Weighting by heritability for detection of quantitative trait loci with microarray estimates of gene expression
© Manly et al.; licensee BioMed Central Ltd. 2005
Received: 25 November 2004
Accepted: 16 February 2005
Published: 28 February 2005
Heritable differences in transcribed RNA levels can be mapped as quantitative trait loci (QTLs). Transcribed RNA levels are often measured by hybridization to microarrays of oligonucleotide probes, in which each transcript is represented by multiple probes. The use of recombinant inbred lines allows an estimate of the heritability of expression measured by individual probes. This heritability varies greatly. We have tested heritability-weighted averages to define expression of a transcript and found that these allow detection of more QTLs than previously described methods.
The steady-state abundance of an RNA species in an organ is, in part, genetically controlled and can be considered a quantitative genetic trait. Microarray methods for estimating RNA sequence abundance , combined with genetic methods for identifying loci affecting quantitative traits [2–4], provide the opportunity to survey tissues for all genetically controlled variation in gene expression. This approach has been called genetical genomics , and its feasibility has been demonstrated in experimental crosses and human populations [6–10].
Genetical genomics is further enhanced by using recombinant inbred lines as a mapping population. The use of recombinant inbred lines allows comparison of gene expression among different tissues and the comparison of gene expression with classical physiological and behavioral traits from the published literature [11, 12]. Public datasets and online software at WebQTL [10, 13] allow free exploration of the characteristics of this form of analysis . In addition, recombinant inbred lines can provide both replicates from genetically identical individuals and samples from different segregants. Data from these define genetic and non-genetic variation, define a measure of heritability for expression of individual genes, and provide the basis for a new method of data reduction for genetical genomics.
Data reduction is an issue because Affymetrix GeneChip oligonucleotide microarrays assay each target mRNA with a set of 11 to 16 pairs of 25-nucleotide DNA probes. Each pair of probes consists of a perfect match (PM) sequence and a mismatch (MM) sequence, the latter intended to estimate nonspecific binding. The Affymetrix software Microarray Suite 4.0 and 5.0 (MAS 4 and MAS 5) estimate expression from the average difference of PM and MM fluorescence. Since the pioneering study of Li and Wong , however, it has been clear that MM binding includes target-specific binding as well as nonspecific binding, and the appropriate use of MM fluorescence has been an open question. In fact, a recent publication shows that it may be more useful to use the sum of PM and MM values instead of their difference . In short, the behavior of oligonucleotide microarrays is not adequately explained by models that only consider base complementarity. More realistic models consider nonspecific binding, saturation, the effects of fluorescent labeling and intramolecular folding of target and probe [15, 17–19].
Several alternative methods have been proposed to combine multiple probe-specific values into a single expression estimate. Three widely used alternatives are robust multiarray average (RMA) , model-based expression index/intensity (MBEI), implemented in dChip software , and positional-dependent nearest-neighbor model (PDNN) . RMA provides statistically robust averaging methods, dChip fits a model that allows probe-specific binding affinities, and PDNN fits a model that allows sequence-specific binding affinities and nearest-neighbor stacking interactions. A weighted-average method is also available, one which weights probe-specific values by a cross-validation procedure ; this method, however, does not take advantage of replicate microarrays and the current implementation in Bioconductor  is too slow for this application. Finally, a method (SUM) based on the sum of PM and MM values has recently been described . The rationale for this method is that MM probes exhibit probe-specific binding as well as nonspecific binding [15, 17] and may therefore be more effective for estimating specific binding than for correcting for nonspecific binding. Indeed, the SUM method outperforms MAS5 in several respects.
We describe here a new method, specifically designed for application to genetical genomics. In this method, called heritability-weighted transform version 1 (HWT1), probe-specific data is combined in a weighted average in which the weights are determined by an estimate of the heritability of the data for each probe.
Heritability is determined by the relative expression variance contributed by dimensions 1 and 2. The HWT1 method described here uses this information from dimensions 1 and 2 to define weights that allow dimension 3 to be collapsed with a weighted average. Dimension 2 is still collapsed with a simple average.
Although estimation methods exist that would avoid these values, the current method is simple and serves the intended purpose if negative heritabilities and those above 1 are adjusted by assigning them values of 0 and 1, respectively. When these adjusted heritabilities are normalized by the average (adjusted) heritability of probes in each probe set, the resulting weights are distributed as shown in the right-hand panels of Figure 2. About 36%, 49%, and 61% (for brain, HSC, and cerebellum, respectively) of probe weights are zero and 55%, 60%, and 66% are less than 1.0. These probes are fully or partly excluded from any weighted average. A small minority of probes, less than 3%, receive weights above half the maximum possible weight, suggesting that they will dominate the average for the probe set to which they belong.
