We use two different data sets downloaded from two databases. The first set is the human Pol II promoter sequences from Release 90 of the Eukaryotic Promoter Database (EPD) 14. The EPD is an annotated non-redundant collection of eukaryotic Pol II promoters, experimentally defined by a transcription start site (TSS) 15. The EPD is a useful database when one wants to deal with the Pol II promoter prediction problem and it is broadly tested by different prediction tools 716171819. A total of 1871 entries of human Pol II promoter sequences with window size of 499 bp upstream and 100 bp downstream of TSS, which is the same as that used in Ref. 16, are obtained from EPD. The sequences containing 'N' are manually filtered out, which results in a total of 1856 sequences. The second set is the non-promoter sequences of the human genome. For this data set, we consider using the Exon/Intron Database (EID), which incorporates information on the exon/intron structure of eukaryotic genes 20 (21, hs35p1.EID.tar.gz). Firstly, the exon/intron sequences with 'n' and length less than 600 are filtered out. Then, we randomly select 1000 intron sequences from the file hs35p1.intrEID and 500 exon sequences from the file hs35p1.exEID. A fragment of length 600 is then selected randomly from each exon/intron sequence with length larger than 600. As the intron sequences are represented by lower-case letters in the file hs35p1.intrEID, we transform them into upper-case letters to be consistent with the promoter and exon sequences.

From the four different methods described in the Methods section, we get a total of 141 parameters. We will test their contributions in the promoter/non-promoter problem. Then we will try to combine some of them to see whether better results can be achieved.

For comparison of various methods, a benchmark should be set up. We use Fisher's linear discriminant algorithm 222324 to calculate the discriminant accuracies. We divide all promoter and non-promoter sequences into two sets randomly. A set of 90% of promoter/non-promoter sequences is regarded as a training set, and the set of remaining 10% of promoter/non-promoter sequences as a test set.

Fisher's discriminant algorithm is used to find a classifier in the parameter space for a training set. The given training set H = {x₁, x₂, ⋯, x_n} is partitioned into n₁ ≤ n training vectors in a subset H₁ and n₂ ≤ n training vectors in a subset H₂, where n₁ + n₂ = n and each x_i is a κ-dimensional vector, represented by one point in the κ-dimensional parameter space. Then H = H₁ ∪ H₂. We need to find a parameter vector w = (w₁, w₂, ⋯, w_κ)^T for the κ-dimensional space such that {yi=wxi}i=1n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaei4EaSNaemyEaK3aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpieqacqWF3bWDcqWF4baEdaWgaaWcbaGae8xAaKgabeaakiabc2ha9naaDaaaleaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGUbGBaaaaaa@3C47@ can be classified into two classes in the space of real numbers. If we denote

m j = 1 n j ∑ x i ∈ H j x i j = 1 , 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaGqabiab=1gaTnaaBaaaleaacqWGQbGAaeqaaOGaeyypa0tcfa4aaSaaaeaacqaIXaqmaeaacqWGUbGBdaWgaaqaaiabdQgaQbqabaaaaOWaaabuaeaacqWF4baEdaWgaaWcbaGae8xAaKgabeaaaeaacqWF4baEdaWgaaadbaGae8xAaKgabeaaliabgIGiolabdIeainaaBaaameaacqWGQbGAaeqaaaWcbeqdcqGHris5aaGcbaGaemOAaOMaeyypa0JaeGymaeJaeiilaWIaeGOmaiJaeiilaWcaaaaa@4700@

S j = ∑ x i ∈ H j ( x i − m j ) ( x i − m j ) T , j = 1 , 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaGqabiab=nfatnaaBaaaleaacqWGQbGAaeqaaOGaeyypa0ZaaabuaeaacqGGOaakcqWF4baEdaWgaaWcbaGae8xAaKgabeaakiabgkHiTiab=1gaTnaaBaaaleaacqWFQbGAaeqaaOGaeiykaKIaeiikaGIae8hEaG3aaSbaaSqaaiab=LgaPbqabaGccqGHsislcqWFTbqBdaWgaaWcbaGae8NAaOgabeaakiabcMcaPmaaCaaaleqabaGaemivaqfaaaqaaiab=Hha4naaBaaameaacqWFPbqAaeqaaSGaeyicI4SaemisaG0aaSbaaWqaaiabdQgaQbqabaaaleqaniabggHiLdGccqGGSaalaeaacqWGQbGAcqGH9aqpcqaIXaqmcqGGSaalcqaIYaGmcqGGSaalaaaaaa@52B5@

