1
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Gong H, Zhou Z, Bu C, Zhang D, Fang Q, Zhang XY, Song Y. Computational dissection of genetic variation modulating the response of multiple photosynthetic phenotypes to the light environment. BMC Genomics 2024; 25:81. [PMID: 38243219 PMCID: PMC10799405 DOI: 10.1186/s12864-024-09968-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2023] [Accepted: 01/03/2024] [Indexed: 01/21/2024] Open
Abstract
BACKGROUND The expression of biological traits is modulated by genetics as well as the environment, and the level of influence exerted by the latter may vary across characteristics. Photosynthetic traits in plants are complex quantitative traits that are regulated by both endogenous genetic factors and external environmental factors such as light intensity and CO2 concentration. The specific processes impacted occur dynamically and continuously as the growth of plants changes. Although studies have been conducted to explore the genetic regulatory mechanisms of individual photosynthetic traits or to evaluate the effects of certain environmental variables on photosynthetic traits, the systematic impact of environmental variables on the dynamic process of integrated plant growth and development has not been fully elucidated. RESULTS In this paper, we proposed a research framework to investigate the genetic mechanism of high-dimensional complex photosynthetic traits in response to the light environment at the genome level. We established a set of high-dimensional equations incorporating environmental regulators to integrate functional mapping and dynamic screening of gene‒environment complex systems to elucidate the process and pattern of intrinsic genetic regulatory mechanisms of three types of photosynthetic phenotypes of Populus simonii that varied with light intensity. Furthermore, a network structure was established to elucidate the crosstalk among significant QTLs that regulate photosynthetic phenotypic systems. Additionally, the detection of key QTLs governing the response of multiple phenotypes to the light environment, coupled with the intrinsic differences in genotype expression, provides valuable insights into the regulatory mechanisms that drive the transition of photosynthetic activity and photoprotection in the face of varying light intensity gradients. CONCLUSIONS This paper offers a comprehensive approach to unraveling the genetic architecture of multidimensional variations in photosynthetic phenotypes, considering the combined impact of integrated environmental factors from multiple perspectives.
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Affiliation(s)
- Huiying Gong
- College of Science, Beijing Forestry University, No. 35, Qinghua East Road, Beijing, 100083, P. R. China
| | - Ziyang Zhou
- College of Science, Beijing Forestry University, No. 35, Qinghua East Road, Beijing, 100083, P. R. China
| | - Chenhao Bu
- College of Biological Sciences and Technology, Beijing Forestry University, No. 35, Qinghua East Road, Beijing, 100083, P. R. China
| | - Deqiang Zhang
- College of Biological Sciences and Technology, Beijing Forestry University, No. 35, Qinghua East Road, Beijing, 100083, P. R. China
| | - Qing Fang
- Faculty of Science, Yamagata University, Yamagata, 990, Japan
| | - Xiao-Yu Zhang
- College of Science, Beijing Forestry University, No. 35, Qinghua East Road, Beijing, 100083, P. R. China.
| | - Yuepeng Song
- College of Biological Sciences and Technology, Beijing Forestry University, No. 35, Qinghua East Road, Beijing, 100083, P. R. China.
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2
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Zhou F, Liu Y, Ren J, Wang W, Wu C. Springer: An R package for bi-level variable selection of high-dimensional longitudinal data. Front Genet 2023; 14:1088223. [PMID: 37091810 PMCID: PMC10117642 DOI: 10.3389/fgene.2023.1088223] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2022] [Accepted: 02/28/2023] [Indexed: 04/09/2023] Open
Abstract
In high-dimensional data analysis, the bi-level (or the sparse group) variable selection can simultaneously conduct penalization on the group level and within groups, which has been developed for continuous, binary, and survival responses in the literature. Zhou et al. (2022) (PMID: 35766061) has further extended it under the longitudinal response by proposing a quadratic inference function-based penalization method in gene–environment interaction studies. This study introduces “springer,” an R package implementing the bi-level variable selection within the QIF framework developed in Zhou et al. (2022). In addition, R package “springer” has also implemented the generalized estimating equation-based sparse group penalization method. Alternative methods focusing only on the group level or individual level have also been provided by the package. In this study, we have systematically introduced the longitudinal penalization methods implemented in the “springer” package. We demonstrate the usage of the core and supporting functions, which is followed by the numerical examples and discussions. R package “springer” is available at https://cran.r-project.org/package=springer.
