Statistical methods for analysis of communities' species structure (with riverine macrozoobenthos as an example)

Species turnover or coherence in species co-occurrence as well as boundary clumping and nestedness in structural composition of ecological communities reflect the extent of determinancy in their organization (Leibold, Mikkelson, 2002). These phenomena may be a consequence of either interactions between species or heterogeneity in spatial distribution of populations density. We have examined statistical patterns of species structure variability using benthic communities of riverine ecosystems as an example. The ecosystems studied are characterized by strongly pronounced linear gradient of landscape features and environmental factors. The results of a long-term hydrobiological survey being conducted at 22 observational stations on the Sok River along with its tributary, the Baytugan River (Lower Volga basin, total watercourse length is 375 km) are involved into the analysis. A spreadsheet for statistical processing of the data included 375 macrozoobenthic taxa contained in 147 samples. An assessment of species structure nestedness in benthic communities at separate sites and along the watercourse as a whole has been carried out using various metrics such as nestedness "temperature" (Patterson, Atmar, 2000), discrepancy measure (Brualdi, Sanderson, 1999), nestedness based on overlap and decreasing fill (NODE--Almeida-Neto et al., 2008) and others. Statistical significance of ecosystems structural determinancy has been tested by means of randomization procedures and standard null models (Gotelli, 2000). The conclusions seem to be ambiguous and dependent on a level and scale of an ecosystem resolution into separate blocks, also on configuration and completion of initial bio-geographical tables. A searching for reliable and representative criteria of nestedness, invariant to various non-ecological modifications of the matrices but sensitive to estimation of analyzed ecological processes and suitable for comparisons of communities, is clearly needed. A quantitative estimation of species turnover and coherence in species cooccurrence has been performed using different indices of unique combinations and checkerboard score (Stone, Roberts, 1992) as well as Schluter's variance test. By means of empirical Bayesian approach (Gotelli, Ulrich, 2010) records of species pairwise combinations are formed where the frequency of species co-occurrence cannot be interpreted as a random value. Positive and negative relationships between taxa in macrozoobenthic communities, which are found out to be statistically significant, in most cases can be explained as being not the consequence of competition for resources but of spatial heterogeneity of biotopical conditions along the whole length of the watercourse.     

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     

The Influence of Matrix Size on Statistical Properties of Co-Occurrence and Limiting Similarity Null Models

Null models exploring species co-occurrence and trait-based limiting similarity are increasingly used to explore the influence of competition on community assembly; however, assessments of common models have not thoroughly explored the influence of variation in matrix size on error rates, in spite of the fact that studies have explored community matrices that vary considerably in size. To determine how smaller matrices, which are of greatest concern, perform statistically, we generated biologically realistic presence-absence matrices ranging in size from 3–50 species and sites, as well as associated trait matrices. We examined co-occurrence tests using the C-Score statistic and independent swap algorithm. For trait-based limiting similarity null models, we used the mean nearest neighbour trait distance (NN) and the standard deviation of nearest neighbour distances (SDNN) as test statistics, and considered two common randomization algorithms: abundance independent trait shuffling (AITS), and abundance weighted trait shuffling (AWTS). Matrices as small as three × three resulted in acceptable type I error rates (p < 0.05) for both the co-occurrence and trait-based limiting similarity null models when exclusive p-values were used. The commonly used inclusive p-value (≤ or ≥, as opposed to exclusive p-values; < or >) was associated with increased type I error rates, particularly for matrices with fewer than eight species. Type I error rates increased for limiting similarity tests using the AWTS randomization scheme when community matrices contained more than 35 sites; a similar randomization used in null models of phylogenetic dispersion has previously been viewed as robust. Notwithstanding other potential deficiencies related to the use of small matrices to represent communities, the application of both classes of null model should be restricted to matrices with 10 or more species to avoid the possibility of type II errors. Additionally, researchers should restrict the use of the AWTS randomization to matrices with fewer than 35 sites to avoid type I errors when testing for trait-based limiting similarity. The AITS randomization scheme performed better in terms of type I error rates, and therefore may be more appropriate when considering systems for which traits are not clustered by abundance.     

