Normalization of microarray data using a spatial mixed model analysis which includes splines

MOTIVATION: Microarray experiments with thousands of genes on a slide and multiple slides used in any experimental set represent a large body of data with many sources of variation. The identification of such sources of variation within microarray experimental sets is critical for correct deciphering of desired gene expression differences. RESULTS: We describe new methods for the normalization using spatial mixed models which include splines and analysis of two-colour spotted microarrays for within slide variation and for a series of slides. The model typically explains 45-85% of the variation on a slide with only approximately 1% of the total degrees of freedom. The results from our methods compare favourably with those from intensity dependent normalization loess methods where we accounted for twice as much uncontrolled and unwanted variation on the slides. We have also developed an index for each EST that combines the various measures of the differential response into a single value that researchers can use to rapidly assess the genes of interest.     

Evaluation of experimental designs and spatial analyses in wheat breeding trials

Thirty-three wheat breeding trials were conducted from 1994 to 1996 in the Northern Grains Region (QLD and Northern NSW) of Australia to evaluate the influence of experimental designs and spatial analyses on the estimation of genotype effects for yield and their impact on selection decisions. The relative efficiency of the alternative designs and analyses was best measured by the average standard error of difference between line means. Both more effective designs and spatial analyses significantly improved the efficiency relative to the randomised complete block model, with the preferred model (which combined the design information and spatial trends) giving an average relative efficiency of 138% over all 33 trials. When the Czekanowski similarity coefficient was used, none of the studied models were in full agreement with the randomised complete block model in the selection of the top lines. The agreement was influenced by selection proportions. Hence, the use of these methodologies can impact on the selection decisions in plant breeding. Received: 17 December 1998 / Accepted: 29 July 1999     

Analyzing spatial structure of communities using the two-dimensional poisson lognormal species abundance model

The joint spatial and temporal fluctuations in the community structure of tropical butterflies are analyzed by fitting the bivariate Poisson lognormal distribution to a large number of observations in space and time. By applying multivariate dependent diffusions for describing the fluctuations in the abundances, the environmental variance is estimated to be very large and so is the strength of local density regulation. The variance in the lognormal species abundance distribution is partitioned into components expressing the heterogeneity between the species, independent noise components for the different species, a demographic stochastic component, and a component due to overdispersion in the sampling. In disagreement with the neutral theory, the estimates show that the heterogeneity component is the dominating one, representing 81% of the total variance in the lognormal model. Different spatial components of diversity, the alpha, beta, and gamma diversity, are also estimated. The spatial scale of the autocorrelation function for the community is of order 1 km, while sampling of a quadrat would need to be 10 km on a side to yield the total diversity for the community.     

Classifying genotypic data from plant breeding trials: a preliminary investigation using repeated checks

Summary Several subjective choices must be made when classifying genotypes based on data from plant breeding trials. One choice involves the method used to weight the contribution each environment makes to the classification. A second involves the use of either genotype-means for each environment or genotypevalues for each block, i.e., considering each block to be a different environment. Another involves whether environments (or blocks) in which genotypes are nonsignificantly different should be included or excluded from such classifications. An alternative to the use of raw or standardized data, is proposed in which each environment is weighted by a discrimination index (DI) that is based on the concept of repeatability. In this study the effect of three weighting methods (raw, standardized and DI), the choice of using environments or blocks, and the choice of including or excluding environments or blocks in which genotypic effects were not significant, were considered in factorial combination to give 12 options. A data set comprised of five check cultivars each repeated six times in each of three blocks at six environments was used. The effect of these options on the ability of a hierarchical clustering technique to correctly classify the repeats into five groups, each consisting of all the six repeats of a particular check cultivar, was investigated. It was found that the DI weighting method generally led to better recovery of the known structure. Using block data rather than environmental data also improved structure recovery for each of the three weighting methods. The exclusive use of environments in which genotypic effects were significant decreased structure recovery while the contrary generally occurred for blocks. The best structure recovery was obtained from the DI weighting applied to blocks (whether genotypes were significant or not).     

