Consensus Principal Components

From the polar forms of the principal components corresponding with each of a set of covariance (or correlation) matrices, a linear combination based on their inner products is defined as the polar form of the consensus. The corresponding eigenvectors form an orthogonal matrix which rotates each of the covariance matrices to approximate diagonal form. From the norms of the polar forms, these eigenvectors can be used to estimate a common covariance matrix. These procedures are illustrated by a numerical example.     

The Method of Random Skewers

Ecological and evolutionary studies are often concerned with the properties of covariance matrices. The method of random skewers (RS method) has been used compare a matrix to an a priori vector or to compare two matrices. The method involves multiplying a matrix by many random vectors drawn from a uniform distribution over all possible vector directions. The comparisons are usually made using the average angle (or cosine) of the response vectors to an a priori vector or to the response vectors corresponding from another matrix. Angles are usually constrained to the interval 0°–90° because the distribution of response vectors is bipolar bimodal. The size of the average angle or cosine depends strongly on the relative sizes of the eigenvalues (especially the first). The distribution of angles between pairs of response vectors from two covariance matrices is more complicated because it depends on the differences in orientation of the eigenvectors and the relative sizes of the eigenvalues of the both matrices. The average absolute value of the angles between these pairs of response vectors depends on the relative sizes of the eigenvalues of the matrices making it difficult to interpret its meaning without knowledge of the eigenvalues and eigenvectors of the two matrices. Thus, it is simpler to just directly compare matrices in terms of these quantities.     

Ischemia episode detection in ECG using kernel density estimation, support vector machine and feature selection

ABSTRACT: BACKGROUND: Myocardial ischemia can be developed into more serious diseases. Early Detection of the ischemic syndrome inelectrocardiogram (ECG) more accurately and automatically can prevent it from developing into a catastrophicdisease. To this end, we propose a new method, which employs wavelets and simple feature selection. METHODS: For training and testing, the European ST-T database is used, which is comprised of 367 ischemic ST episodes in90 records. We first remove baseline wandering, and detect time positions of QRS complexes by a method basedon the discrete wavelet transform. Next, for each heart beat, we extract three features which can be used fordifferentiating ST episodes from normal: 1) the area between QRS offset and T-peak points, 2) the normalizedand signed sum from QRS offset to effective zero voltage point, and 3) the slope from QRS onset to offset point.We average the feature values for successive five beats to reduce effects of outliers. Finally we apply classifiersto those features. RESULTS: We evaluated the algorithm by kernel density estimation (KDE) and support vector machine (SVM) methods.Sensitivity and specificity for KDE were 0.939 and 0.912, respectively. The KDE classifier detects 349 ischemicST episodes out of total 367 ST episodes. Sensitivity and specificity of SVM were 0.941 and 0.923, respectively.The SVM classifier detects 355 ischemic ST episodes. CONCLUSIONS: We proposed a new method for detecting ischemia in ECG. It contains signal processing techniques of removingbaseline wandering and detecting time positions of QRS complexes by discrete wavelet transform, and featureextraction from morphology of ECG waveforms explicitly. It was shown that the number of selected featureswere sufficient to discriminate ischemic ST episodes from the normal ones. We also showed how the proposedKDE classifier can automatically select kernel bandwidths, meaning that the algorithm does not require anynumerical values of the parameters to be supplied in advance. In the case of the SVM classifier, one has to selecta single parameter.     

Solving the Differential Biochemical Jacobian from Metabolomics Covariance Data

High-throughput molecular analysis has become an integral part in organismal systems biology. In contrast, due to a missing systematic linkage of the data with functional and predictive theoretical models of the underlying metabolic network the understanding of the resulting complex data sets is lacking far behind. Here, we present a biomathematical method addressing this problem by using metabolomics data for the inverse calculation of a biochemical Jacobian matrix, thereby linking computer-based genome-scale metabolic reconstruction and in vivo metabolic dynamics. The incongruity of metabolome coverage by typical metabolite profiling approaches and genome-scale metabolic reconstruction was solved by the design of superpathways to define a metabolic interaction matrix. A differential biochemical Jacobian was calculated using an approach which links this metabolic interaction matrix and the covariance of metabolomics data satisfying a Lyapunov equation. The predictions of the differential Jacobian from real metabolomic data were found to be correct by testing the corresponding enzymatic activities. Moreover it is demonstrated that the predictions of the biochemical Jacobian matrix allow for the design of parameter optimization strategies for ODE-based kinetic models of the system. The presented concept combines dynamic modelling strategies with large-scale steady state profiling approaches without the explicit knowledge of individual kinetic parameters. In summary, the presented strategy allows for the identification of regulatory key processes in the biochemical network directly from metabolomics data and is a fundamental achievement for the functional interpretation of metabolomics data.     