Null models for species co-occurrence patterns: assessing bias and minimum iteration number for the sequential swap

The analysis of co-occurrence matrices is a common practice to evaluate community structure. The observed data are compared with a "null model", a randomised co-occurrence matrix derived from the observation by using a statistic, e.g. the C-score, sensitive to the pattern investigated. The most frequently used algorithm, "sequential swap", has been criticised for not sampling with equal frequencies thereby calling into question the results of earlier analysis. The bias of the "sequential swap" algorithm when used with the C-score was assessed by analysing 291 published presence-absence matrices. In 152 cases, the true p-value differed by >5% from the p-value generated by an uncorrected "sequential swap". However, the absolute value of the difference was rather small. Out of the 291 matrices, there were only 5 cases in which an incorrect statistical decision would have been reached by using the uncorrected p-value (3 at the p<0.05 and 2 at the p<0.01 level), and in all 5 of these cases, the true p-value was close to the significance level. Our results confirm analytical studies of Miklos and Podani which show that the uncorrected swap gives slightly conservative results in tests for competitive segregation. However, the bias is very small and should not distort the ecological interpretation. We also estimated the number of iterations needed for the "sequential swap" to generate accurate p-values. While most authors do not exceed a number of 10⁴ iterations, the suggested minimum number of swaps for 29 out of the 291 tested matrices is greater than 10⁴. We recommend to use 30 000 "sequential swaps" if the required sample size is not assessed otherwise.     

Toward ecologically explicit null models of nestedness

A community is "nested" when species assemblages in less rich sites form nonrandom subsets of those at richer sites. Conventional null models used to test for statistically nonrandom nestedness are under- or over-restrictive because they do not sufficiently isolate ecological processes of interest, which hinders ecological inference. We propose a class of null models that are ecologically explicit and interpretable. Expected values of species richness and incidence, rather than observed values, are used to create random presence-absence matrices for hypothesis testing. In our examples, based on six datasets, expected values were derived either by using an individually based random placement model or by fitting empirical models to richness data as a function of environmental covariates. We describe an algorithm for constructing unbiased null matrices, which permitted valid testing of our null models. Our approach avoids the problem of building too much structure into the null model, and enabled us to explicitly test whether observed communities were more nested than would be expected for a system structured solely by species-abundance and species-area or similar relationships. We argue that this test or similar tests are better determinants of whether a system is truly nested; a nested system should contain unique pattern not already predicted by more fundamental ecological principles such as species-area relationships. Most species assemblages we studied were not nested under these null models. Our results suggest that nestedness, beyond that which is explained by passive sampling processes, may not be as widespread as currently believed. These findings may help to improve the utility of nestedness as an ecological concept and conservation tool.     

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).     

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.     

Island-sharing by archipelago species

Summary Diamond (1975) formulated assembly rules for avian species on islands in an archipelago, which made a successful colonisation depend essentially on which other species were present. Critically examining these rules, Connor and Simberloff (1979) maintained that, in the Vanuatu (New Hebrides) archipelago, the field data on species distribution was quite compatible with a null hypothesis, in which species colonise at random with no species interaction. Their work was in turn criticised (Diamond and Gilpin (1982), Gilpin and Diamond (1982)) and a vigorous controversy has ensued.Here we contribute a method in which a simple but hitherto neglected statistic is used as a probe: the number of islands shared by a pair of species, with its first and second moments. The matrix of these sharing values is given as a simple product of the incidence matrix, and its properties are examined — first, for the field data, and then in the random ensemble used by Connor and Simberloff (1979). It is shown that their constraints hold constant the mean number shared, so that any fall in the number that one pair of species share, due to their excluding each other, must imply a rise in the number shared by some other species pair-i.e., an aggregation.Turning to the second moment of the numbers shared, it is shown that its value in the Vanuatu field data exceeds the largest value to be found in a sample of 1000 matrices, these latter being constructed so that they obey the Connor and Simberloff constraints but are otherwise random. This indicates that exclusion and/or aggregation effects are present in the actual distribution of species, which are not catered for by the null hypothesis.The observed distribution thus emerges as much more exceptional than found by Connor and Simberloff (1979), and even by Diamond and Gilpin (1982), when examining the same ensemble. The reason for this disagreement are sought, and some cautions are offered, supported by numerical evidence, concerning the use of the chi-square test when the data points involved are mutually dependent.     

A new algorithm to calculate the nestedness temperature of presence–absence matrices

