Design of microarray experiments for genetical genomics studies 2006

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Copyright 2006 by the Genetics Society of AmericaDOI: 10.1534/genetics.106.057281

Design of Microarray Experiments for Genetical Genomics Studies

Julio S. S. Bueno Filho,* Steven G. Gilmour and Guilherme J. M. Rosa,1

* Departamento de Ciencias Exatas, Universidade Federal de Lavras, Lavras, Minas Gerais 37200-000, Brazil, School of MathematicalSciences, Queen Mary, University of London, London E1 4NS, United Kingdom and Department of Animal Science and

Department of Fisheries and Wildlife, Michigan State University, East Lansing, Michigan 48824-1225

Manuscript received February 15, 2006 Accepted for publication July 19, 2006

ABSTRACT

Microarray experiments have been used recently in genetical genomics studies, as an additional toolto understand the genetic mechanisms governing variation in complex traits, such as for estimatingheritabilities of mRNA transcript abundances, for mapping expression quantitative trait loci, and forinferring regulatory networks controlling gene expression. Several articles on the design of microarrayexperiments discuss situations in which treatment effects are assumed fixed and without any structure. Inthe case of two-color microarray platforms, several authors have studied reference and circular designs.Here, we discuss the optimal design of microarray experiments whose goals refer to specific geneticquestions. Some examples are used to illustrate the choice of a design for comparing fixed, structuredtreatments, such as genotypic groups. Experiments targeting single genes or chromosomic regions (such

as with transgene research) or multiple epistatic loci (such as within a selective phenotyping context) arediscussed. In addition, microarray experiments in which treatments refer to families or to subjects (withinfamily structures or complex pedigrees) are presented. In these cases treatments are more appropriatelyconsidered to be random effects, with specific covariance structures, in which the genetic goals relate tothe estimation of genetic variances and the heritability of transcriptional abundances.

MICROARRAY technology allows the monitoring ofmessenger RNA (mRNA) abundance in cells forthousands of genes simultaneously (Schena et al. 1995;Lockhart et al. 1996). Microarray experiments havebeen used extensively to provide a surrogate measure ofgene activity to compare expression levels of treatmentgroups (within either experimental or observationalcontexts), such as different tissues, cell types (e.g., dis-ease and normal cell) in a specific tissue, different de-

velopmental stages, or experimental conditions (e.g.,different drugs or stress conditions).

The application of this technology however, especiallyfor two-color platforms (either cDNA or long oligonu-cleotide arrays), poses some experimental design chal-lenges. As samples are assayed in a pairwise competitivehybridization fashion on each slide, microarray experi-ments using two-color systems impose a block structure

with blocks of size two. Whenever more than two groups

are compared, this defines an incomplete block design,in which estimated treatment effects and differencesbetween slides are partially confounded. In addition, aseach sample is labeled with a different dye on each slide,microarray experiments involve an additional blockingfactor to be accountedfor, which leads to a rowcolumn,or two-way blocking, structure (slidedye combination).

Many articles on design for two-color microarray ex-periments make some comparison of efficiency orpower (Kerr and Churchill 2001; Yang and Speed2002; Kerr 2003; Rosa et al. 2005; Steibel and Rosa2005; Tempelman 2005; Vinciotti et al. 2005) of dif-ferent designs. Most of them focus on the comparison ofa discrete set of designs, such as reference and circularstructures. Wit et al. (2005) present a more generalapproach, searching for optimal designs given specificexperimental conditions and optimality criteria. It isshown that the optimal microarray design for a spe-cific optimality criterion may refer to a more complexexperimental layout than simple reference or circularstructures.

The objective of the experiment guides the choice ofthe treatment structure, whereas the definition of theexperimental units leads to the blocking structure ofthe experiment. As discussed throughout this article,

the treatment choice in microarray experiments is gen-eral for any platform (e.g., high-density short oligos,cDNA, or long oligonucleotide technologies), whereasthe blocking structure depends on the microarray tech-nology, such as two-color arrays (in which slides refer toa blocking factor) or single-color arrays such as Affymet-rix (in which each sample is hybridized to a differentslide). The treatment structure (as well as the number ofreplications) depends also on the practical/logisticalconstraints of the experiment, such as limited biologicalsamples (limited either by the number of individuals to

1Corresponding author:456 Animal Sciences Bldg., 1675 ObservatoryDr.,University of Wisconsin, Madison, WI 53706. E-mail: [email protected]

Genetics 174: 945957 (October 2006)


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be assayed or by the amount of mRNA from each in-dividual) or number of slides available. The choice ofthe experimental unit, which can be an individual (suchas a plant or animal, hereinafter denotedas a subject)ora group of subjects (such as a family or mRNA pool),may impose some spatial or genetic structure amongunits, which should be taken into account when assign-ing treatments to units.

Microarray experiments have been recently used ongenetical genomics studies, as an additional tool tounderstand the genetic mechanisms governing varia-tion in complex traits, such as disease susceptibility inhuman medicine or production traits in agriculture.Microarray experiments using related subjects can beutilized, for example, to infer heritabilities of mRNAtranscript abundances (Monks et al. 2004). In addition,gene expression profiling can be combined with markergenotype information for mapping expression quanti-tative trait loci (eQTL; Jansen and Nap 2001; Bremet al. 2002; Darvasi 2003; Schadt et al. 2003) and toinfer regulatory networks controlling gene expression

(Chesler et al. 2004; Bystrykh et al. 2005).In this article we discuss the optimal design of micro-

array experiments whose goals refer to specific geneticquestions, such as the comparison of expression levelsof different genotypic groups, eQTL studies, or estima-tion of heritabilities of mRNA transcript abundances.This goes beyond the scope of earlier work on fixed,unstructured treatments.

