Differential expression for a given gene can result from trans-regulation by other genes or from cis-regulation due to variation in the region of the gene itself (altering functional motifs in the promoter region, changing the stability of the mRNA, or modifying the gene product in such a way that the feedback loop is shifted; Jansen and Nap 2004). In either case signal differences are supposed to be rather stable across probes (Figure 1A). The differences may, however, also change from one probe to another, due to known or unknown single-nucleotide polymorphisms (SNPs) or microdeletions between mRNA transcripts of different samples (Figure 1B; see also Doss et al. 2005), due to misdesigned probes (the majority of probe sets in Figure 1 are sequence verified; see also Mecham et al. 2004) or due to other known or unknown factors (e.g., alternative transcription). In such cases computing a single expression value per probe set, as in the current methods, leads to a loss of biologically relevant information. This will also hold for future alternative transcription/splicing arrays with probes located in different exons and not in the last exon or 3′-untranslated region only, as in the MG-U74Av2 array used in our experiment (Sharov et al. 2005). When the differences in signal between probes match with information in alternative splicing databases, they indicate that alternative transcription/splicing is the cause, and not an SNP.

In a genetical genomics analysis we search with the aid of molecular markers for a genome position where the difference in expression between mice carrying the B6 marker allele and mice carrying the D2 marker allele is (most) significant; this genome position is commonly denoted “expression quantitative trait locus” (eQTL or just QTL). In our study the 30 mice have been expression profiled by using probe sets of size 16. The 30 × 16 signals in a given probe set are decomposed intowhere y_ij is the signal of the jth probe of the ith mouse, m is the average signal, B_i is the average effect of the batch to which the ith mouse belongs (three batches), P_j is the average effect of the jth probe, PB_ij is the interaction effect between probe and batch, A_i is the average effect of the allele (B6 or D2) carried by the ith mouse at a given genome position, PA_ij is the interaction effect between probe and allele type, e_i is an error term per mouse, and, finally, e_ij is a probe-specific error term per mouse. The model is computed at each of 705 marker positions to find the position (QTL) with the most significant allele effect A_i; probe-by-QTL interaction is derived from the corresponding PA_ij. Standard F-values are computed: F_1,25 at the mouse stratum for the QTL and F_15,390 at the lowest stratum for the probe-by-QTL interaction.

Genes colocalizing within 20 Mb of their QTL are termed here cis genes, and genes mapping elsewhere are termed trans genes. The cis genes show many more probe-specific QTL effects than the trans genes do (Figure 1, C and D). It is expected that probe sets carrying the more influential polymorphisms between probe and transcript will be picked up as cis-acting with probe-specific QTL effects. Indeed 10 cis-acting genes carry currently known SNPs in one or more of their probes and the directions of the probe-specific QTL effects are in agreement with the SNPs; i.e., the mouse allele perfectly matching the probes on the array has the higher signal (allowing us to use the array for genotyping as well; see also Jansen and Nap 2001; Rostoks et al. 2005). Further lab research (e.g., sequencing of the D2 strain) will make clear how many of the cis-acting genes are caused by SNPs, alternative transcription, or other (hidden) factors.

Our analysis separates probe sets that are “consistently” cis across probes from those that are more “probe-specific” cis and should be investigated in more detail in silico or in the lab. Figure 1C shows that P-values for probe-by-QTL interaction can be very extreme. At the one hand probe-specific QTL effects can be large relative to e_ij (and thus statistically significant), at the other hand they can still be small relative to the average QTL effect (and thus biologically not really relevant). The biologist can best inspect figures (Figure 1B and alike) to see how many probes underlie probe-by-QTL interaction and to decide which genes merit closer scrutiny in silico or in the lab.

Some genomic regions showed up as “master” regulators in trans of many genes, but only if the factor batch was excluded from the model (particularly chromosome 4, see Figure 1E). The 30 samples have been profiled in three batches and samples in the same batch generally showed a very similar profile across the 16 probes, whereas samples in different batches showed different profiles. Spurious linkage between batch and these genome regions leads to “ghost” regulators, and, since batch and probe-specific batch effects are likely to occur in experiments with multiple arrays, it underpins the need for careful statistical modeling. It is important to note that standard permutation tests in QTL analysis without batch effects will not protect against this batch artifact (permutation should be carried out within batch-probe combinations).