Effect of pooling samples on the efficiency of comparative studies using microarrays

MOTIVATION: Many biomedical experiments are carried out by pooling individual biological samples. However, pooling samples can potentially hide biological variance and give false confidence concerning the data significance. In the context of microarray experiments for detecting differentially expressed genes, recent publications have addressed the problem of the efficiency of sample pooling, and some approximate formulas were provided for the power and sample size calculations. It is desirable to have exact formulas for these calculations and have the approximate results checked against the exact ones. We show that the difference between the approximate and the exact results can be large. RESULTS: In this study, we have characterized quantitatively the effect of pooling samples on the efficiency of microarray experiments for the detection of differential gene expression between two classes. We present exact formulas for calculating the power of microarray experimental designs involving sample pooling and technical replications. The formulas can be used to determine the total number of arrays and biological subjects required in an experiment to achieve the desired power at a given significance level. The conditions under which pooled design becomes preferable to non-pooled design can then be derived given the unit cost associated with a microarray and that with a biological subject. This paper thus serves to provide guidance on sample pooling and cost-effectiveness. The formulation in this paper is outlined in the context of performing microarray comparative studies, but its applicability is not limited to microarray experiments. It is also applicable to a wide range of biomedical comparative studies where sample pooling may be involved.     

Some preliminary considerations concerning concentration of oxygen in tissue

An approximate expression for the average concentration of oxygen in vascular tissue is derived on the basis of certain assumptions. These assumptions are then analyzed, and the inadequacies of the approximate derivation are pointed out; possibilities as to the method and results of more exact derivations are briefly discussed.     

Power of the classical twin design revisited.

Statistical power of the classical twin design was revisited. The approximate sampling variances of a least-squares estimate of the heritability in a univariate analysis and estimate of the genetic correlation coefficient in a bivariate analysis were derived analytically for the ACE model. Statistical power to detect additive genetic variation under the ACE model was derived analytically for least-squares, goodness-of-fit and maximum likelihood-based test statistics. The noncentrality parameter for the likelihood ratio test statistic is shown to be a simple function of the MZ and DZ intraclass correlation coefficients and the proportion of MZ and DZ twin pairs in the sample. All theoretical results were validated using simulation. The derived expressions can be used to calculate power of the classical twin design in a simple and rapid manner.     

Selecting an efficient design for assessing exposure-disease relationships in an assembled cohort

We develop approximate methods to compare the efficiencies and to compute the power of alternative potential designs for sampling from a cohort before beginning to collect exposure data. Our methods require only that the cohort be assembled, meaning that the numbers of individuals Nkj at risk at pairs of event times tk and tj greater than or equal to tk are available. To compute Nkj, one needs to know the entry, follow-up, censoring, and event history, but not the exposure, for each individual. Our methods apply to any "unbiased control sampling design," in which cases are compared to a random sample of noncases at risk at the time of an event. We apply our methods to approximate the efficiencies of the nested case-control design, the case-cohort design, and an augmented case-cohort design, compared to the full cohort design, in an assembled cohort of 17,633 members of an insurance cooperative who were followed for mortality from prostatic cancer. The assumptions underlying the approximation are that exposure is unrelated both to the hazard of an event and to the hazard for censoring. The approximations performed well in simulations when both assumptions held and when the exposure was moderately related to censoring.     

Mass action kinetics of virus-cell aggregation and fusion.

A simple approximate solution for the mass action kinetics of small particles (viruses or vesicles) binding to large particles (cells) and their subsequent fusion has been derived. The solution is evaluated in terms of the measurable fluorescence changes expected when the virus or vesicles are labeled with fluorescent probes, which are diluted into the cellular membrane by fusion. Comparison with numerical integrations shows that the approximate solution is extremely accurate. Analytic simplifications for a variety of special cases of this general problem are also shown.     

Fast approximate motif statistics.