The experimenters interest might be in specifictreatment comparisons, e.g., referring to additive anddominance effects. Treatments may refer to families orto subjects (within family structures or complex pedi-grees) in a given experiment. In these cases treatmentsare more appropriately considered to be random ef-fects, with specific covariance structures. These covari-ance structures should then be taken into account whenplanning microarray experiments, so that the efficiencyof broad inferences to the population of interest ismaximized.

The high cost of microarrays greatly limits the samplesizes of heritability or gene-mapping studies of mRNAabundance traits (Jin et al. 2004). Therefore, a carefuldesign of such experiments is extremely important tomake the most of the limited number of slides that aregenerally available in a single experiment.

This article is organized as follows. In the next sectionwe review some experimental design concepts, such astreatment choice, treatment-to-unit allocation, designoptimality criteria, and some useful methods and resultson optimal designs. The third section presents geneticresearch goals that can motivate microarray studies

with two or more treatments, such as genotypic groups.Treatments are considered fixed, but the genetic com-ponent that defines the treatments provides a naturalstructure (which is translated into genetic parametersof interest, such as additive and dominance effects),

which should be explored when designing the experi-ments. In the fourth section, the design of microarrayexperiments with random treatments is discussed. Treat-ments in this case may refer to related individuals orfamilies, and the main goal of the experiments relates toinference on genetic variances and heritabilities. Fi-nally, some concluding remarks are presented in the lastsection.

DESIGN PRINCIPLES AND CRITERIA

The general statistical principles of the design of ex-periments apply to experiments of any kind, includingmicroarray studies. The general procedure for choosinga good design involves the following steps (see alsoBailey 1981, 1997; Mead 1990):

1. The objectives of the experiment lead to the choiceof a set of treatments (groups to compare).

2. The experimental units (here the combination ofslides and dyes) that can be used are identified andany expected patterns of variability among them leadto a choice of blocking structure (here a rowcolumnstructure).

3. Consideration of any restrictions on which treat-ments can be applied to which units might lead tocomplex structures such as split-plot designs.

4. Given 13, a combinatorial, algorithmic or ad hocmethod is found to construct a design.

The block structure is already clear and we do not dis-cuss complex structures here, so the two main tasks indeveloping a design are treatment choice and treatment-to-unit allocation.

Treatment choice: When the experimental resourcesare limited, a choice of the treatments that will berepresented in the experiment, from a larger set ofcandidate treatments, might be needed. In some sit-uations particular treatments cannot be produced, orthe number of candidate treatments is too big for all ofthem to be implemented. For example, suppose we wantto study isolines for transgenic versions of three genes(A/a, B/b, and C/c) from some parental line. Evenrestricting each locus to express just the homozygotegenotypes, it is unlikely that all seven desired isolines

will be obtained by transformation events or any other

currently known gene manipulation techniques. Start-ing from the nontransgenic genotype (AABBCC), forinstance, a single mutation may give rise to the geno-types AABBcc, AAbbCC, or aaBBCC; however, two orthree mutations are needed to obtain the genotypes

AAbbcc, aaBBcc, and aabbCC or aabbcc, respectively.Another and more obvious example is given by partialdiallels, in which some crossings cannot be done. Wediscuss later in this article some multilocus analysisproblems. In these cases, we need to define a model forthe treatment effects and an optimality criterion related

946 J. S. S. Bueno Filho, S. G. Gilmour and G. J. M. Rosa


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to the variances of the estimated parameters of themodel. Then we must find a method of searching fora design that optimizes the criterion, e.g., using theoptimal continuous (approximate) design theory de-

veloped from Kieferand Wolfowitz (1960) or a com-puter algorithm that searches for the optimal exactdesign, such as those developed from Fedorov (1972).

As stated earlier, the treatment choice in microarray

experiments is independent of the array platform, sothe same procedure can be used for single- or two-colortechnologies.

Treatment-to-unit allocation:A given set of treatments(and their respective replication numbers) should becombined in an optimal way within a given experi-mental setup. In the context of two-color microarrayexperiments, for example, this setup is given by thenumber of slides and the two labels (Cy3 and Cy5 dyes).

Again, construction of the design might require optimaldesign theory, this time most likely using a computersearch.

The treatment-to-unit allocation applications dis-

cussed in this article are specific to two-color array plat-forms (cDNA or long oligos), in which samples shouldbe allocated in pairs to the slides (blocks). With single-color microarrays (such as Affymetrix), on the otherhand, each sample is hybridized to a different slide,so this is not an issue and allocation should be donecompletely at random.