Developments in the malting and brewing trials with sorghum

A comparative study of sorghum and barley grains reveals structural and physiological differences in their aleurone, embryo and starchy endosperm cells. These differences are responsible for the observed differences in their malting characteristics. Reports on brewing trials with sorghum favour its use as an adjunct to barley malt. Nevertheless, a recent successful resolution of the incompatibility of the gelatinization and saccharification temperatures of sorghum starch through the adoption of a new mashing technique has greatly improved its extract yield. A similar report on the discovery of a Nigerian-grown sorghum variety with improved β-amylase activity as well as high diastatic power will obviously ensure better fermentable extracts. Further research on the development and trial of new sorghum varieties is strongly recommended. The purpose of this review, however, is to highlight research efforts aimed at alleviating the problems of sorghum as a brewing material.     

Modelling spatial patterns in harbour porpoise satellite telemetry data using maximum entropy

The distribution of harbour porpoises in EU waters is poorly understood, and modelled predictions of their distributions could inform the strategic spatial planning of future exploitation of the marine environment to avoid potential conflicts. We analysed satellite telemetry data from 39 harbour porpoises Phocoena phocoena in inner Danish waters using a modelling tool rooted in maximum entropy: Maxent. Maxent does not require absence data and has been shown to be effective for data characterised by small sample size, sampling bias and locational errors. For each season we used an iterative bootstrapping procedure to randomly select among the most precise records from each of the 39 tagged individuals, and ran Maxent on pooled records based on explanatory environmental variables hypothesised to serve as good proxies for harbour porpoise prey abundance. Among our environmental variables, distance to coast and bottom salinity had the most explanatory power, and their response shapes were relatively consistent across most seasons. The predictive power of the models (assessed by ROC‐AUC) ranged from 0.70 to 0.86 within seasons. The southern Kattegat, the Belt Seas, most western part of the Baltic Sea and the Sound were predicted to have relatively high probabilities of occurrence across seasons. In contrast, the central part of Kattegat and the Baltic Sea south and east of Limhamn and Darss Ridge consistently showed low probabilities of occurrence. Areas with the lowest probabilities of occurrence were generally characterised by high predictive uncertainty. Our methods have implications for the analyses of satellite tagged animals in terrestrial and marine environments. By coupling a bootstrapping procedure with Maxent we circumvented some of the statistical challenges presented by satellite telemetry data to generate spatial predictions within the inner Danish waters.     

Diurnal growth trends, water potential, and osmotic adjustment of maize and sorghum leaves in the field

The daily cycle of leaf elongation rate, water potential, and solute potential of maize and sorghum, as well as temperature, were monitored in the field. Major climatic features were high radiation and a minimum air temperature of about 12 C. Leaf elongation of both crops was slowest at night, presumably because of low temperature. Peak elongation rates were in daytime when leaf water potential (Ψ) was low. Solute potential also decreased during daylight, thus permitting the maintenance of appreciable turgor pressure, a critical parameter for cell expansion.     

Modelling relative recency discrimination tasks using a stochastic working memory model

Relative recency discrimination (RD) task is typically used to assess the temporal organization function of the prefrontal cortex (PFC). Subjects look at a series of cards (with words or drawings on them) and on seeing a test card determine which of the two items was seen more recently. Results show that patients with damage to the prefrontal cortex are severely impaired on this task. We propose a memory trace-priming mechanism, based on automatic time-marking process hypothesis (Schacter, D.L., 1987. Memory, amnesia, and frontal lobe dysfunction: a critique and interpretation. Psychobiology 15, 21-36), to offer a computational account of the results. In this model, successive words seen by subjects leave decaying memory traces in PFC, which subsequently prime the representations in higher sensory areas such as inferior temporal Cortex (IT) during discrimination judgements. The paper focuses on the evaluation of a probabilistic pre-frontal trace mechanism using a pool of clusters of neurons with self-sustained firing that ends at a random time. The results show that the probabilistic behavior of subjects can be accounted for by the stochasticity of the trace model. A good fit to experimental data is obtained with a PFC memory persistence probability with a decay time constant of tau = approximately 30 s. The model allows for a distributed representation in IT and PFC, but the best fit suggests a sparse representation. It is concluded that further data are needed on representations in IT and PFC, on the connectivity between these two areas, and on the statistical and dynamic properties of memory neurons in PFC.     

