We consider the upstream region of each open reading frame (ORF), and we fix the maximum length K of the upstream sequence to be considered. The choice of K depends on the typical location of most regulatory sites: in general K is a number between several hundred and a few thousand. For each ORF g, the actual length of the sequence we consider is K_g defined as the minimum between K and the available upstream sequence before the coding region of the previous gene.

For each word w of length l (6 ≤ l ≤ 8 in this study), and for each ORF g we compute the number m_g(w) of occurrences of w in the upstream region of g. Non palindromic words are counted on both strands: therefore we define the effective number of occurrences n_g(w) as

where is the reverse complement of w.

We define the global frequency p(w) of each word w as

where, in order to count correctly the available space for palindromic and non palindromic words,

p(w) is therefore the frequency with which the word w appears in the upstream regions of the whole genome: it is the "background frequency" against which occurrences in the upstream regions of the individual genes are compared to determine which words are overrepresented.

For each ORF g and each word w we compute the probability b_g(w) of finding n_g(w) or more occurrences of w based on the global frequency p(w):

We define a maximum probability P, depending in general on the length l of the words under consideration, and consider, for each w, the set

of the ORFs in which the word w is overepresented compared to the frequency of w in the upstream regions of the whole genome. That is, w is considered overrepresented in the upstream region of g if the probability of finding n_g(w) or more instances of w based on the global frequency is less than P.

This completes the construction of the sets S(w). Two free parameters have to be fixed: the length K of the upstream region to be considered and the probability cutoff P for each length l of words considered. A result in Ref. 2 suggests suitable choices of these two numbers: the authors list the 34 ORFs of S. cerevisiae that have 3 or more occurrences of the word GATAAG in their 500 bp upstream region. 23 out of these 34 ORFs correspond to a gene with known function, and 20 out of these 23 are regulated by nitrogen. This result suggests to choose K = 500 for the upstream length, and a value of the probability cutoff such that three or more instances of GATAAG in the 500 bp upstream region of an ORF are considered significant. Any choice of P between 0.018 and 0.1 would satisfy this criterion, and we chose P = 0:02. Tentatively, we kept the same value of P for all values of l. With this choice, the number of instances of a word that are necessary to be considered overrepresented in a 500 bp upstream sequence can be as high as six for common 6-letter words and as low as one for rare 8-letter words. In particular, our set S(GATAAG) almost¹ coincides with the one discussed in 2. However the word GATAAG will not turn out to be significant in our study.

As noted above, it would be natural to make the probability cutoff P depend on the word length, simply because the number of possible words increases with their length: For example one could take the cutoff for each word length to be inversely proportional to the number of independent words of such length. However it turns out that this procedure tends to construct sets that are less significant when tested for correlation with expression. Therefore we chose to fix the cutoff at 0.02 for all word lengths. It is important to keep in mind that no statistical significance whatsoever is attributed to the sets per se: The only sets that are retained at the end of the analysis are the ones that show significant correlation with expression. Therefore the choice of the cutoff in the construction of the sets can be based on such a pragmatic approach without jeopardizing the statistical relevance of the final result.

The second step of our procedure consists in studying, for each set S(w) defined as above, the expression profiles of the ORFs belonging to S(w) in DNA microarray experiments. The idea is that if the average expression profile in the set S(w) for a certain experiment is significantly different from the average expression for the same experiment computed on the whole genome, then it is likely that some of the ORFs in S(w) are coregulated and that the word w is a binding site for the common regulating factor.

To look for such instances we consider the gene expression profiles during the diauxic shift, i.e. the metabolic shift from fermentation to respiration, as measured with DNA microarrays techniques in Ref. 1. In the experiment gene expression levels were measured for virtually all the genes of S. Cerevisiae at seven time-points while such metabolic shift took place. The experimental results are publicly available from the web supplement to Ref. 1.

We considered each time-point as a single experiment, and for each gene g we defined the quantity r_g(i) (1 = 1,..., 7) as the log₂ of the ratio between the mRNA levels for the gene g at time-point i and the initial mRNA level. Therefore e.g. r_g(i) = 1 means a two-fold increase in expression at timepoint i compared to initial expression.

For each time-point i we computed the genome-wide average expression R(i) and its standard deviation σ (i). These are reported in Tab. 1, where N(i) is the number of genes with available expression value for each timepoint. Then for each word w we compute the average expression in the subset of S_w given by the genes for which an experimental result is available at timepoint i (in most cases this coincides with S_w):

where N(i, w) is the number of ORFs in S_w for which an experimental result at timepoint i is available, and the difference

ΔR_w(i) is the discrepancy between the genome-wide average expression at time-point i and the average expression at the same time-point of the ORFs that share an abundance of the word w in their upstream region. A significance index sig(i, w) is defined as

and the word w is considered significantly correlated with expression at time point i if

In this work we chose Λ = 6: this means that we consider meaningful a deviation of R_w(i) by six s.d.'s from its expected value. The sign of sig(i, w) indicates whether w acts as an enhancer or an inhibitor of gene expression.

Nine significant words can be associated to the PAC motif 847, all of them with rather high scores. They are shown in Tab. 2, where, as in all the following tables, significativity indices are shown only for those timepoints where they exceed our threshold jsigj > 6. Given the perfect alignment of these words, it is not surprising that these sets largely overlap each other: The union af all the nine sets contains a total of 96 genes. As an example, in Fig. 1 we show the average expression for the genes associated with the word GATGAG as a function of the time, compared to the average expression computed over the whole genome. Fig. 2 shows the significance index for the same set. The genes in this set are shown in Tab. 3 togather with their expression profiles.

