A Higher-Order Generalized Singular Value Decomposition for Comparison of Global mRNA Expression from Multiple Organisms

The number of high-dimensional datasets recording multiple aspects of a single phenomenon is increasing in many areas of science, accompanied by a need for mathematical frameworks that can compare multiple large-scale matrices with different row dimensions. The only such framework to date, the generalized singular value decomposition (GSVD), is limited to two matrices. We mathematically define a higher-order GSVD (HO GSVD) for N≥2 matrices , each with full column rank. Each matrix is exactly factored as Di = UiΣiVT, where V, identical in all factorizations, is obtained from the eigensystem SV = VΛ of the arithmetic mean S of all pairwise quotients of the matrices , i≠j. We prove that this decomposition extends to higher orders almost all of the mathematical properties of the GSVD. The matrix S is nondefective with V and Λ real. Its eigenvalues satisfy λk≥1. Equality holds if and only if the corresponding eigenvector vk is a right basis vector of equal significance in all matrices Di and Dj, that is σi,k/σj,k = 1 for all i and j, and the corresponding left basis vector ui,k is orthogonal to all other vectors in Ui for all i. The eigenvalues λk = 1, therefore, define the “common HO GSVD subspace.” We illustrate the HO GSVD with a comparison of genome-scale cell-cycle mRNA expression from S. pombe, S. cerevisiae and human. Unlike existing algorithms, a mapping among the genes of these disparate organisms is not required. We find that the approximately common HO GSVD subspace represents the cell-cycle mRNA expression oscillations, which are similar among the datasets. Simultaneous reconstruction in the common subspace, therefore, removes the experimental artifacts, which are dissimilar, from the datasets. In the simultaneous sequence-independent classification of the genes of the three organisms in this common subspace, genes of highly conserved sequences but significantly different cell-cycle peak times are correctly classified

Global effects of DNA replication and DNA replication origin activity on eukaryotic gene expression

This report provides a global view of how gene expression is affected by DNA replication. We analyzed synchronized cultures of Saccharomyces cerevisiae under conditions that prevent DNA replication initiation without delaying cell cycle progression. We use a higher-order singular value decomposition to integrate the global mRNA expression measured in the multiple time courses, detect and remove experimental artifacts and identify significant combinations of patterns of expression variation across the genes, time points and conditions. We find that, first, ∼88% of the global mRNA expression is independent of DNA replication. Second, the requirement of DNA replication for efficient histone gene expression is independent of conditions that elicit DNA damage checkpoint responses. Third, origin licensing decreases the expression of genes with origins near their 3′ ends, revealing that downstream origins can regulate the expression of upstream genes. This confirms previous predictions from mathematical modeling of a global causal coordination between DNA replication origin activity and mRNA expression, and shows that mathematical modeling of DNA microarray data can be used to correctly predict previously unknown biological modes of regulation

Tensor Decomposition Reveals Concurrent Evolutionary Convergences and Divergences and Correlations with Structural Motifs in Ribosomal RNA

Evolutionary relationships among organisms are commonly described by using a hierarchy derived from comparisons of ribosomal RNA (rRNA) sequences. We propose that even on the level of a single rRNA molecule, an organism's evolution is composed of multiple pathways due to concurrent forces that act independently upon different rRNA degrees of freedom. Relationships among organisms are then compositions of coexisting pathway-dependent similarities and dissimilarities, which cannot be described by a single hierarchy. We computationally test this hypothesis in comparative analyses of 16S and 23S rRNA sequence alignments by using a tensor decomposition, i.e., a framework for modeling composite data. Each alignment is encoded in a cuboid, i.e., a third-order tensor, where nucleotides, positions and organisms, each represent a degree of freedom. A tensor mode-1 higher-order singular value decomposition (HOSVD) is formulated such that it separates each cuboid into combinations of patterns of nucleotide frequency variation across organisms and positions, i.e., “eigenpositions” and corresponding nucleotide-specific segments of “eigenorganisms,” respectively, independent of a-priori knowledge of the taxonomic groups or rRNA structures. We find, in support of our hypothesis that, first, the significant eigenpositions reveal multiple similarities and dissimilarities among the taxonomic groups. Second, the corresponding eigenorganisms identify insertions or deletions of nucleotides exclusively conserved within the corresponding groups, that map out entire substructures and are enriched in adenosines, unpaired in the rRNA secondary structure, that participate in tertiary structure interactions. This demonstrates that structural motifs involved in rRNA folding and function are evolutionary degrees of freedom. Third, two previously unknown coexisting subgenic relationships between Microsporidia and Archaea are revealed in both the 16S and 23S rRNA alignments, a convergence and a divergence, conferred by insertions and deletions of these motifs, which cannot be described by a single hierarchy. This shows that mode-1 HOSVD modeling of rRNA alignments might be used to computationally predict evolutionary mechanisms

