INTRODUCTION

Top Abstract Introduction RESULTS DISCUSSION References

It is well established that a eukaryotic cell's genomic DNA and expressed mRNA are present in a variety of abundance classes (Britten and Kohn 1968; Galau et al. 1977; Hames and Higgins 1985). Very wide differences in gene expression level, that is, in intracellular mRNA copy number or in amount of gene product, are possible within the same cell. For example, it has been estimated that the copy numbers of expressed genes can vary from 1 to ~200,000 (Patanjali et al. 1991). For many genes, such differences in abundances can be detected coarsely through experimental rehybridization kinetics or can be estimated from the rates of protein synthesis of specific enzymes (Galau et al. 1977). More modern abundance detection techniques are also available (Singer and Berg 1991; Adams 1994; Wilkins et al. 1997). The same cell type, as well as different cell types, may exhibit different patterns of gene expression when exposed to different conditions (Singer and Berg 1991; Adams 1994; Lodish et al. 1995; Wilkins et al. 1997). Assessing differences in expression patterns, therefore, can be used to gauge differences in cell physiology and tissue behavior, intrinsically or in response to many different kinds of stimuli.

Because gene expression is central to modern biology, it is expected that delineations of patterns of gene or protein expression among normal and diseased states will have increasing importance in medical diagnostics and therapy (Anderson et al. 1984; Anderson and Seilhamer 1997). The conjunction of large-scale biology technologies (e.g., genomic sequencing or proteomics) and the need for new pharmaceutical targets has motivated the development of computational methods for detecting unusual expression patterns. Among the patterns of interest is selective expression, in which the expression (mRNA or protein) in a specific tissue is at a significantly different level than the other tissues. Selective expression is of particular interest because it may be correlated with fundamental biological phenomena or disease processes.

Two stereotypical selective expression situations are possible: up, in which expression is elevated in a specific tissue when compared with the levels in other tissues; and down, in which the expression in a specific tissue is reduced significantly when compared with the levels in other tissues. Although mixed situations are possible, they will not be discussed in this article (see Fig. 1 for diagrammatic representations). The extension of the techniques for identifying up or down selective expression to the mixed cases is straightforward.

View larger version (14K): [in this window] [in a new window]   Figure 1   Selective expression types. Examples of selective expression stereotypesup, down, and mixedare diagrammed. Intensities vs. sources from a source set are plotted in arbitrary order. In each diagram, it is presumed that the outstanding intensities are exceptionally large or exceptionally small, when compared against the other intensities; hence, selective expression. Selectively expressed intensities are indicated by encircled symbols.

Up selective expression may be an important indication that the gene has been activated specifically, up-regulated, or its product elevated differentially in association with certain phenomena or agents affecting a particular tissue's biology. Similarly, down selective expression is either a significant down-regulation or a nearly complete inactivation of the gene (e.g., tumor-suppressor loss of function) in association with specific biological events. Such broad phenomena as morphogenesis, differentiation, metabolic alteration, mutagenesis, bacterial and viral infection, physiological stress, disease, drugs and therapeutic interventions, etc. can manifest or cause selective expression effects. A particular example at SmithKline Beecham Pharmaceuticals was the discovery of a novel cathepsin gene (cathepsin K) being selectively up-expressed in an osteoclast library and subsequently shown to be involved in bone resorption (Bossard et al. 1996; Drake et al. 1996; Zhao et al. 1997). This experimental finding significantly motivated the development of this selective expression detection computational method.

This article presents a robust computational method that identifies genes or proteins that are expressed selectively (see Fig. 2 for a representation of selective expression in real data). The method is not restricted to gene or protein expression data, though such expression data is among the more natural contexts for this approach. The method is generally applicable to any kind of intensity data in which a distinguishable data source (e.g., tissue, library, assay, drug, dose, time, etc.) can be associated with each intensity value (e.g., gene or protein abundance, activity, binding strength, fluorescence, etc.). If assessments of reliabilities, that is, confidences, in the sources are available, these can be utilized to make more reliable predictions. The intensities can be experimentally determined values, computationally derived values [e.g., from expressed sequence tag (EST) data (Myers 1994)], or combinations. The method is indifferent to the experimental or computational lineages of the data to be analyzed. All that is required are triples of associated values: intensity, source, and source confidence. Conveniently, a set of values can be organized generally as a two-dimensional matrix of intensities, in which each column corresponds to each source (with its associated source confidence), and each row corresponds to each entity for which intensities are assessed, for example, intensity in row versus column, corresponding respectively to abundance of gene versus library, binding strength of drug versus tissue, biological activity of drug versus dose, and so forth. The method is applicable as well to intensities from the same source (column), for example, different genes' abundances (row) in a particular library (column), binding strength of different drugs (row) in the same tissue (column), biological activity of different drugs (row) at the same dose in the same tissue (column).