Aim The nestedness temperature of presence–absence matrices is currently calculated with the nestedness temperature calculator (NTC). In the algorithm implemented by the NTC: (1) the line of perfect order is not uniquely defined, (2) rows and columns are reordered in such a way that the packed matrix is not the one with the lowest temperature, and (3) the null model used to determine the probabilities of finding random matrices with the same or lower temperature is not adequate for most applications. We develop a new algorithm, BINMATNEST (binary matrix nestedness temperature calculator), that overcomes these difficulties. Methods BINMATNEST implements a line of perfect order that is uniquely defined, uses genetic algorithms to determine the reordering of rows and columns that leads to minimum matrix temperature, and provides three alternative null models to calculate the statistical significance of matrix temperature. Results The NTC performs poorly when the input matrix has checkerboard patterns. The more efficient packing of BINMATNEST translates into matrix temperatures that are lower than those computed with the NTC. The null model implemented in the NTC is associated with a large frequency of type I error, while the other null models implemented in BINMATNEST (null models 2 and 3) are conservative. Overall, null model 3 provides the best performance. The nestedness temperature of a matrix is affected by its size and fill, but the probability that such a temperature is obtained by chance is not. BINMATNEST reorders the input matrix in such a way that, if fragment size/isolation plays a role in determining community structure, there will be a significant rank correlation between the size/isolation of the fragments and the way that they are ordered in the packed matrix. Main conclusions The nestedness temperature of presence–absence matrices should not be calculated with the NTC. The algorithm implemented by BINMATNEST is more robust, allowing for across‐study comparisons of the extent to which the nestedness of communities departs from randomness. The sequence in which BINMATNEST reorders habitat fragments provides information about the causal role of immigration and extinction in shaping the community under study.     

An Intriguing Controversy over Protein Structural Class Prediction

A recent report by Bahar et al. [(1997), Proteins 29, 172–185] indicates that the coupling effects among different amino acid components as originally formulated by K. C. Chou [(1995), Proteins 21, 319–344] are important for improving the prediction of protein structural classes. These authors have further proposed a compact lattice model to illuminate the physical insight contained in the component-coupled algorithm. However, a completely opposite result was concluded by Eisenhaber et al. [(1996), Proteins 25, 169–179], using a different dataset constructed according to their definition. To address such an intriguing controversy, tests were conducted by various approaches for the datasets from an objective database, the SCOP database [Murzin et al. (1995), J. Mol. Biol. 247, 536–540]. The results obtained by both self-consistency and jackknife tests indicate that the overall rates of correct prediction by the algorithm incorporating the coupling effect among different amino acid components are significantly higher than those by the algorithms without counting such an effect. This is fully consistent with the physical reality that the folding of a protein is the result of a collective interaction among its constituent amino acid residues, and hence the coupling effects of different amino acid components must be incorporated in order to improve the prediction quality. It was found by a revisiting the calculation procedures by Eisenhaber et al. that there was a conceptual mistake in constructing the structural class datasets and a systematic mistake in applying the component-coupled algorithm. These findings are informative for understanding and utilizing the component-coupled algorithm to study the structural classes of proteins.     

The checkerboard score and species distributions

Summary There has been an ongoing controversy over how to decide whether the distribution of species is random — i.e., whether it is not greatly different from what it would be if species did not interact. We recently showed (Roberts and Stone (1990)) that in the case of the Vanuatu (formerly New Hebrides) avifauna, the number of islands shared by species pairs was incompatible with a random null hypothesis. However, it was difficult to determine the causes or direction of the community's exceptionality. In this paper, the latter problem is examined further. We use Diamond's (1975) notion of checkerboard distributions (originally developed as an indicator of competition) and construct a C-score statistic which quantifies checkerboardedness. This statistic is based on the way two species might colonise a pair of islands; whenever each species colonises a different island this adds 1 to the C-score. Following Connor and Simberloff (1979) we generate a control group of random colonisation patterns (matrices), and use the C-score to determine their checkerboard characteristics. As an alternative mode of enquiry, we make slight alterations to the observed data, repeating this process many times so as to obtain another control group. In both cases, when we compare the observed data for the Vanuatu avifauna and the Antillean bat communities with that given by their respective control group, we find that these communities have significantly large checkerboard distributions, making implausible the hypothesis that their species distributions are a product of random colonisation.     

Modified nonparametric approaches to detecting differentially expressed genes in replicated microarray experiments

MOTIVATION: An important goal in analyzing microarray data is to determine which genes are differentially expressed across two kinds of tissue samples or samples obtained under two experimental conditions. Various parametric tests, such as the two-sample t-test, have been used, but their possibly too strong parametric assumptions or large sample justifications may not hold in practice. As alternatives, a class of three nonparametric statistical methods, including the empirical Bayes method of Efron et al. (2001), the significance analysis of microarray (SAM) method of Tusher et al. (2001) and the mixture model method (MMM) of Pan et al. (2001), have been proposed. All the three methods depend on constructing a test statistic and a so-called null statistic such that the null statistic's distribution can be used to approximate the null distribution of the test statistic. However, relatively little effort has been directed toward assessment of the performance or the underlying assumptions of the methods in constructing such test and null statistics. RESULTS: We point out a problem of a current method to construct the test and null statistics, which may lead to largely inflated Type I errors (i.e. false positives). We also propose two modifications that overcome the problem. In the context of MMM, the improved performance of the modified methods is demonstrated using simulated data. In addition, our numerical results also provide evidence to support the utility and effectiveness of MMM.     

