## Abstract

Whole-genome profiling of gene expression in a segregating population has the potential to identify the regulatory consequences of natural allelic variation. Costs of such studies are high and require that resources—microarrays and population—are used as efficiently as possible. We show that current studies can be improved significantly by a new design for two-color microarrays. Our “distant pair design” profiles twice as many individuals as there are arrays, cohybridizes individuals with dissimilar genomes, gives more weight to known regulatory loci if wished, and therewith maximizes the power for decomposing expression variation into regulatory factors. It can also exploit a large population (larger than twice the number of available microarrays) as a useful resource to select the most dissimilar pairs of individuals from. Our approach identifies more regulatory factors than alternative strategies do in computer simulations for realistic genome sizes, and similar promising results are obtained in an application on *Arabidopsis thaliana*. Our results will aid the design and analysis of future studies on gene expression and will help to shed more light on gene regulatory networks.

THE combined study of gene expression and molecular marker data was proposed in 2001 as a novel strategy for the analysis of regulatory networks (Jansen and Nap 2001). The strategy has attracted attention both conceptually (Darvasi 2003; Kraft and Horvath 2003; Jansen and Nap 2004) and practically in a number of real-life applications in yeast, mouse, rat, maize, and human (Brem*et al.* 2002; Yvert*et al.* 2003; Schadt*et al.* 2003, 2005; Morley*et al.* 2004; Bystrykh*et al.* 2005; Chesler*et al.* 2005; Hubner*et al.* 2005). The genetics of gene expression have been studied in 40 and 86 haploid yeast segregants (Brem*et al.* 2002; Yvert*et al.* 2003), 111 F_{2} mice (Schadt*et al.* 2003), 30 or 34 recombinant inbred mice (Bystrykh*et al.* 2005; Chesler*et al.* 2005), 76 F_{3} maize lines (Schadt*et al.* 2003, 2005), 56 and 94 human familial individuals (Schadt*et al.* 2003; Morley*et al.* 2004), and 30 recombinant inbred rats (Hubner*et al.* 2005), by using the common reference design in two-color microarrays (Brem*et al.* 2002; Yvert*et al.* 2003; Schadt*et al.* 2003) or by using single-color microarrays (Morley*et al.* 2004; Bystrykh*et al.* 2005; Chesler*et al.* 2005; Hubner*et al.* 2005). Despite the remarkable results obtained in these initial studies, it is noted that the power of differential expression analysis is still limited because of the relatively small population sizes. It would be much better to profile several hundred individuals (Churchill*et al.* 2004), but then the costs may become prohibitive and require that resources—microarrays and population—are used as efficiently as possible. Can we generate more informative data with two-color microarrays than with single-color microarrays? Two-color microarrays can provide twice as much data as single-color microarrays, at least in principle. Is there any room for improving on the well-known common reference and loop designs for two-color arrays where samples are compared to a common reference sample or to each other in a loop order (Churchill 2002; Jansen 2003b)? Ratio-based analysis minimizes the risk of bias as a result of spot or array effect (Wit and McClure 2004); signal-based analysis obtains information about genotype effects also from the differences between spots (Wolfinger*et al.* 2001). Which method of data analysis should one prefer? We develop answers to all of the questions posed. Particularly, we demonstrate that a new strategy for two-color microarrays, coined the “distant pair design,” and analysis can outperform alternative strategies. The structure of this article is as follows. First we focus on two-color microarrays and we argue that standard methods for design and analysis of expression data can be applied but that we should be able to do much better by fine tuning to the genetics. Then we elaborate on the computational details underlying these arguments, and we show how to design and analyze an improved ratio-based strategy. Next we report results from a real study with *Arabidopsis thaliana* and results from computer simulations, evaluating the power of alternative strategies for different sizes of genome and population. Thereafter we compare ratio-based to signal-based analysis and two-color to single-color microarrays. Finally we summarize and discuss the consequences for experimental planning of a genetic study of gene expression.