Linear mixed models for microarray experiments:The outcomes of a microarray experiment are luminousintensities of thousands of genes, which are propor-tional to their mRNA transcriptional abundances. Aftersuitable normalization, the response intensities fromgene g in a two-color microarray experiment can berepresented by the model

ygijk mg1 dgi1 agj1 tgk1 egij; 1

(Wolfinger et al. 2001), where ygijk is the normalizedintensity of the gene expression measured by the dye(label) i of the array j that received treatment k, mg isa common experimental constant, dgi is the effect of dyei(green or red), agj is the effect of array (or slide) j, tgkis the effect of the treatment k, and egij is a residualterm. We consider a single gene at a time for designpurposes, as the same model is generally used to analyzeall genes in a single experiment, and so the subscript g

is dropped. Different experimental layouts may needextra terms in the model for the appropriate modelingof the data. With Affymetrix data, for example, there isno dye effect in the model, but a hierarchical structurecould be used to model intensities at the probe level,by including the effects of probe sets (clones) and ofprobe pairs within clones. Model choice should then beclosely connected to treatment and block structures ofthe experiments (Rosa et al. 2005).

As additional assumptions of model (1), residualterms can be modeled as being normally distributed

with variance s2e; likewise, as arrays come from a pop-ulation of possible assembled arrays, their effects can bemodeled as random normal variables centered aroundzero, with variance s2a. The term treatment here isused in a very broad context, representing either experi-mental groups (such as genotypic classes) or families(and subjects within families) in quantitative geneticsstudies to estimate heritabilities. Therefore, treatment

effects can be modeled as fixed or random, dependingon the objective of the experiment.The usual matrix notation for linear mixed models

such as model (1) is

y Xb1Zu1 e; 2

where y and e are vectors with elements yijk and eij,respectively,b is the vector of fixed effects, Xis a known(design) matrix that relates y to b, u is the vector ofrandom effects, and Z is the matrix that relatesyto u. Anextra assumption is that u $ N0; U, where U is thecovariance matrix of the random effects.

Note that the treatment effects can be taken as beingeither fixed or random and so they can be contained ineither b or u, depending on the experimental context.In the examples discussed in this article, whenevertreatment effects are considered random, they are as-sumed normally distributed; i.e., t $ N0; Gs2t, whereG is a matrix of known constants representing, for ex-ample, genetic relationships among the elements in t(such as a numerator relationship matrix A; Henderson1976), and s2t is the variance among treatments, such asan additive genetic variance.

Design criteria: The covariance matrix of the param-eter estimates for model (2) is the inverse of the in-

formation matrix M, where

M}X9X Z9XX9Z Z9Z1s2eU

1

:

Almost any sensible criterion defining design efficiencyis a function of the information matrix M.

Costs are important when comparing alternative ex-perimental designs. In this article, however, designsare generally compared for a given number of slides. Aspointed out by Yang and Speed (2002), the number ofslides is generally the most important limiting factorin microarray experiments. In addition, for two-color

microarray platforms, as the variation between slides isgenerally larger than the variation within slides, block-ing by slides is crucial. Sincethe ratio of between-slide to

within-slide variation is unknown, it is usual to designexperiments for the worst case, i.e., where this ratio is

very large, in which case the design problem becomesequivalent to one with fixed blocks. Note, however, thatafter the experiment is run, in the mixed-models dataanalysis, we will estimate the variance components; iftheir ratio turns out to be large, we will estimate thetreatment effects with the precision expected when we

Design of Genetical Genomics Studies 947


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chose the design; if theirratio turnsout to be small, thenthe interblock information will give us estimates oftreatment effects better than expected when we chosethe design.

In this context, KerrandChurchill (2001) used theso-called A-optimality criterion to compare block de-signs for microarray studies with simple treatmentstructures, and Yang and Speed (2002) discussed the

efficiency of alternative designs for specific contrasts ofinterest in factorial and time course experiments. Kerr(2003) later argued in favor of double reference designsas being suboptimal but more robust when compared toloop designs. Glonek and Solomon (2004) studiedsome alternative designs for factorial treatment struc-tures. More general work on two-level factorial designsin blocks of size two, whichhas some relevance, has beendone by Draper and Guttman (1997), Yang andDraper (2003), Wang (2004), and Kerr (2006).

Similar sorts of experiments are now being used forunderstanding the genetic basis of mRNA transcriptabundances, with a variety of experimental settings. For

example, fixed treatment effect models arise in geneticanalysis when comparing multilocus genotypic groups,including interaction (epistasis) terms. In addition,random treatment effect models may be used in quan-titative genetics to infer genetic variances and heritabil-ities. No work, however, has been published on optimaldesign for microarray experiments in situations wherethe treatments have any of these structures.

In each of the following sections we present differentexperiments and discuss possible parameter contrastsof interest, as well as the variances of their estimates,sometimes weighted to correct for scale problems. Ineach situation, alternative optimality criteria are dis-cussed. We assume that all power comparisons have amonotonic (inverse) relation with the variances of thedesired comparisons and describe treatment structuresin microarray experiments for some common goals ingenetic studies. The resulting designs are depicted bya setof arrows representing theslides,with heads andtailsdenoting alternative dye labeling (Cy3 and Cy5). Arrows(slides) refer always to different biological replicationsfrom each experimental group, unless otherwise stated.

GENETIC STUDIES WITH FIXED TREATMENTEFFECTS

Two treatment groups in microarray experiments:Genetic motivation: One of the simplest scenarios thatresults in two-treatment experiments arises with a singlecandidate gene (with alleles B and b) and a backcross(BC) or double-haploid (DH) experiment. In these cir-cumstances, individuals belong to one of two possiblegenotypic groups, namely Bb or bb if in a BC experi-ment and BB or bb if in a DH scenario. In bothsituations one is interested in the additive effect a ofthe candidate gene on the expression of each gene

represented in the microarray slide, i.e., how polymor-phisms at a specific site of the genome affect transcrip-tion activity of other genes in the genome. Such effectsare generally referred to as trans-acting factors ( Jansenand Nap 2001).

