A given protein sequence of N residues is viewed as a collection of N – n + 1 overlapping words of n letters, termed n-grams (also called n-tuples), composed of n contiguous amino acids along the sequence. For each protein, we consider all 3-grams (n = 3) or triplets, by sliding a window of three amino acids along the sequence, thus leading to a total of (N – 2) triplets. The natural frequency of a given 3-gram of type i (1 ≤ i ≤ 20³) is defined as

p i u n i = C i ∑ C j MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCdaqhaaWcbaGaemyAaKgabaGaemyDauNaemOBa4MaemyAaKgaaOGaeyypa0ZaaSaaaeaacqWGdbWqdaWgaaWcbaGaemyAaKgabeaaaOqaamaaqaeabaGaem4qam0aaSbaaSqaaiabdQgaQbqabaaabeqab0GaeyyeIuoaaaaaaa@3C31@

where C_i is the number of occurrence in the UniProt, and the summation is performed over all 20³ types of 3-grams in the 172,233 sequences compiled as of March 1, 2005, which contain a total number C_tot = 62,256,868 of 3-grams. Each of the 20³ distinctive types is represented in UniProt. Their counts vary in the range 170 <C_i < 67,264, with the lower and upper bounds corresponding to the 3-grams MCW and AAA, respectively. Figure 1 displays the distribution of the counts C_i. The inset shows a closer view of the percentage of 3-grams types whose UniProt counts fall in successive grids of size Δ C_i = 500, for the range C_i≤ 1.5 10⁴. We note that less than 1% of the 20³ types have UniProt counts C_i lower than 500, and 14.97% has C_i < 2,000. Rare 3-grams are defined as those in the lower end of the histogram, located one standard deviation away from the mean. They comprise 6.0% of all 3-gram types, and 0.595% of all 3-grams counts in the UniProt (see Figure 1 caption).

The unbiased probability of a given 3-gram is p₀ = 1/20³ = 0.000125, and the corresponding count is C₀ = C_tot/20³ = 7782 in the UniProt. The 3-grams whose count is lower than this value by a factor of f = 2 compose 10% of all the counts C_tot in UniProt, and those lower by f = 3 amount to 4% of C_tot.

The frequency piuni MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCdaqhaaWcbaGaemyAaKgabaGaemyDauNaemOBa4MaemyAaKgaaaaa@33D0@ given by eq (1) will also be termed the 'observed' or 'natural' probability. We will associate with each 3-gram type a scarcity score given by

s i u n i = − ln ⁡ [ p i u n i ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemyAaKgabaGaemyDauNaemOBa4MaemyAaKgaaOGaeyypa0JaeyOeI0IagiiBaWMaeiOBa4Maei4waSLaemiCaa3aa0baaSqaaiabdMgaPbqaaiabdwha1jabd6gaUjabdMgaPbaakiabc2faDbaa@4247@

which provides a measure of the information content each 3-gram carries. The distribution of scarcity scores is given in Figure 2. They vary in the range 6.83 ≤ siuni MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemyAaKgabaGaemyDauNaemOBa4MaemyAaKgaaaaa@33D6@ ≤ 12.81. Panel A displays the scores sorted in descending order, plotted against 3-gram type index, and panel B displays their histogram (distribution among the 20³ types), based on grids of size Δsiuni MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemyAaKgabaGaemyDauNaemOBa4MaemyAaKgaaaaa@33D6@ = 0.25. The arrows display the threshold adopted for defining rare 3-grams.

The 'expected' probability of 3-gram i = {XYZ} on the other hand, based on the natural occurrences of individual amino acids, assuming the probabilistic occurrence of amino acids to be independent of their sequential neighbors, is

p i e x p = p 0 X p 0 Y p 0 Z MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCdaqhaaWcbaGaemyAaKgabaacbiGae8xzauMae8hEaGNae8hCaahaaOGaeyypa0JaemiCaa3aaWbaaSqabeaacqaIWaamaaGccqWGybawcqWGWbaCdaahaaWcbeqaaiabicdaWaaakiabdMfazjabdchaWnaaCaaaleqabaGaeGimaadaaOGaemOwaOfaaa@403C@

where p⁰ X is the natural frequency of amino acid of type X (for details see Table S1 in Additional file 1). The enrichment in a given 3-gram count will be described by the enhancement factor

a i = p i u n i / p i e x p MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGHbqydaWgaaWcbaGaemyAaKgabeaakiabg2da9iabdchaWnaaDaaaleaacqWGPbqAaeaacqWG1bqDcqWGUbGBcqWGPbqAaaGccqGGVaWlcqWGWbaCdaqhaaWcbaGaemyAaKgabaacbiGae8xzauMae8hEaGNae8hCaahaaaaa@3FC7@