S_w = S₁ + S₂,

then the parameter vector w is estimated as Sw−1(m1−m2) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacbeGae83uam1aa0baaSqaaiabdEha3bqaaiabgkHiTiabigdaXaaakiabcIcaOiab=1gaTnaaBaaaleaacqaIXaqmaeqaaOGaeyOeI0Iae8xBa02aaSbaaSqaaiabikdaYaqabaGccqGGPaqkaaa@383E@23. As a result, Fisher's discriminant rule becomes: "assign x to H₁ if Z(x)=(m1−m2)TSw−1[x−12(m1+m2)]>0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacbeGae8NwaOLaeiikaGIae8hEaGNaeiykaKIaeyypa0JaeiikaGIae8xBa02aaSbaaSqaaiabigdaXaqabaGccqGHsislcqWFTbqBdaWgaaWcbaGaeGOmaidabeaakiabcMcaPmaaCaaaleqabaGaemivaqfaaOGae83uam1aa0baaSqaaiabdEha3bqaaiabgkHiTiabigdaXaaakiabcUfaBjab=Hha4jabgkHiTKqbaoaalaaabaGaeGymaedabaGaeGOmaidaaOGaeiikaGIae8xBa02aaSbaaSqaaiabigdaXaqabaGccqGHRaWkcqWFTbqBdaWgaaWcbaGaeGOmaidabeaakiabcMcaPiabc2faDjabg6da+iabicdaWaaa@500E@ and to H₂ otherwise" 22.

The discriminant accuracies for resubstitution analysis are defined as

p c = The number of all correct promoter discriminations The number of promoter sequences in the training set , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemiCaa3aaSbaaSqaaiabdogaJbqabaGccqGH9aqpjuaGdaWcaaqaaiabbsfaujabbIgaOjabbwgaLjabbccaGiabb6gaUjabbwha1jabb2gaTjabbkgaIjabbwgaLjabbkhaYjabbccaGiabb+gaVjabbAgaMjabbccaGiabbggaHjabbYgaSjabbYgaSjabbccaGiabbogaJjabb+gaVjabbkhaYjabbkhaYjabbwgaLjabbogaJjabbsha0jabbccaGiabbchaWjabbkhaYjabb+gaVjabb2gaTjabb+gaVjabbsha0jabbwgaLjabbkhaYjabbccaGiabbsgaKjabbMgaPjabbohaZjabbogaJjabbkhaYjabbMgaPjabb2gaTjabbMgaPjabb6gaUjabbggaHjabbsha0jabbMgaPjabb+gaVjabb6gaUjabbohaZbqaaiabbsfaujabbIgaOjabbwgaLjabbccaGiabb6gaUjabbwha1jabb2gaTjabbkgaIjabbwgaLjabbkhaYjabbccaGiabb+gaVjabbAgaMjabbccaGiabbchaWjabbkhaYjabb+gaVjabb2gaTjabb+gaVjabbsha0jabbwgaLjabbkhaYjabbccaGiabbohaZjabbwgaLjabbghaXjabbwha1jabbwgaLjabb6gaUjabbogaJjabbwgaLjabbohaZjabbccaGiabbMgaPjabb6gaUjabbccaGiabbsha0jabbIgaOjabbwgaLjabbccaGiabbsha0jabbkhaYjabbggaHjabbMgaPjabb6gaUjabbMgaPjabb6gaUjabbEgaNjabbccaGiabbohaZjabbwgaLjabbsha0baacqGGSaalaaa@B4C7@

p n c = The number of all correct non-promoter discriminations The number of non-promoter sequences in the training set . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@C048@

For the test analysis, the discriminant accuracies q_c and q_nc are defined similarly by changing "training set" to "test set" in Eqs. (4) and (5), respectively.

We first divide the data into training and test sets randomly, then we use the above algorithm to calculate the discriminant accuracies for different methods. The results are listed in Table 1.

Firstly, seven groups of parameters are derived from the four methods: (i) 9 parameters from fractal methods (MFA and AMFA); (ii) 9 parameters from ZC representing the codon-position-dependent frequencies of mononucleotides; (iii) 12 parameters from ZC representing the frequencies of phase-specific dinucleotides (codon positions 1–2); (iv) 12 parameters from ZC representing the frequencies of phase-specific dinucleotides (codon positions 2–3); (v) 15 parameters for the phase-independent mononucleotides and dinucleotides from ZC; (vi) 36 parameters from GD; (vii) 48 parameters for the frequencies of phase-independent tri-nucleotides from ZC. From Table 1, it is seen that the results from the multifractal analyses seem to be better than that from ZC with an equal number of parameters, namely 9. We have successfully applied multifractal analyses in the clustering of large protein structures 925 and the distinction of coding and non-coding sequences in complete genomes 26, where the length of protein sequences and coding and non-coding sequences are larger than 300. It is well-known that the promoter sequences are highly diverse, which makes it notoriously difficult to generate patterns and rules for promoter prediction. It is expected that multifractal analyses can unfold some useful information on promoter sequences. The results from the frequencies of phase-specific dinucleotides at codon positions 2–3 in ZC indicate a better performance than that at codon positions 1–2. In addition, the accuracies from ZC with the frequencies of phase-independent mononucleotides and dinucleotides are improved but the number of parameters is increased to 15. The GD method shown in boldface in Table 1, denoted as M1, turns out to be especially useful as the accuracies are all larger that 85%. Compared with this, the results from the 48 parameters in ZC are not as good even though the number of parameters is increased.

