Extending the loop design for two-channel microarray experiments

The loop design of Kerr and Churchill is a clever application of incomplete blocks of size 2 to two-channel microarray experiments. In this paper, we extend the loop design to include more replicates, biological and technical replication, multi-factor experiments, and blocking. Loop and extended loop designs are shown to be more efficient than the reference design for any given number of arrays. We also show that adding new treatments to a loop design requires the same number of additional arrays as adding treatments to a reference design, with a greater gain in power. Given the flexibility of extended loop designs and their power, we propose that these should be the designs of choice for most experiments using two-channel microarrays.     

Blocking Approach for Identification of Rare Variants in Family-Based Association Studies

With the advent of next-generation sequencing technology, rare variant association analysis is increasingly being conducted to identify genetic variants associated with complex traits. In recent years, significant effort has been devoted to develop powerful statistical methods to test such associations for population-based designs. However, there has been relatively little development for family-based designs although family data have been shown to be more powerful to detect rare variants. This study introduces a blocking approach that extends two popular family-based common variant association tests to rare variants association studies. Several options are considered to partition a genomic region (gene) into “independent” blocks by which information from SNVs is aggregated within a block and an overall test statistic for the entire genomic region is calculated by combining information across these blocks. The proposed methodology allows different variants to have different directions (risk or protective) and specification of minor allele frequency threshold is not needed. We carried out a simulation to verify the validity of the method by showing that type I error is well under control when the underlying null hypothesis and the assumption of independence across blocks are satisfied. Further, data from the Genetic Analysis Workshop are utilized to illustrate the feasibility and performance of the proposed methodology in a realistic setting.     

The design of in vivo multifraction experiments to estimate the alpha-beta ratio

Using statistical methods, the designs of multifraction experiments which are likely to give the most precise estimate of the alpha-beta ratio in the linear-quadratic model are investigated. The aim of the investigation is to try to understand what features of an experimental design make it efficient for estimating alpha/beta rather than to recommend a specific design. A plot of the design on an nd2 versus nd graph is suggested, and this graph is called the design plot. The best designs are those which have a large spread in the isoeffect direction in the design plot, which means that a wide range of doses per fraction should be used. For binary response assays, designs with expected response probabilities near to 0.5 are most efficient. Furthermore, dose points with expected response probabilities outside the range 0.1 to 0.9 contribute negligibly to the efficiency with which alpha/beta can be estimated. For "top-up" experiments, the best designs are those which replace as small a portion as possible of the full experiment with the top-up scheme. In addition, from a statistical viewpoint, it makes no difference whether a single large top-up dose or several smaller top-up doses are used; however, other considerations suggest that two or more top-up doses may be preferable. The practical realities of designing experiments as well as the somewhat idealized statistical considerations are discussed.     

Construction of resolvable spatial row-column designs

Resolvable row-column designs are widely used in field trials to control variation and improve the precision of treatment comparisons. Further gains can often be made by using a spatial model or a combination of spatial and incomplete blocking components. Martin, Eccleston, and Gleeson presented some general principles for the construction of robust spatial block designs which were addressed by spatial designs based on the linear variance (LV) model. In this article we define the two-dimensional form of the LV model and investigate extensions of the Martin et al. principles for the construction of resolvable spatial row-column designs. The computer construction of efficient spatial designs is discussed and some comparisons made with designs constructed assuming an autoregressive variance structure.     

A Simple Procedure for Constructing Experiment Designs with Incomplete Blocks of Sizes 2 and 3

In order to maximize control of heterogeneity within complete blocks, an experimenter could use incomplete blocks of size k = 2 or 3. In certain situations, incomplete blocks of this nature would eliminate the need for such spatial types of analyses as nearest neighbor. The intrablock efficiency factors for such designs are relatively low. However, with recovery of interblock information, FEDERER and SPEED (1987) have presented measures of design efficiency factors which demonstrate that efficiency factors approach unity for certain ratios of the intrablock and interblock variance components. Hence with recovery of interblock information, even incomplete block designs with k = 2 or 3 have relatively high efficiency factors. The reduction in the intrablock error variance over the complete block error variance in many situations will provide designs with high efficiency. A simple procedure for constructing incomplete blocks of sizes 2 and 3 is presented. It is shown how to obtain additional zero-one association confounding arrangements when v = 4 t, t an integer, and for v = pk, k ≤ p. It is indicated how to do the statistical analysis for these designs.     