Figure 3A compares the observed and expected counts of 3-grams. While the slope gives an average enhancement factor <a_i> = 1.02 with a correlation coefficient of 0.96, a more detailed examination of expected and observed 3-grams Figure 3C shows that the two distributions exhibit statistically significant departure (χ², P < 0.001). The expected histogram is more broadly distributed than that observed. Some 3-grams that show substantial departures from their expected frequencies (i.e. enhancement factor a_i≠ 1) are labeled in panel B. Among over-represented rare 3-grams we observe CCC, WWN, WYW, CWC, CHC and WWW. We also note that AAA and LLL are distinguished by their high piuni MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCdaqhaaWcbaGaemyAaKgabaGaemyDauNaemOBa4MaemyAaKgaaaaa@33D0@, and HHH and QQQ by their enhancement a_i ≥ 6. The HHH enrichment may, however, arise from tagged histidines, rather than reflecting a natural preference.

The present approach permits us to assign a scarcity score to each amino acid in a given sequence without recourse to sequence alignment. In the calculations below we have performed similar analyses for protein families, where the occurrence of rare 3-grams at specific locations where examined with regard to their conservation properties. Given an alignment of homologues sequences, an average scarcity score was assigned to each position j along a sequence k, based on the minimum value, puni|jk=min{p    auni,p    bunip    cuni}jk MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGaemiCaa3aaWbaaSqabeaacqWG1bqDcqWGUbGBcqWGPbqAaaGccqGG8baFdaWgaaWcbaGaemOAaOMaem4AaSgabeaakiabg2da9GWaciaa=1gacaWFPbGaa8NBaiabcUha7jabdchaWnaaDaaaleaacqqGGaaicqqGGaaicqqGGaaicqqGGaaicqqGHbqyaeaacqqG1bqDcqqGUbGBcqqGPbqAaaGccqGGSaalcqWGWbaCdaqhaaWcbaGaeeiiaaIaeeiiaaIaeeiiaaIaeeiiaaIaeeOyaigabaGaeeyDauNaeeOBa4MaeeyAaKgaaOGaemiCaa3aa0baaSqaaiabbccaGiabbccaGiabbccaGiabbccaGiabbogaJbqaaiabbwha1jabb6gaUjabbMgaPbaakiabc2ha9naaBaaaleaacqWGQbGAcqWGRbWAaeqaaaaa@706E@, of the three consecutive 3-grams of types (a, b, c) that contain residue j, using

< s j u n i > = − ln ⁡ [ < p j u n i > ] = − ln ⁡ [ m − 1 Σ k p u n i | j k ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGH8aapcqWGZbWCdaqhaaWcbaGaemOAaOgabaGaemyDauNaemOBa4MaemyAaKgaaOGaeyOpa4Jaeyypa0JaeyOeI0IagiiBaWMaeiOBa4Maei4waSLaeyipaWJaemiCaa3aa0baaSqaaiabdQgaQbqaaiabdwha1jabd6gaUjabdMgaPbaakiabg6da+iabc2faDjabg2da9iabgkHiTiGbcYgaSjabc6gaUjabcUfaBjabd2gaTnaaCaaaleqabaGaeyOeI0IaeGymaedaaOGaeu4Odm1aaSbaaSqaaiabdUgaRbqabaGccqWGWbaCdaahaaWcbeqaaiabdwha1jabd6gaUjabdMgaPbaakiabcYha8naaBaaaleaacqWGQbGAcqWGRbWAaeqaaOGaeiyxa0faaa@5E71@

where the summation is performed over the homologous sequences (1 ≤ k ≤ m) in the alignment.