## MATERIALS AND METHODS

#### Recombinant inbred lines:

To illustrate the design issues involved, we consider expression profiling a population of recombinant inbred lines (RILs). RILs are homozygous individuals, which result from repeated self–self mating or sibling mating, starting from an F_{1} of two homozygous parents, carrying alleles of type *A* and type *B*, respectively. The genome of a RIL is therefore a mosaic of the “founder” genomes, which can be viewed with the aid of molecular markers (Figure 1). The aim of a “genetical genomics” experiment is to test along the genome for differences in average expression between the individuals carrying genetic variant *A* and those carrying *B*. In this way quantitative trait loci (QTL) underlying expression variation can be mapped and classified as *cis*-acting (if gene and QTL colocalize) and *trans*-acting (no colocalization).

#### Alternative experimental designs for two-color microarrays:

Now assume that funding allows buying or making *n* two-color microarray slides. Per microarray two samples can be compared by using a red and a green dye label (the red label for one sample, the green for the other one), so that 2*n* samples can be profiled in total. The idea of the so-called common reference design is to compare all samples to one and the same reference: each of the *n* RILs is profiled once, and half of the microarray resource goes into the reference—not an ideal situation (Figure 1). In a more attractive alternative, the loop design, the first RIL cohybridizes with a second RIL on one array, this second RIL with a third RIL on a second array, and so on. This way *n* RILs can be profiled, not just once as in the common reference design, but twice—a serious improvement in the use of microarray resources (Figure 1). On second thought, however, one is not interested in comparing the first RIL to the second one or any other pair of RILs: there is no real need to loop RILs. Moreover, technical replicates should not be used in place of biological replicates (Speed 2003; Wit and McClure 2004). One could equally well use a random pair design, where the first RIL is compared to a randomly chosen second RIL, a third RIL to a fourth RIL, and so on, so that in total 2*n* genetically different RILs can be profiled (Figure 1). In each direct comparison between RIL *i* and RIL *j* with red and green dye, respectively, there are four possible combinations at a given marker: *A*/*B* (RIL *i* carries allele *A*, RIL *j* carries allele *B*), *B*/*A* (RIL *i* carries allele *B*, RIL *j* carries *A*), and *A*/*A* or *B*/*B* (they carry the same alleles). These four combinations occur with equal probability in a random pair design. We are primarily interested in detecting differential expression between *A* and *B*; thus *A*/*B* and *B*/*A* are of interest, and *A*/*A* and *B*/*B* are not. Having realized this, a natural next step is to improve the random pair design in a way that the number of *A*/*B* and *B*/*A* comparisons is maximized, preferably in equal ratio to balance red and green dye (Dobbin*et al.* 2005a,b), and with minimal extra variation of total numbers of *A*/*B* and *B*/*A* across the different markers. We therefore propose to cohybridize RILs that show to be genetically distant according to their molecular marker fingerprints, and we coin our design the distant pair design (Figure 1). The optimal design—a design with as many informative *A*/*B* and *B*/*A* comparisons as possible—is generated using a statistical optimization technique called simulated annealing; the data generated according to this design can be analyzed for significant differential allele expression *A*/*B* and *B*/*A*, using a method of QTL mapping (see the next sections for details).

#### Quantitative trait locus analysis of gene expression variation:

A gene's expression variation may be caused by genetic variation at a regulatory locus (in *cis* if the gene and regulator coincide, in *trans* otherwise): the expression level of the gene under study may be high (low) when the regulator locus has genotype *A* (*B*) or vice versa. We can observe two types of ratio: the informative *A*/*B* and *B*/*A* and the relatively uninformative *A*/*A* and *B*/*B*. The expression ratios of the latter type should be close to unity (zero on log scale), unless there is dye bias. This can be translated into a mathematical model , where *y _{i}* is the gene's log ratio at the

*i*th microarray (

*i*= 1, … ,

*n*); α is the gene-specific differential effect between dyes Cy3 and Cy5 (characterized as consistent across samples in Dobbin