The model (1) for such a simple treatment structurecan be written with tk qka. For the BC situation, qk is adesign index on genotypes assuming values q1 0 (for

genotype bb) and q2 1 (for genotype Bb). In this sit-uation, one wants to estimate a t2 t1 and the vari-ance of this contrast is given by Vart1 t2 Vart11

Vart2 2Covt1; t2. For the DH situation, q1 0 (forgenotype bb) and q2 2 (for genotype BB). In thissituation, one wants to estimate a 12t2 t1, and the

variance of this comparison is given by one-quarter ofthe right-hand side of the above variance. As these two

variances are proportional, the same optimal designminimizes both of them.

In a microarray experiment, such a simple two-treatment group situation can arise within a selectivephenotyping approach based on a single locus (or small

chromosomic region), as proposed by Jin et al. (2004),or within a candidate gene context in which one isinterested in the effect of a specific locus (e.g., trans-gene) on the expression of other genes in the genome(see, for example, Noueiry et al. 2000).

Another example is to define two experimental groupsby taking the extremes of the phenotypic distribution ofa quantitative trait of interest (Lander and Botstein1989). By comparing the allelic frequencies of molecu-lar markers between the two groups, candidate regionscontaining genes contributing to variation of the phe-notypic trait can be pinpointed. In the context of mi-croarrays, candidate genes are generated by searchingfor differential transcript abundances between the twogroups. In any situation, the proportions of the selectedindividuals should be made small enough so that thetwo phenotypic groups are quite distinct (Darvasi andSoller1992).

Microarray experiments with two experimental groupsdefine rowcolumn designs obtained from multiplecopies of a 2 3 2 Latin square (in the case of two-colorsystems), for which extensive literature is available.

Three treatment groups in microarray experiments:Genetic motivation: In some situations there are threepossible genotypes in a segregating population, for ex-

ample, with F2 populations and codominant markers.The genotypes for each marker can then be describedby an additivedominance model. Here we discuss thecase of a single locus (such as a candidate gene sit-uation) and its trans-acting effect on the expression ofother genes. The case of multiple loci is discussed laterin this article, with factorial and fractional factorialtreatment structures.

Treatment choice: An additivedominance treatmenteffect of a candidate gene in an F2 population can beexpressed as tk qka1 1 jqkjd, where qk 1; 0; 1,



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for genotypes bb, Bb, and BB, respectively. Denote byrk the proportion of the n observations that representeach genotypic group (k 1, 2, 3). The model can berewritten as y m1n311Q

ad1 e, where 1n31 is a col-

umn vector of ones and Q gives the incidences ofgenotypes bb, Bb, and BB as

Q9

1 1 1 0 0 0 1 1 1

0 0 0 1 1 1 0 0 0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}n3r1

|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}n3r2

|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}n3r3

2664 3775:

Both parameters can be optimally estimated using thesame proportion for each homozygous genotype (bband BB), in which case r1 r3 rand r2 1 2r. Theinformation matrix is then given by

M 1 0 1 2r0 2r 0

1 2r 0 1 2r

24

35 ns2e

and the covariance matrix is

M1

12r

0 12r0 12r 0

12r 0 1

2r2r1

264

375s

2e

n:

Figure 1 plots changes in the variances of the pa-rameter estimates with the proportion (r) of eachhomozygous group. If one is interested solely in the(trans-acting) additive effects, just homozygotes shouldbe considered in the design. To estimate (trans-acting)

dominance only, half of the observations should betaken from the heterozygotes and one-quarter fromeach of the homozygotes. To estimate additive and dom-inance effects simultaneously, it is necessary to define anoptimality criterion. Throughout this article, we use the

A-efficiency on this scale, i.e., the mean of the variancesof the estimates, but in any real experiment carefulthought should be given to whether a weighted mean ofthe variances would be a more appropriate criterion.The A-optimal proportion of each homozygote is theirrational number ropt 1

ffiffiffi2

p=2. As no possible num-

ber of arrays satisfies this optimum exactly, an approx-imation is needed. It is reasonable to use $0.293 ofthe observations for each homozygote and the remain-ing 0.414 of the observations for the heterozygotegroup.

Suppose, for example, that we are interested in thebest design that can be made with only 10 slides using atwo-color microarray platform. We proceed as follows.To estimate just additive effects, each homozygous

genotype (BB or bb) will have 10 replications. The het-erozygote group, in this case, is not included in theexperiment. To estimate only dominance effects there

will be 5 replications of each homozygous genotypeand 10 replications of the heterozygote group. Finally, forestimation of both additive and dominance effects, the A-optimal design has 6 replications for each homozygoteand the remaining 8 replications for the heterozygote.

Treatment-to-unit allocation: After the treatments havebeen chosen (i.e., the number of replications for each ge-notypic group), the next step in planning the microarray

Figure 1.Average variances of the estimatesof genetic parameters for a single-locus experi-ment as a function of the proportion of each ho-mozygous group (r), when the experiment goal isto estimate the additive effect (A), the domi-nance effect (B), or both additive and dominanceeffects simultaneously (C).



