A note on estimation of dynamics of multiple gene expression based on singular value decomposition

Recently, data on multiple gene expression at sequential time points were analyzed, using singular value decomposition (SVD) as a means to capture dominant trends, called characteristic modes, followed by fitting of a linear discrete-time dynamical system in which the expression values at a given time point are linear combinations of the values at a previous time point. We attempt to address several aspects of the method. To obtain the model we formulate a non-linear optimization problem and present how to solve it numerically using standard MATLAB procedures. We use publicly available data to test the approach. For reader's convenience, we provide a straightforward, ready-to-use, procedure in MATLAB, which employs its standard features to analyze data of this kind. Then, we investigate the sensitivity of the method to missing measurements and its possibilities to reconstruct missing data. Also, we discuss the possible consequences of data regularization, called sometimes 'polishing', on the outcome of analysis, especially when model is to be used for prediction purposes. Summarizing we point out that approximation of multiple gene expression data preceded by SVD provides some insight into the dynamics but may also lead to unexpected difficulties, like overfitting problems.     

Pathway level analysis of gene expression using singular value decomposition

Background  

A promising direction in the analysis of gene expression focuses on the changes in expression of specific predefined sets of genes that are known in advance to be related (e.g., genes coding for proteins involved in cellular pathways or complexes). Such an analysis can reveal features that are not easily visible from the variations in the individual genes and can lead to a picture of expression that is more biologically transparent and accessible to interpretation. In this article, we present a new method of this kind that operates by quantifying the level of 'activity' of each pathway in different samples. The activity levels, which are derived from singular value decompositions, form the basis for statistical comparisons and other applications.     

Observation of multiple intermediates in alpha-synuclein fibril formation by singular value decomposition analysis

One of the most well known characteristics for Parkinson's disease (PD) is a polymerization of wild-type or mutant alpha-synuclein into aggregates and fibrils, commonly observed as Lewy bodies and Lewy neuritis in PD patients. Although numerous studies on alpha-synuclein fibrillation have been reported, the molecular mechanisms of aggregation and fibrillation are not well understood yet. In the present study, structural properties and propensities to form fibrils of wild-type, A30P, E46K, and A53T alpha-synucleins were investigated using fluorescence and circular dichroism (CD) methods. The results from these studies were analyzed using singular value decomposition (SVD) method which estimates a number of conformationally independent species for a given process. The time-dependent CD spectra of the wild-type alpha-synuclein indicated a multi-step process in the fibril formation, and SVD analysis using the time-dependent CD spectra revealed that five or nine intermediates were formed at the early stage of fibrillation.     

Resampling methods for variance estimation of singular value decomposition analyses from microarray experiments

Microarray experiments offer the ability to generate gene expression measurements for thousands of genes simultaneously. Work has begun recently on attempting to reconstruct genetic networks based on analyses of microarray experiments in time-course studies. An important tool in these analyses has been the singular value decomposition method. However, little work has been done on assessing the variability associated with singular value decomposition analyses. In this report, we discuss use of the bootstrap as a method of obtaining standard errors for singular value decomposition analyses. We consider use of this method both when there are replicates and when no replicates exist. The proposed methods are illustrated with an application to two datasets: one involving a human foreskin study, the other involving yeast. Electronic Publication     

Mapping gene expression quantitative trait loci by singular value decomposition and independent component analysis

Background  

The combination of gene expression profiling with linkage analysis has become a powerful paradigm for mapping gene expression quantitative trait loci (eQTL). To date, most studies have searched for eQTL by analyzing gene expression traits one at a time. As thousands of expression traits are typically analyzed, this can reduce power because of the need to correct for the number of hypothesis tests performed. In addition, gene expression traits exhibit a complex correlation structure, which is ignored when analyzing traits individually.     

Mining gene expression profiles: an integrated implementation of kernel principal component analysis and singular value decomposition

The detection of genes that show similar profiles under different experimental conditions is often an initial step in inferring the biological significance of such genes. Visualization tools are used to identify genes with similar profiles in microarray studies. Given the large number of genes recorded in microarray experiments, gene expression data are generally displayed on a low dimensional plot, based on linear methods. However, microarray data show nonlinearity, due to high-order terms of interaction between genes, so alternative approaches, such as kernel methods, may be more appropriate. We introduce a technique that combines kernel principal component analysis (KPCA) and Biplot to visualize gene expression profiles. Our approach relies on the singular value decomposition of the input matrix and incorporates an additional step that involves KPCA. The main properties of our method are the extraction of nonlinear features and the preservation of the input variables (genes) in the output display. We apply this algorithm to colon tumor, leukemia and lymphoma datasets. Our approach reveals the underlying structure of the gene expression profiles and provides a more intuitive understanding of the gene and sample association.     