The panels of Figure 3 compare QTLs detected after averaging with Affymetrix MAS 5.0 software and QTLs detected with three variations of heritability-weighted averaging. These variations differ in their use of MM probes. As transcript binding to MM probes seems to include both nonspecific and target-specific binding [15, 17, 18], we tested both subtracting MM values from PM (to remove nonspecific signal) and adding MM values to PM (to add target-specific signal). Figure 3a shows results obtained by calculating heritability from PM probes and averaging only those probes, Figure 3b shows results obtained by calculating heritability from and averaging PM - MM differences and Figure 3c shows results obtained by calculating heritability from and averaging all probes (PM and MM) together. Using 20% false-discovery rate as a significance cutoff, each of the heritability-weighting methods yields more QTLs than MAS 5.0. With this dataset, using only PM probes yielded more QTLs than the other two weighting methods.
When weighted expression averages were randomly permuted among the recombinant inbred (RI) strains before mapping, no QTLs were detected at 20% false-discovery rate (data not shown). Since heritability estimates are unaffected by permutation, permuting data after weighted averaging is equivalent to permuting before averaging. Furthermore, simulation showed that heritable variation alone is not sufficient to define QTLs. Simulated datasets were generated with heritable variation distributed among probes in various ways, including one in which all heritable variation was generated for a single probe of each probe set. In these simulated datasets all variation was independent of marker genotypes. No QTLs were detected from these simulated datasets after heritability-weighting and QTL mapping (data not shown).
There is little relationship between the abundance of transcripts and the likelihood of detecting a QTL (data not shown). If anything, strong QTLs tend to be found among transcripts of moderate abundance. This tendency might be explained if apparent interstrain variation, necessary for QTL detection, is reduced when abundance is extreme, either near the lower limit of detection or high enough to saturate some oligonucleotide probes.
These curves show that maximum heritability is more effective than average in predicting a detectable QTL. Because probe sets that do not define a significant QTL greatly outnumber those that do, probe sets defining a QTL are still a minority among probe sets selected for heritability. This situation is illustrated by three points that are circled in the figure. The right-hand circled point shows that selecting for maximum heritability greater than 0.35 selected 77% of probe sets; 2% of these yielded QTLs composing 99% of all QTLs. The center circled point shows that a threshold of 0.525 selected 17% of probe sets, of which 8% yielded QTLs composing 90% of QTLs. The left-hand circled point shows that a threshold of 0.675 selected 4% of probe sets, of which 32% yielded QTLs composing 75% of QTLs.
QTLs for gene expression can be classified according to the chromosomal location of the QTL relative to the location of the gene being expressed. Those for which the location of the QTL and gene are tightly linked are characterized as cis QTLs; those for which the locations are different are trans. In this study the location of a QTL is defined by the location of the marker achieving the highest likelihood ratio statistic (LRS), a marker defined by a simple-sequence repeat whose location is known in the mouse sequence. Cis QTLs are, somewhat arbitrarily, defined as those for which this marker is within 10 megabases (Mb) of the location of the probe sequence by which the gene expression is measured.
QTLs can also be classified according to the direction of the effect on gene expression. We adopt the convention that QTLs are labeled '+' if the DBA/2J allele is associated with higher apparent expression and '-' if the C57BL/6J allele is associated higher apparent expression. Assuming that Affymetrix probe sequences were largely designed for the C57BL/6 sequence, sequence differences between C57BL/6 and DBA/2 in the sequence recognized by a probe will tend to make DBA/2 hybridize more poorly than C57BL/6. That is, variation in sequences complementary to probe sequences can create artifactual QTLs, reflecting a difference in hybridization rather than a difference in expression. Such artifactual QTLs would be expected to be cis -.
For all methods in all datasets, cis - QTLs outnumber cis + QTLs, in some cases by two- or threefold. This excess could be explained by polymorphisms in sequences targeted by Affymetrix probes, polymorphisms reducing the hybridization of DBA/2J RNA. For Brn and HSC the weighting procedure made some attempt to reduce this type of artifact by assigning a weight of 0 to 614 probes having known single-nucleotide polymorphisms (SNPs) in the probe target sequence. The excess of cis - QTLs remaining in Brn and HSC in spite of this procedure suggests that there may be additional effects from polymorphisms not included in our list.