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Affiliation(s)
- Fei Zhou
- Department of Statistics, Kansas State University, Manhattan, KS, United States
| | - Yuwen Liu
- Department of Statistics, Kansas State University, Manhattan, KS, United States
| | - Jie Ren
- Department of Biostatistics and Health Data Sciences, Indiana University School of Medicine, Indianapolis, IN, United States
| | - Weiqun Wang
- Department of Food, Nutrition, Dietetics and Health, Kansas State University, Manhattan, KS, United States
| | - Cen Wu
- Department of Statistics, Kansas State University, Manhattan, KS, United States
- *Correspondence: Cen Wu,
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3
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Cao X, Dong A, Kang G, Wang X, Duan L, Hou H, Zhao T, Wu S, Liu X, Huang H, Wu R. Modeling spatial interaction networks of the gut microbiota. Gut Microbes 2022; 14:2106103. [PMID: 35921525 PMCID: PMC9351588 DOI: 10.1080/19490976.2022.2106103] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 02/04/2023] Open
Abstract
How the gut microbiota is organized across space is postulated to influence microbial succession and its mutualistic relationships with the host. The lack of dynamic or perturbed abundance data poses considerable challenges for characterizing the spatial pattern of microbial interactions. We integrate allometric scaling theory, evolutionary game theory, and prey-predator theory into a unified framework under which quasi-dynamic microbial networks can be inferred from static abundance data. We illustrate that such networks can capture the full properties of microbial interactions, including causality, the sign of the causality, strength, and feedback loop, and are dynamically adaptive along spatial gradients, and context-specific, characterizing variability between individuals and within the same individual across time and space. We design and conduct a gut microbiota study to validate the model, characterizing key spatial determinants of the microbial differences between ulcerative colitis and healthy controls. Our model provides a sophisticated means of unraveling a complete atlas of how microbial interactions vary across space and quantifying causal relationships between such spatial variability and change in health state.
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Affiliation(s)
- Xiaocang Cao
- Department of Gastroenterology and Hepatology, Tianjin Medical University General Hospital, Tianjin Medical University, Tianjin, China,Xiaocang Cao Department of Gastroenterology and Hepatology Tianjin Medical University General Hospital, Tianjin Medical UniversityGeneral Hospital, Tianjin Medical University, TianjinHebeiChina
| | - Ang Dong
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing, China
| | - Guangbo Kang
- School of Chemical Engineering and Technology, Frontiers Science Center for Synthetic Biology and Key Laboratory of Systems Bioengineering (Ministry of Education), Tianjin University, Tianjin, China
| | - Xiaoli Wang
- Department of Gastroenterology and Hepatology, Tianjin Medical University General Hospital, Tianjin Medical University, Tianjin, China
| | - Liyun Duan
- Department of Gastroenterology and Hepatology, Tianjin Medical University General Hospital, Tianjin Medical University, Tianjin, China
| | - Huixing Hou
- Department of Gastroenterology and Hepatology, Tianjin Medical University General Hospital, Tianjin Medical University, Tianjin, China
| | - Tianming Zhao
- Department of Gastroenterology and Hepatology, Tianjin Medical University General Hospital, Tianjin Medical University, Tianjin, China
| | - Shuang Wu
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing, China
| | - Xinjuan Liu
- Department of Gastroenterology, Beijing Chaoyang Hospital, Capital Medical University, Beijing, China,Xinjuan Liu Department of Gastroenterology Beijing Chaoyang Hospital, Capital Medical University, BeijingHebeiChina
| | - He Huang
- School of Chemical Engineering and Technology, Frontiers Science Center for Synthetic Biology and Key Laboratory of Systems Bioengineering (Ministry of Education), Tianjin University, Tianjin, China,He Huang School of Chemical Engineering and Technology Frontiers Science Center for Synthetic Biology and Key Laboratory of Systems Bioengineering (Ministry of Education), Tianjin University, TianjinHebeiChina
| | - Rongling Wu
- Center for Statistical Genetics, Departments of Public Health Sciences and Statistics, The Pennsylvania State University, Hershey, PA, USA,CONTACT Rongling Wu Center for Statistical Genetics, Departments of Public Health Sciences and Statistics, the Pennsylvania State University, Hershey, PA17033, USA
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4
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Zhou F, Lu X, Ren J, Fan K, Ma S, Wu C. Sparse group variable selection for gene-environment interactions in the longitudinal study. Genet Epidemiol 2022; 46:317-340. [PMID: 35766061 DOI: 10.1002/gepi.22461] [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: 07/19/2021] [Revised: 01/31/2022] [Accepted: 03/15/2022] [Indexed: 11/06/2022]
Abstract
Penalized variable selection for high-dimensional longitudinal data has received much attention as it can account for the correlation among repeated measurements while providing additional and essential information for improved identification and prediction performance. Despite the success, in longitudinal studies, the potential of penalization methods is far from fully understood for accommodating structured sparsity. In this article, we develop a sparse group penalization method to conduct the bi-level gene-environment (G × $\times $ E) interaction study under the repeatedly measured phenotype. Within the quadratic inference function framework, the proposed method can achieve simultaneous identification of main and interaction effects on both the group and individual levels. Simulation studies have shown that the proposed method outperforms major competitors. In the case study of asthma data from the Childhood Asthma Management Program, we conduct G × $\times $ E study by using high-dimensional single nucleotide polymorphism data as genetic factors and the longitudinal trait, forced expiratory volume in 1 s, as the phenotype. Our method leads to improved prediction and identification of main and interaction effects with important implications.