View larger version (9K): [in this window] [in a new window]   Figure 2   Selective expression example from real data. The results shown in this example are for an actual assembly, which is highly homologous to mRNA for a 23-kD highly basic protein. Intensities (abundances) are plotted vs. sources (libraries) in arbitrary order. The single exceptionally large intensity compared to the other intensities in the source set was detected by the algorithm as a strong selective expression.

Foundations

Before presenting the approach, we delineate four essential concepts to provide a conceptual footing for the selective expression identification method:

1.	"Intensity" is a non-negative numerical quantity that is representative of the phenomenon of interest. For example, intensity could be a drug's binding affinity, a compound's activity in a screen, or a gene's "abundance," such as the gene product's copy number (molecules or concentration of mRNA) or amount of protein expressed, and so forth. Intensity can be either an experimentally measured quantity or a quantity that is calculated from analyses of EST assemblies. [Assemblies are computational constructs comprising EST components from sequenced libraries that have been combined to represent putative genes (Adams et al. 1994; Burks et al. 1994; Myers 1994).]
2.	"Source" is the identification of the source of the "intensity" data. It can be, for example, a cDNA library or a tissue that provides a set of expression intensity or abundance values (Anderson et al. 1984; Adams et al. 1994; Anderson and Seilhamer 1997) of the genes being compared. If a source is manipulated experimentally or edited in any way, for example, a subtracted or normalized cDNA library, it should not be included in the analysis lest its pattern of expressed genes is artificially skewed. This exclusion principle can be relaxed if all the sources being compared have been manipulated in the same way.
	In general, sources are considered independent of each other with no intrinsic ordering. However, this is not always true when the sources can be ranked according to an external experimentally controlled variable, e.g., time, age, dose, disease staging, treatment concentration, etc. In these situations it is natural, although not required, to order the sources by ascending control variable values.
3.	"Source set" comprises a subset of the sources among which intensities can be compared. In particular, selective expression is a marked difference of intensity in a single source from a baseline level of expression established by the gene's intensities in a particular source set (see Fig. 1 for stereotypical examples). The selective expression concept does not require, however, that comparisons be made against all known sources. Instead, a carefully chosen subset of the known sources can be considered, especially as selective expression is a relative, not an absolute, assessment. Care must be exercised in choosing source sets so as not to bias the selective expression prediction. Recognizing this, choice of source set enables the biological context for expression comparisons to be tailored to the biological questions being askedorgan systems versus one another, tissues versus one another (e.g., endothelium versus smooth muscle or fibroblast), drug dose responses versus one another, effects of compounds versus one another, human versus nonhuman species, and so forth. It is the careful matching of source set to the relevant biological question that should minimize artifactual biases being introduced.
	All the intensities from the same source are scaled by the source's maximum observed intensity. This is done to make intensities from different sources within the source set commensurably comparable, which is necessary if intensity patterns across sources are to be identified.
4.	"Source confidence" represents the quality, reliability, and knowledge of error or the relative trust that can be attributed to the source. When source confidences are quantitated in some fashion, we call them "source quality weights." For example, a larger weight would be associated with a carefully harvested and rapidly preserved tissue (high quality) than with a poorly captured or slowly preserved one (low quality). A cDNA library sequenced in depth is a more reliable source, hence has a larger weight, than the same library sequenced to a lesser depth. An edited or normalized cDNA library should be considered a low confidence source unless all the sources in the source set have been manipulated equivalently. We note that any consistent source quality weighting scheme can be used, but care must be exercised. If the weights are not faithful to the reliabilities of the sources, any results dependent upon them may be improperly distorted. As explained in the Algorithm and Details sections (below), a selective expression assessment can take into account the weights of the different sources constituting the source set.

