Design of Microarray Experiments for Genetical Genomics Studies

Transcription

1 Genetics: Published Aticles Ahead of Pint, published on August 3, 6 as.534/genetics Design of Micoaay Expeiments fo Genetical Genomics Studies Júlio S. S. Bueno Filho *, Steven G. Gilmou and Guilheme J. M. Rosa, * Depatamento de Ciências Exatas, Univesidade Fedeal de Lavas, Lavas, Minas Geais 37-, Bazil School of Mathematical Sciences, Queen May, Univesity of London, London, E 4NS, UK Depatment of Animal Science and Depatment of Fisheies & Wildlife, Michigan State Univesity, East Lansing, MI , USA July 7, 6

2 Key wods: Optimal design, Genetical genomics, Micoaay, Selective phenotyping, Mixed models Running head: Design of Genetical Genomics Studies Coesponding autho s cuent addess: Guilheme J. M. Rosa, 456 Animal Sciences Building, 675 Obsevatoy D., Univesity of Wisconsin - Madison, Madison, WI Phone (68) Fax: (68) gosa@wisc.edu

3 ABSTRACT Micoaay expeiments have been used ecently in genetical genomics studies, as an additional tool to undestand the genetic mechanisms govening vaiation in complex taits, such as fo estimating heitabilities of mrna tanscipt abundances, fo mapping expession quantitative tait loci and fo infeing egulatoy netwoks contolling gene expession. Seveal papes on the design of micoaay expeiments discuss situations in which teatment effects ae assumed fixed and without any stuctue. In the case of two-colo micoaay platfoms, seveal authos study efeence and cicula designs. Hee, we discuss the optimal design of micoaay expeiments whose goals efe to specific genetic questions. Some examples ae used to illustate the choice of a design fo compaing fixed, stuctued teatments, such as genotypic goups. Expeiments tageting single genes o chomosomic egions (such as with tansgene eseach) o multiple epistatic loci (such as within a selective phenotyping context) ae discussed. In addition, micoaay expeiments in which teatments efe to families o to subjects (within family stuctues o complex pedigees) ae pesented. In these cases teatments ae moe appopiately consideed to be andom effects, with specific covaiance stuctues, in which the genetic goals elate to the estimation of genetic vaiances and the heitability of tansciptional abundances. Micoaay technology allows the monitoing of messenge RNA (mrna) abundance in cells fo thousands of genes simultaneously (SCHENA et al. 995, LOCKHART et al. 996). Micoaay expeiments have been used extensively to povide a suogate measue of gene activity to compae expession levels of teatment goups (within eithe expeimental o obsevational contexts), such as diffeent tissues, cell types (e.g., disease and nomal cell) in a specific tissue, diffeent developmental stages, o expeimental conditions (e.g., diffeent dugs o stess conditions). The application of this technology howeve, especially fo two-colo platfoms (eithe cdna o long oligonucleotide aays), poses some expeimental design challenges. As samples ae assayed in a paiwise competitive hybidization fashion on each slide, micoaay expeiments using two-colo systems impose a block stuctue with blocks of size two. Wheneve moe than two goups ae compaed, this defines an incomplete 3

4 block design, in which estimated teatment effects and diffeences between slides ae patially confounded. In addition, as each sample is labeled with a diffeent dye on each slide, micoaay expeiments involve an additional blocking facto to be accounted fo, which leads to a ow-column, o two-way blocking, stuctue (slide-dye combination). Many papes on design fo two-colo micoaay expeiments make some compaison of efficiency o powe (KERR and CHURCHILL, YANG and SPEED, KERR 3, TEMPELMAN 5, ROSA et al. 5, STEIBEL and ROSA 5, VINCIOTTI et al. 5) of diffeent designs. Most of them focus on the compaison of a discete set of designs, such as efeence and cicula stuctues. WIT et al. (5) pesent a moe geneal appoach, seaching fo optimal designs given specific expeimental conditions and optimality citeia. It is shown that the optimal micoaay design fo a specific optimality citeion may efe to a moe complex expeimental layout than simple efeence o cicula stuctues. The objective of the expeiment guides the choice of the teatment stuctue, wheeas the definition of the expeimental units leads to the blocking stuctue of the expeiment. As discussed thoughout this pape, the teatment choice in micoaay expeiments is geneal fo any platfom (e.g. high-density shot oligos, cdna, o long oligonucleotide technologies), wheeas the blocking stuctue dependents on the micoaay technology, such as two-colo aays (in which slides efe to a blocking facto) o single-colo aays such as Affymetix (in which each sample is hybidized to a diffeent slide). The teatment stuctue (as well as the numbe of eplications) depends also on the pactical/logistical constaints of the expeiment, such as limited biological samples (limited eithe by the numbe of individuals to be assayed o by the amount of mrna fom each individual) o numbe of slides available. The choice of the expeimental unit, which can be an individual (such as a plant o animal, heeinafte denoted as a subject) o a goup of subjects (such as a family o mrna pool), may impose some spatial o genetic stuctue among units, which should be taken into account when assigning teatments to units. Micoaay expeiments have been ecently used on genetical genomics studies, as an additional tool to undestand the genetic mechanisms govening vaiation in complex taits, such as disease susceptibility in human medicine o poduction taits in agicultue. Micoaay expeiments using elated subjects can be utilized, fo example, to infe 4