The biological significance of rare 3-grams has been examined by analyzing the results with regard to the dataset of 59 non-redundant enzyme/ligand complexes compiled by Gutteridge and Thornton 24, shortly referred to as the GT dataset (Table 1). Two non-overlapping subsets of 3-grams were retrieved from the GT dataset. The first is composed of all 3-grams located at the active sites, and the second includes all other residues. The two subsets comprise 3,514 and 17,450 3-grams, respectively. Active site 3-grams are defined as those containing at least one heavy atom located within 5Å from the ligand.

The UniProt frequencies of these two groups of 3-grams yielded the histograms displayed in Figure 4A. To obtain these histograms, the ensemble of 3-grams in the respective subsets 1 and 2 of active site residues (gray) and other residues (black) have been organized into groups of different UniProt frequencies, and the counts of 3-grams in each group have been determined. The abscissa shows the ranges of UniProt counts, divided into intervals of Δ C_i = 1000, starting from C_i≤ 1,000 (labeled 1000), and the ordinate represents the population (or probability) of each group. Not all 20³ types of 3-grams were represented in this set of 59 enzymes.

Comparison of the two histograms immediately reveals the tendency of the active site 3-grams to populate the lower UniProt counts. For example, let us consider the rare 3-grams (with C_i < 1,140 or siuni MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemyAaKgabaGaemyDauNaemOBa4MaemyAaKgaaaaa@33D6@ > 10,908; see Methods and Figure 2). 11.67% of active site 3-grams fall in this range, whereas the percentage drops to 4.35% when 3-grams at other regions are considered. Therefore, rare 3-grams are over-represented in the active sites by a factor of 2.67. The difference between the two distributions shown in panel A is statistically significant (χ², P << 0.001).

Similarly, we compared the histogram obtained for the subset of active site 3-grams with that calculated for solvent-exposed 3-grams (see Methods). Figure 4B shows the histograms for active site (gray) and exposed (black) residues. The active sites are again distinguished by the propensity of rare 3-grams. Rare 3-grams are 5.4 times more frequent at active sites compared to exposed regions (χ², P << 0.001) . The difference between the two histograms is more pronounced than that observed in panel A.

Next, we examined the identity of 3-grams at the active sites, and focused in particular on the rare 3-grams that are recruited. Given that not all of the 20³ types of 3-grams were represented in the GT dataset, not all types of 3-grams might be either seen at the active sites. We first analyzed the amino acid composition of active sites. For each amino acid, the probabilities of being located at the active site versus elsewhere were calculated. The ratio between these two probabilities, termed the active site propensity, is shown in Figure 5A. Four amino acids, Cys, His, Gly, and Trp, are distinguished by their high active site propensities. This is in agreement with previous work 25 where His and Cys were shown to have the highest propensity for serving as catalytic residues via their side chains, while Gly has the highest propensity of catalyzing reactions via its main chain. We also note that, in the present study, Arg and Glu exhibit relatively low tendencies to locate at active sites, while these two amino acids were reported by Bartlett et al 25 to have relatively high catalytic propensities. This difference presumably arises from the different definitions adopted for active sites in the two studies: we refer to all amino acids located in the neighborhood of the ligand, while Bartlett et al 25 referred to residues directly involved in catalytic activity.

Cys and His, two residues that have the highest active site propensity (both in terms of their ligand-binding and catalytic activities) are among the least frequent amino acids in terms of their natural occurrence. Figure 5A shows the relation between the frequencies of amino acids in UniProt and their active site propensities. While the data are scattered, there is a discernible anti-correlation between the two sets of data, consistent with Figure 4; rare amino acids have a higher tendency to be in active sites. Clearly this preference becomes more pronounced when 3-grams, as opposed to monograms are examined, as a can be seen from the skewed distribution of the frequency of active site residues towards lower counts in Figure 4.

Rare 3-grams may arise from three possibilities: (1) these may be words in which the individual letters have low natural frequencies themselves (i.e. a_i ≈ 1), (2) these may be rare 'words', i.e. rare combinations of particular amino acids (a_i << 1); or (3) these may be rare despite their enhancement (a_i >>1) due to the extremely low natural frequencies of the individual amino acids that compose them. Counterparts of Figure 3 plotted for the active site 3-grams and other 3-grams showed that both groups fall in the first category (a_i ≈ 1). Their observed counts plotted against the expected ones yield a slope of 1.075 ± 0.008. The rare 3-grams belonging to the two groups, on the other hand, exhibited a slight enhancement of a_i = 1.11 and 1.07, for active site and other residues, respectively. Thus, 3-grams recruited at active sites are usually combinations of residues which already have relatively low natural frequencies of occurrence. Yet, particular 3-grams exhibit significant departure from their expected occurrence probabilities, e.g. SSS, GGG, RRR, GSG (over-represented), and LEL (under-represented). See Table S2 and S3 in Additional files for details. Among rare 3-grams at active sites, also exhibiting a large enhancement in the UniProt (case 3, above), we distinguish WWH and HCW.