The focus of this work is not on the molecular details of the processes of gene expression or protein synthesis. Rather, we are interested in comparing relative levels of mRNA transcripts or protein products. Despite the inherent difficulties in measuring precisely which mRNA species are translated and in what relative proportions, reliable enough information on expression levels can be obtained (Adams 1994; Anderson and Seilhamer 1997; Herbert et al. 1997). Moreover, the established experimental techniques of cDNA and EST sequencing, especially when employed on a large scale, can provide ESTs that can be combined computationally into assemblies (Adams et al. 1994). Assemblies can be interpreted as putative expressed genes, though with widely varying levels of confidence in the assignments to genes (Burks et al. 1994; Myers 1994). Abundances of expressed genes or assemblies obtained from sampling are dependent on the depth of the sampling (Lewins and Joanes 1984; Bunge and Fitzpatrick 1993) and contribute to inaccuracies in the computed intensities (Myers 1994). We note that this sampling depth issue (Audic and Claverie 1997) can be viewed as a source reliability problem.

Statistical Considerations

In general, the problem of identifying selectively expressed intensities in multisource data can be viewed as an outlier identification problem in a different guise. We choose to adopt the view of outlier employed by Barnett and Lewis (1978a) and others (Grubbs 1969): "But what characterizes the 'outlier' is its impact on the observer (it appears extreme in some way)" (Barnett and Lewis 1978b). Identification of an outlying intensity is not the endpoint of the analysis. Rather, statistical outlier identification is one component of a larger analysis scheme that includes additional (e.g., biological) considerations.

Statistical objectivity can be introduced through the concept of discordancy: "... an observation will be termed discordant if it is statistically unreasonable on the basis of some prescribed probability model." (Barnett and Lewis 1978c). This relies on choosing a probability model for the data so quantitative assessments of outliers against the model can be done. We choose to interpret a resulting discordancy significance probability as a relative strength of confidence in the statistical identification. An advantage of this interpretation is that it makes it possible to consistently order the results from statistically insignificant, to weak, to strong. Thus, the rank order of the relative confidences of identification is preserved. This confidence ordering is critical because there is little knowledge of the extent to which the statistical test's probability model may be inappropriate.

At the current state of biological understanding, the actual form of any underlying probability distribution (if one exists at all) is entirely unknown (Singer and Berg 1991; Lewin 1994). Moreover, even if the concept of an underlying intra-source intensity distribution is appropriate, very little can be said about distributions that may be governing inter-source intensity comparisons. Nonetheless, it could be assumed that certain distributions can reasonably describe inter-source intensity comparisons. Because intensities are nonnegative, well-known distributions on the nonnegative real axis could be chosen, e.g., exponential, gamma, log normal. However, the discordancy tests associated with these distributions suffer from the necessity to estimate parameters (Barnett and Lewis 1978a). Neither the available intensity data nor theoretical knowledge support the estimation of distribution parameters in this inter-source sampling problem. As expression intensities are bounded, a practical way exists for the choice of a distribution: Assume a distribution defined on a finite domain, whose shape is simple, and for which a discordancy test independent of the distribution's parameters is available. The Dixon discordancy test for uniform distributions meets these criteria (Barnett and Lewis 1978a). Uniform is a reasonable choice because it confers only a very weak bias in distribution shape or in central tendency. How we employ this test for selective expression is described in the Details subsection of Results.

Outline of the Selective Expression Algorithm

The previous discussions have set the stage for the identification of selective expression in data comprising triples: intensity, source, and source quality weight. First, an overview of the algorithm is given. Then, the mathematical details of the key steps (steps 4-7) are explained in the Details section.

Step 1Minimum Source Quality Criterion

Step 2Minimum Number of Sources Criterion

View larger version (27K): [in this window] [in a new window]   Figure 3   Dependence of the Dixon theoretical statistical significance probabilities on the sample separation ratio at fixed number of samples taken. The logarithms of the Dixon theoretical statistical significance probabilities for discordancy in uniform samples (Barnett and Lewis 1978a) are plotted against the sample separation ratio   [0.8, 0.995], (equations 1-6). Each theoretical curve (right) is at a different fixed number n of samples, n = [10, 20, ... , 100], respectively.  near 1 reflects a largest sample being widely separated from the next largest sample when compared to the separation of the largest and smallest samples (see Fig. 6).