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experiment is the treatment-to-unit allocation, i.e., de-

scribing which treatment should be applied to eachunit, when array and color effects are in the model. Theoptimal design for estimating just additive effects with10 slides is shown in Figure 2A, in which all slides havethe two homozygous genotypes only, with alternatingdirections of the Cy3 and Cy5 labeling. If one is con-cerned with estimating dominant effects, all of theslides should have one of the homozygous genotypescohybridized with the heterozygous genotype, as shownin Figure 2B. Finally, if both additive and dominanceeffects are to be estimated, combinations of the two

types of slide should be used (Figure 2C). It is worthmentioning that the designs of Figure 2, AC, are,respectively, $100, 50, and 9% more efficient than theclassical loop structure. When compared to the refer-ence design, their relative efficiencies are 6.6, 4.7, and3.4, respectively.

Generalization of these results for multiple loci:The effect of multiple candidate genes on the overall

genome transcriptional activity can be investigated atonce in a single microarray study. With simple line-crossing experiments, each gene (locus) may have ei-ther two or three levels (possible genotypes), dependingon the mating system used (BC or DH and F2, re-spectively). Therefore, the treatment structure for Llociis a factorial of the series 2Lor 3Lfor two or threepossiblegenotypes in each locus, respectively. A suitable modelfor the expression of the treatments in such a system,including epistasis, is given by

tk1 k2;...;kL XLl1

qklal1XLl1

1 jqkljdl1XL1l1

XLl9.l

qklqkl9aall9

1XL1l1

XLl9.l

qkl1 jqkl9jadll91XL1l1

XLl9.l

1 jqkljqkl9dall9

1

XL1l1

XLl9.l

1 jqklj1 jqkl9jddll9:

If necessary, higher-order interloci interactions (ad-ditional epistatic effects) can also be included in themodel. A parsimonious version of this model can be ob-tained by reducing the order of the estimable epistaticeffects. For example, in a situation with two loci withthree possible genotypes each, only five parameters

would be needed to model phenotypes (i.e., transcrip-

tional profiling) from the nine possible genotypes if noepistasis is to be estimated.

A matrix representation of a two-locus version (allelesA/a and B/b, respectively) without epistasis is given by

y m1n311Q1a1d1

1Q2

a2d2

1 e;

where Ql l 1; 2 is a representation of the designmatrix for each locus. If total sampling control of thepopulation of genotypes is possible, equal representa-tion of each homozygote genotype for each locus couldbe considered. Letrjlrepresent the proportion of geno-type j for locus l. In this case, assuming equal propor-tions of homozygotes within each locus, we have r11 r31 r:1 and r12 r32 r:2. Table 1 presents the num-ber of observations for each genotype and the expectedphenotypic values.

The information matrix for this model is given by

M n

s2e

1 0 1 2r:1 0 1 2r:20 2r:1 0 0 0

1 2r:1 0 1 2r:1 0 1 2r:11 2r:20 0 0 2r:2 0

1 2r:2 0 1 2r:11 2r:2 0 1 2r:2

2666664

3777775

Figure 2.Optimal designs for estimating trans-acting ad-ditive effects (A), dominance effects (B), and both additiveand dominance effects (C) from one gene, with 10 slides ina two-color microarray experiment.



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and so the covariance matrix of the estimated modelparameters is

M1s2en

r:11r:22r:1r:2

2r:1r:2

0 12r

:10 1

2r:2

0 12r

:10 0 0

12r

:10 1

2r:12r:11

0 0

0 0 0 12r

:20

12r

:20 0 0 1

2r:22r:21

2666666664

3777777775:

The optimal designs for this situation are a straight-forward extension of the optimal proportions for eachgenotype: r

:l 12, r:l

14, and r:l 1

ffiffiffi2

p=2, if one is

interested in estimating additive, dominance, or botheffects for each locus l, respectively.

As an example, Table 2 gives the optimal number ofreplications of each genotype in a two-locus experiment

with 10 slides. For estimating additive effects only, thetreatments compose a 22 factorial set, equally replicated.For estimating dominance or both additive and domi-nance effects, the treatments correspond to 32 factorialdesigns with different replications in the two cases. It

can be seen that the design obtained by continuoustechniques can be directly implemented only for ad-ditive effects; in the other two situations an approxima-tion is needed. For dominant effects, the continuousoptimal design gives 0.0625 3 20 1.25 and 0.0125 320 2.5, which were approximated by 1 and 3, re-spectively. This reflects a decrease in the relative im-portance of homozygote combinations as compared to

values for the optimallarge sample design. On the otherhand, the genotype AaBb has only four replicates (in-stead of 0.25 3 20 5). This specific treatment choicedoes not allow a straightforward application of the con-tinuous optimal design theory and the design chosenmight not actually be optimal. The same problem hap-pens when estimating both additive and dominanteffects: either the approximation of 0.0858 3 20 1.716 % 2 or the truncation of 0.1213 3 20 2.426 % 2forces the final treatment choice away from the large-sample optimal genotype proportions.

After the treatment choice is performed, the designof the experiment is completed with the treatment-to-

unit allocation step. Optimal two-color microarray de-signs (with 10 slides) for inferring additive, dominance,or both effects are shown in Figure 3. Similarly to thesituation with a single locus, a clear pattern of allowingonly the important contrasts within each slide arisesfrom these pictures as well. It can be seen that all thecomparisons for estimating additive effects are doneamong homozygotesfor eachlocus.For estimatingdom-inance effects, all slides bring a comparison of a homo-zygous with the heterozygous genotype, for both loci.Moreover, for the simultaneous estimation of additiveand dominance effects, the two types of comparison areneeded. The relative efficiencies of the designs of Figure3, AC, are, respectively, 1.5, 2.9, and 2.1 when com-pared with the loop design and $5.1 compared with thereference design.