The cerebellum dataset yielded a large number of significant QTLs. In part this yield was expected because the number of probe sets for M430 microarrays is 3.6-fold larger than for U74Av2. However, the QTL yield for the cerebellum data is about 10-fold higher than for brain or HSC, or about 2.7-fold higher relative to the number of genes represented on the microarrays. As discussed further below, the cerebellum data were obtained in two unbalanced batches, and a difference between these batches might create artifactual QTLs on chromosome 2. However, although 475 significant QTLs, 16% of the total, appear on chromosome 2, this number is too small to fully explain the large number of the cerebellum QTLs.
The number of probes that contribute to weighted averages varies considerably between probe sets. The effective number of probes can be defined, as described in Materials and methods, by a measure which is the reciprocal of a weighted average of the weights. The measure varies from 1.0, if all weights but one are zero, to the number of probes (usually 11.0 or 16.0), if all probes are weighted equally.
The data in Figure 8 allow three conclusions. First, a substantial fraction of probes contribute to weighted averages that define QTLs. In each case, the central half of QTLs falls into the 7- to 13-probe interval. Although the groups do not differ significantly, there is a possible tendency for + QTLs to involve more probes than - QTLs. Finally, only the cis - group includes QTLs defined by fewer than four probes. QTLs that depend on so few probes are most likely to be artifactual QTLs caused by polymorphisms in the probe target sequences.
The heritability-weighted averaging method described here successfully summarizes oligonucleotide microarray measurements of gene expression in a way that facilitates detection of QTLs affecting that expression. It is a heuristic method, one that is not derived from an explicit statistical model. Nevertheless, the rationale is simple and rests on three facts: first, heritable variation is necessary (but not sufficient) to define a QTL; second, probes within a probe set differ greatly in the heritability of their expression estimates; and third, probes within a probe set differ greatly in their ability to detect a QTL. These facts suggested that a simple weighted average would summarize probe set data without obscuring the signal of those probes which could detect a QTL.
HWT1 is designed specifically for QTL mapping. In its present form, it does not apply to the more common experimental situation designed to estimate expression differences between samples. In that experimental situation, this method would be circular, weighting probes according to an estimate of the quantity to be estimated. QTL mapping, in contrast, does not depend directly on the differences between samples, but on the correlation of those differences with a genetic marker. Indeed, the data of Figure 4 imply the existence of a few probes with high heritability that nevertheless yield no significant QTL.
Although we designed this weighting to reflect heritability, it may, depending on the experimental design, involve more than heritability. The heritability estimate is based on the variance between strains (which includes genetically determined variance) and the variance within strains, as an estimate of non-genetic variance. This estimate is closely related to other size-of-effect measures, such as repeatability, ω2, η2, or ε2 [26–29]. Although we have not tested weighting with these alternative measures, we expect any of them would provide a similar benefit for QTL mapping. However, the optimum weighting for this application is not yet determined.
The frequencies of cis QTLs detected in this study (31-77%) fall within the wide range of frequencies detected in other studies. The most closely comparable study is that of mouse liver transcription, in which the frequency of cis QTLs varied from 34% for moderately significant QTLs (log odds score (LOD) > 4.3) to 71% for more significant QTLs (LOD > 7.1) . However those results were based on microarrays of 60-nucleotide probes, which would be expected to be less sensitive than Affymetrix probes to the effects of single-nucleotide polymorphisms. The same study reported a frequency of 80% for the more significant QTLs (LOD > 7.0) for maize leaves. For yeast transcription assayed with cDNA arrays, Brem and co-workers estimated 36% cis QTLs , and for a human cell line assayed with Affymetrix arrays Morley and co-workers reported 18% .
Variance within strains usually includes non-genetic biological variation, but that was not true for the HSC dataset, for which replicates were derived from a single biological sample. In that dataset, heritability estimates were presumably higher than if replicates had been derived from separate biological samples. Nevertheless, HWT1 weighting was clearly useful for detecting QTLs in this set.
Systematic differences among strains can affect weighting in either of two ways. Batch effects that are balanced within strains (partly true in the cerebellum data) will contribute to the within-strain variance and will deflate heritability estimates. This effect may explain why cerebellum raw probe weights include many more negative values than do brain or HSC (Figure 2). On the other hand, systematic non-genetic differences between strains (such as the batch effect in HSC data) will inflate heritability estimates. For heritability estimates, the HSC batch effect was avoided by using data from one batch.