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Affiliation(s)
- Fei Zhou
- Department of Statistics, Kansas State University, Manhattan, Kansas, 66506, USA
| | - Xi Lu
- Department of Statistics, Kansas State University, Manhattan, Kansas, 66506, USA
| | - Jie Ren
- Department of Biostatistics and Health Data Sciences, Indiana University School of Medicine, Indianapolis, Indiana, 46202, USA
| | - Kun Fan
- Department of Statistics, Kansas State University, Manhattan, Kansas, 66506, USA
| | - Shuangge Ma
- Department of Biostatistics, Yale University, New Haven, Connecticut, 06520, USA
| | - Cen Wu
- Department of Statistics, Kansas State University, Manhattan, Kansas, 66506, USA
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5
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Zhang M, Lu N, Jiang L, Liu B, Fei Y, Ma W, Shi C, Wang J. Multiple dynamic models reveal the genetic architecture for growth in height of Catalpa bungei in the field. TREE PHYSIOLOGY 2022; 42:1239-1255. [PMID: 34940852 DOI: 10.1093/treephys/tpab171] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2021] [Accepted: 12/19/2021] [Indexed: 06/14/2023]
Abstract
Growth in height (GH) is a critical determinant for tree survival and development in forests and can be depicted using logistic growth curves. Our understanding of the genetic mechanism underlying dynamic GH, however, is limited, particularly under field conditions. We applied two mapping models (Funmap and FVTmap) to find quantitative trait loci responsible for dynamic GH and two epistatic models (2HiGWAS and 1HiGWAS) to detect epistasis in Catalpa bungei grown in the field. We identified 13 co-located quantitative trait loci influencing the growth curve by Funmap and three heterochronic parameters (the timing of the inflection point, maximum acceleration and maximum deceleration) by FVTmap. The combined use of FVTmap and Funmap reduced the number of candidate genes by >70%. We detected 76 significant epistatic interactions, amongst which a key gene, COMT14, co-located by three models (but not 1HiGWAS) interacted with three other genes, implying that a novel network of protein interaction centered on COMT14 may control the dynamic GH of C. bungei. These findings provide new insights into the genetic mechanisms underlying the dynamic growth in tree height in natural environments and emphasize the necessity of incorporating multiple dynamic models for screening more reliable candidate genes.
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Affiliation(s)
- Miaomiao Zhang
- State Key Laboratory of Tree Genetics and Breeding, Key Laboratory of Tree Breeding and Cultivation of State Forestry Administration, Research Institute of Forestry, Chinese Academy of Forestry, Beijing 100091, China
| | - Nan Lu
- State Key Laboratory of Tree Genetics and Breeding, Key Laboratory of Tree Breeding and Cultivation of State Forestry Administration, Research Institute of Forestry, Chinese Academy of Forestry, Beijing 100091, China
| | - Libo Jiang
- School of Life Sciences and Medicine, Shandong University of Technology, Zibo 255049, China
| | - Bingyang Liu
- State Key Laboratory of Tree Genetics and Breeding, Key Laboratory of Tree Breeding and Cultivation of State Forestry Administration, Research Institute of Forestry, Chinese Academy of Forestry, Beijing 100091, China
| | - Yue Fei
- State Key Laboratory of Tree Genetics and Breeding, Key Laboratory of Tree Breeding and Cultivation of State Forestry Administration, Research Institute of Forestry, Chinese Academy of Forestry, Beijing 100091, China
| | - Wenjun Ma
- State Key Laboratory of Tree Genetics and Breeding, Key Laboratory of Tree Breeding and Cultivation of State Forestry Administration, Research Institute of Forestry, Chinese Academy of Forestry, Beijing 100091, China
| | - Chaozhong Shi
- State Key Laboratory of Tree Genetics and Breeding, Key Laboratory of Tree Breeding and Cultivation of State Forestry Administration, Research Institute of Forestry, Chinese Academy of Forestry, Beijing 100091, China
| | - Junhui Wang
- State Key Laboratory of Tree Genetics and Breeding, Key Laboratory of Tree Breeding and Cultivation of State Forestry Administration, Research Institute of Forestry, Chinese Academy of Forestry, Beijing 100091, China
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6
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Inferring multilayer interactome networks shaping phenotypic plasticity and evolution. Nat Commun 2021; 12:5304. [PMID: 34489412 PMCID: PMC8421358 DOI: 10.1038/s41467-021-25086-5] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2021] [Accepted: 07/12/2021] [Indexed: 02/07/2023] Open
Abstract
Phenotypic plasticity represents a capacity by which the organism changes its phenotypes in response to environmental stimuli. Despite its pivotal role in adaptive evolution, how phenotypic plasticity is genetically controlled remains elusive. Here, we develop a unified framework for coalescing all single nucleotide polymorphisms (SNPs) from a genome-wide association study (GWAS) into a quantitative graph. This framework integrates functional genetic mapping, evolutionary game theory, and predator-prey theory to decompose the net genetic effect of each SNP into its independent and dependent components. The independent effect arises from the intrinsic capacity of a SNP, only expressed when it is in isolation, whereas the dependent effect results from the extrinsic influence of other SNPs. The dependent effect is conceptually beyond the traditional definition of epistasis by not only characterizing the strength of epistasis but also capturing the bi-causality of epistasis and the sign of the causality. We implement functional clustering and variable selection to infer multilayer, sparse, and multiplex interactome networks from any dimension of genetic data. We design and conduct two GWAS experiments using Staphylococcus aureus, aimed to test the genetic mechanisms underlying the phenotypic plasticity of this species to vancomycin exposure and Escherichia coli coexistence. We reconstruct the two most comprehensive genetic networks for abiotic and biotic phenotypic plasticity. Pathway analysis shows that SNP-SNP epistasis for phenotypic plasticity can be annotated to protein-protein interactions through coding genes. Our model can unveil the regulatory mechanisms of significant loci and excavate missing heritability from some insignificant loci. Our multilayer genetic networks provide a systems tool for dissecting environment-induced evolution.