Accurate critical constants for the one-sided approximate likelihood ratio test of a normal mean vector when the covariance matrix is estimated

Tang, Gnecco, and Geller (1989, Biometrika 76, 577-583) proposed an approximate likelihood ratio (ALR) test of the null hypothesis that a normal mean vector equals a null vector against the alternative that all of its components are nonnegative with at least one strictly positive. This test is useful for comparing a treatment group with a control group on multiple endpoints, and the data from the two groups are assumed to follow multivariate normal distributions with different mean vectors and a common covariance matrix (the homoscedastic case). Tang et al. derived the test statistic and its null distribution assuming a known covariance matrix. In practice, when the covariance matrix is estimated, the critical constants tabulated by Tang et al. result in a highly liberal test. To deal with this problem, we derive an accurate small-sample approximation to the null distribution of the ALR test statistic by using the moment matching method. The proposed approximation is then extended to the heteroscedastic case. The accuracy of both the approximations is verified by simulations. A real data example is given to illustrate the use of the approximations.     

Recognition of general patterns using neural networks

The Hopfield model of neural network stores memory in its symmetric synaptic connections and can only learn to recognize sets of nearly orthogonal patterns. A new algorithm is put forth to permit the recognition of general (non-orthogonal) patterns. The algorithm specifies the construction of the new network's memory matrix T _ij, which is, in general, asymmetrical and contains the Hopfield neural network (Hopfield 1982) as a special case. We find further that in addition to this new algorithm for general pattern recognition, there exists in fact a large class of T _ij memory matrices which permit the recognition of non-orthogonal patterns. The general form of this class of T _ij memory matrix is presented, and the projection matrix neural network (Personnaz et al. 1985) is found as a special case of this general form. This general form of memory matrix extends the library of memory matrices which allow a neural network to recognize non-orthogonal patterns. A neural network which followed this general form of memory matrix was modeled on a computer and successfully recognized a set of non-orthogonal patterns. The new network also showed a tolerance for altered and incomplete data. Through this new method, general patterns may be taught to the neural network.     

An evaluation of randomization models for nested species subsets analysis

Randomization models, often termed “null” models, have been widely used since the 1970s in studies of species community and biogeographic patterns. More recently they have been used to test for nested species subset patterns (or nestedness) among assemblages of species occupying spatially subdivided habitats, such as island archipelagoes and terrestrial habitat patches. Nestedness occurs when the species occupying small or species-poor sites have a strong tendency to form proper subsets of richer species assemblages. In this paper, we examine the ability of several published simulation models to detect, in an unbiased way, nested subset patterns from a simple matrix of site-by-species presence-absence data. Each approach attempts to build in biological realism by following the assumption that the ecological processes that generated the patterns observed in nature would, if they could be repeated many times over using the same species and landscape configuration, produce islands with the same number of species and species present on the same number of islands as observed. In mathematical terms, the mean marginal totals (column and row sums) of many simulated matrices would match those of the observed matrix. Results of model simulations suggest that the true probability of a species occupying any given site cannot be estimated unambiguously. Nearly all of the models tested were shown to bias simulation matrices toward low levels of nestedness, increasing the probability of a Type I statistical error. Further, desired marginal totals could be obtained only through ad-hoc manipulation of the calculated probabilities. Paradoxically, when such results are achieved, the model is shown to have little statistical power to detect nestedness. This is because nestedness is determined largely by the marginal totals of the matrix themselves, as suggested earlier by Wright and Reeves. We conclude that at the present time, the best null model for nested subset patterns may be one based on equal probabilities of occurrence for all species. Examples of such models are readily available in the literature. Received: 3 February 1997 / Accepted: 21 September 1997     