Secondly, we want to test whether the results can be improved by increasing the number of parameters. It is not possible to test all the subsets of the 141 parameters but we can test all the combinations of the above seven methods (120 altogether). In our test, the accuracies do not simply increase as the number of parameters becomes larger, which indicates there might be some redundancy/correlation among the 141 parameters. For example, the accuracies with the 141 parameters are similar to those with only 117 parameters, suggesting the information from the mononucleotides and phase-independent dinucleotides in ZC is contained in the other methods. Therefore, all these parameters are not really needed. Nevertheless, in some circumstances the results do improve when the number of parameters is increased. Especially, among the 120 combinations, the results are relatively satisfactory in the cases of 96 and 117 parameters, which is shown in boldface in Table 1. We denote them by M2 and M3, respectively. In order to see whether multifractal analysis brings out useful information, we remove the 9 parameters of MFA and AMFA from M3 and test the results for such new combination. The p_c, p_nc, q_c, and q_nc calculated from this combination are: 86.05% 92.67%, 86.02% and 92.00% respectively. They are similar to those from M3 (86.89%, 93.11%, 86.02% and 92.67%), which demonstrates that multifractal analysis does not significantly improve the performance in M3.

In order to evaluate the correct prediction rate and reliability of a predictive method, the sensitivity (S_n), specificity (S_p), accuracy (A_c) and correlation coefficient (CC) are also used 1:

S_n = T P/(T P + F N),

S_p = T P/(T P + F P),

A_c = (S_n+ S_p)/2,

C C = ( T P × T N ) − ( F P × F N ) ( T P + F P ) × ( T N + F N ) × ( T P + F N ) × ( T N + F P ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaem4qamKaem4qamKaeyypa0tcfa4aaSaaaeaacqGGOaakcqWGubavcqWGqbaucqGHxdaTcqWGubavcqWGobGtcqGGPaqkcqGHsislcqGGOaakcqWGgbGrcqWGqbaucqGHxdaTcqWGgbGrcqWGobGtcqGGPaqkaeaadaGcaaqaaiabcIcaOiabdsfaujabdcfaqjabgUcaRiabdAeagjabdcfaqjabcMcaPiabgEna0kabcIcaOiabdsfaujabd6eaojabgUcaRiabdAeagjabd6eaojabcMcaPiabgEna0kabcIcaOiabdsfaujabdcfaqjabgUcaRiabdAeagjabd6eaojabcMcaPiabgEna0kabcIcaOiabdsfaujabd6eaojabgUcaRiabdAeagjabdcfaqjabcMcaPaqabaaaaOGaeiilaWcaaa@6569@

where TP denotes the number of correctly recognized promoter sequences, FN the number of promoter sequences recognized as non-promoter sequences, FP the number of non-promoter sequences recognized as promoter sequences, TN the number of correctly recognized non-promoter sequences.

From Fisher's discriminant algorithm, we calculate the four quantities defined above. The results related to Table 1 by the "order" mark are listed in Table 2.

Overall, from Tables 1 and 2, when the methods are used independently, we can see that M1 is the best one. The combined methods M2 and M3 improve the results. However, the number of parameters is too high in M3. Taking this aspect into account, a preferred method would be M1 or M2.

Some studies suggested that various properties, such as stability, bendability and curvature, of the region immediately upstream of the TSS differ from that of downstream region 63839. The upstream region is less stable, more rigid and more curved than the downstream region. Kan-here and Bansal 6 predicted the prokaryotic promoter based on such difference in DNA stability. We convert the original sequences into new numeric sequences according to the free energy of dinucleotides. A sliding window with size of 2nt is used and moved one base pair forward each time. The numeric sequences can be smoothed with a larger window size. For more details on the smoothing method, one can refer to Ref. 40. The free energy values corresponding to the 10 unique dinucleotides are taken from the unified parameters proposed in Ref. 8. They are: AA/TT = -1.00 kcal/mol, AT/TA = -0.88 kcal/mol, TA/AT = -0.58 kcal/mol, CA/GT = -1.45 kcal/mol, CT/GA = -1.44 kcal/mol, GT/CA = -1.28 kcal/mol, GA/CT = -1.30 kcal/mol, CG/GC = -2.17 kcal/mol, GC/CG = -2.24 kcal/mol, GG/CC = -1.84 kcal/mol. The ten values are added by 2.24 kcal/mol (the negative of the smallest free energy) so that all the values are larger than or equal to zero in order to construct a measure from the time series for the multifractal method in the following analysis. For example, the free energy sequence for one of the promoter sequences with a sliding window of size 2nt is given in Figure 1.