GSVD Comparison of Patient-Matched Normal and Tumor aCGH Profiles Reveals Global Copy-Number Alterations Predicting Glioblastoma Multiforme Survival

Despite recent large-scale profiling efforts, the best prognostic predictor of glioblastoma multiforme (GBM) remains the patient's age at diagnosis. We describe a global pattern of tumor-exclusive co-occurring copy-number alterations (CNAs) that is correlated, possibly coordinated with GBM patients' survival and response to chemotherapy. The pattern is revealed by GSVD comparison of patient-matched but probe-independent GBM and normal aCGH datasets from The Cancer Genome Atlas (TCGA). We find that, first, the GSVD, formulated as a framework for comparatively modeling two composite datasets, removes from the pattern copy-number variations (CNVs) that occur in the normal human genome (e.g., female-specific X chromosome amplification) and experimental variations (e.g., in tissue batch, genomic center, hybridization date and scanner), without a-priori knowledge of these variations. Second, the pattern includes most known GBM-associated changes in chromosome numbers and focal CNAs, as well as several previously unreported CNAs in 3% of the patients. These include the biochemically putative drug target, cell cycle-regulated serine/threonine kinase-encoding TLK2, the cyclin E1-encoding CCNE1, and the Rb-binding histone demethylase-encoding KDM5A. Third, the pattern provides a better prognostic predictor than the chromosome numbers or any one focal CNA that it identifies, suggesting that the GBM survival phenotype is an outcome of its global genotype. The pattern is independent of age, and combined with age, makes a better predictor than age alone. GSVD comparison of matched profiles of a larger set of TCGA patients, inclusive of the initial set, confirms the global pattern. GSVD classification of the GBM profiles of an independent set of patients validates the prognostic contribution of the pattern

Mathematical Modelling of DNA Replication Reveals a Trade-off between Coherence of Origin Activation and Robustness against Rereplication

Eukaryotic genomes are duplicated from multiple replication origins exactly once per cell cycle. In Saccharomyces cerevisiae, a complex molecular network has been identified that governs the assembly of the replication machinery. Here we develop a mathematical model that links the dynamics of this network to its performance in terms of rate and coherence of origin activation events, number of activated origins, the resulting distribution of replicon sizes and robustness against DNA rereplication. To parameterize the model, we use measured protein expression data and systematically generate kinetic parameter sets by optimizing the coherence of origin firing. While randomly parameterized networks yield unrealistically slow kinetics of replication initiation, networks with optimized parameters account for the experimentally observed distribution of origin firing times. Efficient inhibition of DNA rereplication emerges as a constraint that limits the rate at which replication can be initiated. In addition to the separation between origin licensing and firing, a time delay between the activation of S phase cyclin-dependent kinase (S-Cdk) and the initiation of DNA replication is required for preventing rereplication. Our analysis suggests that distributive multisite phosphorylation of the S-Cdk targets Sld2 and Sld3 can generate both a robust time delay and contribute to switch-like, coherent activation of replication origins. The proposed catalytic function of the complex formed by Dpb11, Sld3 and Sld2 strongly enhances coherence and robustness of origin firing. The model rationalizes how experimentally observed inefficient replication from fewer origins is caused by premature activation of S-Cdk, while premature activity of the S-Cdk targets Sld2 and Sld3 results in DNA rereplication. Thus the model demonstrates how kinetic deregulation of the molecular network governing DNA replication may result in genomic instability

Discovery of Mechanisms from Mathematical Modeling of DNA Microarray Data: Computational Prediction and Experimental Verification