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7
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Wang H, Ye M, Fu Y, Dong A, Zhang M, Feng L, Zhu X, Bo W, Jiang L, Griffin CH, Liang D, Wu R. Modeling genome-wide by environment interactions through omnigenic interactome networks. Cell Rep 2021; 35:109114. [PMID: 33979624 DOI: 10.1016/j.celrep.2021.109114] [Citation(s) in RCA: 18] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/30/2020] [Revised: 03/11/2021] [Accepted: 04/21/2021] [Indexed: 10/21/2022] Open
Abstract
How genes interact with the environment to shape phenotypic variation and evolution is a fundamental question intriguing to biologists from various fields. Existing linear models built on single genes are inadequate to reveal the complexity of genotype-environment (G-E) interactions. Here, we develop a conceptual model for mechanistically dissecting G-E interplay by integrating previously disconnected theories and methods. Under this integration, evolutionary game theory, developmental modularity theory, and a variable selection method allow us to reconstruct environment-induced, maximally informative, sparse, and casual multilayer genetic networks. We design and conduct two mapping experiments by using a desert-adapted tree species to validate the biological application of the model proposed. The model identifies previously uncharacterized molecular mechanisms that mediate trees' response to saline stress. Our model provides a tool to comprehend the genetic architecture of trait variation and evolution and trace the information flow of each gene toward phenotypes within omnigenic networks.
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Affiliation(s)
- Haojie Wang
- Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Meixia Ye
- Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Yaru Fu
- Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Ang Dong
- Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Miaomiao Zhang
- Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Li Feng
- Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Xuli Zhu
- Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Wenhao Bo
- Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Libo Jiang
- Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Christopher H Griffin
- Applied Research Laboratory, The Pennsylvania State University, University Park, PA 16802, USA
| | - Dan Liang
- Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Rongling Wu
- Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China; Center for Statistical Genetics, Departments of Public Health Sciences and Statistics, The Pennsylvania State University, Hershey, PA 17033, USA.
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8
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Wang Z, Wang N, Wang Z, Jiang L, Wang Y, Li J, Wu R. HiGwas: how to compute longitudinal GWAS data in population designs. Bioinformatics 2020; 36:4222-4224. [PMID: 32502244 DOI: 10.1093/bioinformatics/btaa294] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2019] [Revised: 02/15/2020] [Accepted: 06/01/2020] [Indexed: 11/14/2022] Open
Abstract
SUMMARY Genome-wide association studies (GWAS), particularly designed with thousands and thousands of single-nucleotide polymorphisms (SNPs) (big p) genotyped on tens of thousands of subjects (small n), are encountered by a major challenge of p ≪ n. Although the integration of longitudinal information can significantly enhance a GWAS's power to comprehend the genetic architecture of complex traits and diseases, an additional challenge is generated by an autocorrelative process. We have developed several statistical models for addressing these two challenges by implementing dimension reduction methods and longitudinal data analysis. To make these models computationally accessible to applied geneticists, we wrote an R package of computer software, HiGwas, designed to analyze longitudinal GWAS datasets. Functions in the package encompass single SNP analyses, significance-level adjustment, preconditioning and model selection for a high-dimensional set of SNPs. HiGwas provides the estimates of genetic parameters and the confidence intervals of these estimates. We demonstrate the features of HiGwas through real data analysis and vignette document in the package. AVAILABILITY AND IMPLEMENTATION https://github.com/wzhy2000/higwas. CONTACT rwu@phs.psu.edu. SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.
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Affiliation(s)
- Zhong Wang
- School of Software Technology, Dalian University of Technology, Dalian 116023, China.,Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, Beijing Forestry University, Beijing 100083, China.,Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Nating Wang
- Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, Beijing Forestry University, Beijing 100083, China.,Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Zilu Wang
- The College of Arts & Sciences, Cornell University, Ithaca, NY 14850, USA
| | - Libo Jiang
- Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, Beijing Forestry University, Beijing 100083, China.,Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Yaqun Wang
- Department of Biostatistics, Rutgers University, New Brunswick, NJ 08901, USA
| | - Jiahan Li
- Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA
| | - Rongling Wu
- Beijing Advanced Innovation Center for Tree Breeding by Molecular Design, Beijing Forestry University, Beijing 100083, China.,Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China.,Center for Statistical Genetics, Departments of Public Health Sciences and Statistics, The Pennsylvania State University, Hershey, PA 17033, USA
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9
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Xu H, Li X, Yang Y, Li Y, Pinheiro J, Sasser K, Hamadeh H, Steven X, Yuan M. High-throughput and efficient multilocus genome-wide association study on longitudinal outcomes. Bioinformatics 2020; 36:3004-3010. [PMID: 32096821 DOI: 10.1093/bioinformatics/btaa120] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2019] [Revised: 01/16/2020] [Accepted: 02/18/2020] [Indexed: 11/12/2022] Open
Abstract
MOTIVATION With the emerging of high-dimensional genomic data, genetic analysis such as genome-wide association studies (GWAS) have played an important role in identifying disease-related genetic variants and novel treatments. Complex longitudinal phenotypes are commonly collected in medical studies. However, since limited analytical approaches are available for longitudinal traits, these data are often underutilized. In this article, we develop a high-throughput machine learning approach for multilocus GWAS using longitudinal traits by coupling Empirical Bayesian Estimates from mixed-effects modeling with a novel ℓ0-norm algorithm. RESULTS Extensive simulations demonstrated that the proposed approach not only provided accurate selection of single nucleotide polymorphisms (SNPs) with comparable or higher power but also robust control of false positives. More importantly, this novel approach is highly scalable and could be approximately >1000 times faster than recently published approaches, making genome-wide multilocus analysis of longitudinal traits possible. In addition, our proposed approach can simultaneously analyze millions of SNPs if the computer memory allows, thereby potentially allowing a true multilocus analysis for high-dimensional genomic data. With application to the data from Alzheimer's Disease Neuroimaging Initiative, we confirmed that our approach can identify well-known SNPs associated with AD and were much faster than recently published approaches (≥6000 times). AVAILABILITY AND IMPLEMENTATION The source code and the testing datasets are available at https://github.com/Myuan2019/EBE_APML0. SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.