Simulations of a protein translocation pore: SecY

SecY is the central channel protein of the SecYEbeta translocon, the structure of which has been determined by X-ray diffraction. Extended (15 ns) MD simulations of the isolated SecY protein in a phospholipid bilayer have been performed to explore the relationship between protein flexibility and the mechanisms of channel gating. In particular, principal components analysis of the simulation trajectory has been used to probe the intrinsic flexibility of the isolated SecY protein in the absence of the gamma-subunit (SecE) clamp. Analysis and visualization of the principal eigenvectors support a "plug and clamshell" model of SecY channel gating. The simulation results also indicate that hydrophobic gating at the central pore ring prevents leakage of water and ions through the channel in the absence of a translocating peptide.     

AN EIGENVECTOR METHOD FOR ESTIMATING PHYLOGENETIC INERTIA

We propose a new method to estimate and correct for phylogenetic inertia in comparative data analysis. The method, called phylogenetic eigenvector regression (PVR) starts by performing a principal coordinate analysis on a pairwise phylogenetic distance matrix between species. Traits under analysis are regressed on eigenvectors retained by a broken-stick model in such a way that estimated values express phylogenetic trends in data and residuals express independent evolution of each species. This partitioning is similar to that realized by the spatial autoregressive method, but the method proposed here overcomes the problem of low statistical performance that occurs with autoregressive method when phylogenetic correlation is low or when sample size is too small to detect it. Also, PVR is easier to perform with large samples because it is based on well-known techniques of multivariate and regression analyses. We evaluated the performance of PVR and compared it with the autoregressive method using real datasets and simulations. A detailed worked example using body size evolution of Carnivora mammals indicated that phylogenetic inertia in this trait is elevated and similarly estimated by both methods. In this example, Type I error at α = 0.05 of PVR was equal to 0.048, but an increase in the number of eigenvectors used in the regression increases the error. Also, similarity between PVR and the autoregressive method, defined by correlation between their residuals, decreased by overestimating the number of eigenvalues necessary to express the phylogenetic distance matrix. To evaluate the influence of cladogram topology on the distribution of eigenvalues extracted from the double-centered phylogenetic distance matrix, we analyzed 100 randomly generated cladograms (up to 100 species). Multiple linear regression of log transformed variables indicated that the number of eigenvalues extracted by the broken-stick model can be fully explained by cladogram topology. Therefore, the broken-stick model is an adequate criterion for determining the correct number of eigenvectors to be used by PVR. We also simulated distinct levels of phylogenetic inertia by producing a trend across 10, 25, and 50 species arranged in “comblike” cladograms and then adding random vectors with increased residual variances around this trend. In doing so, we provide an evaluation of the performance of both methods with data generated under different evolutionary models than tested previously. The results showed that both PVR and autoregressive method are efficient in detecting inertia in data when sample size is relatively high (more than 25 species) and when phylogenetic inertia is high. However, PVR is more efficient at smaller sample sizes and when level of phylogenetic inertia is low. These conclusions were also supported by the analysis of 10 real datasets regarding body size evolution in different animal clades. We concluded that PVR can be a useful alternative to an autoregressive method in comparative data analysis.     

Electrocardiogram reference values for the buzzard in Spain.

Electrocardiographic reference values were established on apparently healthy buzzards (Buteo buteo) in Lugo (Spain) from March 1997 to June 1999. All birds were anesthetized with isofluorane and placed in dorsal recumbence. The standard and augmented unipolar limb leads electrocardiograms were recorded in 65 buzzards. The wave forms were analyzed in lead II at 50 mm/sec and at 1 cm = 1 mV to determine P, PR, QRS, T and QT durations and P, QRS and T amplitudes. The polarity of each wave form was tabulated in all leads. The mean electrical axis (MEA) for the frontal plane was calculated using leads II and III. The mean heart rate was 325.2 +/- 52.9 beats/min. In lead II, the P wave was positive, the dominant pattern of QRS complex was QS and the T wave was always positive. The average value of the MEA was -99.2 +/- 7.7 degrees. Establishment of normal electrocardiogram (EKG) values will facilitate a better understanding of EKG changes seen in many diseases of these birds.     