5 heitabilities of mrna tanscipt abundances (MONKS et al. 4). In addition, gene expession pofiling can be combined with make genotype infomation fo mapping expession quantitative tait loci (eqtl; JANSEN and NAP, BREM et al., DARVASI 3, SCHADT et al. 3), and to infe egulatoy netwoks contolling gene expession (CHESLER et al. 4, BYSTRYKH et al. 5). In this pape we discuss the optimal design of micoaay expeiments whose goals efe to specific genetic questions, such as the compaison of expession levels of diffeent genotypic goups, eqtl studies, o estimation of heitabilities of mrna tanscipt abundances. This goes beyond the scope of ealie wok on fixed, unstuctued teatments. The expeimentes inteest might be in specific teatment compaisons, e.g. efeing to additive and dominance effects. Teatments may efe to families o to subjects (within family stuctues o complex pedigees) in a given expeiment. In these cases teatments ae moe appopiately consideed to be andom effects, with specific covaiance stuctues. These covaiance stuctues should then be taken into account when planning micoaay expeiments, so that the efficiency of boad infeences to the population of inteest is maximized. The high cost of micoaays geatly limits the sample sizes of heitability o gene mapping studies of mrna abundance taits (JIN et al. 4). Theefoe, a caeful design of such expeiments is extemely impotant to make the most of the limited numbe of slides that ae geneally available in a single expeiment. The manuscipt is oganized as follows. In the next section we eview some expeimental design concepts, such as teatment choice, teatment to unit allocation, design optimality citeia and some useful methods and esults on optimal designs. The thid section pesents genetic eseach goals that can motivate micoaay studies with two o moe teatments, such as genotypic goups. Teatments ae consideed fixed, but the genetic component that defines the teatments povides a natual stuctue (which is tanslated into genetic paametes of inteest, such as additive and dominance effects), which should be exploed when designing the expeiments. In the fouth section, the design of micoaay expeiments with andom teatments is discussed. Teatments in this case may efe to elated individuals o families, and the main goal of the expeiments elates to infeence on genetic vaiances and heitabilities. Finally, some concluding 5

6 emaks ae pesented in the last section. DESIGN PRINCIPLES AND CRITERIA The geneal statistical pinciples of the design of expeiments apply to expeiments of any kind, including micoaay studies. The geneal pocedue fo choosing a good design involves the following steps (see also BAILEY 98, 998 and MEAD 99): ) The objectives of the expeiment lead to the choice of a set of teatments (goups to compae). ) The expeimental units (hee the combination of slides and dyes) that can be used ae identified and any expected pattens of vaiability among them lead to a choice of blocking stuctue (hee a ow-column stuctue). 3) Consideation of any estictions on which teatments can be applied to which units might lead to complex stuctues such as split-plot designs. 4) Given -3, a combinatoial, algoithmic o ad hoc method is found to constuct a design. The block stuctue is aleady clea and we will not discuss complex stuctues hee, so the two main tasks in developing a design ae teatment choice and teatment to unit allocation. Teatment choice: When the expeimental esouces ae limited, a choice of the teatments that will be epesented in the expeiment, fom a lage set of candidate teatments, might be needed. In some situations paticula teatments cannot be poduced, o the numbe of candidate teatments is too big fo all of them to be implemented. Fo example, suppose we want to study isolines fo tansgenic vesions of thee genes (A/a, B/b and C/c) fom some paental line. Even esticting each locus to expess just the homozygote genotypes, it is unlikely that all seven desied isolines will be obtained by tansfomation events o any othe cuently known gene manipulation techniques. Stating fom the non-tansgenic genotype (AABBCC), fo instance, a single mutation may give ise to the genotypes AABBcc, AAbbCC, o aabbcc; howeve, two o thee mutations ae needed to obtain the genotypes AAbbcc, aabbcc and aabbcc, o aabbcc, 6

7 espectively. Anothe and moe obvious example is given by patial diallels, in which some cossings cannot be done. We will discuss late in this pape some multilocus analysis poblems. In these cases, we need to define a model fo the teatment effects and an optimality citeion elated to the vaiances of the estimated paametes of the model. Then we must find a method of seaching fo a design that optimizes the citeion, e.g. using the optimal continuous (appoximate) design theoy developed fom KIEFER and WOLFOWITZ (96) o a compute algoithm that seaches fo the optimal exact design, such as those developed fom FEDOROV (97). As stated ealie, the teatment choice in micoaay expeiments is independent of the aay platfom, so the same pocedue can be used fo single- o two-colo technologies. Teatment to unit allocation: A given set of teatments (and thei espective eplication numbes) should be combined in an optimal way within a given expeimental set-up. In the context of two-colo micoaay expeiments, fo example, this set-up is given by the numbe of slides and the two labels (Cy3 and Cy5 dyes). Again, constuction of the design might equie optimal design theoy, this time most likely using a compute seach. The teatment to unit allocation applications discussed in this pape ae specific to two-colo aay platfoms (cdna o long oligos), in which samples should be allocated in pais to the slides (blocks). With single colo micoaays (such as Affymetix), on the othe hand, each sample is hybidized to a diffeent slide, so this is not an issue and allocation should be done completely at andom. Linea Mixed Models fo Micoaay Expeiments The outcomes of a micoaay expeiment ae luminous intensities of thousands of genes, which ae popotional to thei mrna tansciptional abundances. Afte suitable nomalization, the esponse intensities fom gene g in a two-colo micoaay expeiment can be epesented by the model (WOLFINGER et al. ) y = µ + d + a + t + e, () gijk g gi gj gk gij whee y gijk is the nomalized intensity of the gene expession measued by the dye (label) i of the aay j that eceived teatment k, µ g is a common expeimental constant, 7