We present in this article a fast approximate method for computing the statistics of a number of non-self-overlapping matches of motifs in a random text in the nonuniform Bernoulli model. This method is well suited for protein motifs where the probability of self-overlap of motifs is small. For 96% of the PROSITE motifs, the expectations of occurrences of the motifs in a 7-million-amino-acids random database are computed by the approximate method with less than 1% error when compared with the exact method. Processing of the whole PROSITE takes about 30 seconds with the approximate method. We apply this new method to a comparison of the C. elegans and S. cerevisiae proteomes.     

Thermodynamics of carbohydrate isomerization reactions. The conversion of aqueous allose to psicose

The thermodynamics of the conversion of aqueous D-psicose to D-allose has been investigated using high-pressure liquid chromatography. The reaction was carried out in phosphate buffer at pH 7.4 over the temperature range 317.25-349.25 K. The following results are obtained for the conversion process at 298.15 K: DeltaG degrees = - 1.41 +/- 0.09 kJ mol(-1), DeltaH degrees = 7.42 +/- 1.7 kJ mol(-1), and DeltaC(p) degrees = 67 +/- 50 J mol(-1) K(-1). An approximate equilibrium constant of 0.30 is obtained at 333.15 K for the conversion of aqueous D-psicose to D-altrose. Available thermodynamic data for isomerization reactions involving aldohexoses and aldopentoses are summarized.     

Looking at proteins: representations, folding, packing, and design. Biophysical Society National Lecture, 1992.

Looking at proteins is an active process of interpretation and selection, emphasizing some features and deleting others. Multiple representations are needed, for such purposes as showing motions or conveying both the chain connectivity and the three-dimensional shape simultaneously. In studying and comparing protein structures, ideas are suggested about the determinants of tertiary structure and of folding (e.g., that Greek key beta barrels may fold up two strands at a time). The design and synthesis of new proteins "from scratch" provides a route toward the experimental testing of such ideas. It has also been a fruitful new perspective from which to look at structures, requiring such things as statistics on very narrowly defined structural categories and explicit attention to "negative design" criteria that actively block unwanted alternatives (e.g., reverse topology of a helix bundle, or edge-to-edge aggregation of beta sheets). Recently, the field of protein design has produced a rather unexpected general result: apparently we do indeed know enough to successfully design proteins that fold into approximately correct structures, but not enough to design unique, native-like structures. The degree of order varies considerably, but even the best designed material shows multiple conformations by NMR, more similar to a "molten globule" folding intermediate than to a well ordered native tertiary structure. In response to this conclusion, we are now working on systems that test useful questions with approximate structures (such as determining which factors most influence the choice of helix-bundle topology) and also analyzing how natural proteins achieve unique core conformations (e.g., for side chains on the interior side of a beta sheet, illustrated in the kinemages).     

Finding approximate tandem repeats in genomic sequences.

An efficient algorithm is presented for detecting approximate tandem repeats in genomic sequences. The algorithm is based on a flexible statistical model which allows a wide range of definitions of approximate tandem repeats. The ideas and methods underlying the algorithm are described and its effectiveness on genomic data is demonstrated.     

Confidence Intervals and Tests of Hypotheses on Variance Components in an Unbalanced Two-fold Nested Design

Confidence intervals and tests of hypotheses on variance components are required in studies that employ a random effects design. The unbalanced random two-fold nested design is considered in this paper and confidence intervals are proposed for the variance components σ2/A and σ2/B. Computer simulation is used to show that even in very unbalanced designs, these intervals generally maintain the stated confidence coefficient. The hypothesis test for σ2/A based on the lower bound of the recommended confidence interval is shown to be better than previously proposed approximate tests.     

A continuous,multistage tower fermentor. I. Design and performance tests

The design of a continuous column fermentor with a multiple staging effect is described. The column is divided into four compartments by horizontal perforated plates and is provided with a central agitator shaft driving an impeller in each compartment. A tube at the center of each plate forms a liquid seal around the shaft and also acts as a “downcomer.” The fermentor is normally operated with counter-current flow of gas and medium. Fresh medium is added to the top stage and product is withdrawn from the bottom. The effect of plate and agitator design on fermentor performance was studied in terms of factor such as oxygen transfer rate, gas holdup, and interstage mixing. By proper choice of the design parameters, the fermentor was made to approximate a perfect four-stage cascade in terms of reactor performance. Preliminary experiments were performed with air-water systems, but a more realistic picture of fermentor performance was obtained in experience involving propagation of Escherichia coli. Data for business and substrate concentrations in each stage confirmed the staging effect of the apparatus. The fermentor operated in a stable manner for periods of more than two weeks.     

