Volume 8 Supplement 6

## Otto Warburg International Summer School and Workshop on Networks and Regulation

# Dissecting complex transcriptional responses using pathway-level scores based on prior information

- Harmen J Bussemaker
^{1, 2}Email author, - Lucas D Ward
^{1}and - Andre Boorsma
^{3}

**8(Suppl 6)**:S6

https://doi.org/10.1186/1471-2105-8-S6-S6

© Bussemaker et al; licensee BioMed Central Ltd. 2007

**Published: **27 September 2007

## Abstract

### Background

The genomewide pattern of changes in mRNA expression measured using DNA microarrays is typically a complex superposition of the response of multiple regulatory pathways to changes in the environment of the cells. The use of prior information, either about the function of the protein encoded by each gene, or about the physical interactions between regulatory factors and the sequences controlling its expression, has emerged as a powerful approach for dissecting complex transcriptional responses.

### Results

We review two different approaches for combining the noisy expression levels of multiple individual genes into robust pathway-level differential expression scores. The first is based on a comparison between the distribution of expression levels of genes within a predefined gene set and those of all other genes in the genome. The second starts from an estimate of the strength of genomewide regulatory network connectivities based on sequence information or direct measurements of protein-DNA interactions, and uses regression analysis to estimate the activity of gene regulatory pathways. The statistical methods used are explained in detail.

### Conclusion

By avoiding the thresholding of individual genes, pathway-level analysis of differential expression based on prior information can be considerably more sensitive to subtle changes in gene expression than gene-level analysis. The methods are technically straightforward and yield results that are easily interpretable, both biologically and statistically.

## Introduction

Many of the popular methods for analyzing DNA microarray expression data, from clustering [1] to more sophisticated machine-learning approaches [2–5], require expression data over a large number of different conditions as input. However, it is common to only have expression data for a few different strains and/or conditions. In this case, what is of interest are the changes in mRNA abundance for each gene, usually represented as the logarithm of the fold-change between test and control. The traditional way of analyzing such data is to first identify significantly up- and down-regulated genes, and subsequently to characterize these sets in terms of enrichment for functional annotation [6] or upstream promoter elements [7–9]. However, by requiring statistically significant differential expression at the level of individual genes, a lot of information about differential expression will be lost that could have been detected using analysis methods working at the level of pathways.

To understand this, assume that we are comparing two conditions and that the measurement error for the fold-change of individual genes is 20%. Now consider a specific pathway consisting of 100 genes that are all upregulated by 10%. This level of differential expression is well within the noise for individual genes, none of which will therefore be classified as significantly induced. However, the error in the *average* expression of 100 *randomly* chosen genes will be on the order of 20%/$\sqrt{100}$ = 2%. The 10% change in expression at the level of the whole pathway therefore corresponds to five units of standard error and is highly statistically significant.

*categorical*information of boolean type ("true" or "false"), which tells us whether or not a gene belongs to a specific gene set; and

*quantitative*information, e.g., the mRNA expression log-ratio between two conditions for a gene or the ChIP-chip [11] fold enrichment for the gene's promoter region. Given any two distinct features characterizing each gene, their genomewide statistical association can be scored using an appropriate statistical test (Table 1).

When to use which statistical test.

First Feature | Second Feature | (Non)-Parametric? | APPROPRIATE TEST |
---|---|---|---|

categorical | categorical | non-parametric | hypergeometric |

quantitative | quantitative | parametric | Pearson |

quantitative | quantitative | non-parametric | Spearman, Kendall |

categorical | quantitative | parametric | two-sample t-test |

categorical | quantitative | non-parametric | Wilcoxon-Mann-Whitney, Kolmogorov-Smirnov |

### The traditional approach: scoring over-representation of predefined gene sets

*overlap*between both gene sets. Let the total number of genes in set

*A*be

*a*, the total number of genes in set

*B*be

*b*, and the total number of genes in the genome be

*n*. Furthermore, let the overlap

*x*denote the number of genes shared between

*A*and

*B*. If the two sets are chosen randomly and independently, the average overlap will be:

*b/n*of all genes belongs to set

*B*then the expected fraction of genes in set

*A*that also belongs to set

*B*equals

*x/a*. In the case of over-representation, when

*x*> <

*x*>, the P-value that quantifies how likely it is to get at least the same number of overlapping genes by chance, is given by

*H*is the hypergeometric distribution given by

*x*< <

*x*>). In that case, the P-value is given by

This use of the cumulative hypergeometric distribution is also known as "Fisher's exact test." The test is by nature non-parametric because both input features are non-parametric. Under specific conditions the hypergeometric distribution may be approximated by the binomial or chi-square distribution. Several implementations of this approach are reviewed by Khatri and Draghici [12]. Since typically a large number of gene sets are scored in parallel, the p-values must be corrected for multiple testing. Grossman et al. [13] recently addressed technical complications arising from the strong overlap between the hierarchically organized Gene Ontology categories.