*et al.*2005b); β is the effect of differential allele expression between

*A*and

*B*at a regulatory locus (or nearby marker) under study; and

*x*takes the following values: 1 for

_{i}*A*/

*B*, −1 for

*B*/

*A*, and 0 for

*A*/

*A*and

*B*/

*B*. In matrix form , where . All standard analysis of variance theory applies;

*e.g.*, the estimate of

*b*is , the variance–covariance matrix of is , and the test for differential allele expression follows an

*F*-distribution (null hypothesis β = 0

*vs.*alternative hypothesis β ≠ 0). The ratios

*A*/

*B*and

*B*/

*A*are converted into differences log(

*A*/

*B*) − log(

*B*/

*A*) = 2{log(

*A*) − log(

*B*)} and used for estimating β and calculating the genotype mean squares in the numerator of the

*F*-test. All four ratios

*A*/

*B*,

*B*/

*A*,

*A*/

*A*, and

*B*/

*B*are used for calculating the error mean squares in the denominator of the

*F*-test. The model can be regarded as a ratio-based modification of the standard QTL model; missing or incomplete marker data can be handled likewise (Jansen 2003a; Sen

*et al.*2005). The model can also be extended,

*e.g.*, , to dissect the expression variation into the effects of two or more different regulatory loci or QTL on the genome. In a full genome scan, differential allele expression is tested at all (combinations of) markers along the genome to detect one or multiple QTL.

#### Finding the optimal design:

We define a distant pair design to be optimal for a given marker if it minimizes the sum of the variances of . This is equivalent to minimizing the sum of the diagonal elements (the trace) of the matrix . In our model ; this reduces to finding the minimum of . Note that this favors a design with large and small . Here represents the total number of informative *A*/*B* and *B*/*A* comparisons (which should be large), and represents the difference between the number of *A*/*B* comparisons and the number of *B*/*A* comparisons (which should be small, *i.e.*, dyes should be well balanced).

We define a distant pair design to be optimal over all markers if the sum over all markers of the variances of is minimized. We denote this sum by *S* hereafter. It reads , where *j* refers to the *j*th marker (*j* = 1, … , *m*).

We used a technique called simulated annealing to find a design that is optimal or close to optimal (Kirkpatrick*et al.* 1983; Wit and McClure 2004). The search was iterative and at any particular iteration step compared the current design with a slightly modified version: samples of two randomly chosen pairs {*a*, *b*} and {*c*, *d*} in the current design were randomly repaired in the new one. Or elements in a randomly chosen pair {*a*, *b*} were replaced by one or two samples randomly drawn from the unused part of a larger resource population (selective phenotyping). The new design was accepted if it was better (has a lower value of *S*) than the old one. It is useful also to accept worse designs with a certain probability to be able to move away from “locally optimal” designs. This probability was , where *T* was a tuning parameter that was slowly lowered to zero during the iterative process. One can start from any arbitrary design, but starting at a good initial design can clearly save a lot of computational time. We used a simple heuristic approach to select a reasonable design to be used as a starting point for the simulated annealing procedure: we iteratively selected distant pairs without replacement (starting with a random RIL, selecting a second and distant RIL, randomly taking a third RIL, and so on), taking dye balance into account.

#### Arabidopsis experiment:

The Arabidopsis RIL population consists of 159 individuals from a cross of Landsberg *erecta* (L*er*) and Cape Verde Islands (Cvi) (Alonso-Blanco*et al.* 1998). The individuals were paired using a core set of 99 markers, evenly spread over the five chromosomes. Samples were expression profiled using 80 two-color microarrays [QIAGEN–Operon (Valencia, CA/Alameda, CA) Arabidopsis genome oligo set Version 2.10.2]. Data were corrected for background and dye effect and normalized within and between arrays according to methods described in Yang*et al.* (2001). We analyzed the normalized data in two ways: ratio based (described above) and signal based (described below).