On the selection of phylogenetic eigenvectors for ecological analyses

Among the statistical methods available to control for phylogenetic autocorrelation in ecological data, those based on eigenfunction analysis of the phylogenetic distance matrix among the species are becoming increasingly important tools. Here, we evaluate a range of criteria to select eigenvectors extracted from a phylogenetic distance matrix (using phylogenetic eigenvector regression, PVR) that can be used to measure the level of phylogenetic signal in ecological data and to study correlated evolution. We used a principal coordinate analysis to represent the phylogenetic relationships among 209 species of Carnivora by a series of eigenvectors, which were then used to model log‐transformed body size. We first conducted a series of PVRs in which we increased the number of eigenvectors from 1 to 70, following the sequence of their associated eigenvalues. Second, we also investigated three non‐sequential approaches based on the selection of 1) eigenvectors significantly correlated with body size, 2) eigenvectors selected by a standard stepwise algorithm, and 3) the combination of eigenvectors that minimizes the residual phylogenetic autocorrelation. We mapped the mean specific component of body size to evaluate how these selection criteria affect the interpretation of non‐phylogenetic signal in Bergmann's rule. For comparison, the same patterns were analyzed using autoregressive model (ARM) and phylogenetic generalized least‐squares (PGLS). Despite the robustness of PVR to the specific approaches used to select eigenvectors, using a relatively small number of eigenvectors may be insufficient to control phylogenetic autocorrelation, leading to flawed conclusions about patterns and processes. The method that minimizes residual autocorrelation seems to be the best choice according to different criteria. Thus, our analyses show that, when the best criterion is used to control phylogenetic structure, PVR can be a valuable tool for testing hypotheses related to heritability at the species level, phylogenetic niche conservatism and correlated evolution between ecological traits.     

Local feature analysis: a statistical theory for reproducible essential dynamics of large macromolecules

Multivariate statistical methods are widely used to extract functional collective motions from macromolecular molecular dynamics (MD) simulations. In principal component analysis (PCA), a covariance matrix of positional fluctuations is diagonalized to obtain orthogonal eigenvectors and corresponding eigenvalues. The first few eigenvectors usually correspond to collective modes that approximate the functional motions in the protein. However, PCA representations are globally coherent by definition and, for a large biomolecular system, do not converge on the time scales accessible to MD. Also, the forced orthogonalization of modes leads to complex dependencies that are not necessarily consistent with the symmetry of biological macromolecules and assemblies. Here, we describe for the first time the application of local feature analysis (LFA) to construct a topographic representation of functional dynamics in terms of local features. The LFA representations are low dimensional, and like PCA provide a reduced basis set for collective motions, but they are sparsely distributed and spatially localized. This yields a more reliable assignment of essential dynamics modes across different MD time windows. Also, the intrinsic dynamics of local domains is more extensively sampled than that of globally coherent PCA modes.     

Molecular bases of immune complex pathology

The binding of antigens with antibodies forms immune complexes in the body. Usually these complexes are eliminated by the system of mononuclear phagocytes without development of pathological changes. This review highlights principal mechanisms responsible for safe removal of immune complexes in primates and humans. Special attention is given to diseases known as “immune complex diseases”, when antigen-antibody complexes induce inflammatory reactions. The review considers key experimental works that significantly contributed to current knowledge of etiology and pathogenesis of type III hypersensitivity. Some factors of the development of immune complex syndrome such as level of humoral immune response to antigen, isotype and affinity of forming antibodies, the amount of immune complexes, and the consequences of their interaction with the complement system and Fc-receptors are analyzed based on the molecular mechanisms involved. The review contains a retrospective analysis of the most significant scientific achievements in immune complex pathology investigation within the last 100 years.     

Structure determination of a DNA octamer in solution by NMR spectroscopy. Effect of fast local motions.

NMR structures of biomolecules are primarily based on nuclear Overhauser effects (NOEs) between protons. For the interpretation of NOEs in terms of distances, usually the assumption of a single rotational correlation time corresponding to a rigid molecule approximation is made. Here we investigate the effect of fast internal motions of the interproton vectors in the context of the relaxation matrix approach for structure determination of biomolecules. From molecular dynamics simulations generalized order parameters were calculated for the DNA octamer d(GCGTTCGC).d(CGCAACGC), and these were used in the calculation of NOE intensities. The magnitudes of the order parameters showed some variation for the different types of interproton vectors. The lowest values were observed for the interresidue base H6/H8-H2" proton vectors (S2 = 0.60), while the cytosine H5-H6 interproton vectors were among the most motionally restricted (S2 = 0.92). Inclusion of the motion of the interproton vectors resulted in a much better agreement between theoretically calculated NOE spectra and the experimental spectra measured by 2D NOE spectroscopy. The interproton distances changed only slightly, with a maximum of 10%; nevertheless, the changes were significant and resulted in constraints that were better satisfied. The structure of the DNA octamer was determined by using restrained molecular dynamics simulations with H2O as a solvent, with and without the inclusion of local internal motions. Starting from A- or B-DNA, the structures showed a high local convergence (0.86 A), while the global convergence for the octamer was ca. 2.6 A.     