The Application of Growth Analysis to Structured Experimental Designs and a New Procedure for Estimating Unit Leaf Rate and its Variance

A method is presented for estimating relative growth rates andother growth parameters using data from randomized block designs,prior to conventional or functional growth analysis. This methodcan be extended to other experimental designs where an additivemodel is applicable. The effects of this approach are illustratedusing the most commonly derived growth quantities, relativegrowth rate, leaf area ratio and unit leaf rate (net assimilationrate), from an extensive data set available for the growth ofthe biennial Arctium tomentosum. Comparisons are made betweendifferent growth analytical techniques and a new procedure fordetermining unit leaf rate, with a rigorous statistical treatment,is presented for the particular case when the logarithms ofthe variates for leaf area and total dry weight are linearlyrelated. Arctium tomentosum, burdock, growth analysis, randomized block design, unit leaf rate, relation of leaf area and total dry weight     

Factorial and time course designs for cDNA microarray experiments

Microarrays are powerful tools for surveying the expression levels of many thousands of genes simultaneously. They belong to the new genomics technologies which have important applications in the biological, agricultural and pharmaceutical sciences. There are myriad sources of uncertainty in microarray experiments, and rigorous experimental design is essential for fully realizing the potential of these valuable resources. Two questions frequently asked by biologists on the brink of conducting cDNA or two-colour, spotted microarray experiments are 'Which mRNA samples should be competitively hybridized together on the same slide?' and 'How many times should each slide be replicated?' Early experience has shown that whilst the field of classical experimental design has much to offer this emerging multi-disciplinary area, new approaches which accommodate features specific to the microarray context are needed. In this paper, we propose optimal designs for factorial and time course experiments, which are special designs arising quite frequently in microarray experimentation. Our criterion for optimality is statistical efficiency based on a new notion of admissible designs; our approach enables efficient designs to be selected subject to the information available on the effects of most interest to biologists, the number of arrays available for the experiment, and other resource or practical constraints, including limitations on the amount of mRNA probe. We show that our designs are superior to both the popular reference designs, which are highly inefficient, and to designs incorporating all possible direct pairwise comparisons. Moreover, our proposed designs represent a substantial practical improvement over classical experimental designs which work in terms of standard interactions and main effects. The latter do not provide a basis for meaningful inference on the effects of most interest to biologists, nor make the most efficient use of valuable and limited resources.     

Two-dimensional polyacrylamide gel electrophoresis characterization of APz, a sperm protein involved in zona binding in the pig and evidence for its binding to specific zona glycoproteins

A boar sperm integral plasma membrane protein (APz) involved in the adhesion of uncapacitated and capacitated sperm to the porcine zona pellucida (ZP) has been characterized by two-dimensional polyacrylamide gel electrophoresis (PAGE) and tested for its ability to bind to various zona glycopeptides. APz shows microheterogeneity and focuses over a wide pH range, with predominant forms focusing above pH 7. The protein, when excised from nonreducing polyacrylamide gels, inhibited sperm-egg binding and bound heat-solubilized zonae preventing these zonae from blocking sperm binding to eggs. In an indirect assay, a polyclonal monovalent antibody, which blocks sperm-egg binding and which is absorbed by APz, was used to determine the ability of zona glycopeptides to prevent the sperm-egg blocking activity of the antibody from being absorbed by intact sperm. When whole heat-solubilized ZP was added to sperm at doses that block sperm-egg binding and the excess ZP was removed, the sperm-egg blocking activity of the antibody was not absorbed by these sperm, and antibody-containing supernatants blocked the binding of untreated sperm to eggs as effectively as antibody that was not mixed with fresh sperm. When alpha ZP3 was used in the same manner, sperm-egg blocking activity again was not absorbed by antibody-treated cells. Beta ZP3, however, failed to block sperm-egg binding and failed to absorb the sperm-egg blocking activity of the antibody. These findings support the argument that the action of APz is physiologically significant and involves specific binding sites on the ZP3 component of the ZP.     