#### The simulation study:

To better understand the relative performance of alternative experimental designs, we mimicked the above Arabidopsis experiment in a computer simulation, assuming availability of 160 RILs and a genome size of 500 cM. We also simulated experiments with larger genome sizes. The two-color data were simulated in a number of steps. First, marker genotypes of 160 RILs were generated for a genome of a given size (500, 1000, 2000, and 4000 cM, with chromosomes of size 100 cM, each chromosome carrying 21 equidistant markers). Then, assuming availability of 80 two-color microarrays, 80 hybridization combinations were selected using the loop design (loop), the random pair design (ranPair), the distant pair design (disPair), and the common reference design (comRef with 80 arrays). The latter design was also generated for 160 two-color microarrays (comRef with 160 arrays). Finally, hybridization signals were simulated for all pairs of RILs. The hybridization signal *s _{ij}* for a RIL at a given spot on the

*i*th microarray (

*i*= 1, … ,

*n*) with the

*j*th dye (

*j*= 1, 2) was simulated using the model log(

*s*) =

_{ij}*m*+

*G*+

_{ij}*B*+

_{ij}*T*

_{1i}+

*T*

_{2ij}, where

*m*is the overall mean,

*G*is the genotype effect,

_{ij}*B*is the biological sampling error,

_{ij}*T*

_{1i}is the common technical error for observations at the same spot, and

*T*

_{2ij}is the remaining technical error. The model was also used to simulate single-color microarray data (let

*j*take only the value 1).

Genotype effect was simulated for a single quantitative trait locus (QTL) located at 35 cM on the first chromosome and with allele substitution effect of 0.15 (+0.15 for those RIL homozygous *A* at the QTL and −0.15 for those homozygous *B*, median QTL effect in the Arabidopsis experiment, explaining 6.4% of the phenotypic variance).

Errors were randomly and independently drawn from normal distributions. Biological sampling error was generated with standard deviation 0.2, technical error *T*_{1i} with standard deviation 0.5, and technical error *T*_{2ij} per hybridization with standard deviation 0.2 (levels of noise as observed in the Arabidopsis experiment). In the common reference design, one biological reference sample with one and the same genotype effect and biological sampling error in all hybridizations (*G _{i}*

_{2}≡

*G*

_{2},

*B*

_{i}_{2}≡

*B*

_{2}) was used (

*i.e.*, no biological replication, just “pure” technical replication). In the loop design we used

*B*≡

_{ij}*B*

_{i}_{′j′}for the observations on the same RIL (again just pure technical replication).

We simulated 80 two-color microarrays to 80 or 160 single-color microarrays, assuming that the two technologies would both suffer from the same sources of technical noise *T*_{1i} and *T*_{2ij} at different levels: up to four times less noise (in terms of variance) in single-color arrays than in two-color microarrays.

#### Analysis of simulated data:

Ratio-based analysis of common reference data was standard: for each array we calculated the expression ratios *y _{i} =* log(

*s*

_{i}_{1}) − log(

*s*

_{i}_{2}) between a RIL and the common reference and then compared at each marker the ratios from the RILs carrying the

*A*allele to those carrying the

*B*allele using a standard

*F*-test. Ratio-based analysis of data from the other designs requires some further elaboration. First we calculated log ratios

*y*= log(

_{i}*s*

_{i}_{1}/

*s*

_{i}_{2}) = log(

*s*

_{i}_{1}) − log(

*s*

_{i}_{2}) =

*m*+

*G*

_{i}_{1}+

*B*

_{i}_{1}+

*T*

_{11}+

*T*

_{2i1}− (

*m*+

*G*

_{i}_{2}+

*B*

_{i}_{2}+

*T*

_{12}+

*T*

_{2i2}) =

*G*

_{i}_{1}−

*G*

_{i}_{2}

*+ B*

_{i}_{1}−

*B*

_{i}_{2}+

*T*

_{2i1}−

*T*

_{2i2}for the

*i*th pair of cohybridized RILs. Note that this eliminates the

*T*

_{1i}error. Then we translated this into the ratio-based model that we have developed above. Here, the

*x*depends on the RIL's genotypes

_{i}*G*

_{i}_{1}and

*G*

_{i}_{2}at the marker under study:

*x*= 1 for

_{i}*A*/

*B*, 0 for

*A*/

*A*or

*B*/

*B*, and −1 for

*B*/

*A*. The

*e*'s represent the composite error

_{i}*B*

_{i}_{1}−

*B*

_{i}_{2}plus

*T*

_{2i1}−

*T*

_{2i2}and these

*e*'s are independent in the random and distant pair designs, but not in the loop design.

