological processes at the molecular level. Indeed, the detection of the network communities can enhance our understanding of the molecular basis of disease pathology, and promote drug discovery and disease treatment in personalized medicine. This work gives an overview of recent computational methods for the detection of protein complexes and functional modules in protein-protein interaction networks, also providing a focus on some of its applications. We propose a systematic reformulation of frequently adopted taxonomies for these methods, also proposing new categories to keep up with the most recent research. We review the literature of the last five years (2017-2021) and provide links to existing data and software resources. Finally, we survey recent works exploiting module identification and analysis, in the context of a variety of disease processes for biomarker identification and therapeutic target detection. Our review provides the interested reader with an up-to-date and self-contained view of the existing research, with links to state-of-the-art literature and resources, as well as hints on open issues and future research directions in complex detection and its applications.
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3
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Xuan C, Wang Y, Zhang B, Wu H, Ding T, Gao J. scBPGRN: Integrating single-cell multi-omics data to construct gene regulatory networks based on BP neural network. Comput Biol Med 2022; 151:106249. [PMID: 36335815 DOI: 10.1016/j.compbiomed.2022.106249] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2022] [Revised: 10/14/2022] [Accepted: 10/22/2022] [Indexed: 12/27/2022]
Abstract
The deterioration and metastasis of cancer involve various aspects of genomic changes, including genomic DNA changes, epigenetic modifications, gene expression, and other complex interactions. Therefore, integrating single-cell multi-omics data to construct gene regulatory networks containing more omics information is of great significance for understanding the pathogenesis of cancer. In this article, an algorithm integrating single-cell RNA sequencing data and DNA methylation data to construct a gene regulatory network based on the back-propagation (BP) neural network (scBPGRN) is proposed. This algorithm uses biweight extreme correlation coefficients to measure the correlation between factors and uses neural networks to calculate generalized weights to construct gene regulation networks. Finally, the node strength is calculated to identify the genes associated with cancer. We apply the scBPGRN algorithm to hepatocellular carcinoma (HCC) data. We construct a regulatory network and identify top-ranked genes, such as MYCBP, KLHL35, PRKCZ, and SERPINA6, as the key HCC-related genes. We analyze the top 100 genes, and the HCC-related genes are concentrated in the top 20. In addition, the single cell data is found to consist of two subpopulations. We also apply scBPGRN to two subpopulations. We analyze the top 50 genes in them, and the HCC-related genes are concentrated in the top 20. The consequences of functional enrichment analysis indicate that the gene regulatory network we have constructed is valid. Our results have been verified in several pieces of literature. This study provides a reference for the integration of single-cell multi-omics data to construct gene regulatory networks.
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Affiliation(s)
- Chenxu Xuan
- School of Science, Jiangnan University, Wuxi, Jiangsu, 214122, China
| | - Yan Wang
- School of Science, Jiangnan University, Wuxi, Jiangsu, 214122, China
| | - Bai Zhang
- School of Science, Jiangnan University, Wuxi, Jiangsu, 214122, China
| | - Hanwen Wu
- School of Science, Jiangnan University, Wuxi, Jiangsu, 214122, China
| | - Tao Ding
- School of Mathematics Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK
| | - Jie Gao
- School of Science, Jiangnan University, Wuxi, Jiangsu, 214122, China.
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4
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Robin V, Bodein A, Scott-Boyer MP, Leclercq M, Périn O, Droit A. Overview of methods for characterization and visualization of a protein–protein interaction network in a multi-omics integration context. Front Mol Biosci 2022; 9:962799. [PMID: 36158572 PMCID: PMC9494275 DOI: 10.3389/fmolb.2022.962799] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/06/2022] [Accepted: 08/16/2022] [Indexed: 11/26/2022] Open
Abstract
At the heart of the cellular machinery through the regulation of cellular functions, protein–protein interactions (PPIs) have a significant role. PPIs can be analyzed with network approaches. Construction of a PPI network requires prediction of the interactions. All PPIs form a network. Different biases such as lack of data, recurrence of information, and false interactions make the network unstable. Integrated strategies allow solving these different challenges. These approaches have shown encouraging results for the understanding of molecular mechanisms, drug action mechanisms, and identification of target genes. In order to give more importance to an interaction, it is evaluated by different confidence scores. These scores allow the filtration of the network and thus facilitate the representation of the network, essential steps to the identification and understanding of molecular mechanisms. In this review, we will discuss the main computational methods for predicting PPI, including ones confirming an interaction as well as the integration of PPIs into a network, and we will discuss visualization of these complex data.