View larger version (18K): [in this window] [in a new window]   Figure 4   Dependence of Dixon theoretical statistical significance probabilities on the number of samples taken at fixed separation ratios. The logarithms of the Dixon test theoretical statistical significance probabilities (equation 6) are plotted against the number of points n in a sample, [10, 11, ... , 100]. Each theoretical line (right) is at a different fixed sample separation ratio  = [0.5, 0.8, 0.9, 0.95, 0.99], respectively. From the family of lines, the rapid decrease in statistical significance probability with increasing sample size n is apparent. This effect is stronger the closer  is to unity.

Step 3Discordancy Statistical Test

Step 4Adjustment Due to Intensity Baseline

View larger version (29K): [in this window] [in a new window]   Figure 5   Discordancy statistical significance adjusted for baseline position. Synthetic intensity data vs. source for several different baselines, [0.25, 0.5, 0.75, and 0.9], are plotted. In each case, the number of samples (n = 22), the maximum intensity (xn = 1), and the traditional Dixon significance probability [log10(sp) = 20] are kept fixed. Constant Dixon statistical significance, regardless of baseline position, is achieved deliberately in these synthetic data by adjusting xn 1 according to equations 1-6. Hence, the gap (xn  xn  1) necessarily decreases as the baseline increases; yet, the traditional Dixon statistical significance remains unchanged. The closer the baseline is to the maximum allowed intensity (i.e., 1), the less statistical confidence we can have in an outlier assessment. As can be seen in each panel, the baseline adjusted statistical significance decreases as the baseline increases toward the allowed maximum. The erosion of statistical confidence from the traditional Dixon significance as baselines are continuously increased toward the allowed maximum is plotted in Fig. 7 (see also Table 1).

Step 5Minimum Intensity Gap Criterion

View larger version (7K): [in this window] [in a new window]   Figure 6   Separation of a largest value from the others and the basic measures for the Dixon test. This diagram displays the separation of a largest value from the others when the values being compared are sorted from smallest to largest, xi  1  xi. The separation ratio , is the distance between the largest and the next-to-largest values (gap = xn  xn  1) divided by the distance between the largest and smallest values (xn  x1). The ratio  is fundamental in the Dixon test for discordancy (Barnett and Lewis 1978a), which assesses statistically how exceptionally large the largest value is compared to the others. See equations 1-6 in Algorithm Details. In the examples, fmax = 1.

Step 6Overall Confidence in Selective Expression Determination

	RESULTS

Top Abstract Introduction RESULTS DISCUSSION References

Here we present the mathematical details of the algorithms's key steps (3, 4, and 6), as well as examples of the algorithm applied to synthetic and real data.

Details of the Key Steps of the Algorithm

Step 3The Discordancy Test for Identifying Exceptional Intensities

Step 3Mathematical Details

(1a)

(1b)

(1c)

i  1

(2)

(3)

(4)

(5)

(6)

(7)

n1

n1

Step 4Details of the Baseline Adjustment

(8)

(9)

(10)

(11)

n1

n1

(12)

View larger version (12K): [in this window] [in a new window]   Figure 7   Erosion of confidence increases as the baseline position increases toward the maximum. The number of samples (n = 22), the maximum intensity (xn = 1), and the traditional Dixon significance probability [log10(sp) = 20) are kept fixed throughout this example as in Fig. 5. The resulting erosion of confidence, i.e., the loss of statistical significance log10(sp) from the traditional Dixon value, is plotted continuously as the baseline increases toward the allowed maximum intensity (see also Table 1).

20

Step 6Decision Function

(13)

(15)

1

View larger version (28K): [in this window] [in a new window]   Figure 8   Decision function for selective expression of overall confidence. The decision function [d(g,s), equation 15] is plotted as constant d contours in the space of (g,s) coordinates, each 0 and 1. (g,s) Respective linear transformations of gap and baseline adjusted log10(sp) between the weak thresholds and strong limits (equations 13 and 14). Overall confidence of a selective expression is assessed by the value of the decision function, weak being associated with d near 0 and strong associated with d near 1. Typically, we assign overall confidences as weak when 0  d < 0.33, moderate when 0.33  d < 0.66, and strong when 0.66  d 1.

Application of the Selective Expression Algorithm

Selective expression detection results will be illustrated through a set of examples. The examples are shown in Figure 9, with source qualities and corresponding intensities in Table 3, and computed numerical results summarized in Table 4. Though these intensity and source quality data are synthetic, they are representative of real examples derived from a large database of gene abundances and library qualities.