Mechanism of action of monoclonal antibodies that block the light activation of the guanyl nucleotide-binding protein, transducin

Seven monoclonal antibodies to the alpha subunit (G alpha) of the frog photoreceptor guanyl nucleotide-binding protein (transducin or G-protein) have been characterized as to their effect on G-protein function, and this has been correlated in the accompanying paper (Deretic, D., and Hamm, H. E. (1987) J. Biol. Chem. 262, 10839-10847) with the antibody-binding sites on G alpha tryptic fragments. Antibodies 4A, 7A, 7B, 7C, and 7D are members of a class of antibodies that block G-protein activation by light and therefore also block activation of the cGMP phosphodiesterase. All these blocking antibodies also block the interaction of G-protein with rhodopsin as measured by the light-scattering "binding signal," and as measured by the stabilization of meta-rhodopsin II by bound G-protein (extra-meta-rhodopsin II). The antibodies (or Fab fragments) also solubilize G alpha beta gamma from the membrane in the dark under isosmotic conditions and thus interfere with G alpha interaction with the membrane. Antibody 4A also blocks the extra-meta-rhodopsin II generated by G-protein-rhodopsin interaction in detergent solubilized membranes. Thus, even in the absence of phospholipids, antibody 4A blocks G-protein-rhodopsin interaction. Therefore, we suggest that the antibodies recognize a region of G alpha involved with binding to rhodopsin. An alternative hypothesis is that this antigenic site is a region of interaction between the alpha and beta gamma subunits, disruption of this interaction leading to removal of both the alpha and beta gamma subunits from the membrane and blocking interaction with rhodopsin. This does not seem to be the case because the antibodies immunoprecipitate the alpha beta gamma complex, and not just the alpha subunit. Other antibodies, 4C and 4H, do not block phosphodiesterase activation, the light-scattering signal, extra-meta-rhodopsin II formation, or interaction with the membrane in the dark and therefore recognize other sites on G alpha.     

Analysis of Partial Diallel Crosses in Incomplete Blocks

With a large number of lines in a diallel cross experiment, the number of crosses becomes unmanageable to be accommodated in homogeneous blocks. To overcome this problem, a sample of crosses, known as partial diallel cross (PDC) is often used. The selection of a PDC is based on the criterion of high efficiency for the estimation of general combining ability (gca) effects. Even with a moderately large number of crosses, the use of incomplete blocks is necessary to obtain homogeneous experimental units. The analysis of data from a general PDC grown in general incomplete block designs is being described. An iterative scheme is being developed for obtaining a generalized inverse of the information matrix used in estimating gca effects. Properties such as connectedness and efficiency of mating designs embedded in environment designs are being examined. The paper also examines the universal optimality of some designs in a class of designs. An illustration of the numerical procedure is also presented.     

Polymorphism and inheritance of gliadin components controlled by chromosome 6A of spring durum wheat

The gliadin composition of 78 spring durum wheat varieties has been studied by one-dimensional (Al-lactate, pH 3.1) and two-dimensional (first dimension, Al-lactate, pH 3.1; second dimension, sodium dodecyl sulfate-polyacrylamide gel) electrophoresis. Analysis of hybrids has shown that all components of the alpha zone of gliadin spectra are inherited together as blocks and are, probably, coded for by a cluster of tightly linked genes located on chromosome 6A. Fourteen variants of gliadin blocks have been identified, which can be classified into five families on the basis of component composition. All families but one have analogues among chromosome 6A-controlled blocks of bread wheat. The results indicate that some of the genome A diploid genotypes that were ancestors of durum wheats were also ancestors of bread wheats and that polyploid wheats were produced by repeated allopolyploidization events, as has been suggested earlier.     