ROCK: a spreadsheet-based program for the generation and analysis of random oligonucleotide primers used in PCR.

Oligonucleotide primers used to amplify target DNA regions via PCR should meet certain design criteria to maximize the potential for efficient priming. The Random Oligonucleotide Construction Kit (ROCK), a spreadsheet-based program that runs under Microsoft Excel 97 or later version for Microsoft Windows, was developed to facilitate the design of efficient random oligonucleotide primers. Primer sequences are generated that meet user-defined criteria with regard to G + C content, size of a 3' GC clamp, maximum intramolecular/intermolecular complementation potential, and maximum intersequence similarity. The user can analyze the intramolecular/intermolecular complementation potential of program-generated primer sequences or of sequences entered manually. The latter may contain any of the standard nucleotide symbols, including ambiguous bases. Primer sequence length, GC%, individual base composition, molecular weight, approximate melting temperature, and mass/volume/concentration relationships can be determined for any sequence generated by ROCK or entered manually.     

Active site of trypanothione reductase. A target for rational drug design.

The X-ray crystal structure of the enzyme trypanothione reductase, isolated from the trypanosomatid organism Crithidia fasciculata, has been solved by molecular replacement. The search model was the crystal structure of human glutathione reductase that shares approximately 40% sequence identity. The trypanosomal enzyme crystallizes in the tetragonal space group P4(1) with unit cell lengths of a = 128.9 A and c = 92.3 A. The asymmetric unit consists of a homodimer of approximate molecular mass 108 kDa. We present the structural detail of the active site as derived from the crystallographic model obtained at an intermediate stage of the analysis using diffraction data to 2.8 A resolution with an R-factor of 23.2%. This model has root-mean-square deviations from ideal geometry of 0.026 A for bond lengths and 4.7 degrees for bond angles. The trypanosomid enzyme assumes a similar biological function to glutathione reductase and, although similar in topology to human glutathione reductase, has an enlarged active site and a number of amino acid differences, steric and electrostatic, which allows it to process only the unique substrate trypanothione and not glutathione. This protein represents a prime target for chemotherapy of several debilitating tropical diseases caused by protozoan parasites belonging to the genera Trypanosoma and Leishmania. The structural differences between the parasite and host enzymes and their substrates thus provides a rational basis for the design of new drugs active against trypanosomes. In addition, our model explains the results of site-directed mutagenesis experiments, carried out on recombinant trypanothione reductase and glutathione reductases, designed by consideration of the crystal structure of human glutathione reductase.     

On predicting extinction in simple population models. I. Stochastic linearization

The technique of stochastic linearization derived by Bartlett is used to give an approximate solution to each of three stochastic models of predation. By defining the existence of a quasi-stationary equilibrium distribution of population sizes, the approximate variances and covariances of the joint distribution of population sizes are calculated. The results are used to predict whether prey or predators are likely to become extinct first and the predictions are tested against simulation data. The linearization technique is a good predictor of the outcome of extinction but not of how long it takes.     

Population growth under the influence of random dispersal

A population is considered which grows according to the logistic equation while spreading out at random. An approximate method is used to obtain transient and steady-state values for various simple boundary conditions such as that of a population started in an infinite one- or two-dimensional region with or without reflecting or absorbing barriers. An approximate steady-state solution is given for the one-dimensional case of two neighboring regions having different growth rates, mobilities, and degrees of attractiveness.     

Crystallization of recombinant Crithidia fasciculata tryparedoxin.