View larger version (22K): [in this window] [in a new window]   Figure 9   Examples of synthetic, yet realistic, intensity (abundances) vs. source (library) data for genes. (A) Example 2 () and example 1 () are compared; (B) example 3 () vs. example 1 (); (C) source qualities corresponding to the intensities. In A and B, the putative selective expression occurs in the third source. The numerical values of the data along with the computed baseline estimate baseline and separation ratios  are presented in Table 3. The accompanying selective expression algorithm summarized calculations are in Table 4.

To convey the effects of various components of the algorithm, each example of Figure 9 and Table 3 is constructed deliberately to have very similar qualitative patterns of intensity versus source. Yet, the examples are different in overall confidence of selective expression. Each example has the same source set (size n = 15) and, moreover, exactly the same separation ratio ( = 0.67) before any adjustments are made for baselines. Hence, these examples have by design exactly the same traditional Dixon statistical significance probability before baseline compression adjustment.

Table 4 shows the effects of the algorithm's components. For example, the effects of adjusting the statistical significance probability for baseline can be seen by comparing each example after adjustment for baseline (case b) against its respective case unadjusted for baseline (case a). Example 3 is the only case in which statistical significance probability is changed nonnegligibly by baseline adjustment. This can be appreciated by observing in case 3b the effects of baseline on , hence on , when compared against the reference case 1a.

Examples 2 and 3, however, have markedly smaller gaps than does example 1. These diminutive gaps are responsible for the respective decision function values being much smaller even though the discordancy statistical significance probabilities (with or without baseline adjustments) are not changed much. The exception is case 3a, which has an ample loss of significance probability due to baseline adjustment. Though the 3b gap is the same as 3a, 3b's decision function is zero because baseline adjustment of its significance probability has resulted in its log₁₀(sp_adjusted) not meeting the minimum statistical significance criterion of log₁₀(sp)_thresh = 5.

Taken together, these examples illustrate how qualitatively similar intensity versus source patterns can have different overall confidences of selective expression determination. The decision function values depend on the baseline of the data and the size of the gap, even when the expression patterns have essentially identical unadjusted discordancy significance probabilities. By analyzing these examples, it can be seen how the qualitatively stronger overall confidence of selective expression of example 1 as compared to examples 2 and 3 (which is informally conveyed in Fig. 9) is quantitated through the decision function.

Also, to convey the appearances of stereotypical selective expression patterns in real gene expression data, intensity versus source plots of some actual examples of algorithmically detected extremely strong, strong, and weak overall confidence selective gene expression are shown in Figure 10. Calculations corresponding to these examples are shown in Table 5. In these particular examples, baseline adjustment has no effect because the baselines are well below 0.5 of the maximum intensity. Hence, the discordancy statistical significance probabilities are the same as the unadjusted ones.

View larger version (21K): [in this window] [in a new window]   Figure 10   Stereotypical examples of selective expression in real abundance data as detected by the algorithm. Intensities (abundance) vs. source (library) for three assemblies from a real database of sources and assembly abundances are plotted. (A) An extremely strong overall confidence of selective expression (decision function d = 1.0); (B) strong overall confidence of selective expression (d = 0.75); (C) weak overall confidence of selective expression (d = 0.31). Calculations from the algorithm are summarized in Table 5.

From Table 5, the  are decreasing from example A to C, with the larger decrease being from example B to C. That the statistical significance probabilities decrease so dramatically with this series of  values is because of the considerable size of the n involved. The marked difference in log₁₀(sp) between A and B is caused much more by the difference in n than in . However, the substantial difference in log₁₀(sp) between B and C is caused by the difference in  more than the difference in n. These differences are not surprising given the nonlinear dependence of log₁₀(sp) on  in equation 6 (visualized in Figs. 3 and 4). Clearly, A is an extremely strong overall confidence selective expression determination, as can be recognized visually in Figure 10 and quantitatively in Table 5. That the d for C is half that for B is due to both the gap and the log₁₀(sp) in combination being weaker in C than B.

To understand the data it can be useful to dissect the various contributions to the decision function as done above. However, the real power of the decision function is its utility in qualitatively ranking selective expression patterns in large-scale data in a way that is not only easily automated, but objective and consistent.

To convey the application of the algorithm to large-scale data, results from a large database of assemblies are depicted in Figure 11 (see equations 13-15). Each assembly that is identified by the algorithm as being selectively expressed is plotted (bubb