_{i}Alternatively, the signal-based model log(*s _{ij}*) =

*m*+

*G*+

_{ij}*B*+

_{ij}*T*

_{1i}+

*T*

_{2ij}can be used as the starting point for a signal-based analysis of two-color microarray data. We treated the error components

*B*,

_{ij}*T*

_{1i}, and

*T*

_{2ij}as random (Wolfinger

*et al.*2001). The two error components

*B*and

_{ij}*T*

_{2ij}are confounded in the random pair and distant pair designs (but not in the loop design) and in these cases the model used in the analysis reduces to log(

*s*) =

_{ij}*m*+

*G*+

_{ij}*T*

_{1i}+

*e*, where

_{ij}*e*represents the composite errors

_{ij}*B*and

_{ij}*T*

_{2ij}. We used this signal-based model also to analyze our Arabidopsis data. The loop design can be analyzed by using the full model log(

*s*) =

_{ij}*m*+

*G*+

_{ij}*B*+

_{ij}*T*

_{1i}+

*T*

_{2ij}, where

*B*≡

_{ij}*B*

_{i}_{′j′}for observations on the same RIL.

The signal-based model log(*s _{ij}*) =

*m*+

*G*+

_{ij}*B*+

_{ij}*T*

_{1i}+

*T*

_{2ij}can also be used to analyze single-color microarray data (

*j*takes only the value 1). However, in this case the three error components

*B*,

_{ij}*T*

_{1i}, and

*T*

_{2ij}are confounded and the model used for analysis reduces to log(

*s*) =

_{ij}*m*+

*G*+

_{ij}*e*, where

_{ij}*e*represents the composite error terms

_{ij}*B*,

_{ij}*T*

_{1i}, and

*T*

_{2ij}.

For all designs and methods of analysis, differential expression was tested at each marker using the *F*-test, and in all simulations the minimum *P*-value along the first chromosome was stored and its distribution over 10,000 runs per simulation setting was plotted [as −log10(*P*)].

## RESULTS

#### Case study on Arabidopsis:

We designed the experiment for a recent study of genetical genomics in *A. thaliana*. The Arabidopsis RIL population contains 159 individuals. In a random pair design, one would expect half the cohybridizations to give informative ratios *A*/*B* or *B*/*A*: 40 in the Arabidopsis study. The distant pair approach increased the number to on average 55 (exceeding average values of 40 under the random pair design; Figure 2). Thus, our strategy of pairing distant genotypes led to a significant increase of informative *A*/*B* and *B*/*A* cohybridizations over the random pair design. We analyzed our Arabidopsis data with the QTL approach. A total of 2829 and 3011 of 24,065 expressed genes showed significant QTL at *P* < 10^{−5} in the ratio-based and signal-based analyses, respectively, with strong overlap between the two sets.

#### Simulation study—mimicking Arabidopsis genome size:

We performed a computer simulation study with 160 RILs and a genome consisting of five chromosomes of 100 cM each, *i.e.*, comparable to our real-life Arabidopsis experiment. Figure 3A shows that the distant pair design (solid line) outperforms both the random pair design (dotted line) and the common reference design (dashed line) with 80 arrays (1.6 and 3.9 times more QTL detected using threshold *P* = 10^{−5}, respectively). The difference in power between the random pair design and the distant pair design can be explained by the increased number of informative comparisons *A*/*B* and *B*/*A* in the latter design. The difference in power between these two designs and the common reference design can be explained by the fact that both the random pair design and the distant pair design allow for direct comparison of *A* to *B* in the cohybridizations *A*/*B* and *B*/*A*, while this comparison is indirect (calculated via the reference *R*) in the common reference design with cohybridizations *A*/*R* and *B*/*R* (the technical error from the reference *R* adds noise to the comparison of *A* to *B*). The common reference design with 160 arrays had more power than the common reference design with 80 arrays, because the first design provided twice as many data points as the latter. However, the common reference design with 160 arrays gave results slightly worse than the distant pair design with 80 arrays, whereas the costs for arraying were twice as high for the first method as for the second.