Note that these designs are still optimal for estimatingthea-and d-effects for one locus (say for gene A/a) evenif the other locus considered (say B/b) has no effect at

TABLE 1

Expected phenotypic value and number of observationsfor each genotype in a two-locus experiment with

an F2 population

Genotype Expected phenotypic value No. of observations

AABB m1a11a2 r.1r.2nAABb m1a11 d2 r.1(1 2r.2)n

AAbb m1a1 a2 r.1r.2nAaBB m1 d11a2 (1 2r.1)r.2nAaBb m1 d11 d2 (1 2r.1)(1 2r.2)nAabb m1 d1 a2 (1 2r.1)r.2naaBB m a11a2 r.1r.2naaBb m a11 d2 r.1(1 2r.2)naabb m a1 a2 r.1r.2n

TABLE 2

Optimal number of replications of each genotype in a two-locus experiment with no epistasis,using 20 experimental units

Parameters of interestGenotype Optimal proportions as ds (a and d)s

AABB r.1r.2 0.25n 5 0.0625n % 1 %0.0858n % 2AABb r.1(1 2r.2) 0n 0 0.125n % 3 %0.1213n % 2AAbb r.1r.2 0.25n 5 0.0625n % 1 %0.0858n % 2 AaBB (1 2r.1)r.2 0n 0 0.125n % 3 %0.1213n % 2 AaBb (1 2r.1)(1 2r.2) 0n 0 0.25n % 4 %0.1716n % 4 Aabb (1 2r.1)r.2 0n 0 0.125n % 3 %0.1213n % 2aaBB r.1r.2 0.25n 5 0.0625n % 1 %0.0858n % 2aaBb r.1(1 2r.2) 0n 0 0.125n % 3 %0.1213n % 2aabb r.1r.2 0.25n 5 0.0625n % 1 %0.0858n % 2



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all! Similar conclusions can be generalized to situations

with more than two loci and larger numbers of slides.It is interesting to note how different the designs

of Figure 3, which explore a natural structure on treat-ment groups, are from the optimal designs for two-colormicroarray experiments involving unstructured treat-ments, which tend toward symmetry whenever possible.For example, Figure 4 depicts two experiments, oneinvolving four treatments and 10 slides (Figure 4A) andthe other with nine treatments and 9 slides (Figure 4B).The experiment with nine treatments is an example ofthe so-called loop (or circular) design commonly found

in the literature, whereas the experiment with fourtreatments consists of two loops with reverse-dye label-

ing plus two additional interwoven connections.Multiple loci with epistatic effects: Suppose that inthis situation (two loci with three genotypes each and 10slides of a two-color microarray system), we would likealso to estimate trans-acting epistasis (i.e., how the jointeffect of two loci changes the expression of other genesin the genome) with the following model:

tk1 k2;...;kL qk1a11qk2a211 jqk1 jd111 jqk2 jd21qk1 qk2aa121qk1 1 jqk2 jad1211 jqk1 jqk2da1211 jqk1 j1 jqk2 jdd12:

In selecting an optimal design, careful thought must

be given to which classes of designs are considered ascandidates. In this situation, it is reasonable to insist onchoosing a treatment set that is symmetrical with respectto the two gene loci, i.e., that is genewise balanced. Thebest genewise balanced design is given in Figure 5. Notethat in all slides one locus has its genotype repeated

while the other varies. This allows us to infer eachgenomic expression and interaction independently butat the expense of decreasing precision for the maineffects parameters previously of interest (additive anddominance effects).

Figure 3.Optimal designs for estimating trans-acting ad-ditive effects (A), dominance effects (B), and both additiveand dominant effects (C) of two nonepistatic genes, with10 slides in a two-color microarray experiment. Dashed arrowsconnect the homozygous genotypes in the corners.

Figure 4.Optimal designs for two-color microarray ex-periments with four (A) and nine (B) unstructured treat-ments of fixed effects and 10 and 9 slides, respectively.



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GENETIC STUDIES WITH RANDOM TREATMENTEFFECTS

Multiple-treatment microarray experiments: Apartfrom those situations in which 2L or 3L factorials arise

from targeting L genes, there are other genetic objec-tives that need many treatments. One common goal ingenetic studies is to obtain good estimates of genetic var-iances and heritabilities. Within a genetical genomicscontext this means studying the heritability of themRNA abundance of thousands of genes, having onlya sample ofnsubjects (with a specific relationship struc-ture among them) to look at.

In genetic models with random treatments, the treat-ment effects are often the subject effects. This is thecase, for example, with the so-called animal models(Lynch and Walsh 1998), which have been extensivelyused in animal breeding and more recently in plantbreeding (mainly in forestry). For a comprehensive pre-sentation of modern analytical techniques for suchmodels refer to Sorensen and Gianola (2002).

For these situations we have the problem of findinggood designs to obtain estimates using the best linearunbiased predictors of breeding values, given a knownpedigree or family structure. The best designs are veryspecific for each pedigree and estimation objective, asdiscussed below.

