Calculating as many fluxes as possible in underdetermined metabolic networks

A frequently occurring problem in Metabolic Flux Analysis is that the linear equation systems are underdetermined. A procedure for determining which reaction velocities can be calculated in underdetermined metabolic systems from measured rates and computing these velocities is given. The method is based on the null-space matrix to the stoichiometry matrix corresponding to the reactions with unknown velocities. Moreover, an elementary representation of the null-space is presented. This representation enables one to find those sets of measurable velocities that allow calculation of a certain non-measurable rate. The approach is illustrated by an example from monosaccharide metabolism.     

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.     

Confidence intervals for a variance ratio, or for heritability, in an unbalanced mixed linear model

A procedure is presented for constructing an exact confidence interval for the ratio of the two variance components in a possibly unbalanced mixed linear model that contains a single set of m random effects. This procedure can be used in animal and plant breeding problems to obtain an exact confidence interval for a heritability. The confidence interval can be defined in terms of the output of a least squares analysis. It can be computed by a graphical or iterative technique requiring the diagonalization of an m X m matrix or, alternatively, the inversion of a number of m X m matrices. Confidence intervals that are approximate can be obtained with much less computational burden, using either of two approaches. The various confidence interval procedures can be extended to some problems in which the mixed linear model contains more than one set of random effects. Corresponding to each interval procedure is a significance test and one or more estimators.     

Null matrices and the analysis of species co-occurrences

Patterns in species occurrences on islands have been analyzed by several authors. At issue is the number of non-occurring pairs of species (also known as checkerboards). Previous authors have suggested that if the number of checkerboards differs from what is expected by chance, then island communities might have been structured by competition. Investigators have pursued this problem by first generating random (or null) matrices and then testing a metric derived from the collection of null matrices against the metric calculated from the actual species co-occurrence matrix. The random matrices were constrained by requiring the number of species on each island, and the number of islands on which each species occurred to be equal to their observed values. We show that results from previous studies are generally flawed. We present a fast, efficient algorithm to generate null matrices for any set of fixed row and column sums, and propose a modification of a previously proposed metric as a test statistic. We evaluated the efficacy of our construction method for null creation and our metric using incidence matrices from the avifauna of Vanuatu (formerly New Hebrides). Received: 31 March 1997 / Accepted: 8 April 1998     

A Two-Dimensional Pooling Strategy for Rare Variant Detection on Next-Generation Sequencing Platforms

We describe a method for pooling and sequencing DNA from a large number of individual samples while preserving information regarding sample identity. DNA from 576 individuals was arranged into four 12 row by 12 column matrices and then pooled by row and by column resulting in 96 total pools with 12 individuals in each pool. Pooling of DNA was carried out in a two-dimensional fashion, such that DNA from each individual is present in exactly one row pool and exactly one column pool. By considering the variants observed in the rows and columns of a matrix we are able to trace rare variants back to the specific individuals that carry them. The pooled DNA samples were enriched over a 250 kb region previously identified by GWAS to significantly predispose individuals to lung cancer. All 96 pools (12 row and 12 column pools from 4 matrices) were barcoded and sequenced on an Illumina HiSeq 2000 instrument with an average depth of coverage greater than 4,000×. Verification based on Ion PGM sequencing confirmed the presence of 91.4% of confidently classified SNVs assayed. In this way, each individual sample is sequenced in multiple pools providing more accurate variant calling than a single pool or a multiplexed approach. This provides a powerful method for rare variant detection in regions of interest at a reduced cost to the researcher.     

Analysis of protein circular dichroism spectra for secondary structure using a simple matrix multiplication

Inverse circular dichroism (CD) spectra are presented for each of the five major secondary structures of proteins: alpha-helix, antiparallel and parallel beta-sheet, beta-turn, and other (random) structures. The fraction of the each secondary structure in a protein is predicted by forming the dot product of the corresponding inverse CD spectrum, expressed as a vector, with the CD spectrum of the protein digitized in the same way. We show how this method is based on the construction of the generalized inverse from the singular value decomposition of a set of CD spectra corresponding to proteins whose secondary structures are known from X-ray crystallography. These inverse spectra compute secondary structure directly from protein CD spectra without resorting to least-squares fitting and standard matrix inversion techniques. In addition, spectra corresponding to the individual secondary structures, analogous to the CD spectra of synthetic polypeptides, are generated from the five most significant CD eigenvectors.     