8 d gi is the effect of dye i (geen o ed), a gj is the effect of aay (o slide) j, t gk is the effect of the teatment k and e gij is a esidual tem. We will conside a single gene at a time fo design puposes, as the same model is geneally used to analyze all genes in a single expeiment, and so the subscipt g will be dopped. Diffeent expeimental layouts may need exta tems in the model fo the appopiate modeling of the data. With Affymetix data, fo example, thee is no dye effect in the model, but a hieachical stuctue could be used to model intensities at the pobe level, by including the effects of pobe sets (clones) and of pobe pais within clones. Model choice should then be closely connected to teatment and block stuctues of the expeiments (ROSA et al. 5). As additional assumptions of model (), esidual tems can be modeled as being nomally distibuted with vaiance σ e ; likewise, as aays come fom a population of possible assembled aays, thei effects can be modeled as andom nomal vaiables centeed aound zeo, with vaiance σ a. The tem teatment hee is used in a vey boad context, epesenting eithe expeimental goups (such as genotypic classes) o families (and subjects within families) in quantitative genetics studies to estimate heitabilities. Theefoe, teatment effects can be modeled as fixed o andom, depending on the objective of the expeiment. The usual matix notation fo linea mixed models such as model () is y = Xβ + Zu + e, () whee y and e ae vectos with elements y ijk and e ij, espectively, β is the vecto of fixed effects, X is a known (design) matix that elates y to β, u is the vecto of andom effects and Z is the matix that elates y to u. An exta assumption is that u ~ N (, U), whee U is the covaiance matix of the andom effects. Note that the teatment effects can be taken as being eithe fixed o andom and so they can be contained in eithe β o u, depending on the expeimental context. In the examples discussed in this pape, wheneve teatment effects ae consideed andom, they will be assumed nomally distibuted, i.e. t ~ N(, Gσ ), whee G is a matix of known constants epesenting, fo example, genetic elationships among the elements in t (such as a numeato elationship matix A, HENDERSON 976), and among teatments, such as an additive genetic vaiance. t σ t is the vaiance 8

9 Design Citeia The covaiance matix of the paamete estimates fo model () is the invese of the infomation matix M, whee XX ' Z' X M. XZ ' Z' Z + σeu Almost any sensible citeion defining design efficiency is a function of the infomation matix M. Costs ae impotant when compaing altenative expeimental designs. In this pape, howeve, designs will be geneally compaed fo a given numbe of slides. As pointed out by YANG and SPEED (), the numbe of slides is geneally the most impotant limiting facto in micoaay expeiments. In addition, fo two-colo micoaay platfoms, as the vaiation between slides is geneally lage than the vaiation within slides, blocking by slides is cucial. Since the atio of between-slide to within-slide vaiation is unknown, it is usual to design expeiments fo the wost case, i.e. whee this atio is vey lage, in which case the design poblem becomes equivalent to one with fixed blocks. Note, howeve, that afte the expeiment is un, in the mixed models data analysis, we will estimate the vaiance components; if thei atio tuns out to be lage, we will estimate the teatment effects with the pecision expected when we chose the design; if thei atio tuns out to be small, then the inte-block infomation will give us estimates of teatment effects bette than expected when we chose the design. In this context, KERR and CHURCHILL () and YANG and SPEED () used the so-called A-optimality citeion to compae block designs fo micoaay studies with simple teatment stuctues. KERR (3) late agued in favo of double efeence designs as being suboptimal but moe obust when compaed to loop designs. GLONEK and SOLOMON (4) studied some altenative designs fo factoial teatment stuctues. Moe geneal wok on two-level factoial designs in blocks of size two, which has some elevance, has been done by DRAPER and GUTTMAN (997), YANG and DRAPER (3), WANG (4) and KERR (6). Simila sots of expeiments ae now being used fo undestanding the genetic basis of mrna tanscipt abundances, with a vaiety of expeimental settings. Fo example, fixed teatment effect models aise in genetic analysis when compaing multiloci 9

10 genotypic goups, including inteaction (epistasis) tems. In addition, andom teatment effect models may be used in quantitative genetics to infe genetic vaiances and heitabilities. No wok howeve has been published on optimal design fo micoaay expeiments in situations whee the teatments have any of these stuctues. In each of the following sections we pesent diffeent expeiments, and discuss possible paamete contasts of inteest, as well as the vaiances of thei estimates, sometimes weighted to coect fo scale poblems. In each situation, altenative optimality citeia ae discussed. We assume that all powe compaisons have a monotonic (invese) elation with the vaiances of the desied compaisons and descibe teatment stuctues in micoaay expeiments fo some common goals in genetic studies. The esulting designs ae depicted by a set of aows epesenting the slides, with heads and tails denoting altenative dye labeling (Cy3 and Cy5). Aows (slides) efe always to diffeent biological eplications fom each expeimental goup, unless othewise stated. GENETIC STUDIES WITH FIXED TREATMENT EFFECTS Two Teatment Goups in Micoaay Expeiments Genetic motivation: One of the simplest scenaios that esults in two-teatment expeiments aises with a single candidate gene (with alleles B and b) and a backcoss (BC) o double haploid (DH) expeiment. In these cicumstances, individuals belong to one of two possible genotypic goups, namely Bb o bb if in a BC expeiment, and BB o bb if in a DH scenaio. In both situations one is inteested in the additive effect (α ) of the candidate gene on the expession of each gene epesented in the micoaay slide, i.e. how polymophisms at a specific site of the genome affect tansciption activity of othe genes in the genome. Such effects ae geneally efeed to as tans-acting factos (JANSEN and NAP ). The model () fo such a simple teatment stuctue can be witten with t k = q k α. Fo the BC situation, q k is a design index on genotypes assuming values q = (fo genotype bb) and q (fo genotype Bb). In this situation, one wants to estimate =