Such batch effects may also affect QTL mapping, causing a higher frequency of false positives in areas of the genome where a batch effect fortuitously correlates with marker alleles. In fact, if the batch number in cerebellum is treated as a trait, it associates with three areas on chromosome 2 (none of which, however, reaches a suggestive level of significance). These effects could be controlled by using batch as a cofactor, both in the analysis of variance that estimates heritability and in the subsequent QTL mapping. However, these refinements go beyond what is needed to introduce the HWT1 method. Thus, in the cerebellum dataset, QTLs mapping to chromosome 2 may include false positives caused by a difference in microarray processing batch. This batch effect, however, cannot explain the exceptional number of QTLs detected in the cerebellum dataset. The excess number of QTLs detected for cerebellum (compared with brain or HSC) greatly exceeds the total number of QTLs on chromosome 2.
The comparison of heritability-weighting with other data reduction methods (Figure 7) should be considered as preliminary because they are based on results from only one set of data. More important, that comparison does not imply anything about their suitability for other purposes. In addition, modifications of any of those methods might make them more suitable for QTL mapping.
It is not clear why probes of a single probe set should vary so greatly in the heritability of their expression estimates. We suggest three possibilities. First, changes in RNA concentration will result in greatest changes in fluorescence if RNA concentrations are close to the effective binding constant for a probe. Since effective binding constants of probes vary [17–19], sensitivity to changes will vary. Second, nonspecific hybridization of probes with RNA species that do not vary among strains will reduce specific hybridization that might define a QTL. If probes differ in nonspecific hybridization, they will differ in their ability to define a QTL. Third, since probes assay different parts of the target transcript, alternative splicing and differential degradation will affect probes differently.
The QTLs described in this report were detected by fitting a single-QTL model, a statistical model assuming that all QTLs contribute to a trait with independent effects. This model can be misleading if linked and/or interacting QTLs contribute to a trait. Nevertheless, since many traits are largely controlled by one QTL or few unlinked QTLs, these results are reliable and useful. They further suggest that it may be fruitful to adapt the principle of heritability-weighting to QTL searches with multi-QTL models.
To summarize expression data for individual transcripts, the HWT1 method combines probe-specific data in a weighted average in which weights are determined by the heritability of the probe-specific data. It provides a useful way to summarize datasets for genetical genomics because it places weight on probe-specific data having variation that could define a quantitative trait locus.
Materials and methods
Brain RNA was obtained from 32 strains of BXD recombinant inbred mice, the parental strains C57BL/6J and DBA/2J, and (C57BL/6 × DBA/2)F1 hybrid. Data from parental and F1 animals were included in the heritability estimates but were not used for QTL mapping. Each individual array experiment used a pool of brain tissue (forebrain plus the midbrain, but without the olfactory bulb) that was taken from three adult animals usually of the same age. More detailed information is available at WebQTL . All results derive from the 100-array December 2003 data freeze.
Hematopoietic stem cell (HSC) RNA
Bone marrow cells were stained with lineage-specific antibodies and purified by flow cytometry. A stem-cell population was defined as the 5% cells showing least lineage-specific fluorescence . Replicate samples of RNA were separately amplified from a single cell preparation for each BXD strain, and these samples were processed in two batches of 22 and eight strains. These data are described at WebQTL  as the March 2004 data freeze.
Each individual microarray assay used Affymetrix MOE 430A and MOE430B GeneChip pairs to assay RNA from a pool of intact whole cerebella taken from three adult animals usually of the same age. RNA samples were processed in two large batches. The first batch consisted of single samples from 17 BXD strains. The second batch consisted of biological replicates for 10 strains, additional technical replicates for two strains, single samples for four additional strains, and duplicate samples for five additional strains. RNA was extracted at the University of Tennessee Health Science Center and all samples were processed at the Hartwell Center (St. Jude Children's Research Hospital, Memphis). These data are described at WebQTL  as the SJUT Cerebellum January 2004 data freeze.
Brain and HSC data were obtained from Affymetrix U74Av2 microarrays, which provide more than 12,000 probe sets, almost all of which are represented by 16 PM probes and 16 MM probes. The cerebellum data were obtained from Affymetrix 430A and 430B microarrays, which provide more than 45,000 probe sets, almost all of which are represented by 11 PM probes and 11 MM probes.