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Affiliation(s)
- Huang Xu
- Department of Statistics and Finance, University of Science and Technology of China, Hefei 230026, China
| | - Xiang Li
- Janssen Research and Development, Raritan, NJ 08869, USA
| | - Yaning Yang
- Department of Statistics and Finance, University of Science and Technology of China, Hefei 230026, China
| | - Yi Li
- Department of Statistics and Finance, University of Science and Technology of China, Hefei 230026, China
| | - Jose Pinheiro
- Janssen Research and Development, Raritan, NJ 08869, USA
| | | | | | - Xu Steven
- Genmab US, Inc., Princeton, NJ 08540, USA
| | - Min Yuan
- School of Public Health Administration, Anhui Medical University, Hefei 230032, China
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10
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Guo H, Yu Z, An J, Han G, Ma Y, Tang R. A Two-Stage Mutual Information Based Bayesian Lasso Algorithm for Multi-Locus Genome-Wide Association Studies. ENTROPY 2020; 22:e22030329. [PMID: 33286103 PMCID: PMC7516787 DOI: 10.3390/e22030329] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/01/2020] [Revised: 02/28/2020] [Accepted: 03/10/2020] [Indexed: 12/31/2022]
Abstract
Genome-wide association study (GWAS) has turned out to be an essential technology for exploring the genetic mechanism of complex traits. To reduce the complexity of computation, it is well accepted to remove unrelated single nucleotide polymorphisms (SNPs) before GWAS, e.g., by using iterative sure independence screening expectation-maximization Bayesian Lasso (ISIS EM-BLASSO) method. In this work, a modified version of ISIS EM-BLASSO is proposed, which reduces the number of SNPs by a screening methodology based on Pearson correlation and mutual information, then estimates the effects via EM-Bayesian Lasso (EM-BLASSO), and finally detects the true quantitative trait nucleotides (QTNs) through likelihood ratio test. We call our method a two-stage mutual information based Bayesian Lasso (MBLASSO). Under three simulation scenarios, MBLASSO improves the statistical power and retains the higher effect estimation accuracy when comparing with three other algorithms. Moreover, MBLASSO performs best on model fitting, the accuracy of detected associations is the highest, and 21 genes can only be detected by MBLASSO in Arabidopsis thaliana datasets.
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Affiliation(s)
- Hongping Guo
- Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education and Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, China; (H.G.); (G.H.); (Y.M.); (R.T.)
- School of Mathematics and Computer Science, Hanjiang Normal University, Shiyan 442000, China
| | - Zuguo Yu
- Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education and Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, China; (H.G.); (G.H.); (Y.M.); (R.T.)
- School of Electrical Engineering and Computer Science, Queensland University of Technology, Brisbane, QLD 4001, Australia
- Correspondence:
| | - Jiyuan An
- Centre for Tropical Crops and Biocommodities, Queensland University of Technology, Brisbane, QLD 4001, Australia;
| | - Guosheng Han
- Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education and Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, China; (H.G.); (G.H.); (Y.M.); (R.T.)
| | - Yuanlin Ma
- Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education and Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, China; (H.G.); (G.H.); (Y.M.); (R.T.)
| | - Runbin Tang
- Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education and Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, China; (H.G.); (G.H.); (Y.M.); (R.T.)
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11
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Wang Q, Gan J, Wei K, Berceli SA, Gragnoli C, Wu R. A unified mapping framework of multifaceted pharmacodynamic responses to hypertension interventions. Drug Discov Today 2019; 24:883-889. [PMID: 30690194 PMCID: PMC6492935 DOI: 10.1016/j.drudis.2019.01.009] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/13/2018] [Revised: 01/03/2019] [Accepted: 01/17/2019] [Indexed: 02/04/2023]
Abstract
The personalized therapy for hypertension needs comprehensive knowledge about how blood pressures (BPs; systolic and diastolic) and their pulsatile and steady components are controlled by genetic factors. Here, we propose a unified pharmacodynamic (PD) functional mapping framework for identifying specific quantitative trait loci (QTLs) that mediate multivariate response-dose curves of BP. This framework can characterize how QTLs govern pulsatile and steady components through jointly regulating systolic and diastolic pressures. The model can quantify the genetic effects of individual QTLs on maximal drug effect, the maximal rate of drug response, and the dose window of maximal drug response. This unified mapping framework provides a tool for identifying pharmacological genes potentially useful to design the right medication and right dose for patients.