Can we do better if a larger population (say, 1000 RILs) is available but no more than 80 microarrays? Selective phenotyping has been suggested as a strategy to increase the power (*e.g.*, Jin*et al.* 2004). Our optimization strategy actually selected the best 80 pairs from the population to maximize the number of informative *A*/*B* and *B*/*A* comparisons. The computer simulation demonstrated that the number of *A*/*B* and *B*/*A* comparisons can be increased to 66 and that power was increased likewise (Figure 3A, disPair^{+}).

#### Simulation study—mimicking other genome sizes:

It is more likely to find distant pairs for organisms with small genomes (*e.g.*, Arabidopsis) than for ones with large genomes (*e.g.*, human). Nevertheless, as shown in Figure 3B, distant pairing is still advantageous for genomes of larger sizes. In all cases the number of desired hybridizations *A*/*B* and *B*/*A* exceeded the value of 40, which can be obtained on average in a random pair design. The genomewide average number of desired *A*/*B* and *B*/*A* hybridizations was 47 (49, 53, 58) for a genome size of 4000 (2000, 1000, 500) cM, respectively.

It may not always be the purpose of the researcher to give equal weight to all regions of the genome: emphasis may be on one or more regions on the genome known to carry genes (or QTL) of particular interest. For example, with special interest on two unlinked regions of 20 cM, the researcher can increase the power for differential analysis enormously by using markers only in these two regions in the distant pairing algorithm (Figure 3B, 40 cM in total). The number of *A*/*B* and *B*/*A* comparisons in the two 20-cM regions increased to 72, close to the upper limit of 80.

#### Simulation study—the alternatives:

Figure 3C shows that ratio-based and signal-based analyses of the simulated data for the distant pair design are almost equally powerful, the latter performing slightly better by obtaining additional information about genotype effect from the *A*/*A* and *B*/*B* observations. Figure 3C also shows that the distant pair design outperforms the loop design and the random pair design in signal-based analysis. Figure 3D shows that a distant pair design with 80 two-color microarrays is slightly worse than a straightforward design with 160 single-color microarrays if the technical sources are up to four times noisier (in terms of variance) in the two-color platform than in the one-color platform.

## DISCUSSION

A good experimental strategy provides better interpretable and more reliable results at the same or lower costs than alternative strategies do. Central to any gene expression study is the choice of a microarray platform, for which there are various two-color and single-color technologies as options. At first sight two-color microarray technologies have two clear advantages over single-color ones. First, two-color microarrays can generate twice as much hybridization data as single-color ones for the same number of microarrays. Second, two-color microarrays can cohybridize deliberately selected pairs of samples and therefore offer greater flexibility in experimental design than single-color microarrays do. In this article we investigated how to optimize the experimental design for two-color microarrays in genetic studies on gene expression and compared the alternative experimental designs for two-color microarrays to the straightforward design for single-color microarrays.