Orly Alter of the Department of Biomedical Engineering,Institute for Cellular and Molecular Biology, and Institute for Computational Engineering and Sciences, University of Texas at Austin presented a lecture on February 16, 2010 at 2:00 pm in room 1116 of the Klaus Advanced Computing Building on the Georgia Tech campus.Runtime: 56:47 minutesFuture discovery and control in biology and medicine will come from the mathematical modeling of large-scale molecular biological data, such as DNA microarray data, just as Kepler discovered the laws of planetary motion by using mathematics to describe trends in astronomical data [1]. In this talk, I will demonstrate that mathematical modeling of DNA microarray data can be used to correctly predict previously unknown mechanisms that govern the activities of DNA and RNA. First, I will describe the computational prediction of a mechanism of regulation, by developing generalizations of the matrix and tensor computations that underlie theoretical physics and using them to uncover a genome-wide pattern of correlation between DNA replication initiation and RNA expression during the cell cycle [2,3]. Second, I will describe the recent experimental verification of this computational prediction, by analyzing global expression in synchronized cultures of yeast under conditions that prevent DNA replication initiation without delaying cell cycle progression [4]. Third, I will describe the use of the singular value decomposition to uncover "asymmetric Hermite functions," a generalization of the eigenfunctions of the quantum harmonic oscillator, in genome-wide mRNA lengths distribution data [5]. These patterns might be explained by a previously undiscovered asymmetry in RNA gel electrophoresis band broadening and hint at two competing evolutionary forces that determine the lengths of gene transcripts. Finally, I will describe ongoing work in the development of tensor algebra algorithms (as well as viusal correlation tools), the integrative and comparative modeling of DNA microarray data (as well as rRNA sequence data), and the discovery of mechanisms that regulate cell division, cancer and evolution. 1. Alter, PNAS 103, 16063 (2006). 2. Alter & Golub, PNAS 101, 16577 (2004). 3. Omberg, Golub & Alter, PNAS 104, 18371 (2007). 4. Omberg, Meyerson, Kobayashi, Drury, Diffley & Alter, Nature MSB 5, 312 (2009). 5. Alter & Golub, PNAS 103, 11828 (2006)

Platform-Independent Genome-Wide Pattern of DNA Copy-Number Alterations Predicting Astrocytoma Survival and Response to Treatment Revealed by the GSVD Formulated as a Comparative Spectral Decomposition

<div><p>We use the generalized singular value decomposition (GSVD), formulated as a comparative spectral decomposition, to model patient-matched grades III and II, i.e., lower-grade astrocytoma (LGA) brain tumor and normal DNA copy-number profiles. A genome-wide tumor-exclusive pattern of DNA copy-number alterations (CNAs) is revealed, encompassed in that previously uncovered in glioblastoma (GBM), i.e., grade IV astrocytoma, where GBM-specific CNAs encode for enhanced opportunities for transformation and proliferation via growth and developmental signaling pathways in GBM relative to LGA. The GSVD separates the LGA pattern from other sources of biological and experimental variation, common to both, or exclusive to one of the tumor and normal datasets. We find, first, and computationally validate, that the LGA pattern is correlated with a patient’s survival and response to treatment. Second, the GBM pattern identifies among the LGA patients a subtype, statistically indistinguishable from that among the GBM patients, where the CNA genotype is correlated with an approximately one-year survival phenotype. Third, cross-platform classification of the Affymetrix-measured LGA and GBM profiles by using the Agilent-derived GBM pattern shows that the GBM pattern is a platform-independent predictor of astrocytoma outcome. Statistically, the pattern is a better predictor (corresponding to greater median survival time difference, proportional hazard ratio, and concordance index) than the patient’s age and the tumor’s grade, which are the best indicators of astrocytoma currently in clinical use, and laboratory tests. The pattern is also statistically independent of these indicators, and, combined with either one, is an even better predictor of astrocytoma outcome. Recurring DNA CNAs have been observed in astrocytoma tumors’ genomes for decades, however, copy-number subtypes that are predictive of patients’ outcomes were not identified before. This is despite the growing number of datasets recording different aspects of the disease, and due to an existing fundamental need for mathematical frameworks that can simultaneously find similarities and dissimilarities across the datasets. This illustrates the ability of comparative spectral decompositions to find what other methods miss.</p></div