Design criterion: Bueno Filho and Gilmour(2003)argued that a suitable criterion for selecting amongrelated treatments can be obtained as a weighted

average of the variances of all pairwise contrasts. Thiscriterion has a broad relevance and estimating genetic

variance components and heritabilities is one of thepossible goals. In the following examples we apply thiscriterion to two small experimental situations to illus-trate some features of optimal designs. In Table 3, twosituations are presented, involving nine animals com-ing from either a hierarchical or a factorial matingsystem.

These two different pedigrees result in the additive(numerator) relationship matrices

Ah

1 0:25 0:25 0 0 0 0 0 00:25 1 0:25 0 0 0 0 0 00:25 0:25 1 0 0 0 0 0 0

0 0 0 1 0:25 0:25 0 0 00 0 0 0:25 1 0:25 0 0 00 0 0 0:25 0:25 1 0 0 00 0 0 0 0 0 1 0:25 0:250 0 0 0 0 0 0:25 1 0:250 0 0 0 0 0 0:25 0:25 1

26666666666664

37777777777775

and

Af

1 0:25 0:25 0:25 0 0 0:25 0 00:25 1 0:25 0 0:25 0 0 0:25 00:25 0:25 1 0 0 0:25 0 0 0:250:25 0 0 1 0:25 0:25 0:25 0 0

0 0:25 0 0:25 1 0:25 0 0:25 00 0 0:25 0:25 0:25 1 0 0 0:25

0:25 0 0 0:25 0 0 1 0:25 0:250 0:25 0 0 0:25 0 0:25 1 0:25

0 0 0:25 0 0 0:25 0:25 0:25 1

26666666666664

37777777777775

(Lynch and Walsh 1998), where the indices h and frefer to the hierarchical and factorial mating systems,respectively.

Restricting our discussion to the estimation of addi-tive variance components, it can be seen that in suchcases we need to find both the best replication for eachtreatment and the best treatment-to-unit allocation. Itis worth noting that in this case treatments and rep-lications can refer either to families and sibs withinfamilies or to biological individuals and technical

samples from the same individual.As an illustration, consider two replications of each

treatment (individual). For this kind of group-divisibletreatment set, it is suggested (Bueno Filho andGilmour2003) that the best designs are in the class ofregular graph designs (e.g., loop designs), but the allo-cation of treatments to treatment labels is important.The best design is one that avoids relatives (in this case,paternal or maternal half-sibs) in the same array, asmuch as possible. Therefore, the resulting optimal de-signs are more restrictive for the factorial than for the

Figure 5.Optimal design for estimating trans-acting ep-istatic effects between two genes, with 10 slides in a two-colormicroarray experiment.

TABLE 3

Examples of hierarchical and factorial mating systems withnine progeny

Hierarchical Factorial

Progeny Dam Sire Dam Sire

1 1 1 1 12 2 1 2 1

3 3 1 3 14 4 2 1 25 5 2 2 26 6 2 3 27 7 3 1 38 8 3 2 39 9 3 3 3



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hierarchical setup. In Figure 6 optimal designs for thesetwo situations are depicted.

More complex situations can be found in practice, inwhich larger numbers of individuals can be consideredand some other important measures can be taken.Lynch and Walsh (1998) discuss simple pedigrees in

which analytical results are possible and suggest the best

number of families and individuals within families touse in the experiment. For more complex pedigrees,however, a general search algorithm should be utilized,as sometimes it isthe only way to get a useful design. Thealgorithm we have used is a simple modification ofthe exchange algorithm of Fedorov (1972). This hasthe following general form:

1. List the possible candidate treatments.2. Choose a random set ofntreatments, which becomes

the currentdesign.

3. If the current design does not allow the model to befitted, return to step 2.

4. Systematically exchange each treatment in the cur-rent design with each (different) treatment in thecandidate set. Accept any exchanges that improvethe criterion to get a new current design, continuinguntil no exchange improves the design.

5. When no improvement is possible, the currentdesign becomes the final design returned by thealgorithm.

It is common practice to restart the algorithm withseveral random starting designs and to choose the bestdesign found overall.

As an illustration, consider a treatment choice prob-lem with a microarray experiment with a limited num-ber of slides, for which a subset of 40 subjects needs tobe chosen among a population of 80 animals with thepedigree givenin Figure 7. An R function was developedto search for optimal designs when treatment effectsare random, with any covariance structure among them(i.e., any pedigree setup), such as in this situation. Thetreatments (animals) selected in the resulting designare indicated by solid circles in Figure 7. Such a designis good only for this very specific situation and it isnot possible to produce deep generalizations. The only

clear message with messy pedigrees is that a searchalgorithm may be the only way to reach a good (if notoptimal) design for these experiments.

In another very common situation in animal exper-imentation, related individuals are subjected to differ-ent experimental procedures (such as different diets,drugs, etc.). In these circumstances both random andfixed treatment effects are present in the experiment.Such experiments may have multiple objectives and

variance component (or heritability) estimation canbe one of them. One should be aware, however, that

Figure 6.Optimal design for a microarray experimentwith nine treatments, having random effects, representingnine progenies from hierarchical (A) and factorial (B) matingsystems (Table 3). (A) Different circle shadings refer to differ-

ent paternal half-sibs. (B) Each shape and shading denote adifferent dam and sire, respectively. In both cases, each geo-metric figure represents a subject (biological replication),

which is assayed in two slides each (technical replication).

Figure 7.Three-generation pedigree with 80 individualsin two genetically disconnected groups, from which 40 indi-

viduals (solid circles) were selected for a microarray experi-ment to infer mRNA abundance heritabilities.