11 α = t t and the vaiance of this contast is given by Va( tˆ -tˆ ) = Va( tˆ ) + Va( tˆ )-Cov( tˆ,t ˆ ). Fo the DH situation, q = (fo genotype bb) and = q (fo genotype BB). In this situation, one wants to estimate α = ( t t ), and the vaiance of this compaison is given by one quate of the ight-hand side of the above vaiance. As these two vaiances ae popotional, the same optimal design minimizes both of them. In a micoaay expeiment, such a simple two teatment goup situation can aise within a selective phenotyping appoach based on a single locus (o small chomosomic egion), as poposed by JIN et al. (4), o within a candidate gene context in which one is inteested on the effect of a specific locus (e.g. tansgene) on the expession of othe genes in the genome (see fo example NOUEIRY et al. ). Anothe example is to define two expeimental goups by taking the extemes of the phenotypic distibution of a quantitative tait of inteest (LANDER and BOTSTEIN 989). By compaing the allelic fequencies of molecula makes between the two goups, candidate egions containing genes contibuting to vaiation of the phenotypic tait can be pinpointed. In the context of micoaays, candidate genes ae geneated by seaching fo diffeential tanscipt abundances between the two goups. In any situation, the popotions of the selected individuals should be made small enough so that the two phenotypic goups ae quite distinct (DARVASI and SOLLER 99). Micoaay expeiments with two expeimental goups define ow-column designs obtained fom multiple copies of a Latin squae (in the case of two-colo systems), fo which an extensive liteatue is available. Thee Teatment Goups in Micoaay Expeiments Genetic motivation: In some situations thee ae thee possible genotypes in a segegating population, fo example with F populations and co-dominant makes. The genotypes fo each make can then be descibed by an additive-dominance model. In this section we discuss the case of a single locus (such as a candidate gene situation), and its tans-acting effect on the expession of othe genes. The case of multiple loci is discussed late in this pape, with factoial and factional factoial teatment stuctues. Teatment choice: An additive-dominance teatment effect of a candidate gene in an

12 F population can be expessed as t = qkα + ( - qk ) δ, whee q k = -,,, fo genotypes k bb, Bb, and BB, espectively. Denote by k the popotion of the n obsevations that will epesent each genotypic goup (k =,, 3). The model can be ewitten as α y = µ n + Q + e, whee nx is a column vecto of ones and Q gives the incidences δ of genotypes bb, Bb and BB as L L L Q ' = L L L. 4 n n n 3 Both paametes can be optimally estimated using the same popotion fo each homozygous genotype (bb and BB), in which case = 3 = and =. The infomation matix is then given by and the covaiance matix is M = n σ σ e M =. n ( ) Figue plots changes in the vaiances of the paamete estimates with the popotion () of each homozygous goup. If one is solely inteested in the (tans-acting) additive effects, just homozygotes should be consideed in the design. To estimate (tans-acting) dominance only, half of the obsevations should be taken fom the heteozygotes and one quate fom each of the homozygotes. To estimate additive and dominance effects simultaneously, it is necessay to define an optimality citeion. Thoughout this pape, we use the A-efficiency on this scale, i.e. the mean of the vaiances of the estimates, but in any eal expeiment caeful thought should be given to whethe a weighted mean of the vaiances would be a moe appopiate citeion. The A-optimal popotion of each homozygote is the iational numbe =. As thee is no possible numbe of opt e

13 aays that satisfies this optimum exactly, an appoximation is needed. It is easonable to use about.93 of the obsevations fo each homozygote and the emaining.44 of the obsevations fo the heteozygote goup. <Please inset Figue hee> Suppose, fo example, that we ae inteested in the best design that can be made with only slides using a two-colo micoaay platfom. We poceed as follows. To estimate just additive effects, each homozygous genotype (BB o bb) will have eplications. The heteozygote goup, in this case, is not included in the expeiment. To estimate only dominance effects thee will be five eplications of each homozygous genotype and eplications of the heteozygote goup. Finally, fo estimation of both additive and dominance effects, the A-optimal design has six eplications fo each homozygote and the emaining eight eplications fo the heteozygote. Teatment to unit allocation: Afte the teatments have been chosen (i.e. the numbe of eplications fo each genotypic goup), the next step in planning the micoaay expeiment is the teatment-to-unit allocation, i.e. descibing which teatment should be applied to each unit, when aay and colo effects ae in the model. The optimal design fo estimating just additive effects with slides ae shown in Figue a, in which all slides have the two homozygous genotypes only, with altenating diections of the Cy3 and Cy5 labeling. If one is concened with estimating dominant effects, all of the slides should have one of the homozygous genotypes co-hybidized with the heteozygous genotype, as shown in Figue b. Finally, if both additive and dominance effects ae to be estimated, combinations of the two types of slide should be used (Figue c). It is woth mentioning that the designs of Figues a, b and c ae, espectively, about %, 5% and 9% moe efficient than the classical loop stuctue. When compaed to the efeence design, thei elative efficiencies ae 6.6, 4.7 and 3.4, espectively. <Please inset Figue hee> 3

14 Genealization of These Results fo Multiple Loci The effect of multiple candidate genes on the oveall genome tansciptional activity can be investigated at once in a single micoaay study. With simple line cossing expeiments, each gene (locus) may have eithe two o thee levels (possible genotypes), depending on the mating system used (BC o DH, and F, espectively). Theefoe the teatment stuctue fo L loci is a factoial of the seies L o 3 L fo two o thee possible genotypes in each locus, espectively. A suitable model fo the expession of the teatments in such a system, including epistasis, is given by t k k L k L = L l= q k l α l + L l= L L l= l' > l L L ( - qk ) δl + qk qk αα ' + ( ' ll qk - q l l l l kl ' l= l' > l ) αδ ll' + L L l= l' > l L L ( - qk ) qk δ α ' + ( )( ) ' ll qk qk δδ l l l l ' ll'. l= l' > l If necessay, highe ode inte-loci inteactions (additional epistatic effects) can also be included in the model. A pasimonious vesion of this model can be obtained by educing the ode of the estimable epistatic effects. Fo example, in a situation with two loci with thee possible genotypes each, only five paametes would be needed to model phenotypes (i.e. tansciptional pofiling) fom the nine possible genotypes if no epistasis is to be estimated. A matix epesentation of a two-locus vesion (alleles A/a and B/b, espectively) without epistasis is given by: α α y = µ n + Q + Q + e, δ δ whee Q ( l =, ) is a epesentation of the design matix fo each locus. If total l sampling contol of the population of genotypes is possible, equal epesentation of each homozygote genotype fo each locus could be consideed. Let jl epesent the popotion of genotype j fo locus l. In this case, assuming equal popotions of homozygotes within each locus, we have = 3 = and = 3 =. Table pesents the numbe of obsevations fo each genotype and the expected phenotypic values. <Please inset Table hee> 4