Functional connectivity metrics during stroke recovery

We explore functional connectivity in nine subjects measured with 1.5T fMRI-BOLD in a longitudinal study of recovery from unilateral stroke affecting the motor area (Small et al., 2002). We found that several measures of complexity of covariance matrices show strong correlations with behavioral measures of recovery. In Schmah et al. (2010), we applied Linear and Quadratic Discriminants (LD and QD) computed on a principal components (PC) subspace to classify the fMRI volumes into "early" and "late" sessions. We demonstrated excellent classification accuracy with QD but not LD, indicating that potentially important differences in functional connectivity exist between the early and late sessions. Motivated by Mclntosh et al. (2008), who showed that EEG brain-signal variability and behavioral performance both increased with age during development, we investigated complexity of the covariance matrix for this longitudinal stroke recovery data set. We used three complexity measures: the sphericity index described by Abdi (2010); "unsupervised dimensionality", which is the number of PCs that minimizes unsupervised generalization error of a covariance matrix (Hansen et al., 1999); and "QD dimensionality", which is the number of PCs that minimizes the classification accuracy of QD. Although these approaches measure different kinds of complexity, all showed strong correlations with one or more behavioral tests: nine-hole peg test, hand grip test and pinch test. We could not demonstrate that either sphericity or unsupervised dimensionality were significantly different for the "early" and "late" sessions using a paired Wilcoxon test. However, the amount of relative behavioral improvement was correlated with sphericity of the overall covariance matrix (pooled across all sessions), as well as with the divergence of the eigenspectra between the "early" and "late" covariance matrices. Complexity measures that use the number of PCs (which optimize QD classification or unsupervised generalization) were correlated with the behavioral performance of the final session, but not with the relative improvement. These are suggestive, but limited, results given the sample size, restricted behavioral measurements and older 1.5T BOLD data sets. Nevertheless, they indicate one potentially fruitful direction for future data-driven fMRI studies of stroke recovery in larger, better-characterized longitudinal stroke data sets recorded at higher field strength. Finally, we produced sensitivity maps (Kjems et al., 2002) corresponding to both linear and quadratic discriminants for the "early" vs. "late" classification. These maps measure the influence of each voxel on the class assignments for a given classifier. Differences between the scaled sensitivity maps for the linear and quadratic discriminants indicate brain regions involved in changes in functional connectivity. These regions are highly variable across subjects, but include the cerebellum and the motor area contralateral to the lesion.     

Interaction of MAR-sequences with nuclear matrix proteins.

The recent discovery of DNA sequences responsible for the specific attachment of chromosomal DNA to the nuclear skeleton (MARs/SARs) was an important step towards our understanding of the functional and structural organization of eukaryotic chromatin [Mirkovitch et al.: Cell 44:273-282, 1984; Cockerill and Garrard: Cell 44:273-282, 1986]. A most important question, however, remains the nature of the matrix proteins involved in the specific binding of the MARs. It has been shown that topoisomerase II and histone H1 were capable of a specific interaction with SARs by the formation of precipitable complexes [Adachi et al.: EMBO J8:3997-4006, 1989; Izaurralde et al.: J Mol Biol 210:573-585, 1989]. Here, applying a different approach, we were able to "visualize" some of the skeletal proteins recognizing and specifically binding MAR-sequences. It is shown that the major matrix proteins are practically the same in both salt- and LIS-extracted matrices. However, the relative MAR-binding activity of the individual protein components may be different, depending on the method of matrix preparation. The immunological approach applied here allowed us to identify some of the individual MAR-binding matrix proteins. Histone H1 and nuclear actin are shown to be not only important components of the matrix, but to be involved in a highly efficient interaction with MAR-sequences as well. Evidence is presented that proteins recognized by the anti-HMG antibodies also participate in MAR-interactions.     

Changes in distribution of the long form of type XII collagen during chicken corneal development.

The expression and distribution of the long form of Type XII collagen were investigated histochemically during chicken corneal development using a monoclonal antibody (P3D11) raised against the N-terminal domain of chicken Type XII collagen. Specificity of the antibody was confirmed by immunoprecipitation before and after bacterial collagenase digestion. Immunofluorescent microscopic studies showed that during chicken cornea formation, the long form of Type XII collagen is initially detected on Day 3 embryo (stage 19) in the sub-epithelial matrix of the corneal periphery and in the matrix around the optic cup. On Day 5 embryo (stage 27) the long form was expressed in the primary stroma. Thereafter, as the secondary stroma was formed, the long form localized in the sub-epithelial and sub-endothelial matrices and in the anterior region of the limbus (corneoscleral junction) before the formation of Descemet's and Bowman's membranes. After hatching, the immunoreactivity decreased predominantly in the sub-epithelial and sub-endothelial matrices but remained at the anterior region of the limbus. Immunoelectron microscopic examination demonstrated that the long form localizes in the Descemet's and Bowman's membranes and along the collagen fibrils in the stroma with a periodic repeat. Based on the distribution of the long form of Type XII collagen in the sub-epithelial and sub-endothelial matrices and limbus, it was suggested that the long form of Type XII collagen is involved in formation of the Descemet's and Bowman's membranes and in stabilization of the limbus.     

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.