Scarcity scores were calculated for all 3-grams in each of the 59 proteins in the GT dataset. The postulate that rare 3-grams (above a threshold scarcity score) preferentially locate in active sites was examined for a series of threshold scores. For each enzyme, the rates of true positives (TP) and false positives (FP) were evaluated as a function of threshold scarcity scores, TP rates being defined as the fraction of active site 3-grams that exhibit scarcity score above the threshold value. Table 2 shows the average ratios between the rates of TPs and FPs for seven different thresholds. The uppermost threshold (s_max = 11) shows more than threefold enhancement in the rate of TPs relative to FPs, while the discriminative power of the scarcity score vanishes as we lower the threshold. A detailed list of TP and FP rates along with the top ranking 3-grams are provided for each enzyme in the Additional files Table S4.

The functional role of rare 3-grams at proteins' functional sites is illustrated here for three test cases, a Src tyrosine kinase, hemoglobin and a tyrosyl-tRNA synthetase. For each case, a set of homologous proteins has been compiled and aligned, and the scarcity score <sjuni MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemOAaOgabaGaemyDauNaemOBa4MaemyAaKgaaaaa@33D8@> of amino acid(s) at each position has been calculated. The analysis reveals that stretches of rare amino acids are involved in functional roles even beyond ligand binding sites.

In the above three examples the scarcity score was calculated by averaging p^uni|_jk over a set of homologous proteins. However, in principle, scarcity score calculations can be equally performed for a single sequence as the ingredient is the UniProt frequencies for triplets along the sequence. The alignments are simply used to magnify (increase the accuracy of) the signals that could, otherwise, be extracted from single sequence analysis. As shown in the figures S1–S3 in the Additional files, the sjuni MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemOAaOgabaGaemyDauNaemOBa4MaemyAaKgaaaaa@33D8@ profiles based on single sequences closely approximate those derived from multiple sequence alignments. For hemoglobin subunit α, for example, six stretches of amino acids are distinguished by their high scarcity scores: W₁₄GK₁₆, R₃₁MF₃₃, F₄₃PHF₄₆, D₇₅MPN₇₈, H₈₇AH₈₉, S₁₀₂HCL₁₀₅, in accord with the results presented above. The only stretch of amino acids that differs from the above analysis is D₇₅MPN₇₈. Thus, when sequence conservation algorithms fail, the 3-gram analysis of single chains can be advantageously resorted to for identifying potentially functional sites.

Present calculations are based on the data extracted from UniProt 20 version 4.2. This database was chosen because it is manually curated with minimal level of redundancy and still comprehensive. Furthermore, UniProt contains entries from multiple organisms and it is therefore less susceptible to biases in 3-gram distributions of specific species as noted by Ganpathiraju et al 54.

The biological implications of the 3-grams distinguished by high scarcity scores were examined using the dataset of 59 enzyme/ligand complexes compiled by Gutteridge and Thornton 24 (GT dataset; see Table 1). We performed three case studies: C-Src kinase, hemoglobin subunits α and β, and tyrosyl-tRNA synthetase. Homologous sequences were retrieved in each case by a PBLAST search 55 of the SwissProt database. Sequences with E-values lower than 10^-57 were extracted, leading to m = 111 homologous sequences for c-Src, 138 and 224 sequences for hemoglobin α- and β-subunits, respectively, and 24 sequences for tyrosyl tRNA synthetase. The sequences were aligned with ClustalX using default parameters 56.

The Accessible surface area (ASA) of each amino acid in the GT database was calculated using the program NACCESS 57. Amino acids were defined as exposed if their relative exposed surface are was > 20%. 3-grams with two or more exposed amino acids were defined as exposed ones.