Thinning strategies for Eucalyptus dunnii population: balance between breeding and conservation using spatial variation and competition model

Tree Genetics & Genomes - Malus baccata (L.) Borkh. and Malus toringo (Siebold) Siebold ex de Vriese of the genus Malus Mill. (Rosaceae) are wild crabapples occurring in temperate East Asia....     

Analysing yield stability of maize genotypes using a spatial model

Summary The yield stability of some CIMMYT tropical maize (Zea mays L.) populations of early and intermediate maturity, measured by the performance of varieties derived from them, was determined. Results of the stability analyses, conducted over international environments from 1980 to 1983, indicated that selections from Mezcla Amarilla exhibited good stability in high yielding sites. Varieties derived from Antigua-Republica Dominicana tended to be more stable in unfavourable environments, whereas selections from Blanco Cristalino-1 and Blanco Dentado-2 were stable in both low and high production sites. The combination of enviromental factors in the specific test locations, namely Poza Rica (Mexico), Tocumen (Panama), Islamabad (Pakistan), and Ferkessedougou (Ivory Coast), allowed selection of varieties that are very stable in other regions of the world. The varieties formed on the basis of multilocational data do not seem to be any more stable than those formed using data from a single location.     

Study on multicellular systems using a phase field model

A model of multicellular systems with several types of cells is developed from the phase field model. The model is presented as a set of partial differential equations of the field variables, each of which expresses the shape of one cell. The dynamics of each cell is based on the criteria for minimizing the surface area and retaining a certain volume. The effects of cell adhesion and excluded volume are also taken into account. The proposed model can be used to find the position of the membrane and/or the cortex of each cell without the need to adopt extra variables. This model is suitable for numerical simulations of a system having a large number of cells. The two-dimensional results of cell division, cell adhesion, rearrangement of a cell cluster, chemotaxis, and cell sorting as well as the three-dimensional results of cell clusters on the substrate are presented.     

An assessor-specific Bayesian multi-threshold mixed model for analyzing ordered categorical traits in tree breeding

Many traits of biological interest in tree breeding are assessed using more than two ordered discrete categories. These scores have a more or less arbitrary and subjective assignment by the assessors, which could lead to a strong departure from the Gaussian distribution. Different assessors may also use different regions of the available scale. This study describes the use of the multi-threshold mixed model proposed by Varona et al. (J Anim Sci 87:1210–1217, 2009), which allows different thresholds for different assessors on an underlying Gaussian distribution. This method was applied to a six-point score for stem quality in an open-pollinated progeny trial of Prosopis alba Griseb. Four mixed models were used: (1) a linear mixed model with observed score (LMM); (2) a linear mixed model with transformed “normal scores” (LMM_NS); (3) a threshold mixed model (TMM); and (4) an assessor-specific multi-threshold mixed model (MTMM). Dispersion parameters were estimated using Bayesian techniques via the Gibbs sampling with a data augmentation step. The proposed MTMM produced higher posterior mean heritabilities (0.096) than the commonly used LMM (0.077). Posterior mean heritabilities from LMM_NS (0.094) and TMM (0.097) were comparable to those obtained using MTMM; however, MTMM yielded slightly more precise estimates than TMM. Although correlations of the estimated breeding values were high between different models (from 0.88 to 0.99), the heterogeneity in the estimated posterior means of the thresholds between the three assessors caused notable changes in the top 10 families between TMM and MTMM. The proposed model is helpful in fitting subjective ordered categorical traits assessed by different assessors in tree breeding.     