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Affiliation(s)
- Vivian Robin
- Molecular Medicine Department, CHU de Québec Research Center, Université Laval, Québec, QC, Canada
| | - Antoine Bodein
- Molecular Medicine Department, CHU de Québec Research Center, Université Laval, Québec, QC, Canada
| | - Marie-Pier Scott-Boyer
- Molecular Medicine Department, CHU de Québec Research Center, Université Laval, Québec, QC, Canada
| | - Mickaël Leclercq
- Molecular Medicine Department, CHU de Québec Research Center, Université Laval, Québec, QC, Canada
| | - Olivier Périn
- Digital Sciences Department, L'Oréal Advanced Research, Aulnay-sous-bois, France
| | - Arnaud Droit
- Molecular Medicine Department, CHU de Québec Research Center, Université Laval, Québec, QC, Canada
- *Correspondence: Arnaud Droit,
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5
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Pardo-Diaz J, Poole PS, Beguerisse-Díaz M, Deane CM, Reinert G. Generating weighted and thresholded gene coexpression networks using signed distance correlation. NETWORK SCIENCE (CAMBRIDGE UNIVERSITY PRESS) 2022; 10:131-145. [PMID: 36217370 PMCID: PMC7613200 DOI: 10.1017/nws.2022.13] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/16/2023]
Abstract
Even within well-studied organisms, many genes lack useful functional annotations. One way to generate such functional information is to infer biological relationships between genes or proteins, using a network of gene coexpression data that includes functional annotations. Signed distance correlation has proved useful for the construction of unweighted gene coexpression networks. However, transforming correlation values into unweighted networks may lead to a loss of important biological information related to the intensity of the correlation. Here we introduce a principled method to construct weighted gene coexpression networks using signed distance correlation. These networks contain weighted edges only between those pairs of genes whose correlation value is higher than a given threshold. We analyse data from different organisms and find that networks generated with our method based on signed distance correlation are more stable and capture more biological information compared to networks obtained from Pearson correlation. Moreover, we show that signed distance correlation networks capture more biological information than unweighted networks based on the same metric. While we use biological data sets to illustrate the method, the approach is general and can be used to construct networks in other domains. Code and data are available on https://github.com/javier-pardodiaz/sdcorGCN.
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Affiliation(s)
| | - Philip S Poole
- Department of Plant Sciences, University of Oxford, Oxford OX1 3RB, UK
| | | | | | - Gesine Reinert
- Department of Statistics, University of Oxford, Oxford OX1 3LB, UK
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6
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Hrovatin K, Fischer DS, Theis FJ. Toward modeling metabolic state from single-cell transcriptomics. Mol Metab 2022; 57:101396. [PMID: 34785394 PMCID: PMC8829761 DOI: 10.1016/j.molmet.2021.101396] [Citation(s) in RCA: 20] [Impact Index Per Article: 10.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 08/05/2021] [Revised: 10/21/2021] [Accepted: 11/09/2021] [Indexed: 12/31/2022] Open
Abstract
BACKGROUND Single-cell metabolic studies bring new insights into cellular function, which can often not be captured on other omics layers. Metabolic information has wide applicability, such as for the study of cellular heterogeneity or for the understanding of drug mechanisms and biomarker development. However, metabolic measurements on single-cell level are limited by insufficient scalability and sensitivity, as well as resource intensiveness, and are currently not possible in parallel with measuring transcript state, commonly used to identify cell types. Nevertheless, because omics layers are strongly intertwined, it is possible to make metabolic predictions based on measured data of more easily measurable omics layers together with prior metabolic network knowledge. SCOPE OF REVIEW We summarize the current state of single-cell metabolic measurement and modeling approaches, motivating the use of computational techniques. We review three main classes of computational methods used for prediction of single-cell metabolism: pathway-level analysis, constraint-based modeling, and kinetic modeling. We describe the unique challenges arising when transitioning from bulk to single-cell modeling. Finally, we propose potential model extensions and computational methods that could be leveraged to achieve these goals. MAJOR CONCLUSIONS Single-cell metabolic modeling is a rising field that provides a new perspective for understanding cellular functions. The presented modeling approaches vary in terms of input requirements and assumptions, scalability, modeled metabolic layers, and newly gained insights. We believe that the use of prior metabolic knowledge will lead to more robust predictions and will pave the way for mechanistic and interpretable machine-learning models.