### An alternative: scoring the distribution of expression levels for predefined gene sets

*S*consisting of

*n*genes and the genomewide mean

*μ*:

Here ${\sigma}_{\overline{x}}$ = *σ*/$\sqrt{n}$ is the standard error of the mean, *σ* being the standard deviation of the genomewide distribution of log-ratios. The same metric is used by the "parametric analysis of gene expression" (PAGE) method of Kim and Volsky [15]. For larger gene sets, however, the standard t-test for the difference between means yields more accurate results [16]. The t-test, in general, scores the statistical association between a categorical and quantitative feature. The categorical feature is used to partition the set of all genes, *G*, into two complementary subsets *S* and S'.
The *t* statistic measures the difference between the means of the two subsets in units of its standard error:

$t=\frac{{\overline{x}}_{S}-{\overline{x}}_{{S}^{\prime}}}{\sigma \sqrt{\frac{1}{\left|S\right|}+\frac{1}{\left|{S}^{\prime}\right|}}}$

_{ S }and ${\overline{x}}_{{S}^{\prime}}$ are the mean expression value of genes in set

*S*and

*S'*, respe ctively, and the standard error of the difference is given by

with *σ*_{
S
}and *σ*_{
S'
}the standard deviation of the expression values of the genes within set *S* and *S'*, respectively. Using a t-distribution with *n* - 2 degrees of freedom, each t-value can be converted to a p-value, which should again be corrected for multiple testing.

Other statistical tests have also been used to detect differential expression of gene sets based on the distribution of expression values. The original version of the "gene set enrichment analysis" (GSEA) method [17] used the Kolmogorov-Smirnov (KS) statistic to test whether the distribution of expression levels in a specific gene set was different from that of all genes; this approach was later found to require a modification to work reliably [18]. The Wilcoxon-Mann-Whitney test, a non-parametric equivalent of the t-test that uses expression values only to rank the genes, has also been applied to this problem [19].

### Beyond gene sets: approaches based on regression analysis

The assignment of genes to gene sets is categorical: Either the gene belongs to the set, or it does not. However, gene sets are often a proxy for regulatory pathways. This is most obvious in the case of the gene sets based on ChIP-chip data [11], which were used by Boorsma et al. [16] to analyze differential mRNA expression using the t-test. The strict delineation of "targets" of a given TF based on thresholding of the ChIP-chip signals is an oversimplification. In reality, the degree to which the transcription rate for a given gene responds to a change in the activity of the TF depends in a continuous fashion on the binding affinity between the TF and the promoter DNA (as well as interactions with co-factors, chromatin, etc.). Thus, if an estimate of this affinity is used as a predictor for changes in transcription rate (and therefore expression), a single parameter that quantifies the global change in TF activity may explain a wide range of transcriptional responses across the genome. This intuition can be formalized in the form of a linear regression model:

*A*_{
g
}= *C* + *FN*_{
g
}

*C*is an intercept and

*F*a slope estimating the change in TF activity. The dependent ("response") variable

*A*

_{ g }is the mRNA expression log-ratio of gene

*g*between conditions. The independent ("predictor") variable

*N*

_{ g }represents the regulatory network connectivity between the TF and the promoter region of gene

*g*. For given

*A*

_{ g }and

*N*

_{ g }, the deviance

*D*between the measured and predicted expression values

and*C* = <*A*> - *F* <*N*>.

*X*> = (1/

*G*) ∑

_{ g }

*X*

_{ g }denotes an average over all genes and

*δX*

_{ g }≡

*X*

_{ g }- <

*X*> denotes the deviation from the genomic mean, so that <

*δX*

^{2}> equals the variance of

*X*. Because we are dealing with

*univariate*regression (a single independent variable), the Pearson correlation coefficient between

*A*and

*N*,

*F*by the following equation:

*R*

^{2}, defined as the fraction of the variance in expression that can be explained by the linear model, is given by the square of Pearson correlation:

*r*due to R.A. Fisher

yields a statistic *t* that is distributed according a t-distribution with *n* - 3 degrees of freedom, and can thus be easily converted to a p-value. Again, multiple testing will need to be accounted for whenever the association with multiple features is scored in parallel.

There are many ways in which the regulatory network connectivities *N*_{
g
}can be chosen. The first application of regression analysis to microarray data, by Bussemaker et al. [20], used integer motif counts in promoter regions. Continuous sequence scores based on position-specific scoring matrices (PSSMs) [21, 22] and position-specific affinity matrices (PSAMs) [23, 24] have also been used. The values for *R*^{2} obtained with such sequence-based predictors are typically in the range of 1–5%. Another possible choice for *N* are ChIP-chip enrichment (log-)ratios [25, 26]. As these values are relatively noisy experimental measurements, the values for *R*^{2} observed in this case are usually smaller (< 1%).

## Conclusion

In this work, rather than providing a comprehensive review of all relevant literature, we have outlined two conceptually different approaches for scoring differential expression at the pathway level. These methods use prior information about how different genes relate to each other to reduce the dimensionality of the problem. This obviates the need to first obtain gene clusters or modules from expression data over multiple conditions, and thereby makes it possible to analyze each differential expression profile by itself in a condition-specific fashion.

## Declarations

### Acknowledgements

We thank members of the Bussemaker Lab for valuable discussions, and Barrett Foat for a critical reading of the manuscript. This work was supported by grants HG003008, CA121852, and GM074105 from the National Institutes of Health and grant APB.5504 from the Netherlands Foundation for Technical Research (STW).

This article has been published as part of *BMC Bioinformatics* Volume 8 Supplement 6, 2007: Otto Warburg International Summer School and Workshop on Networks and Regulation. The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2105/8?issue=S6

## Authors’ Affiliations

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