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Affiliation(s)
- Qian Wang
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Jingwen Gan
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Kun Wei
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China
| | - Scott A Berceli
- Malcom Randall VA Medical Center, Gainesville, FL 32610, USA; Department of Surgery, University of Florida, Box 100128, Gainesville, FL 32610, USA; Department of Biomedical Engineering, University of Florida, Gainesville, FL 32610, USA
| | - Claudia Gragnoli
- Division of Endocrinology, Diabetes, and Metabolic Disease, Translational Medicine, Department of Medicine, Sidney Kimmel Medical College, Thomas Jefferson University, Philadelphia, PA 19107, USA; Department of Public Health Sciences, Penn State College of Medicine, Hershey, PA 17033, USA; Molecular Biology Laboratory, Bios Biotech Multi Diagnostic Health Center, Rome 00197, Italy
| | - Rongling Wu
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing 100083, China; Department of Public Health Sciences, Penn State College of Medicine, Hershey, PA 17033, USA; Center for Statistical Genetics, Departments of Public Health Sciences and Statistics, Pennsylvania State University, Hershey, PA 17033, USA.
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12
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Liu J, Ye M, Zhu S, Jiang L, Sang M, Gan J, Wang Q, Huang M, Wu R. Two-stage identification of SNP effects on dynamic poplar growth. THE PLANT JOURNAL : FOR CELL AND MOLECULAR BIOLOGY 2018; 93:286-296. [PMID: 29168265 DOI: 10.1111/tpj.13777] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2017] [Revised: 10/16/2017] [Accepted: 10/23/2017] [Indexed: 05/23/2023]
Abstract
This project proposes an approach to identify significant single nucleotide polymorphism (SNP) effects, both additive and dominant, on the dynamic growth of poplar in diameter and height. The annual changes in yearly phenotypes based on regular observation periods are considered to represent multiple responses. In total 156,362 candidate SNPs are studied, and the phenotypes of 64 poplar trees are recorded. To address this ultrahigh dimensionality issue, this paper adopts a two-stage approach. First, the conventional genome-wide association studies (GWAS) and the distance correlation sure independence screening (DC-SIS) methods (Li et al., 2012) were combined to reduce the model dimensions at the sample size; second, a grouped penalized regression was applied to further refine the model and choose the final sparse SNPs. The multiple response issue was also carefully addressed. The SNP effects on the dynamic diameter and height growth patterns of poplar were systematically analyzed. In addition, a series of intensive simulation studies was performed to validate the proposed approach.
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Affiliation(s)
- Jingyuan Liu
- Department of Statistics in School of Economics, Wang Yanan Institute for Studies in Economics, Fujian Key Laboratory of Statistical Science, Xiamen University, China
| | - Meixia Ye
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing, 100081, China
| | - Sheng Zhu
- Jiangsu Key Laboratory for Poplar Germplasm Enhancement and Variety Improvement, Nanjing Forestry University, Nanjing, 210037, China
| | - Libo Jiang
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing, 100081, China
| | - Mengmeng Sang
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing, 100081, China
| | - Jingwen Gan
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing, 100081, China
| | - Qian Wang
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing, 100081, China
| | - Minren Huang
- Jiangsu Key Laboratory for Poplar Germplasm Enhancement and Variety Improvement, Nanjing Forestry University, Nanjing, 210037, China
| | - Rongling Wu
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing, 100081, China
- Department of Public Health Sciences, Penn State Hershey College of Medicine, Hershey, PA17033, USA
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13
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Jiang L, Zhang M, Sang M, Ye M, Wu R. Evo-Devo-EpiR: a genome-wide search platform for epistatic control on the evolution of development. Brief Bioinform 2017; 18:754-760. [PMID: 27473062 DOI: 10.1093/bib/bbw062] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2016] [Indexed: 11/14/2022] Open
Abstract
Evo-devo is a theory proposed to study how phenotypes evolve by comparing the developmental processes of different organisms or the same organism experiencing changing environments. It has been recognized that nonallelic interactions at different genes or quantitative trait loci, known as epistasis, may play a pivotal role in the evolution of development, but it has proven difficult to quantify and elucidate this role into a coherent picture. We implement a high-dimensional genome-wide association study model into the evo-devo paradigm and pack it into the R-based Evo-Devo-EpiR, aimed at facilitating the genome-wide landscaping of epistasis for the diversification of phenotypic development. By analyzing a high-throughput assay of DNA markers and their pairs simultaneously, Evo-Devo-EpiR is equipped with a capacity to systematically characterize various epistatic interactions that impact on the pattern and timing of development and its evolution. Enabling a global search for all possible genetic interactions for developmental processes throughout the whole genome, Evo-Devo-EpiR provides a computational tool to illustrate a precise genotype-phenotype map at interface between epistasis, development and evolution.