Assessment of repeatability of surface electromyography signals by singular value decomposition

Singular value decomposition (SVD) of full-wave rectified surface electromyography (sEMG) signals was investigated for repeatability indices of sEMG linear envelopes (LE) during biceps curl. The SVD based repeatability indices were compared with a well-known method, the variance ratio (VR). The usefulness of the offered indices was examined by a simulation and it was applied to the sEMG LEs. The results have shown that the VRs were correlated with the SVD based indices significantly. The usefulness of the offered indices on real world EMG signals practically comes from decreasing amplitudes of the first few singular values of EMG LE matrix. If repeatability is high, singular values decay fast and vice versa.If justified by further researches, the offered indices may be used practically for repeatability measurement of sEMG LEs.     

A thermodynamic comparison of HPr proteins from extremophilic organisms

A thermodynamic stability study of five histidine-containing phosphocarrier protein (HPr) homologues derived from organisms inhabiting diverse environments is described. These HPr homologues are from Bacillus subtilis (Bs), Streptococcus thermophilus (St), Bacillus staerothermophilus (Bst), Bacillus halodurans (Bh), and Oceanobacillus iheyensis (Oi). Analyses of solvent and thermal denaturation experiments provide the cardinal thermodynamic parameters, like deltaG, deltaH, deltaS, T(m), and deltaC(p), that characterize the conformational stability for each homologue. The homologue from Bacillus staerothermophilus (BstHPr) was established as the most thermostable homologue and also the homologue with highest deltaG at all temperatures. A good correlation between habitat temperature of the organism and thermal stability of the protein is also seen. Stability curves (deltaG vs T) for every homologue are also reported; these reveal very similar deltaC(p) and temperature of maximum stability (T(S)) values for all HPr homologues. Stability curves show that the higher thermal stability of some homologues is not a result of change in curvature of the curve or a shift to higher temperature, but rather a displacement of the stability curves to higher deltaG values. Stability curves also allowed estimation of deltaG at habitat temperature of the organisms, and we find good agreement between homologues. Electrostatic contributions to stability of each homologue were investigated by measuring stability as a function of varying pH and NaCl concentration, and our results suggest that most HPr homologues share similar electrostatic contributions to stability.     

Analysis of self-associating proteins by singular value decomposition of solution scattering data

We describe a method by which a single experiment can reveal both association model (pathway and constants) and low-resolution structures of a self-associating system. Small-angle scattering data are collected from solutions at a range of concentrations. These scattering data curves are mass-weighted linear combinations of the scattering from each oligomer. Singular value decomposition of the data yields a set of basis vectors from which the scattering curve for each oligomer is reconstructed using coefficients that depend on the association model. A search identifies the association pathway and constants that provide the best agreement between reconstructed and observed data. Using simulated data with realistic noise, our method finds the correct pathway and association constants. Depending on the simulation parameters, reconstructed curves for each oligomer differ from the ideal by 0.05-0.99% in median absolute relative deviation. The reconstructed scattering curves are fundamental to further analysis, including interatomic distance distribution calculation and low-resolution ab initio shape reconstruction of each oligomer in solution. This method can be applied to x-ray or neutron scattering data from small angles to moderate (or higher) resolution. Data can be taken under physiological conditions, or particular conditions (e.g., temperature) can be varied to extract fundamental association parameters (ΔH_ass, ΔS_ass).     

Application of singular value decomposition to the analysis of time-resolved macromolecular x-ray data

Singular value decomposition (SVD) is a technique commonly used in the analysis of spectroscopic data that both acts as a noise filter and reduces the dimensionality of subsequent least-squares fits. To establish the applicability of SVD to crystallographic data, we applied SVD to calculated difference Fourier maps simulating those to be obtained in a time-resolved crystallographic study of photoactive yellow protein. The atomic structures of one dark state and three intermediates were used in qualitatively different kinetic mechanisms to generate time-dependent difference maps at specific time points. Random noise of varying levels in the difference structure factor amplitudes, different extents of reaction initiation, and different numbers of time points were all employed to simulate a range of realistic experimental conditions. Our results show that SVD allows for an unbiased differentiation between signal and noise; a small subset of singular values and vectors represents the signal well, reducing the random noise in the data. Due to this, phase information of the difference structure factors can be obtained. After identifying and fitting a kinetic mechanism, the time-independent structures of the intermediates could be recovered. This demonstrates that SVD will be a powerful tool in the analysis of experimental time-resolved crystallographic data.     