A consistent metric for nestedness analysis in ecological systems: reconciling concept and measurement

Nestedness has been widely reported for both metacommunities and networks of interacting species. Even though the concept of this ecological pattern has been well-defined, there are several metrics by which it can be quantified. We noted that current metrics do not correctly quantify two major properties of nestedness: (1) whether marginal totals (i.e. fills) differ among columns and/or among rows, and (2) whether the presences (1's) in less-filled columns and rows coincide, respectively, with those found in the more-filled columns and rows. We propose a new metric directly based on these properties and compare its behavior with that of the most used metrics, using a set of model matrices ranging from highly-nested to alternative structures in which no nestedness should be detected. We also used an empirical dataset to explore possible biases generated by the metrics as well as to evaluate correlations between metrics. We found that nestedness has been quantified by metrics that inappropriately detect this pattern, even for matrices in which there is no nestedness. In addition, the most used metrics are prone to type I statistical errors while our new metric has better statistical properties and consistently rejects a nested pattern for different types of random matrices. The analysis of the empirical data showed that two nestedness metrics, matrix temperature and the discrepancy measure, tend to overestimate the degrees of nestedness in metacommunities. We emphasize and discuss some implications of these biases for the theoretical understanding of the processes shaping species interaction networks and metacommunity structure.     

Nestedness and τ-temperature in ecological networks

The τ-temperature is a measure of disorder of bipartite networks that is based on the total Manhattan distance of the adjacency matrix. Two properties of this measure are that it does not depend on permutations of lines or columns that have the same connectivity and it is completely determined by connectivities of lines and columns. The normalisation of τ is done by an uniform random matrix whose elements were previously sorted. τ shows no bias against uniform random matrices of several occupations, ρ, sizes, L, and shapes. The scaling of the total Manhattan distance of a random matrix is D_rand ∝ L³ρ while the same scaling for a full nested matrix is D_nest ∝ L³ρ^3/2. We test τ for a large set of empirical matrices to verify these scalings. The index τ correlates better with the temperature of Atmar than with the NODF index of nestedness. We conclude this work by discussing differences between nestedness indices and order/disorder indices.     

Exact Confidence Bounds for Comparing Two Regression Lines with a Control Regression Line on a Fixed Interval

The problem of finding exact simultaneous confidence bounds for comparing simple linear regression lines for two treatments with a simple linear regression line for the control over a fixed interval is considered. The assumption that errors are iid normal random is considered. It is assumed that the design matrices for the two treatments are equal and the design matrix for the control has the same number of copies of each distinct row of the design matrix for the treatments. The method is based on a pivotal quantity that can be expressed as a function of four t variables. The probability point depends on the size of an angle associated with the interval. We present probability points for various sample sizes and angles. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A radial basis function network approach for the computation of inverse continuous time variant functions

This Paper presents an efficient approach for the fast computation of inverse continuous time variant functions with the proper use of Radial Basis Function Networks (RBFNs). The approach is based on implementing RBFNs for computing inverse continuous time variant functions via an overall damped least squares solution that includes a novel null space vector for singularities prevention. The singularities avoidance null space vector is derived from developing a sufficiency condition for singularities prevention that conduces to establish some characterizing matrices and an associated performance index.     

A data-driven model of the generation of human EEG based on a spatially distributed stochastic wave equation

We discuss a model for the dynamics of the primary current density vector field within the grey matter of human brain. The model is based on a linear damped wave equation, driven by a stochastic term. By employing a realistically shaped average brain model and an estimate of the matrix which maps the primary currents distributed over grey matter to the electric potentials at the surface of the head, the model can be put into relation with recordings of the electroencephalogram (EEG). Through this step it becomes possible to employ EEG recordings for the purpose of estimating the primary current density vector field, i.e. finding a solution of the inverse problem of EEG generation. As a technique for inferring the unobserved high-dimensional primary current density field from EEG data of much lower dimension, a linear state space modelling approach is suggested, based on a generalisation of Kalman filtering, in combination with maximum-likelihood parameter estimation. The resulting algorithm for estimating dynamical solutions of the EEG inverse problem is applied to the task of localising the source of an epileptic spike from a clinical EEG data set; for comparison, we apply to the same task also a non-dynamical standard algorithm.     