Microarray data reduction
Individual probe intensities from Affymetrix U74Av2 microarrays were log2-transformed and normalized to a standard array-wide mean and standard deviation. For each probe, mean squared deviations within strains (MSw) and between strains (MSb) were calculated by analysis of variance of the log-transformed, normalized expression. In the interests of speed, age and sex of animals were not included as cofactors in the analysis of variance. Raw heritability was estimated as (MSb - MSw)/(nMSt), where n is the average number of replicates per strain and MSt is total variance (excluding strains without replicates, if any) . Adjusted heritability was derived from raw heritability by assigning values of 0 and 1, respectively, to raw heritability values below 0.0 or above 1.0. Weights for each probe were calculated by dividing the adjusted heritability by the mean adjusted heritability for all probes in the probe set. Finally, expression estimates for each probe set and strain were calculated by an unweighted average of replicates within each strain and a weighted average of those probe-specific means, using the weights just described. To avoid division by zero, and to avoid using weights based on very small heritabilities, probes in a probe set were assigned a weight of 1.0 if the average adjusted heritability of those probes was less than 0.01. That is, expression for those probe sets was calculated from an unweighted average. The number of probe sets affected by this treatment was 5 (0.04%), 33 (.26%) and 4,178 (9.3%), respectively, for the Brn, HSC and Cer datasets. The large number of affected probe sets for cerebellum is consistent with the high number of negative raw heritability estimates for this dataset.
As explained under Results, polymorphisms between C57BL/6J and DBA/2J in probe target sequences would be expected to affect hybridization of Affymetrix probes, generating an apparent QTL mapping to the location of the transcript. To reduce the effect of this type of artifact, we prepared, from sequence information for the two strains, a list of 614 probes having polymorphisms in target sequences of probes on the U74Av2 microarray. During the weighting procedure described above, these probes were assigned a weight of 0, removing their contribution from any QTL for their probe set. This procedure was not applied to the cerebellum data, which came from a different microarray.
Among the HSC data, a systematic difference between the first and second batches described above would have greatly inflated all heritability estimates. To avoid this problem, heritability estimates were based on the first batch only, but all data were weighted and used for QTL mapping. Among cerebellum data, weighting was necessarily based only on replicated samples, most of which consisted of one sample from each batch. Any systematic batch difference would decrease heritability estimates. As with HSC data, cerebellum data from all strains was included in QTL mapping, weighted according to heritability estimates based on the strains with replicated samples.
Heritability-weighted averages were evaluated by regression against marker genotypes, where alleles at markers were coded as -1 or 1. In the interest of speed, regression was performed only at marker locations, but the limitations of this restriction were minimized by using 779 markers (described as the BXD genotype set at WebQTL ). Although WebQTL includes values for parental lines and F1 related to the BXD RI lines, these were not used in QTL mapping . For each microarray trait value, the locus yielding the maximum LRS  and the LRS itself were retained. An empirical P-value was then calculated for this LRS by a permutation test . Microarray trait values were permuted randomly among the progeny individuals 1,000 times and the regression analysis is repeated for each permuted dataset. If the original LRS fell within the distribution so that at least 10 values from permuted sets were greater, a P-value was calculated from the rank of the original LRS in the distribution. If a P-value could not be calculated, additional permutations are performed, until a P-value could be calculated or until 1,000,000 permutations had been performed. For each microarray trait, four data values were retained, the locus yielding the highest LRS, the LRS and regression coefficient at that locus, and the P-value of the LRS. To evaluate significance, all results from one microarray experiment were sorted by P-value, and the significance of the smallest P-values was determined by the method of Benjamini and Hochberg , using a false-discovery rate of 20%.
Mapping was performed with custom software, called QTL Reaper, written in Python and C for Linux. This software will be described fully in a subsequent publication but is currently available from SourceForge . Calculations were performed on an eight-node Linux cluster, which achieved processing rates of about 5,000 genome scans per cpu-second. Most processing time was spent on the small fraction of probe sets requiring more than 105 permutations.
Effective number of probes
Within a probe set, the weight of each probe may vary from 0 to the number of probes in the set, n. The effective number of probes f in a weighted average is defined as
where w i is the weight of probe i. This index varies from 1 to n. It is equal to k if k of the probes are weighted equally, and it is less than k if k of the probes are weighted unequally (with zero weight for the n - k remaining probes).
The HSC dataset has been placed in GEO. The accession number is GSE2031, and the arrays are GSM36673 to GSM36716. The Brn and Cer datasets are now both accessible from WebQTL .
We gratefully acknowledge the support of The National Institute on Alcohol Abuse and Alcoholism, INIA grants U01AA13499, U24AA13513, and the Human Brain Project P20-MH 62009, funded jointly by the NIMH, NIDA and NSF. Data were generated with funds to R.W.W. from the Dunavant Chair of Excellence, University of Tennessee Health Science Center, Department of Pediatrics. We thank the joint St. Jude Children's Research Hospital-UTHSC Cerebellum Consortium and The Hartwell Center for generating the cerebellum (Cer) dataset. We thank Bing Zhang, Cheng Li and Li Zhang, respectively, for performing the RMA, dChip and PDNN transformations for the brain (Brn) dataset. We thank two anonymous reviewers for specific, constructive comments.
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