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14
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Ren J, He T, Li Y, Liu S, Du Y, Jiang Y, Wu C. Network-based regularization for high dimensional SNP data in the case-control study of Type 2 diabetes. BMC Genet 2017; 18:44. [PMID: 28511641 PMCID: PMC5434559 DOI: 10.1186/s12863-017-0495-5] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/05/2016] [Accepted: 03/25/2017] [Indexed: 12/02/2022] Open
Abstract
Background Over the past decades, the prevalence of type 2 diabetes mellitus (T2D) has been steadily increasing around the world. Despite large efforts devoted to better understand the genetic basis of the disease, the identified susceptibility loci can only account for a small portion of the T2D heritability. Some of the existing approaches proposed for the high dimensional genetic data from the T2D case–control study are limited by analyzing a few number of SNPs at a time from a large pool of SNPs, by ignoring the correlations among SNPs and by adopting inefficient selection techniques. Methods We propose a network constrained regularization method to select important SNPs by taking the linkage disequilibrium into account. To accomodate the case control study, an iteratively reweighted least square algorithm has been developed within the coordinate descent framework where optimization of the regularized logistic loss function is performed with respect to one parameter at a time and iteratively cycle through all the parameters until convergence. Results In this article, a novel approach is developed to identify important SNPs more effectively through incorporating the interconnections among them in the regularized selection. A coordinate descent based iteratively reweighed least squares (IRLS) algorithm has been proposed. Conclusions Both the simulation study and the analysis of the Nurses’s Health Study, a case–control study of type 2 diabetes data with high dimensional SNP measurements, demonstrate the advantage of the network based approach over the competing alternatives.
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Affiliation(s)
- Jie Ren
- Department of Statistics, Kansas State University, 1116 Mid-Campus Drive N., 66506, Manhattan, KS, USA
| | - Tao He
- Department of Mathematics, San Francisco State University, San Francisco, CA, USA
| | - Ye Li
- Department of Biostatistics, Yale University, New Haven, CT, USA
| | - Sai Liu
- Division of Nephrology, School of Medicine, Stanford University, Palo Alto, CA, USA
| | - Yinhao Du
- Department of Statistics, Kansas State University, 1116 Mid-Campus Drive N., 66506, Manhattan, KS, USA
| | - Yu Jiang
- Division of Epidemiology, Biostatistics, and Environmental Health, School of Public Health, University of Memphis, Memphis, TN, USA
| | - Cen Wu
- Department of Statistics, Kansas State University, 1116 Mid-Campus Drive N., 66506, Manhattan, KS, USA.
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15
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Ruffieux H, Davison AC, Hager J, Irincheeva I. Efficient inference for genetic association studies with multiple outcomes. Biostatistics 2017; 18:618-636. [DOI: 10.1093/biostatistics/kxx007] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2016] [Accepted: 02/06/2017] [Indexed: 02/04/2023] Open
Abstract
SUMMARY
Combined inference for heterogeneous high-dimensional data is critical in modern biology, where clinical and various kinds of molecular data may be available from a single study. Classical genetic association studies regress a single clinical outcome on many genetic variants one by one, but there is an increasing demand for joint analysis of many molecular outcomes and genetic variants in order to unravel functional interactions. Unfortunately, most existing approaches to joint modeling are either too simplistic to be powerful or are impracticable for computational reasons. Inspired by Richardson and others (2010, Bayesian Statistics 9), we consider a sparse multivariate regression model that allows simultaneous selection of predictors and associated responses. As Markov chain Monte Carlo (MCMC) inference on such models can be prohibitively slow when the number of genetic variants exceeds a few thousand, we propose a variational inference approach which produces posterior information very close to that of MCMC inference, at a much reduced computational cost. Extensive numerical experiments show that our approach outperforms popular variable selection methods and tailored Bayesian procedures, dealing within hours with problems involving hundreds of thousands of genetic variants and tens to hundreds of clinical or molecular outcomes.
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Affiliation(s)
- Helene Ruffieux
- Nestlé Institute of Health Sciences SA, EPFL Innovation Park, 1015 Lausanne, Switzerland Ecole Polytechnique Fédérale de Lausanne, EPFL SB MATH STAT, Station 8, 1015 Lausanne, Switzerland
| | - Anthony C. Davison
- Ecole Polytechnique Fédérale de Lausanne, EPFL SB MATH STAT, Station 8, 1015 Lausanne, Switzerland
| | - Jorg Hager
- Nestlé Institute of Health Sciences SA, EPFL Innovation Park, 1015 Lausanne, Switzerland
| | - Irina Irincheeva
- Nestlé Institute of Health Sciences SA, EPFL Innovation Park, 1015 Lausanne, Switzerland
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16
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Wang Q, Gosik K, Xing S, Jiang L, Sun L, Chinchilli VM, Wu R. Integration of epigenetic game theory and developmental principles: Reply to comments on "Epigenetic game theory: How to compute the epigenetic control of maternal-to-zygotic transition". Phys Life Rev 2017; 20:166-169. [PMID: 28256424 DOI: 10.1016/j.plrev.2017.01.029] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2017] [Accepted: 01/31/2017] [Indexed: 10/20/2022]
Affiliation(s)
- Qian Wang
- Xinxiang Central Hospital, Henan, China
| | - Kirk Gosik
- Division of Biostatistics and Bioinformatics, Department of Public Health Sciences, Penn State College of Medicine, Hershey, PA, USA
| | | | - Libo Jiang
- Center for Computational Biology, Beijing Forestry University, Beijing, China
| | - Lidan Sun
- Division of Biostatistics and Bioinformatics, Department of Public Health Sciences, Penn State College of Medicine, Hershey, PA, USA; Center for Computational Biology, Beijing Forestry University, Beijing, China
| | - Vernon M Chinchilli
- Division of Biostatistics and Bioinformatics, Department of Public Health Sciences, Penn State College of Medicine, Hershey, PA, USA
| | - Rongling Wu
- Division of Biostatistics and Bioinformatics, Department of Public Health Sciences, Penn State College of Medicine, Hershey, PA, USA.