The philosophical basis of character coding for the inference of phylogenetic hypotheses

The coding of observations of organisms into a data matrix for the inference of phylogenetic hypotheses has suffered from a variety of problems that have precluded development of a uniform approach to the issue. Probably the most notable consequence is that the philosophical basis for coding has been prominently ignored in lieu of emphasis placed on specific coding strategies. From an epistemic standpoint, part of the problem lies with the distinction of ‘characters’ and ‘character states’, which does not accurately convey perceptual beliefs or observation statements. The ability to perceive objects is by the nature of the properties of those objects. One's sense perceptions are of characters, not states; or more appropriately, one observes objects by way of the properties perceived of those objects. The proper distinction is therefore not one of character/state, but one of object/character, as communicated by subject–predicate relations. With observation statements referring to shared similarities among organisms distributed among two or more species, and phylogenetic hypotheses in the form of cladograms serving as at least tentative explanations for those effects, then what dictates the coding of observations is not adherence to a particular coding strategy, but the need to accurately convey the causal questions that address those observations. A phylogenetic data matrix is therefore not composed of columns as ‘characters’ and cells as ‘states’. Rather, column headings indicate the observed subjects that instantiate various characters; columns represent specifiable causal questions based on observations, and cells present the subject–predicate relations of observation statements. Since data matrices must represent one's causal questions, the inclusion of outgroup taxa is justified as components of those questions. With these criteria, the coding strategies summarized by Pleijel (1995) are critiqued. It is shown that advocacy of any one of those approaches is not possible, and that strategies that incorrectly apply the notion of ‘absence’ are especially prone to misrepresent observations.     

A novel definition of the multivariate coefficient of variation

The coefficient of variation CV (%) is widely used to measure the relative variation of a random variable to its mean or to assess and compare the performance of analytical techniques/equipments. A review is made of the existing multivariate extensions of the univariate CV where, instead of a random variable, a random vector is considered, and a novel definition is proposed. The multivariate CV obtained only requires the calculation of the mean vector, the covariance matrix and simple quadratic forms. No matrix inversion is needed which makes the new approach equally attractive in high dimensional as in very small sample size problems. As an illustration, the method is applied to electrophoresis data from external quality assessment in laboratory medicine, to phenotypic characteristics of pocket gophers and to a microarray data set.     

Complexity and stability revisited

Since Robert May's work on random community matrices it has been known that stability tends to decrease with complexity. Recently, it was shown that this is not necessarily true in competitive ecosystems. We investigated the stability of random ecosystems and found that it can largely be predicted by simple matrix statistics such as the mean and the variance of the interaction coefficients. We use this to explain why stability can increase as well as decrease with complexity in ecological communities. We argue that the variance, and to a lesser extent the mean, of the interaction coefficients go a long way in explaining patterns in the stability of ecosystems.     

Disentangling community patterns of nestedness and species co-occurrence

Two opposing patterns of meta‐community organization are nestedness and negative species co‐occurrence. Both patterns can be quantified with metrics that are applied to presence‐absence matrices and tested with null model analysis. Previous meta‐analyses have given conflicting results, with the same set of matrices apparently showing high nestedness (Wright et al. 1998) and negative species co‐occurrence (Gotelli and McCabe 2002). We clarified the relationship between nestedness and co‐occurrence by creating random matrices, altering them systematically to increase or decrease the degree of nestedness or co‐occurrence, and then testing the resulting patterns with null models. Species co‐occurrence is related to the degree of nestedness, but the sign of the relationship depends on how the test matrices were created. Low‐fill matrices created by simple, uniform sampling generate negative correlations between nestedness and co‐occurrence: negative species co‐occurrence is associated with disordered matrices. However, high‐fill matrices created by passive sampling generate the opposite pattern: negative species co‐occurrence is associated with highly nested matrices. The patterns depend on which index of species co‐occurrence is used, and they are not symmetric: systematic changes in the co‐occurrence structure of a matrix are only weakly associated with changes in the pattern of nestedness. In all analyses, the fixed‐fixed null model that preserves matrix row and column totals has lower type I and type II error probabilities than an equiprobable null model that relaxes row and column totals. The latter model is part of the popular nestedness temperature calculator, which detects nestedness too frequently in random matrices (type I statistical error). When compared to a valid null model, a matrix with negative species co‐occurrence may be either highly nested or disordered, depending on the biological processes that determine row totals (number of species occurrences) and column totals (number of species per site).     

Constraints and missing reactions in the urea cycle.