15 5 The infomation matix fo this model is given by = ) )( ( ) )( ( n σ e M and so the covaiance matix of the estimated model paametes is + = ) ( ) ( n σ e M. The optimal designs fo this situation ae a staightfowad extension of the optimal popotions fo each genotype: = / l, 4 = / l and = l, if one is inteested in estimating additive, dominance, o both effects fo each locus l, espectively. As an example, Table gives the optimal numbe of eplications of each genotype in a two-locus expeiment with slides. Fo estimating additive effects only, the teatments compise a factoial set, equally eplicated. Fo estimating dominance o both additive and dominance effects, the teatments coespond to 3 factoial designs with diffeent eplications in the two cases. It can be seen that the design obtained by continuous techniques can be diectly implemented only fo additive effects; in the othe two situations an appoximation is needed. Fo dominant effects, the continuous optimal design gives.65 =.5 and.5 =.5, which wee appoximated by and 3, espectively. This eflects a decease in the elative impotance of homozygote combinations as compaed to values fo the optimal lage sample design. On the othe hand, the genotype AaBb has only fou eplicates (instead of.5 = 5). This specific teatment choice does not allow a staightfowad application of the continuous optimal design theoy and the design chosen might not actually be optimal. The same poblem

16 happens when estimating both additive and dominant effects: eithe the appoximation of.858 =.76 o the tuncation of.3 =.46 foces the final teatment choice away fom the lage-sample optimal genotype popotions. <Please inset Table hee> Afte the teatment choice is pefomed, the design of the expeiment is completed with the teatment-to-unit allocation step. Optimal two-colo micoaay designs (with slides) fo infeing additive, dominance o both effects, ae shown in Figue 3. Similaly to the situation with a single locus, a clea patten of allowing only the impotant contasts within each slide aises fom these pictues as well. It can be seen that all the compaisons fo estimating additive effects ae done among homozygotes fo each locus. Fo estimating dominance effects, all slides bing a compaison of a homozygous with the heteozygous genotype, fo both loci. Moeove, fo the simultaneous estimation of additive and dominance effects, the two types of compaison ae needed. The elative efficiency of the designs of Figue 3a, 3b and 3c ae espectively.5,.9 and. when compaed with the loop design and about 5. compaed with the efeence design. <Please inset Figue 3 hee> Note that these designs ae still optimal fo estimating the α and δ effects fo one locus (say fo gene A/a) even if the othe locus consideed (say B/b) has no effect at all! Simila conclusions can be genealized to situations with moe than two loci and lage numbes of slides. It is inteesting to note how diffeent the designs of Figue 3, which exploe a natual stuctue on teatment goups, ae fom the optimal designs fo two-colo micoaay expeiments involving unstuctued teatments, which tend towad symmety wheneve possible. Fo example, Figue 4 depicts two expeiments, one involving fou teatments and ten slides (Figue 4a) and the othe with nine teatments and nine slides (Figue 4b). The expeiment with nine teatments is an example of the so-called loop (o cicula) design commonly found in the liteatue, wheeas the expeiment with fou teatments consists of two loops with evese dye labeling plus two additional intewoven 6

17 connections. <Please inset Figue 4 hee> Multiple Loci with Epistatic Effects Suppose that in this situation (two loci with thee genotypes each and slides of a two-colo micoaay system), we would like also to estimate tans-acting epistasis (i.e. how the joint effect of two loci changes the expession of othe genes in the genome) with the following model: t k kl k = qk α + qk α + - qk ) δ + (- qk ) δ + qk q L k ( αα + q ( k - q ) ( ) ( )( ) k αδ + - q q δα q q δδ k k + k k. In selecting an optimal design, caeful thought must be given to which classes of designs ae consideed as candidates. In this situation, it is easonable to insist on choosing a teatment set which is symmetical with espect to the two gene loci, i.e. which is genewise balanced. The best genewise balanced design is given in Figue 5. Note that in all slides one locus has its genotype epeated while the othe vaies. This allows us to infe each genomic expession and inteaction independently but at the expense of deceasing pecision fo the main effects paametes peviously of inteest (additive and dominance effects). <Please inset Figue 5 hee> GENETIC STUDIES WITH RANDOM TREATMENT EFFECTS Multiple Teatment Micoaay Expeiments Apat fom those situations in which L o 3 L factoials aise fom tageting L genes, thee ae othe genetic objectives that need many teatments. One common goal in genetic studies is to obtain good estimates of genetic vaiances and heitabilities. Within a genetical genomics context this means studying the heitability of the mrna abundance of thousands of genes, having only a sample of n subjects (with a specific elationship stuctue among them) to look at. 7

18 In genetic models with andom teatments, the teatment effects ae often the subject effects. This is the case fo example with the so-called Animal Models (LYNCH and WALSH 998), which have been extensively used in animal beeding and moe ecently in plant beeding (mainly in foesty). Fo a compehensive pesentation of moden analytical techniques fo such models efe to SORENSEN and GIANOLA (). Fo these situations we have the poblem of finding good designs to obtain estimates using the best linea unbiased pedictos of beeding values, given a known pedigee o family stuctue. The best designs ae vey specific fo each pedigee and estimation objective, as discussed below. Design Citeion BUENO FILHO and GILMOUR (3) agued that a suitable citeion fo selecting among elated teatments can be obtained as a weighted aveage of the vaiances of all paiwise contasts. This citeion has a boad elevance and estimating genetic vaiance components and heitabilities is one of the possible goals. In the following examples we apply this citeion to two small expeimental situations to illustate some featues of optimal designs. In Table 3, two situations ae pesented, involving nine animals coming eithe fom a hieachical o a factoial mating system. <Please inset Table 3 hee> These two diffeent pedigees esult in the additive (numeato) elationship matices (LYNCH and WALSH 998) and A h =