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designs that are optimal for some of these objectivesmight not be so for others. An example of such a sit-uation is given in Table 4. The experiment consists of 20pigs from five full-sib families of size four, to be arrayedon a limited set of 10 two-color system slides. The fourindividuals in each family (litter) refer to two male andtwo female sibs that were selected at random. One ani-mal from each sex was submitted to one of two experi-mental treatments (placebo and drug). Two differentobjectives of the study were considered: (a) estimationof heritability and (b) estimation of the fixed effects ofdrug and sex, as well as the interaction between them.Optimal designs for each of these situations for a spe-cific configuration of variance components are pre-sented in Figure 8.

It can be seen that in the first situation (estimation ofheritability) the search algorithm converged to a designin which all slides connected individuals from differentsibs. Also, all slides have individuals of the same sex

and/or same experimental group (placebo or drug).Conversely, if the experiment goal is to infer the ef-fects of sex, drug, and their interaction, the geneticcovariance among sibs becomes another blocking fac-tor, which the search algorithm tended to confound orcombine with the blocking effect of arrays. In this case,the slides now tend to assay genetically related individ-uals. As expected, to infer the main effects of drug andsex, some of the slides compare individuals of the samesex, but different experimental groups (such as those offamily 5), while others compare individuals of the same

group, but different sex (such as those of family 3). Inaddition, a third set of slides compares individuals ofdifferent sex and experimental groups (e.g., family 1) toinfer the interaction between sex and drug effects.

CONCLUDING REMARKS

This article discusses the design of microarray experi-ments whose goalsrefer to a variety of genetic questions,such as the comparison of expression levels of different

genotypic groups, eQTL studies, or estimation of heri-tabilities of mRNA transcript abundances.

The general approach adopted for choosing a gooddesign consists of a first step oftreatment choice, in whichsubjects (as well as their experimental group labels) areselected to be included in theexperiment. This first stepis general and common to any microarray experiment,regardless of platform. In the case of two-color micro-array systems, however, the design choice is followed bytreatment-to-unit allocation, in which mRNA samples areassigned in an optimal way to slides and dye labeling.

Figure 8.Optimal designs for the estimation of heritabil-ity (A) and fixed effects of drug and sex and their interaction(B), in a two-color microarray experiment with five full-sibswine families of two males (squares) and two females (cir-cles), submitted either to a placebo (open squares and cir-

cles) or to a drug treatment (shaded squares and circles).

TABLE 4

Litter, treatment (drug), and sex of 20 piglets availablefor microarray experiments for inferring the effects

of sex, drug, and the interaction between sexand drug, as well as the heritability

Progeny Litter Drug Sex

1 1 Yes F

2 1 Yes M3 1 No F4 1 No M5 2 Yes F6 2 Yes M7 2 No F8 2 No M9 3 Yes F10 3 Yes M11 3 No F12 3 No M13 4 Yes F14 4 Yes M15 4 No F16 4 No M17 5 Yes F18 5 Yes M19 5 No F20 5 No M



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It has been shown how the design choice should beintimately related to the goal of the experiment, such asthe contrast between fixed effects (e.g., the comparisonof alternative drugs or the expected phenotypic value ofdifferent genotypes), or inference regarding variancecomponents (such as genetic variances), or heritabilityof transcriptional levels.

Sometimes it is possible to have an analytical solution

for the optimality problem, but in many situations anumerical solution is needed. Whenever a numericalsolution is implemented, a complete search (comparingall possible designs) guarantees the optimal finding.

A complete search, however, is often unattainable dueto the size of the search problem involved. In these si-tuations algorithms are needed that can find optimal ornear-optimal designs. R functions were developed forsearching and comparing the different designs discussedin this article; they can be obtained directly from theauthors.

As a final remark, it is important to note that thisarticle considers a single gene at a time for design pur-

poses, as the same model is generally used to analyze allgenes in a single experiment. This is not an issue forfixed-effects models (such as for the examples compar-ing genotypic groups) as the optimal design criteria forfixed-effects models depend only on the design matrix.This is not true in mixed- or random-effects models, asin these cases optimal designs depend on both thedesign matrix and the variance components parame-ters. Since the variance parameters vary among genes,the optimal design for one gene may be different fromthat for another gene. As a consequence, caution isneeded before we simplify the optimal design for themicroarray to an optimal design for a single gene ifrandom- or mixed-effects models are considered.

This is a very interesting topic, which deserves furtherresearch. A possible approach to this issue would be touse an empirical distribution for the genetic variancesor heritabilities to weight the decision function that

yields the design criterion. Alternatively, robust de-signs could be sought by focusing on the high herita-bility genes. This could be achieved by putting more

weight on the higher heritabilities than suggested by anempirical distribution or by even ignoring the region oflow heritability when searching for good designs. Anextreme version of this approach would be by taking

genetic effects as fixed, but in general this would be aninefficient strategy. Moreover, while modeling geneticeffects as fixed may be simple with experiments in-

volving family structures such as half- or full-sibs, it isnot clear how this approach could be easily extendedfor complex pedigrees.

This work was supported by the Michigan State University Intra-

mural Research Grant Program, by grant 2004-33120-15204 from theUnited States Department of Agriculture to G.J.M.R., and by grant474787/2004-4 from the Conselho Nacional de Desenvolvimento

Cientfico e Tecnologico-Brazil to J.S.S.B.F.

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