Synthesis and evaluation of cis-hexahydropyrrolo[3,2-b]pyrrol-3-one peptidomimetic inhibitors of CAC1 cysteinyl proteinases

A stereoselective synthesis of functionalised cis-hexahydropyrrolo[3,2-b]pyrrol-3-ones has been developed through Fmoc and Cbz-protected intermediates 5 and 6. Building blocks 5 and 6 were prepared via the intramolecular cyclisation of anti-epoxide 17. The intramolecular reaction occurred exclusively through the anti-epoxide to provide the 5,5-cis-fused bicycle, whereas the syn-epoxide, which theoretically would provide the 5,5-trans-fused bicycle, remained unchanged. These experimental observations are consistent with a key design element that we have introduced within this novel bicyclic ketone scaffold. Our bicyclic design strategy provides chiral stability to the bridgehead stereocentre that is situated alpha to the ketone because the cis-fused geometry is both thermodynamically and kinetically stable. Building blocks 5 and 6 have been utilised in both solid phase and solution phase syntheses of peptidomimetics 22, 36-40, which exhibit potent in vitro inhibition against a range of CAC1 cysteinyl proteinases. Compound 22, a potent and selective inhibitor of human cathepsin K exhibited good primary DMPK properties along with promising activity in an in vitro cell-based human osteoclast assay of bone resorption.     

Experimental design for gene expression microarrays

We examine experimental design issues arising with gene expression microarray technology. Microarray experiments have multiple sources of variation, and experimental plans should ensure that effects of interest are not confounded with ancillary effects. A commonly used design is shown to violate this principle and to be generally inefficient. We explore the connection between microarray designs and classical block design and use a family of ANOVA models as a guide to choosing a design. We combine principles of good design and A-optimality to give a general set of recommendations for design with microarrays. These recommendations are illustrated in detail for one kind of experimental objective, where we also give the results of a computer search for good designs.     

Planning incomplete block experiments when treatments are genetically related

Selection trials in plant and animal breeding, in incomplete blocks, are described by linear models with random effect parameters associated with treatments with known genetic covariance structure. It is now well known that the information on relatives can improve the analysis and many extensions of this model have been proposed, but no studies have been done on the consequences of this genetical relatedness among treatments for the optimality of block designs. Using a suitable optimality criterion, we show that the knowledge on relatedness may imply that the optimal design is not in the class of designs which are optimal for unrelated treatments. Implications for practical applications are discussed.     

DESIGN: computerized optimization of experimental design for estimating Kd and Bmax in ligand binding experiments. II. Simultaneous analysis of homologous and heterologous competition curves and analysis blocking and of "multiligand" dose-response surfaces

We have developed a computer program, DESIGN, for optimization of ligand binding experiments to minimize the "average" uncertainty in all unknown parameters. An earlier report [G. E. Rovati, D. Rodbard, and P. J. Munson (1988) Anal. Biochem. 174, 636-649] described the application of this program to experiments involving a single homologous or heterologous dose-response curve. We now present several advanced features of the program DESIGN, including simultaneous optimization of two or more binding competition curves optimization of a "multiligand" experiment. Multiligand designs are those which use combinations of two (or more) ligands in each reaction tube. Such designs are an important and natural extension of the popular method of "blocking experiments" where an additional ligand is used to suppress one or more classes of sites. Extending the idea of a dose-response curve, the most general multiligand design would result in a "dose-response surface". One can now optimize the design not only for a single binding curve, but also for families of curves and for binding surfaces. The examples presented in this report further demonstrate the power and utility of the program DESIGN and the nature of D-optimal designs in the context of more complex binding experiments. We illustrate D-optimal designs involving one radioligand and two unlabeled ligands; we consider one example of homogeneous and several examples of heterogeneous binding sites. Further, to demonstrate the virtues of the dose-response surface experiment, we have compared the optimal surface design to the equivalent design restricted to traditional dose-response curves. The use of DESIGN in conjunction with multiligand experiments can improve the efficiency of estimation of the binding parameters, potentially resulting in reduction of the number of observations needed to obtain a desired degree of precision in representative cases.     