19 A f = , whee the indices h and f efe to the hieachical and factoial mating systems, espectively. Resticting ou discussion to the estimation of additive vaiance components, it can be seen that in such cases we need to find both the best eplication fo each teatment and the best teatment to unit allocation. It is woth noting that in this case teatments and eplications can efe eithe to families and sibs within families o to biological individuals and technical samples fom the same individual. As an illustation, conside two eplications of each teatment (individual). Fo this kind of goup-divisible teatment set, it is suggested (BUENO FILHO and GILMOUR 3) that the best designs ae in the class of egula gaph designs (e.g. loop designs), but the allocation of teatments to teatment labels is impotant. The best design is one that avoids elatives (in this case, patenal o matenal half sibs) in the same aay, as much as possible. Theefoe, the esulting optimal designs ae moe estictive fo the factoial than fo the hieachical setup. In Figue 6 optimal designs fo these two situations ae depicted. <Please inset Figue 6> Moe complex situations can be found in pactice, in which lage numbes of individuals can be consideed and some othe impotant measues can be taken. LYNCH and WALSH (998) discuss simple pedigees in which analytical esults ae possible and suggest the best numbe of families and individuals within families to use in the expeiment. Fo moe complex pedigees, howeve, a geneal seach algoithm should be utilized, as sometimes it is the only way to get a useful design. The algoithm we have used is a simple modification of the exchange algoithm of Fedoov (97). This has the 9

20 geneal fom:. List the possible candidate teatments.. Choose a andom set of n teatments, which become the cuent design. 3. If the cuent design does not allow the model to be fitted, etun to. 4. Systematically exchange each teatment in the cuent design with each (diffeent) teatment in the candidate set. Accept any exchanges which impove the citeion to get a new cuent design, continuing until no exchange impoves the design. 5. When no impovement is possible, the cuent design becomes the final design etuned by the algoithm. It is common pactice to estat the algoithm with seveal andom stating designs and to choose the best design found oveall. As an illustation, conside a teatment choice poblem with a micoaay expeiment with a limited numbe of slides, fo which a subset of 4 subjects need to be chosen among a population of 8 animals with the pedigee given in Figue 7. An R function was developed to seach fo optimal designs when teatment effects ae andom, with any covaiance stuctue among them (i.e. any pedigee set-up), such as in this situation. The teatments (animals) selected in the esulting design ae indicated by black cicles in Figue 7. Such a design is only good fo this vey specific situation and it is not possible to poduce deep genealizations. The only clea message with messy pedigees is that a seach algoithm may be the only way to each a good (if not optimal) design fo these expeiments. <Please inset Figue 7 hee> In anothe vey common situation in animal expeimentation, elated individuals ae subjected to diffeent expeimental pocedues (such as diffeent diets, dugs, etc.). In these cicumstances both andom and fixed teatment effects ae pesent in the expeiment. Such expeiments may have multiple objectives and vaiance component (o heitability) estimation can be one of them. One should be awae, howeve, that designs that ae optimal fo some of these objectives might not be so fo othes. An example of such a situation is given in Table 4. The expeiment consists of pigs fom five full-sib

21 families of size fou, to be aayed on a limited set of two-colo system slides. The fou individuals in each family (litte) efe to two male and two female sibs that wee selected at andom. One animal fom each sex was submitted to one of two expeimental teatments (placebo and dug). Two diffeent objectives of the study wee consideed: a) estimation of heitability; and b) estimation of the fixed effects of dug and sex, as well as the inteaction between them. Optimal designs fo each of these situations fo a specific configuation of vaiance components ae pesented in Figue 8. <Please inset Table 4 hee> <Please inset Figue 8 hee> It can be seen that in the fist situation (estimation of heitability) the seach algoithm conveged to a design in which all slides connected individuals fom diffeent sibs. Also, all slides have individuals of the same sex and/o same expeimental goup (placebo o dug). Convesely, if the expeiment goal is to infe the effects of sex, dug and thei inteaction, the genetic covaiance among sibs becomes anothe blocking facto, which the seach algoithm tended to confound o combine with the blocking effect of aays. In this case, the slides now tend to assay genetically elated individuals. As expected, to infe the main effects of dug and sex, some of the slides compae individuals of the same sex, but diffeent expeimental goups (such as those of Family 5), while othes compae individuals of the same goup, but diffeent sex (such as those of Family 3). In addition, a thid set of slides compae individuals of diffeent sex and expeimental goups (e.g. Family ) to infe the inteaction between sex and dug effects. CONCLUDING REMARKS This pape discusses the design of micoaay expeiments whose goals efe to a vaiety of genetic questions, such as the compaison of expession levels of diffeent genotypic goups, eqtl studies, o estimation of heitabilities of mrna tanscipt abundances. The geneal appoach adopted fo choosing a good design consists of a fist step of teatment choice, in which subjects (as well as thei expeimental goup labels) ae