Common reference and loop designs are currently the most frequently used experimental designs for two-color microarrays, but they do not exploit the full power of genetic studies. We developed an alternative design, the distant pair design, in which individuals with dissimilar genetic (marker) fingerprints are paired together on the same microarray. The principles underlying the distant pair design are: first, direct comparisons between target samples are better than indirect ones via reference samples; second, comparisons between individuals carrying different alleles at many genes are informative, whereas comparisons between individuals carrying identical alleles at many genes are less informative and should be avoided; and third, profiling more genetically different individuals (2*n* instead of *n* using the two colors for different RILs without technical replication) increases the number of recombinant points in the study, which may lead to higher power and better resolution in QTL analysis. Our new experimental design and analysis method was outlined for a RIL population, but the three principles of our strategy apply equally well to any other type of population (*e.g.*, F_{2} or pedigree) in many organisms (including the human) and the method of minimizing the summed variances of parameter estimates is generally applicable (*e.g.*, to find the optimal design for detecting dominance, epistasis, or interaction between QTL and environment). In particular cases the power of the method can be increased even further—we studied two such cases. First, one or a few genome regions may be of major interest if they are known to contain important regulatory loci or genes for phenotypic traits and it is possible to increase or maximize the power for assessing differential expression caused by these loci by giving more weight to markers in these regions (*e.g.*, spanning 40 cM) than to the remaining markers (elsewhere in the genome) in the algorithm for finding distant pairs. Second, a large fingerprinted population (say of size 1000) can serve as a useful resource to select the most distant pairs of individuals from (say 80 pairs only), therewith reducing the number of arrays used without sacrificing too much power. Other strategies of selective phenotyping have been described earlier. For example, Jin*et al.* (2004) noted that selecting for equal and higher proportions of genotypes *A* and *B* would optimize power in a common reference design. We simulated the approach by Jin *et al.* and noted an increase in power but not to the extent obtained by the distant pair design (the Jin *et al.* power curve was almost equal to that of the random pair design in Figure 3A; only the curve of the latter design is shown). Note that the approach by Jin *et al.* works only in the case of profiling a subset of a large population, whereas our new strategy optimizes power no matter whether a population is profiled in part or as a whole. Jannink (2005) provided an interesting suggestion related to selective phenotyping. He noted that some progeny result from gametes with a greater number of recombination events than others and that such progeny can be more useful for QTL mapping than others. This idea can be incorporated into our approach simply by adding an extra term to the optimization criteria to favor selection of genotypes with a larger number of recombination events.

In the analysis of two-color microarray data one can calculate ratios between signals of the cohybridized samples at the same spot. Such a “within spots” analysis minimizes the risk of bias as a result of spot or array effects (Wit and McClure 2004). Alternatively, one can also consider and analyze the individual signals by treating the spot or array effect of cohybridized samples as random (Wolfinger*et al.* 2001). This analysis has the potential to obtain additional information about genotype effects from the *A*/*A* and *B*/*B* observations, which can further increase the power of QTL detection. However, distributional assumptions are questionable and bias is risked (Kerr 2003; Wit and McClure 2004). Of course, in our simulations distributional assumptions are known to be correct and therefore there is no risk of bias. For two-color microarrays we recommend using the distant pair design and checking, after the experiment has been performed, how the ratio- and signal-based methods of analysis compare.

We have shown that current studies can be improved significantly by a new design for two-color microarrays. But suppose that one also has the option to buy single-color microarrays. Which technology should one go for? Two-color arrays produce twice as many data as single-color arrays. Both systems suffer from various sources of noise, which may be common or specific for different technology platforms. Our simulations demonstrated that a distant pair design with 80 two-color microarrays can outperform a design with 160 one-color microarrays even if the two-color platform is substantially noisier than the one-color platform. Nevertheless, sources and levels of noise will differ between microarray platforms and even within platforms between laboratories, and only a careful case-to-case evaluation can guarantee the best return on investments.

We conclude that our complete integration of microarrays and genetic analysis—at the level of experimental planning and computational analysis—offers greatly improved perspectives for dissecting the complexity of gene expression regulation. Our results will aid the design and analysis of future genetic studies on gene expression and help researchers in making a choice between two-color and single-color platforms. As in other experimental sciences, the success of a genetic study on gene expression is based a great deal on the experimental planning and analysis.

## Acknowledgments

We thank J. P. Nap, M. Koornneef, I. R. Terpstra, A. F. J. M. van den Ackerveken, J. J. B. Keurentjes, J. E. Kammenga, and two anonymous reviewers for stimulating discussions. We also thank D. J. de Koning and C. S. Haley for valuable comments on an earlier draft of this manuscript. J. F. was supported by Netherlands Organization for Scientific Research-Genomics grant 050-10-029.

## Footnotes

Communicating editor: R. W. Doerge

- Received June 17, 2005.
- Accepted December 5, 2005.

- Copyright © 2006 by the Genetics Society of America