Design of experiments for the precise estimation of dose-response parameters: the Hill equation

Optimal experimental designs were evaluated for the precise estimation of parameters of the Hill model. The optimally effective designs were obtained by using the criterion of D-optimization. For the Hill model, optimal designs replicate 3 sampling points. These points were shown to be quite sensitive to the behavior of the experimental error. Since an investigator is often uncertain about error conditions in biological studies, a practical approach would use the sampling scheme calculated for an intermediate error condition. Thus, if the behavior of error variances is not known, precise parameters of the Hill model are obtained by choosing concentrations which yield fractional responses (responses divided by their asymptotic, maximum value) of 0.086, 0.581 and 1.0. When experimental constraints limit the maximum attainable concentration and response, all design points are lowered. Appropriate designs can be constructed based on the design which is optimal when constraints result in a maximum attainable fractional response of 0.5. The optimal designs were found to be robust when the parameter values assumed by the investigator did not equal their true values. The estimating efficiencies obtained by using two frequently applied designs were assessed. Uniformly spaced concentrations yielded imprecise parameters. Six-point, geometrically spaced designs gave generally good results. However, their estimating efficiency was generally exceeded by the recommended sampling schemes even in the presence of uncertainty about error conditions. The method exemplified in this paper can be used for other models.     

Comparison of microarray designs for class comparison and class discovery

MOTIVATION: Two-color microarray experiments in which an aliquot derived from a common RNA sample is placed on each array are called reference designs. Traditionally, microarray experiments have used reference designs, but designs without a reference have recently been proposed as alternatives. RESULTS: We develop a statistical model that distinguishes the different levels of variation typically present in cancer data, including biological variation among RNA samples, experimental error and variation attributable to phenotype. Within the context of this model, we examine the reference design and two designs which do not use a reference, the balanced block design and the loop design, focusing particularly on efficiency of estimates and the performance of cluster analysis. We calculate the relative efficiency of designs when there are a fixed number of arrays available, and when there are a fixed number of samples available. Monte Carlo simulation is used to compare the designs when the objective is class discovery based on cluster analysis of the samples. The number of discrepancies between the estimated clusters and the true clusters were significantly smaller for the reference design than for the loop design. The efficiency of the reference design relative to the loop and block designs depends on the relation between inter- and intra-sample variance. These results suggest that if cluster analysis is a major goal of the experiment, then a reference design is preferable. If identification of differentially expressed genes is the main concern, then design selection may involve a consideration of several factors.     

Evaluation of experimental designs for two-color cDNA microarrays.

The major goal of two-color cDNA microarray experiments is to measure the relative gene expression level (i.e., relative amount of mRNA) of each gene between samples in studies of gene expression. More specifically, given an N-sample experiment, we need all N(N - 1)/2 relative expression levels of all sample pairs of each gene for identification of the differentially expressed genes and for clustering of gene expression patterns. However, the intensities observed from two-color cDNA microarray experiments do not simply represent the relative gene expression level. They are composed of signal (gene expression level), noise, and other factors. In discussions on the experimental design of two-color cDNA microarray experiments, little attention has been given to the fact that different combinations of test and control samples will produce microarray intensities data with varying intrinsic composition of factors. As a consequence, not all experimental designs for two-color cDNA microarray experiments are able to provide all possible relative gene expression levels. This phenomenon has never been addressed. To obtain all possible relative gene expression levels, a novel method for two-color cDNA microarray experimental design evaluation is necessary that will allow the making of an accurate choice. In this study, we propose a model-based approach to illustrate how the factor composition of microarray intensities changed with different experimental designs in two-color cDNA microarray experiments. By analyzing 12 experimental designs (including 5 general forms), we demonstrate that not all experimental designs are able to provide all possible relative gene expression levels due to the differences in factor composition. Our results indicate that whether an experimental design can provide all possible relative expression levels of all sample pairs for each gene should be the first criterion to be considered in an evaluation of experimental designs for two-color cDNA microarray experiments.