The stoichiometric relations in a series of biochemical reactions are summarized by a stoichiometric number matrix (with a column for each reaction) and a conservation matrix (with a row for each constraint). These two matrices for a series or cycle of biochemical reactions are related because the columns of the stoichiometric number matrix are in the null space of the conservation matrix, and the rows of the transpose of the conservation matrix are in the null space of the transpose of the stoichiometric number matrix. The conservation matrix for a system of biochemical reactions is of interest because it shows the nature of the constraints in addition to the conservation of atoms and groups. Constraints beyond those for the conservation of atoms and groups indicate "missing reactions" that do not occur because the enzymes involved couple reactions that could occur and still conserve atoms and groups. The interpretation of conservation matrices and stoichiometric matrices for a reaction system is complicated by the fact that they are not unique. However, their row-reduced forms are unique, as are their dimensions, which represent the number of reactants and number of independent reactions. Two matrices that look different contain the same information if they have the same row-reduced form. The urea cycle, which involves five enzyme-catalyzed reactions, and its net reaction are discussed in terms of the linear constraints produced by enzyme catalysis. A procedure to obtain a set of conservation equations that will yield the correct net reaction is described.     

The stability–complexity relationship at age 40: a random matrix perspective

Since the work of Robert May in 1972, the local asymptotic stability of large ecological systems has been a focus of theoretical ecology. Here we review May's work in the light of random matrix theory, the field of mathematics devoted to the study of large matrices whose coefficients are randomly sampled from distributions with given characteristics. We show how May's celebrated “stability criterion” can be derived using random matrix theory, and how extensions of the so-called circular law for the limiting distribution of the eigenvalues of large random matrix can further our understanding of ecological systems. Our goal is to present the more technical material in an accessible way, and to provide pointers to the primary mathematical literature on this subject. We conclude by enumerating a number of challenges, whose solution is going to greatly improve our ability to predict the stability of large ecological networks.     

The empirical Bayes approach as a tool to identify non-random species associations

A statistical challenge in community ecology is to identify segregated and aggregated pairs of species from a binary presence–absence matrix, which often contains hundreds or thousands of such potential pairs. A similar challenge is found in genomics and proteomics, where the expression of thousands of genes in microarrays must be statistically analyzed. Here we adapt the empirical Bayes method to identify statistically significant species pairs in a binary presence–absence matrix. We evaluated the performance of a simple confidence interval, a sequential Bonferroni test, and two tests based on the mean and the confidence interval of an empirical Bayes method. Observed patterns were compared to patterns generated from null model randomizations that preserved matrix row and column totals. We evaluated these four methods with random matrices and also with random matrices that had been seeded with an additional segregated or aggregated species pair. The Bayes methods and Bonferroni corrections reduced the frequency of false-positive tests (type I error) in random matrices, but did not always correctly identify the non-random pair in a seeded matrix (type II error). All of the methods were vulnerable to identifying spurious secondary associations in the seeded matrices. When applied to a set of 272 published presence–absence matrices, even the most conservative tests indicated a fourfold increase in the frequency of perfectly segregated “checkerboard” species pairs compared to the null expectation, and a greater predominance of segregated versus aggregated species pairs. The tests did not reveal a large number of significant species pairs in the Vanuatu bird matrix, but in the much smaller Galapagos bird matrix they correctly identified a concentration of segregated species pairs in the genus Geospiza. The Bayesian methods provide for increased selectivity in identifying non-random species pairs, but the analyses will be most powerful if investigators can use a priori biological criteria to identify potential sets of interacting species.     

Testing for phylogenetic signal in phenotypic traits: new matrices of phylogenetic proximities

Abouheif adapted a test for serial independence to detect a phylogenetic signal in phenotypic traits. We provide the exact analytic value of this test, revealing that it uses Moran's I statistic with a new matrix of phylogenetic proximities. We introduce then two new matrices of phylogenetic proximities highlighting their mathematical properties: matrix A which is used in Abouheif test and matrix M which is related to A and biodiversity studies. Matrix A unifies the tests developed by Abouheif, Moran and Geary. We discuss the advantages of matrices A and M over three widely used phylogenetic proximity matrices through simulations evaluating power and type-I error of tests for phylogenetic autocorrelation. We conclude that A enhances the power of Moran's test and is useful for unresolved trees. Data sets and routines are freely available in an online package and explained in an online supplementary file.