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Affiliation(s)
- Karin Hrovatin
- Institute of Computational Biology, Helmholtz Center Munich, Ingolstaedter Landstraße 1, Neuherberg, 85764, Germany; TUM School of Life Sciences Weihenstephan, Technical University of Munich, Alte Akademie 8, Freising, 85354, Germany.
| | - David S Fischer
- Institute of Computational Biology, Helmholtz Center Munich, Ingolstaedter Landstraße 1, Neuherberg, 85764, Germany; TUM School of Life Sciences Weihenstephan, Technical University of Munich, Alte Akademie 8, Freising, 85354, Germany.
| | - Fabian J Theis
- Institute of Computational Biology, Helmholtz Center Munich, Ingolstaedter Landstraße 1, Neuherberg, 85764, Germany; TUM School of Life Sciences Weihenstephan, Technical University of Munich, Alte Akademie 8, Freising, 85354, Germany; Department of Mathematics, Technical University of Munich, Boltzmannstr. 3, Garching bei München, 85748, Germany.
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7
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Zhang C, Gao L, Wang B, Gao Y. Improving Single-Cell RNA-seq Clustering by Integrating Pathways. Brief Bioinform 2021; 22:6262246. [PMID: 33940590 DOI: 10.1093/bib/bbab147] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/30/2020] [Revised: 03/21/2021] [Accepted: 03/26/2021] [Indexed: 01/03/2023] Open
Abstract
Single-cell clustering is an important part of analyzing single-cell RNA-sequencing data. However, the accuracy and robustness of existing methods are disturbed by noise. One promising approach for addressing this challenge is integrating pathway information, which can alleviate noise and improve performance. In this work, we studied the impact on accuracy and robustness of existing single-cell clustering methods by integrating pathways. We collected 10 state-of-the-art single-cell clustering methods, 26 scRNA-seq datasets and four pathway databases, combined the AUCell method and the similarity network fusion to integrate pathway data and scRNA-seq data, and introduced three accuracy indicators, three noise generation strategies and robustness indicators. Experiments on this framework showed that integrating pathways can significantly improve the accuracy and robustness of most single-cell clustering methods.
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Affiliation(s)
- Chenxing Zhang
- Computer Science and Technology at Xidian University, Xi'an 710071, China
| | - Lin Gao
- School of Computer Science and Technology at Xidian University, Xi'an 710071, China
| | - Bingbo Wang
- School of Computer Science and Technology, Xidian University, Xi'an 710071, China
| | - Yong Gao
- Computer Science at the University of British Columbia Okanagan (UBC Okanagan), Canada
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8
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Basu A, Ash PEA, Wolozin B, Emili A. Protein Interaction Network Biology in Neuroscience. Proteomics 2021; 21:e1900311. [PMID: 33314619 PMCID: PMC7900949 DOI: 10.1002/pmic.201900311] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/07/2020] [Revised: 11/27/2020] [Indexed: 01/04/2023]
Abstract
Mapping the intricate networks of cellular proteins in the human brain has the potential to address unsolved questions in molecular neuroscience, including the molecular basis of cognition, synaptic plasticity, long-term potentiation, learning, and memory. Perturbations to the protein-protein interaction networks (PPIN) present in neurons, glia, and other cell-types have been linked to multifactorial neurological disorders. Yet while knowledge of brain PPINs is steadily improving, the complexity and dynamic nature of the heterogeneous central nervous system in normal and disease contexts poses a formidable experimental challenge. In this review, the recent applications of functional proteomics and systems biology approaches to study PPINs central to normal neuronal function, during neurodevelopment, and in neurodegenerative disorders are summarized. How systematic PPIN analysis offers a unique mechanistic framework to explore intra- and inter-cellular functional modules governing neuronal activity and brain function is also discussed. Finally, future technological advancements needed to address outstanding questions facing neuroscience are outlined.
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Affiliation(s)
- Avik Basu
- Center for Network Systems BiologyBoston UniversityBostonMA02118USA
- Department of BiochemistryBoston University School of MedicineBostonMA02118USA
| | - Peter EA Ash
- Department of Pharmacology and Experimental TherapeuticsBoston University School of MedicineBostonMA02118USA
| | - Benjamin Wolozin
- Department of Pharmacology and Experimental TherapeuticsBoston University School of MedicineBostonMA02118USA
| | - Andrew Emili
- Center for Network Systems BiologyBoston UniversityBostonMA02118USA
- Department of BiochemistryBoston University School of MedicineBostonMA02118USA
- Department of BiologyBoston UniversityBostonMA02215USA
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9
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Grønning AGB, Oubounyt M, Kanev K, Lund J, Kacprowski T, Zehn D, Röttger R, Baumbach J. Enabling single-cell trajectory network enrichment. NATURE COMPUTATIONAL SCIENCE 2021; 1:153-163. [PMID: 38217228 DOI: 10.1038/s43588-021-00025-y] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2020] [Accepted: 01/15/2021] [Indexed: 01/15/2024]
Abstract
Single-cell sequencing (scRNA-seq) technologies allow the investigation of cellular differentiation processes with unprecedented resolution. Although powerful software packages for scRNA-seq data analysis exist, systems biology-based tools for trajectory analysis are rare and typically difficult to handle. This hampers biological exploration and prevents researchers from gaining deeper insights into the molecular control of developmental processes. Here, to address this, we have developed Scellnetor; a network-constraint time-series clustering algorithm. It allows extraction of temporal differential gene expression network patterns (modules) that explain the difference in regulation of two developmental trajectories. Using well-characterized experimental model systems, we demonstrate the capacity of Scellnetor as a hypothesis generator to identify putative mechanisms driving haematopoiesis or mechanistically interpretable subnetworks driving dysfunctional CD8 T-cell development in chronic infections. Altogether, Scellnetor allows for single-cell trajectory network enrichment, which effectively lifts scRNA-seq data analysis to a systems biology level.