Fragmented QRS – Its significance

Fragment QRS (fQRS) complex is a myocardial conduction abnormality that indicates myocardial scar. It is defined as additional notches in the QRS complex. Though initially fQRS was defined in the setting of normal QRS duration (<120 m s), later it has been expanded to include conditions with wide QRS complexes as in bundle branch block, ventricular ectopy and paced rhythm, when more than 2 notches are present. It is an important, yet often overlooked marker of mortality and arrhythmic events in many cardiac diseases. The significance of fQRS lies in the fact that it just requires a surface ECG for its recording and the value of information about the condition of the heart it dispenses based on the clinical setting. We review the role of fQRS in predicting adverse cardiac events in various conditions.     

Three-dimensional structure of the KChIP1-Kv4.3 T1 complex reveals a cross-shaped octamer

Brain I(A) and cardiac I(to) currents arise from complexes containing Kv4 voltage-gated potassium channels and cytoplasmic calcium-sensor proteins (KChIPs). Here, we present X-ray crystallographic and small-angle X-ray scattering data that show that the KChIP1-Kv4.3 N-terminal cytoplasmic domain complex is a cross-shaped octamer bearing two principal interaction sites. Site 1 comprises interactions between a unique Kv4 channel N-terminal hydrophobic segment and a hydrophobic pocket formed by displacement of the KChIP H10 helix. Site 2 comprises interactions between a T1 assembly domain loop and the KChIP H2 helix. Functional and biochemical studies indicate that site 1 influences channel trafficking, whereas site 2 affects channel gating, and that calcium binding is intimately linked to KChIP folding and complex formation. Together, the data resolve how Kv4 channels and KChIPs interact and provide a framework for understanding how KChIPs modulate Kv4 function.     

An application of common principal component analysis to cranial morphometry of Microtus californicus and M. ochrogaster (Mammalia, Rodentia): further remarks

We welcome Dr Thorpe's interesting discussion (Thorpe, 1988), and we would like to take this opportunity to clarify some points.
Both MGPCA (multiple group principal component analysis) and CPCA (common principal component analysis) serve essentially the same purpose, namely estimation of principal components simultaneously in several groups, based on the assumption of equality of principal component directions across groups, while eigenvalues may differ between groups. However, CPCA has the distinct advantage that this assumption can actually be tested, using the (CPC) statistic. In analyses involving more than two variables, it is usually difficult to decide, without a formal test, whether or not the assumption of common directions of principal components is reasonable.
There is also a conceptual difficulty with MGPCA. In statistical terms, both methods assume that:
(a) a certain set of parameters (namely those determining the eigenvectors) are common to all groups
(b) there are sets of parameters (namely p eigenvalues per group) which are specific to each group.
CPCA sets up a model that reflects this structure and estimates the parameters accordingly. MGPCA, on the other hand, ignores part (b), at least temporarily, by pooling the variance-covariance matrices and extracting eigenvectors from the single pooled matrix. This may lead to reasonable results, but there is no guarantee that it will indeed do so. The reader may find a more familiar analog in the fitting of regression lines when data are in groups. If it is assumed that all regression lines are parallel, one should set up an appropriate model based on a single slope parameter common to all groups, and groupwise intercepts. One should then estimate the parameters of this model, and not simply apply a technique which is appropriate in the one-group case only.     

Time-oriented hierarchical method for computation of principal components using subspace learning algorithm

Principal Component Analysis (PCA) and Principal Subspace Analysis (PSA) are classic techniques in statistical data analysis, feature extraction and data compression. Given a set of multivariate measurements, PCA and PSA provide a smaller set of "basis vectors" with less redundancy, and a subspace spanned by them, respectively. Artificial neurons and neural networks have been shown to perform PSA and PCA when gradient ascent (descent) learning rules are used, which is related to the constrained maximization (minimization) of statistical objective functions. Due to their low complexity, such algorithms and their implementation in neural networks are potentially useful in cases of tracking slow changes of correlations in the input data or in updating eigenvectors with new samples. In this paper we propose PCA learning algorithm that is fully homogeneous with respect to neurons. The algorithm is obtained by modification of one of the most famous PSA learning algorithms--Subspace Learning Algorithm (SLA). Modification of the algorithm is based on Time-Oriented Hierarchical Method (TOHM). The method uses two distinct time scales. On a faster time scale PSA algorithm is responsible for the "behavior" of all output neurons. On a slower scale, output neurons will compete for fulfillment of their "own interests". On this scale, basis vectors in the principal subspace are rotated toward the principal eigenvectors. At the end of the paper it will be briefly analyzed how (or why) time-oriented hierarchical method can be used for transformation of any of the existing neural network PSA method, into PCA method.     