Systemic metabolic reactions are obtained by singular value decomposition of genome-scale stoichiometric matrices

Genome-scale metabolic networks can be reconstructed. The systemic biochemical properties of these networks can now be studied. Here, genome-scale reconstructed metabolic networks were analysed using singular value decomposition (SVD). All the individual biochemical conversions contained in a reconstructed metabolic network are described by a stoichiometric matrix (S). SVD of S led to the definition of the underlying modes that characterize the overall biochemical conversions that take place in a network and rank-ordered their importance. The modes were shown to correspond to systemic biochemical reactions and they could be used to identify the groups and clusters of individual biochemical reactions that drive them. Comparative analysis of the Escherichia coli, Haemophilus influenzae, and Helicobacter pylori genome-scale metabolic networks showed that the four dominant modes in all three networks correspond to: (1) the conversion of ATP to ADP, (2) redox metabolism of NADP, (3) proton-motive force, and (4) inorganic phosphate metabolism. The sets of individual metabolic reactions deriving these systemic conversions, however, differed among the three organisms. Thus, we can now define systemic metabolic reactions, or eigen-reactions, for the study of systems biology of metabolism and have a basis for comparing the overall properties of genome-specific metabolic networks.     

Application of linear prediction singular value decomposition for processingin vivo NMR data with low signal-to-noise ratio

Summary Low sensitivity of nuclear magnetic resonance (NMR) measurements of living cell composition by conventional methods requires samples with high cell density compared to that in growing cultures. Reasonably accurate intracellular concentration estimates from lower cell density samples can be obtained by treating the time-domain NMR data by linear prediction singular value decomposition (LPSVD) prior to Fourier transformation. Alternatively, application of LPSVD enables intracellular concentration estimates in less NMR acquisition time, improving time resolution in NMR measurements of intracellular transients.     

Reliability of ultrasound speckle tracking with singular value decomposition for quantifying displacement in the carpal tunnel

Inhibited movement patterns of carpal tunnel structures have been found in carpal tunnel syndrome (CTS) patients. Motion analysis on ultrasound images allows us to non-invasively study the (relative) movement of carpal tunnel structures and recently a speckle tracking method using singular value decomposition (SVD) has been proposed to optimize this tracking. This study aims to assess the reliability of longitudinal speckle tracking with SVD in both healthy volunteers and patients with CTS.Images from sixteen healthy volunteers and twenty-two CTS patients were used. Ultrasound clips of the third superficial flexor tendon and surrounding subsynovial connective tissue (SSCT) were acquired during finger flexion-extension. A custom made tracking algorithm was used for the analysis. Intra-class correlation coefficients (ICCs) were calculated using a single measure, two-way random model with absolute agreement and Bland-Altman plots were added for graphical representation.ICC values varied between 0.73 and 0.95 in the control group and 0.66–0.98 in the CTS patients, with the majority of the results classified as good to excellent. Tendon tracking showed higher reliability values compared to the SSCT, but values between the control and CTS groups were comparable.Speckle tracking with SVD can reliably be used to analyze longitudinal movement of anatomical structures with different sizes and compositions within the context of the carpal tunnel in both a healthy as well as a pathological state. Based on these results, this technique also holds relevant potential for areas where ultrasound based dynamic imaging requires quantification of motion.     

Comparative investigation of superoxide trapping by cyclic nitrone spin traps: the use of singular value decomposition and multiple linear regression analysis

The kinetics of the reaction between superoxide and the spin trapping agents 5,5-dimethyl-1-pyrroline N-oxide (DMPO), 5-(diethoxyphosphoryl)-5-methyl-1-pyrroline N-oxide (DEPMPO), and 5-tert-butoxycarbonyl-5-methyl-1-pyrroline N-oxide (BMPO) were re-examined in the superoxide-generating xanthine/xanthine oxidase system, by competition with spontaneous dismutation. The approach used singular value decomposition (SVD), multiple linear regression, and spectral simulation. The experiments were carried out using a two-syringe mixing arrangement with fast scan acquisition of 100 consecutive EPR spectra. Using SVD analysis, the extraction of both temporal and spectral information could be obtained from in a single run. The superoxide spin adduct was the exclusive EPR active species in the case of DEPMPO and BMPO, and the major component when DMPO was used. In the latter case a very low concentration of hydroxyl adduct was also observed, which did not change during the decay of the DMPO-superoxide adduct. This indicates that the hydroxyl radical adduct is not formed from the spontaneous decay of the superoxide radical adduct, as has been previously suggested [correction]. It was established that in short-term studies (up to 100 s) DMPO was the superior spin trapping agent, but for reaction times longer than 100 s the other two spin traps were more advantageous. The second order rate constants for the spin trapping reaction were found to be DMPO (2.4 M(-1)s(-1)), DEPMPO (0.53 M(-1)s(-1)), and BMPO (0.24 M(-1)s(-1)) determined through competition with spontaneous dismutation of superoxide, at pH 7.4 and 20 degrees C.