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17
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Wang Q, Gosik K, Xing S, Jiang L, Sun L, Chinchilli VM, Wu R. Epigenetic game theory: How to compute the epigenetic control of maternal-to-zygotic transition. Phys Life Rev 2017; 20:126-137. [DOI: 10.1016/j.plrev.2016.11.001] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/04/2016] [Revised: 11/01/2016] [Accepted: 11/04/2016] [Indexed: 01/04/2023]
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18
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Xu M, Jiang L, Zhu S, Zhou C, Ye M, Mao K, Sun L, Su X, Pan H, Zhang S, Huang M, Wu R. A computational framework for mapping the timing of vegetative phase change. THE NEW PHYTOLOGIST 2016; 211:750-60. [PMID: 26958803 DOI: 10.1111/nph.13907] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/04/2015] [Accepted: 01/17/2016] [Indexed: 05/24/2023]
Abstract
Phase change plays a prominent role in determining the form of growth and development. Although considerable attention has been focused on identifying the regulatory control mechanisms of phase change, a detailed understanding of the genetic architecture of this phenomenon is still lacking. We address this issue by deriving a computational model. The model is founded on the framework of functional mapping aimed at characterizing the interplay between quantitative trait loci (QTLs) and development through biologically meaningful mathematical equations. A multiphasic growth equation was implemented into functional mapping, which, via a series of hypothesis tests, allows the quantification of how QTLs regulate the timing and pattern of vegetative phase transition between independently regulated, temporally coordinated processes. The model was applied to analyze stem radial growth data of an interspecific hybrid family derived from two Populus species during the first 24 yr of ontogeny. Several key QTLs related to phase change have been characterized, most of which were observed to be in the adjacent regions of candidate genes. The identification of phase transition QTLs, whose expression is regulated by endogenous and environmental signals, may enhance our understanding of the evolution of development in changing environments.
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Affiliation(s)
- Meng Xu
- Co-Innovation Center for Sustainable Forestry in Southern China, Nanjing Forestry University, Nanjing, 210037, China
| | - Libo Jiang
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing, 100083, China
| | - Sheng Zhu
- Co-Innovation Center for Sustainable Forestry in Southern China, Nanjing Forestry University, Nanjing, 210037, China
| | - Chunguo Zhou
- Co-Innovation Center for Sustainable Forestry in Southern China, Nanjing Forestry University, Nanjing, 210037, China
| | - Meixia Ye
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing, 100083, China
| | - Ke Mao
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing, 100083, China
| | - Lidan Sun
- Center for Statistical Genetics, The Pennsylvania State University, Hershey, PA, 17033, USA
| | - Xiaohua Su
- Research Institute of Forestry, Chinese Academy of Forestry, Beijing, 100094, China
| | - Huixin Pan
- Co-Innovation Center for Sustainable Forestry in Southern China, Nanjing Forestry University, Nanjing, 210037, China
| | - Shougong Zhang
- Research Institute of Forestry, Chinese Academy of Forestry, Beijing, 100094, China
| | - Minren Huang
- Co-Innovation Center for Sustainable Forestry in Southern China, Nanjing Forestry University, Nanjing, 210037, China
| | - Rongling Wu
- Center for Computational Biology, College of Biological Sciences and Technology, Beijing Forestry University, Beijing, 100083, China
- Center for Statistical Genetics, The Pennsylvania State University, Hershey, PA, 17033, USA
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19
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Qi J, Sun J, Wang J. E-Index for Differentiating Complex Dynamic Traits. BIOMED RESEARCH INTERNATIONAL 2016; 2016:5761983. [PMID: 27064292 PMCID: PMC4811058 DOI: 10.1155/2016/5761983] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/03/2015] [Revised: 10/28/2015] [Accepted: 02/11/2016] [Indexed: 11/21/2022]
Abstract
While it is a daunting challenge in current biology to understand how the underlying network of genes regulates complex dynamic traits, functional mapping, a tool for mapping quantitative trait loci (QTLs) and single nucleotide polymorphisms (SNPs), has been applied in a variety of cases to tackle this challenge. Though useful and powerful, functional mapping performs well only when one or more model parameters are clearly responsible for the developmental trajectory, typically being a logistic curve. Moreover, it does not work when the curves are more complex than that, especially when they are not monotonic. To overcome this inadaptability, we therefore propose a mathematical-biological concept and measurement, E-index (earliness-index), which cumulatively measures the earliness degree to which a variable (or a dynamic trait) increases or decreases its value. Theoretical proofs and simulation studies show that E-index is more general than functional mapping and can be applied to any complex dynamic traits, including those with logistic curves and those with nonmonotonic curves. Meanwhile, E-index vector is proposed as well to capture more subtle differences of developmental patterns.
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Affiliation(s)
- Jiandong Qi
- School of Information, Beijing Forestry University, Beijing 100083, China
| | - Jianfeng Sun
- School of Information, Beijing Forestry University, Beijing 100083, China
| | - Jianxin Wang
- School of Information, Beijing Forestry University, Beijing 100083, China
- Center for Computational Biology, Beijing Forestry University, Beijing 100083, China
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