22 selected to be included in the expeiment. This fist step is geneal and common to any micoaay expeiment, egadless of platfom. In the case of two-colo micoaay systems, howeve, the design choice is followed by teatment to unit allocation, in which mrna samples ae assigned in an optimal way to slides and dye labeling. It has been shown how the design choice should be intimately elated to the goal of the expeiment, such as the contast between fixed effects (e.g. the compaison of altenative dugs o the expected phenotypic value of diffeent genotypes), o infeence egading vaiance components (such as genetic vaiances) o heitability of tansciptional levels. Sometimes it is possible to have an analytical solution fo the optimality poblem, but in many situations a numeical solution is needed. Wheneve a numeical solution is implemented, a complete seach (compaing all possible designs) guaantees the optimal finding. A complete seach howeve is often unattainable due to the size of the seach poblem involved. In these situations algoithms ae needed that can find optimal o neaoptimal designs. R functions wee developed fo seaching and compaing the diffeent designs discussed in this pape; it can be obtained diectly fom the authos. As a final emak, it is impotant to note that this pape consides a single gene at a time fo design puposes, as the same model is geneally used to analyze all genes in a single expeiment. This is not an issue fo fixed effects models (such as fo the examples compaing genotypic goups) as the optimal design citeia fo fixed-effects models depend only on the design matix. This is not tue in mixed o andom-effects models, as in these cases optimal designs depend on both the design matix and the vaiance components paametes. Since the vaiance paametes vay among genes, the optimal design fo one gene may be diffeent fom that fo anothe gene. As a consequence, caution is needed befoe we simplify the optimal design fo the micoaay to optimal design fo a single gene if andom o mixed-effects models ae consideed. This is a vey inteesting topic, which deseves futhe eseach. A possible appoach to this issue would be to use an empiical distibution fo the genetic vaiances o heitabilities to weight the decision function that yields the design citeion. Altenatively, "obust" designs could be sought by focusing on the high heitability genes. This could be achieved by putting moe weight on the highe heitabilities than suggested by an empiical distibution, o by even ignoing the egion of low heitability

23 when seaching fo good designs. An exteme vesion of this appoach would be by taking genetic effects as fixed, but in geneal this would be an inefficient stategy. Moeove, while modeling genetic effects as fixed may be simple with expeiments involving family stuctues such as half o full sibs, it is not clea how this appoach could be easily extended fo complex pedigees. This wok was suppoted by the Michigan State Univesity - Intamual Reseach Gant Pogam and by gant fom the United States Depatment of Agicultue (USDA) to G.J.M.R., and by gant /4-4 fom the CNPq - Bazil to JSSBF. LITERATURE CITED BAILEY, R. A., 98 A unified appoach to design of expeiments. J. R. Stat. Soc. A 44: 4-3. BAILEY, R. A., 997 Statistics and mathematics: the appopiate use of mathematics within statistics (with discussion). The Statistician 47: 6-7; discussion: BREM, R. B., G. YVERT, R. CLINTON and L. KRUGLYAK, Genetic dissection of tansciptional egulation in budding yeast. Science 96: BUENO FILHO, J. S. S., and S. G. GILMOUR, 3 Planning incomplete block expeiments when teatments ae genetically elated. Biometics 59: BYSTRYKH, L., E. WEERSING, B. DONTJE, et al., 5 Uncoveing egulatoy pathways affecting hematopoietic stem cell function using genetical genomics. Nat. Genet. 37: 5-3. CHESLER, E. J., L. LU, J. WANG, R. W. WILLIAMS, K. F. MANLY, 4 WebQTL: apid exploatoy analysis of gene expession and genetic netwoks fo bain and behavio. Nat. Neuosci. 7: DARVASI, A., 3 Gene expession meets genetics. Natue 4: DARVASI, A., and M. SOLLER, 99 Selective genotyping fo detemination of linkage between a make locus and a quantitative tait locus. Theo. Appl. Genet. 85: DRAPER, N. R., and I. GUTTMAN, 997 Two-level factoial and factional factoial designs in blocks of size two. J. Qual. Technol. 9: FEDOROV, V. V., 97 Theoy of Optimal Expeiments. Academic Pess, New Yok. 3

24 GLONEK G. F. V., and P. J. SOLOMON, 4 Factoial and time couse designs fo cdna micoaay expeiments. Biostatistics 5: 89-. HENDERSON, C. R., 976 Simple method fo computing invese of a numeato elationship matix used in pediction of beeding values. Biometics 3: JANSEN, R.C., and J. NAP, Genetical genomics: the added value fom segegation. Tends Genet. 7: JIN C., H. LAN., A. D. ATTIE, G. A. CHURCHILL, D. BULUTOGLU et al., 4 Selective phenotyping fo inceased efficiency in genetic mapping studies. Genetics 68: KERR, M. K., 3 Design consideations fo efficient and effective micoaay studies. Biometics 59: KERR, M. K., 6 ^k factoials in blocks of size, with application to two-colo micoaay expeiments. UW Biostatistics Woking Pape Seies. Woking Pape 7. ( KERR, M. K., and G. A. CHURCHILL, Statistical design and the analysis of gene expession micoaay data. Genet. Res. 77: 3-8. KIEFER, J. C. and J. WOLFOWITZ, 96 The equivalence of two extemum poblems. Can. J. Math. : LANDER, E. S, and D. BOTSTEIN, 989 Mapping Mendelian factos undelying quantitative taits using RFLP linkage maps. Genetics : LOCKHART, D. J., H. L. DONG, M. C. BYRNE, M. T. FOLLETTIE, M. V. GALLO et al., 996 Expession monitoing by hybidization to high-density oligonucleotide aays. Nat. Biotech. 4: LYNCH, M., and B. WALSH, 998 Genetics and Analysis of Quantitative Taits. Sinaue. MEAD, R., 99 The nonothogonal design of expeiments (with discussion). J. R. Stat. Soc. A 53: 5-. MONKS, S. A., A. LEONARDSON, H. ZHU, et al., 4 Genetic inheitance of gene expession in human cell lines. Am. J. Hum. Genet. 75: NOUEIRY, A. O., J. CHEN and P. AHLQUIST, A mutant allele of essential, geneal tanslation initiation facto DED selectively inhibits tanslation of a vial mrna. Poc. Natl. Acad. Sci. USA 97:

25 ROSA, G. J. M., J. P. STEIBEL and R. J. TEMPELMAN, 5 Reassessing design and analysis of two-colo micoaay expeiments using mixed effects models. Comp. Funct. Genomics 6: 3-3. SCHADT, E. E., S. A. MONKS, T. A. DRAKE, A. J. LUSIS, N. CHE et al., 3 Genetics of gene expession suveyed in maize, mouse and man. Natue 4: SCHENA, M., D. SHALON, R. W. DAVIS and P. O. BROWN, 995 Quantitative monitoing of gene-expession pattens with a complementay-dna micoaay. Science 7: SORENSEN, D., and D. GIANOLA, Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics. Spinge, New Yok. STEIBEL, J. P., and G. J. M. ROSA, 5 On efeence designs fo micoaay expeiments. Stat. Appl. Gen. Mol. Biol. 4(): Aticle 36, online. TEMPELMAN, R. J., 5 Assessing statistical pecision, powe, and obustness of altenative expeimental designs fo two colo micoaay platfoms based on mixed effects models. Vet. Immunol. Immunopathol. 5: VINCIOTTI, V., R. KHANIN, D. D ALIMONTE, X. LIU, N. CATTINI et al., 5 An expeimental evaluation of a loop vesus a efeence design fo two-channel micoaays. Bioinfomatics : WANG, P. C., 4 Designing two-level factional factoial expeiments in blocks of size two. Sankhya 66: WIT, E., A. NOBILE, and R. KHANIN, 5 Nea-optimal designs fo dual channel micoaay studies. Appl. Statist. 54: WOLFINGER, R. D., G. GIBSON, E. D. WOLFINGER, L. BENNETT, H. HAMADEH et al., Assessing gene significance fom cdna micoaay expession data via mixed models. J. Comp. Biol. 8: YANG, Y. H., and T. SPEED, Design issues fo cdna micoaay expeiments. Nat. Rev. Genet. 3: YANG, Y. J., and N. R. DRAPER, 3 Two-level factoial and factional factoial designs in blocks of size two. J. Qual. Technol. 35:

26 TABLE Expected phenotypic value and numbe of obsevations fo each genotype in a twolocus expeiment with an F population Genotype Expected phenotypic value Numbe of obsevations AABB µ + α + α n AABb µ + α + δ (- ) n AAbb µ + α α n AaBB µ + δ + α (- ) n AaBb µ + δ + δ (- ) (- ) n Aabb µ + δ α (- ) n aabb µ α + α n aabb µ α + δ (- ) n aabb µ α α n 6

27 TABLE Optimal numbe of eplications of each genotype in a two-locus expeiment with no epistasis, using expeimental units Genotype Optimal Paametes of inteest Popotions α s δ s (α and δ) s AABB.5n = 5.65 n.858 n AABb (- ) n =.5 n 3.3 n AAbb.5n = 5.65 n.858 n AaBB (- ) n =.5 n 3.3 n AaBb (- )(- ) n =.5 n 4.76 n 4 Aabb (- ) n =.5 n 3.3 n aabb.5 n = 5.65 n.858 n aabb (- ) n =.5 n 3.3 n aabb.5 n = 5.65 n.858 n 7

28 TABLE 3 Examples of hieachical and factoial mating systems with nine pogeny Hieachical Factoial Pogeny Dam Sie Dam Sie

29 TABLE 4 Litte, weight and sex of twenty piglets available fo micoaay expeiments fo infeing the effects of sex, weight and the inteaction between sex and weight, as well as the heitability Pogeny Litte Dug Sex Yes F Yes M 3 No F 4 No M 5 Yes F 6 Yes M 7 No F 8 No M 9 3 Yes F 3 Yes M 3 No F 3 No M 3 4 Yes F 4 4 Yes M 5 4 No F 6 4 No M 7 5 Yes F 8 5 Yes M 9 5 No F 5 No M 9

30 FIGURE.Aveage vaiances of the estimates of genetic paametes fo a single locus expeiment as a function of the popotion of each homozygous goup (), when the expeiment goal is to estimate the additive effect (a), the dominance effect (b), o both additive and dominance effects simultaneously (c). (a) (b) (c) 3

31 FIGURE. Optimal designs fo estimating tans-acting additive effects (a), dominance effects (b) and both additive and dominance effects (c) fom one gene, with slides in a two-colo micoaay expeiment. (a) bb BB Bb (b) bb BB Bb (c) bb BB Bb 3

32 FIGURE 3.Optimal designs fo estimating tans-acting additive effects (a), dominance effects (b) and both additive and dominant effects (c) of two non-epistatic genes, with slides in a two-colo micoaay expeiment. Dashed aows connect the homozygous genotypes in the cones. 3

33 (a) AABB AABb AAbb AaBB AaBb Aabb aabb aabb aabb (b) AABB AABb AAbb AaBB AaBb Aabb aabb aabb aabb (c) AABB AABb AAbb AaBB AaBb Aabb aabb aabb aabb 33

34 FIGURE 4.Optimal designs fo two-colo micoaay expeiments with fou (a) and nine (b) unstuctued teatments of fixed effects and ten and nine slides espectively. (a) I II III IV (b) IX I II VIII III VII IV VI V 34

35 FIGURE 5.Optimal designs fo estimating tans-acting epistatic effects between two genes, with slides in a two-colo micoaay expeiment. AABB AABb AAbb AaBB AaBb Aabb aabb aabb aabb 35

36 FIGURE 6.Optimal design fo a micoaay expeiment with nine teatments, having andom effects, epesenting nine pogenies fom hieachical (a) and factoial (b) mating systems (Table 3). In Panel (a), diffeent cicle shadings efe to diffeent patenal halfsibs; in panel (b), each shape and shading denote a diffeent dam and sie, espectively. In both cases, each geometic figue epesents a subject (biological eplication), which is assayed in two slides each (technical eplication). (a) (b) 36

37 FIGURE 7.Thee-geneation pedigee with 8 individuals in two genetically disconnected goups, fom which 4 individuals (black cicles) wee selected fo a micoaay expeiment to infe mrna abundance heitabilities. 37

38 FIGURE 8.Optimal designs fo the estimation of heitability (a) and fixed effects of dug and sex and thei inteaction (b), in a two-colo micoaay expeiment with five full-sib swine families of two males (squaes) and two females (cicles), submitted eithe to a placebo (in white) o a dug teatment (in gay). (a) (b) 38