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Affiliation(s)
- Alexander G B Grønning
- Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark.
- Novo Nordisk Foundation Center for Basic Metabolic Research, Faculty of Health and Medical Sciences, University of Copenhagen, Copenhagen, Denmark.
| | - Mhaned Oubounyt
- Chair of Experimental Bioinformatics, TUM School of Life Sciences Weihenstephan, Technical University of Munich, Freising, Germany
- Chair of Computational Systems Biology, University of Hamburg, Hamburg, Germany
| | - Kristiyan Kanev
- Division of Animal Physiology and Immunology, TUM School of Life Sciences Weihenstephan, Technical University of Munich, Freising, Germany
| | - Jesper Lund
- Department of Biostatistics and Epidemiology, University of Southern Denmark, Odense, Denmark
| | - Tim Kacprowski
- Division Data Science in Biomedicine, Peter L. Reichertz Institute for Medical Informatics of TU Braunschweig and Hannover Medical School, Brunswick, Germany
| | - Dietmar Zehn
- Division of Animal Physiology and Immunology, TUM School of Life Sciences Weihenstephan, Technical University of Munich, Freising, Germany
| | - Richard Röttger
- Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark
| | - Jan Baumbach
- Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark.
- Chair of Experimental Bioinformatics, TUM School of Life Sciences Weihenstephan, Technical University of Munich, Freising, Germany.
- Chair of Computational Systems Biology, University of Hamburg, Hamburg, Germany.
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10
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Klimm F. Functional change along cellular trajectories. NATURE COMPUTATIONAL SCIENCE 2021; 1:102-103. [PMID: 38217227 DOI: 10.1038/s43588-021-00026-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/15/2024]
Affiliation(s)
- Florian Klimm
- Department of Mathematics, Imperial College London, London, UK.
- Medical Research Council Mitochondrial Biology Unit, University of Cambridge, Cambridge, UK.
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11
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Reyna MA, Chitra U, Elyanow R, Raphael BJ. NetMix: A Network-Structured Mixture Model for Reduced-Bias Estimation of Altered Subnetworks. J Comput Biol 2021; 28:469-484. [PMID: 33400606 DOI: 10.1089/cmb.2020.0435] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022] Open
Abstract
A classic problem in computational biology is the identification of altered subnetworks: subnetworks of an interaction network that contain genes/proteins that are differentially expressed, highly mutated, or otherwise aberrant compared with other genes/proteins. Numerous methods have been developed to solve this problem under various assumptions, but the statistical properties of these methods are often unknown. For example, some widely used methods are reported to output very large subnetworks that are difficult to interpret biologically. In this work, we formulate the identification of altered subnetworks as the problem of estimating the parameters of a class of probability distributions that we call the Altered Subset Distribution (ASD). We derive a connection between a popular method, jActiveModules, and the maximum likelihood estimator (MLE) of the ASD. We show that the MLE is statistically biased, explaining the large subnetworks output by jActiveModules. Based on these insights, we introduce NetMix, an algorithm that uses Gaussian mixture models to obtain less biased estimates of the parameters of the ASD. We demonstrate that NetMix outperforms existing methods in identifying altered subnetworks on both simulated and real data, including the identification of differentially expressed genes from both microarray and RNA-seq experiments and the identification of cancer driver genes in somatic mutation data.
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Affiliation(s)
- Matthew A Reyna
- Department of Biomedical Informatics, Emory University, Atlanta, Georgia, USA
| | - Uthsav Chitra
- Department of Computer Science, Princeton University, Princeton, New Jersey, USA
| | - Rebecca Elyanow
- Department of Computer Science, Princeton University, Princeton, New Jersey, USA
- Department of Computer Science, Brown University, Providence, Rhode Island, USA
| | - Benjamin J Raphael
- Department of Computer Science, Princeton University, Princeton, New Jersey, USA
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