Recombinant tryparedoxin, a thioredoxin homologue from Crithidia fasciculata, has been purified from an Escherichia coli expression system and used in crystallization trials. Orthorhombic needles in space group P212121, with unit cell dimensions of a = 38.63, b = 51. 47, and c = 73.41 A, have been obtained. The crystals present a monomer of approximate molecular mass 16 kDa in the asymmetric unit and diffract to 1.8-A resolution using synchrotron radiation. Structure determination will be carried out to further the understanding of the role tryparedoxin plays in regulating oxidative stress in parasitic trypanosomatids.     

Computer design of bioactive molecules: a method for receptor-based de novo ligand design.

The design of molecules to bind specifically to protein receptors has long been a goal of computer-assisted molecular design. Given detailed structural knowledge of the target receptor, it should be possible to construct a model of a potential ligand, by algorithmic connection of small molecular fragments, that will exhibit the desired structural and electrostatic complementarity with the receptor. However, progress in this area of receptor-based, de novo ligand design has been hampered by the complexity of the construction process, in which potentially huge numbers of structures must be considered. By limiting the scope of the structure-space examined to one particular class of ligands--namely, peptides and peptide-like compounds--the problem complexity has been reduced to the point that successful, de novo design is now possible. The methodology presented employs a large template set of amino acid conformations which are iteratively pieced together in a model of the target receptor. Each stage of ligand growth is evaluated according to a molecular mechanics-based energy function, which considers van der Waals and coulombic interactions, internal strain energy of the lengthening ligand, and desolvation of both ligand and receptor. The search space is managed by use of a data tree which is kept under control by pruning according to the energy evaluation. Ligands grown by this procedure are subjected to follow-up evaluation in which an approximate binding enthalpy is determined. This methodology has proven useful as a precise model-builder and has also shown the ability to design bioactive ligands.     

The Vancouver Lymphadenopathy - AIDS Study: 2. Seroepidemiology of HTLV-III antibody

Testing for antibody to human T-lymphotropic retrovirus (HTLV-III) was carried out in 448 participants in the Vancouver Lymphadenopathy-AIDS (acquired immune deficiency syndrome) Study. The overall prevalence rate of seropositivity was 34%. Of 130 seronegative subjects followed for an average of 8.5 months, 14 became seropositive; thus, the approximate annual seroconversion rate was 15%. More than 100 male sexual partners in one''s lifetime, frequent receptive anal intercourse, fisting, a history of gonorrhea or hepatitis, and frequent sexual contact in clubs were found to be independent risk factors for HTLV-III seropositivity.     

Chemical synthesis and expression of the HIV-1 protease gene in E. coli

The 297bp HIV-1 protease gene was constructed from five discrete synthetic fragments and expressed in E. coli. A soluble protein product of 11.5 Kd was detected by immunoblotting using protease specific antisera. A quantitative assay system, utilizing a synthetic nonapeptide spanning the cleavage site between p17-p24 in the gag polyprotein, was used to measure the specific protease activity in crude extracts. The protease hydrolyzed tyrosyl-proline bonds with an approximate specific activity of 43 pmoles/min/micrograms of total protein. The chemical synthesis of the protease gene and it's expression provides a feasible method for rapid mutant analysis, important for structure-function studies and rational design of potential inhibitors.     

Atomically detailed potentials to recognize native and approximate protein structures

Atomically detailed potentials for recognition of protein folds are presented. The potentials consist of pair interactions between atoms. One or three distance steps are used to describe the range of interactions between a pair. Training is carried out with the mathematical programming approach on the decoy sets of Baker, Levitt, and some of our own design. Recognition is required not only for decoy-native structural pairs but also for pairs of decoy and homologous structures. Performance is tested on the targets of CASP5 using templates from the Protein Data Bank, on two test ab initio decoy sets from Skolnick's laboratory, and on decoy sets from Moult's laboratory. We conclude that the newly derived potentials have significant recognition capacity, comparable to the best models derived from other techniques. The new potentials require a significantly smaller number of parameters. The enhanced recognition capacity extends primarily to the identification of structures generated by ab initio simulation and less to the recognition of approximate shapes created by homology.