Statistical modelling of growth using a mixed model with orthogonal polynomials

In statistical modelling, the effects of single-nucleotide polymorphisms (SNPs) are often regarded as time-independent. However, for traits recorded repeatedly, it is very interesting to investigate the behaviour of gene effects over time. In the analysis, simulated data from the 13th QTL-MAS Workshop (Wageningen, The Netherlands, April 2009) was used and the major goal was the modelling of genetic effects as time-dependent. For this purpose, a mixed model which describes each effect using the third-order Legendre orthogonal polynomials, in order to account for the correlation between consecutive measurements, is fitted. In this model, SNPs are modelled as fixed, while the environment is modelled as random effects. The maximum likelihood estimates of model parameters are obtained by the expectation–maximisation (EM) algorithm and the significance of the additive SNP effects is based on the likelihood ratio test, with p-values corrected for multiple testing. For each significant SNP, the percentage of the total variance contributed by this SNP is calculated. Moreover, by using a model which simultaneously incorporates effects of all of the SNPs, the prediction of future yields is conducted. As a result, 179 from the total of 453 SNPs covering 16 out of 18 true quantitative trait loci (QTL) were selected. The correlation between predicted and true breeding values was 0.73 for the data set with all SNPs and 0.84 for the data set with selected SNPs. In conclusion, we showed that a longitudinal approach allows for estimating changes of the variance contributed by each SNP over time and demonstrated that, for prediction, the pre-selection of SNPs plays an important role.     

Identification of spatial genetic boundaries using a multifractal model in human population genetics

There are two purposes in displaying spatial genetic structure. One is that a visual representation of the variation of the genetic variable should be provided in the contour map. The other is that spatial genetic structure should be reflected by the patterns or the gradients with genetic boundaries in the map. Nevertheless, most conventional interpolation methods, such as Cavalli-Sforza's method in genography, inverse distance-weighted methods, and the Kriging technique, focus only on the first primary purpose because of their arbitrary thresholds marked on the maps. In this paper we present an application of the contour area multifractal model (CAMM) to human population genetics. The method enables the analysis of the geographic distribution of a genetic marker and provides an insight into the spatial and geometric properties of obtained patterns. Furthermore, the CAMM may overcome some of the limitations of other interpolation techniques because no arbitrary thresholds are necessary in the computation of genetic boundaries. The CAMM is built by establishing power law relationships between the area A (> or =rho) in the contour map and the value p itself after plotting these values on a log-log graph. A series of straight-line segments can be fitted to the points on the log-log graph, each representing a power law relationship between the area A (> or =rho) and the cutoff genetic variable value for rho in a particular range. These straight-line segments can yield a group of cutoff values, which can be identified as the genetic boundaries that can classify the map of genetic variable into discrete genetic zones. These genetic zones usually correspond to spatial genetic structure on the landscape. To provide a better understanding of the interest in the CAMM approach, we analyze the spatial genetic structures of three loci (ABO, HLA-A, and TPOX) in China using the CAMM. Each synthetic principal component (SPC) contour map of the three loci is created by using both Han and minority groups data together. These contour maps all present an obvious geographic diversity, which gradually increases from north to south, and show that the genetic differences among populations in different districts of the same nationality are greater than those among different nationalities of the same district. It is surprising to find that both the value of p and the fractal dimension alpha have a clear north to south gradient for each locus, and the same clear boundary between southern and northern Asians in each contour map is still seen in the zone of the Yangtze River, although substantial population migrations have occurred because of war or famine in the last 2,000 or 3,000 years. A clear genetic boundary between Europeans and Asians in each contour map is still seen in northwestern China with a small value of alpha, although the genetic gradient caused by gene flow between Europeans and Asians has tended to show expansion from northwestern China. From the three contour maps another interesting result can be found: The values of alpha north of the Yangtze River are generally less than those south of the Yangtze River. This indicates that the genetic differences among the populations north of the Yangtze River are generally smaller than those in populations south of the Yangtze River.