Effects of co-operative ligand binding on protein amide NH hydrogen exchange

Amide protection factors have been determined from NMR measurements of hydrogen/deuterium amide NH exchange rates measured on assigned signals from Lactobacillus casei apo-DHFR and its binary and ternary complexes with trimethoprim (TMP), folinic acid and coenzymes (NADPH/NADP(+)). The substantial sizes of the residue-specific DeltaH and TDeltaS values for the opening/closing events in NH exchange for most of the measurable residues in apo-DHFR indicate that sub-global or global rather than local exchange mechanisms are usually involved. The amide groups of residues in helices and sheets are those most protected in apo-DHFR and its complexes, and the protection factors are generally related to the tightness of ligand binding. The effects of ligand binding that lead to changes in amide protection are not localised to specific binding sites but are spread throughout the structure via a network of intramolecular interactions. Although the increase in protein stability in the DHFR.TMP.NADPH complex involves increased ordering in the protein structure (requiring TDeltaS energy) this is recovered, to a large extent, by the stronger binding (enthalpic DeltaH) interactions made possible by the reduced motion in the protein. The ligand-induced protection effects in the ternary complexes DHFR.TMP.NADPH (large positive binding co-operativity) and DHFR.folinic acid.NADPH (large negative binding co-operativity) mirror the co-operative effects seen in the ligand binding. For the DHFR.TMP.NADPH complex, the ligand-induced protection factors result in DeltaDeltaG(o) values for many residues being larger than the DeltaDeltaG(o) values in the corresponding binary complexes. In contrast, for DHFR.folinic acid.NADPH, the DeltaDeltaG(o) values are generally smaller than many of those in the corresponding binary complexes. The results indicate that changes in protein conformational flexibility on formation of the ligand complex play an important role in determining the co-operativity in the ligand binding.     

Perils of parsimony: properties of reduced-rank estimates of genetic covariance matrices

Eigenvalues and eigenvectors of covariance matrices are important statistics for multivariate problems in many applications, including quantitative genetics. Estimates of these quantities are subject to different types of bias. This article reviews and extends the existing theory on these biases, considering a balanced one-way classification and restricted maximum-likelihood estimation. Biases are due to the spread of sample roots and arise from ignoring selected principal components when imposing constraints on the parameter space, to ensure positive semidefinite estimates or to estimate covariance matrices of chosen, reduced rank. In addition, it is shown that reduced-rank estimators that consider only the leading eigenvalues and -vectors of the "between-group" covariance matrix may be biased due to selecting the wrong subset of principal components. In a genetic context, with groups representing families, this bias is inverse proportional to the degree of genetic relationship among family members, but is independent of sample size. Theoretical results are supplemented by a simulation study, demonstrating close agreement between predicted and observed bias for large samples. It is emphasized that the rank of the genetic covariance matrix should be chosen sufficiently large to accommodate all important genetic principal components, even though, paradoxically, this may require including a number of components with negligible eigenvalues. A strategy for rank selection in practical analyses is outlined.     

Sequestered water and binding energy are coupled in complexes of lambda Cro repressor with non-consensus binding sequences

We use the osmotic pressure dependence of dissociation rates and relative binding constants to infer differences in sequestered water among complexes of lambda Cro repressor with varied DNA recognition sequences. For over a 1000-fold change in association constant, the number of water molecules sequestered by non-cognate complexes varies linearly with binding free energy. One extra bound water molecule is coupled with the loss of approximately 150 cal/mol complex in binding free energy. Equivalently, every tenfold decrease in binding constant at constant salt and temperature is associated with eight to nine additional water molecules sequestered in the non-cognate complex. The relative insensitivity of the difference in water molecules to the nature of the osmolyte used to probe the reaction suggests that the water is sterically sequestered. If the previously measured changes in heat capacity for lambda Cro binding to different non-cognate sequences are attributed solely to this change in water, then the heat capacity change per incorporated water is almost the same as the difference between ice and water. The associated changes in enthalpies and entropies, however, indicate that the change in complex structure involves more than a simple incorporation of fixed water molecules that act as adaptors between non-complementary surfaces.