Two words can be associated with confidence to the motif RRPE 47, and are shown in Tab. 4. The union of the two sets contains 76 genes. We see that genes containing the motifs PAC and RRPE are repressed at the late stage of the diauxic shift compared to the early stages. This result is in agreement with the expression coherence score data available from the web supplement to Ref. 6: There one can see that (1) of all known regulatory motifs, PAC and RRPE show the highest expression coherence for the diauxic shift and (2) viceversa, of the eight experimental conditions considered in Ref. 6, the diauxic shift is the one in which both the PAC and RRPE motif show the highest expression coherence score.

A total of ten significant words can be associated to the motifs STRE 910 and MIG1 1112. It is well known that these play an important role in glucose repression (see e.g.113 and references therein). Most of these words show comparable scores for the two motifs (due to their similarity) so we decided to show them together in Tab. 5 which shows the two scores for each word. A total of 212 genes belong to the union of all these sets.

Two words are associated to the known UME6 motif, a.k.a. URS1 1415, known to be a pleiotropic regulator implicated in glucose repression 16. They are shown in Tab. 6. The two sets do not overlap, so that a total of 56 genes are associated to this motif.

Three words, shown in Tab. 7, are of uncertain status: for the first one, the set S(ACTTTC) contains only 2 genes, making the statistical significance of the result questionable. The word CCCCTGAA scores best with the PDR motif (0.58): given the low significance of this score, and the fact that PDR does not seem to be relevant for any other word, this is most likely accidental. The word should probably be considered as belonging to the STRE/MIG1 motif (the scores are STRE: 0.46, MIG1: 0.49). Finally the word GCCCCTGA scores best with UME6 (0.55), but its expression pattern is more similar to the STRE/MIG1 motifs (scores: STRE:0.44, MIG1: 0.46).

The genome of S. cerevisiae contains a few families of genes whose coding and upstream regions are identical or nearly identical. Consider for example the COS1 gene (YNL336W): the seven genes COS2-COS8 have both coding sequence and 500 kb upstream sequence coinciding better than 80% with the COS1 sequence. Therefore if the upstream sequence of COS1 contains over-erpresented words, they will likely appear in all of the upstream regions. On the other hand, the expression profiles of all the genes in the family will be the same when measured by a microarray experiment, simply because the experimental apparatus cannot distinguish between the mRNA produced by the various members of the family, due to cross-hybridization between their mRNA. Therefore all of the genes of the family are likely to occur in the sets of the words that are overrepresented in their upstream region, and even a small deviation from the genome-averaged expression acquires a statistical significance.

We found two instances of this in our analysis: the words GACGTAGC and GGTCGCAC appear to be associated to significant enhancement of the corresponding sets of genes at late timepoints in the diauxic shift: however the two sets contain respectively seven out of eight and all of the COS1-COS8 genes. Since the COS genes are mildly overexpressed, this creates a false statistical significance. When one corrects for this, by keeping only one representative of the family, the statistical significance of the two sets disappears.

Finally, the word ATAAGGG/CCCTTAT is a candidate new binding site, since it does not have good comparison scores with any of the known motifs. It scores best with the AFT1 motif, with a 0.52 score which is practically meaningless since 84.9% of all independent 7-letter words score the same or better with at least one motif. It is associated with 13 genes, as shown in Tab. 8, which are overexpressed at late timepoints. The average expression levels for the set and the significance index are shown as a function of time in Figs. 3 and 4.

As stated in the introduction, the method proposed in Ref.5 also allows one to identify regulatory motifs without any previous clustering of gene expression data: a linear dependence of the logarithm of the expression levels on the number of repetitions of each regulatory motifs is postulated, and motifs are ranked according to the reduction in χ² obtained when such dependence is subtracted from the experimental expression levels. Iteration of the procedure produces a model, that is a set of relevant regulatory motifs, for each expression data set.

In Ref. 5 such a model is presented for the 14 min. time point in the α-synchronized cell-cycle experiment of Spellmann et al., Ref. 17. We used our algorithm on the same data set to compare the findings. Let us concentrate on the 7-letter words (the longest considered in 5). We found 9 significant words, reported in Tab. 9. Of these, five coincide with or are very similar to words found by the authors of Ref.5 (see their Tab. 2). The remaining four (AGGCTAA, GGCTAAG, GCTAAGC and CTAAGCG, whose similarity clearly suggests the existence of a longer motif) are of particular interest for the purpose of comparing the two methods: If one looks at the dependence of the expression levels on the number of occurrences of these words in the 500 bp upstream region, one clearly sees the existence of an activation threshold (see Fig. 5, where such dependence is shown for GGC-TAAG). On the other hand, by looking at these data one hardly expects a significant reduction in χ² when trying to describe this dependence with a straight line. This should be compared to the same dependence for the word AAAATTT, shown in Fig. 6, which is found by both algorithms. On the other hand, there are two 7-word motifs found in 5 that do not pass our significativity threshold, that is CCTCGAC and TAAACAA.

We can conclude that the two methods tend to find motifs with a different effect on gene expression: probably the best results can be obtained by using them both on the same data set.