Let T_t, t = 1, 2, ⋯, N, be the numeric sequence of a promoter/non-promoter with length N . First, we define

F t = T t ∑ j = 1 N T j , ( t = 1 , 2 , ⋯ , N ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiabdAeagnaaBaaaleaacqWG0baDaeqaaOGaeyypa0tcfa4aaSaaaeaacqWGubavdaWgaaqaaiabdsha0bqabaaabaWaaabCaeaacqWGubavdaWgaaqaaiabdQgaQbqabaaabaGaemOAaOMaeyypa0JaeGymaedabaGaemOta4eacqGHris5aaaakiabcYcaSaqaaiabcIcaOiabdsha0jabg2da9iabigdaXiabcYcaSiabikdaYiabcYcaSiabl+UimjabcYcaSiabd6eaojabcMcaPaaaaaa@494A@

to be the frequency of T_t. It follows that ∑t=1NFt=1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWaaabCaeaacqWGgbGrdaWgaaWcbaGaemiDaqhabeaakiabg2da9iabigdaXaWcbaGaemiDaqNaeyypa0JaeGymaedabaGaemOta4eaniabggHiLdaaaa@3753@. We define a measure μ on the interval [0, 1) by

μ(dx) = Y(x) dx,

where

Y ( x ) = N × F t = T t 1 N ∑ j = 1 N T j , x ∈ [ t − 1 N , t N ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiabdMfazjabcIcaOiabdIha4jabcMcaPiabg2da9iabd6eaojabgEna0kabdAeagnaaBaaaleaacqWG0baDaeqaaOGaeyypa0tcfa4aaSaaaeaacqWGubavdaWgaaqaaiabdsha0bqabaaabaWaaSaaaeaacqaIXaqmaeaacqWGobGtaaWaaabCaeaacqWGubavdaWgaaqaaiabdQgaQbqabaaabaGaemOAaOMaeyypa0JaeGymaedabaGaemOta4eacqGHris5aaaakiabcYcaSaqaaiabdIha4jabgIGiolabcUfaBLqbaoaalaaabaGaemiDaqNaeyOeI0IaeGymaedabaGaemOta4eaaOGaeiilaWscfa4aaSaaaeaacqWG0baDaeaacqWGobGtaaGccqGGPaqkcqGGUaGlaaaaaa@578B@

We denote the interval [t−1N,tN) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaei4waSvcfa4aaSaaaeaacqWG0baDcqGHsislcqaIXaqmaeaacqWGobGtaaGccqGGSaaljuaGdaWcaaqaaiabdsha0bqaaiabd6eaobaakiabcMcaPaaa@3724@ by I_t. It is easy to see that μ([0, 1)) = 1 and μ(I_t) = F_t. We call μ(x) the measure representation 2641 for the numeric sequence of a promoter/non-promoter.

The most common algorithms of multifractal analysis are the so called fixed-size box-counting algorithms 42. In the one-dimensional case, for a given measure μ with support E ⊂ ℝ, we consider the partition sum

Z ε ( q ) = ∑ μ ( B ) ≠ 0 [ μ ( B ) ] q , q ∈ ℝ , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiabdQfaAnaaBaaaleaaiiGacqWF1oqzaeqaaOGaeiikaGIaemyCaeNaeiykaKIaeyypa0ZaaabuaeaacqGGBbWwcqWF8oqBcqGGOaakcqWGcbGqcqGGPaqkcqGGDbqxdaahaaWcbeqaaiabdghaXbaaaeaacqWF8oqBcqGGOaakcqWGcbGqcqGGPaqkcqGHGjsUcqaIWaamaeqaniabggHiLdGccqGGSaalaeaacqWGXbqCcqGHiiIZtuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab+1risjabcYcaSaaaaaa@54A6@

where the sum runs over all different nonempty boxes B of a given side ε in a grid covering of the support E, that is,

B = [kε, (k + 1)ε).

The mass exponent τ (q) is defined 4344 as

τ ( q ) = lim ⁡ ε → 0 ln ⁡ Z ε ( q ) ln ⁡ ε MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGae8hXdqNaeiikaGIaemyCaeNaeiykaKIaeyypa0ZaaCbeaeaacyGGSbaBcqGGPbqAcqGGTbqBaSqaaiab=v7aLjabgkziUkabicdaWaqabaqcfa4aaSaaaeaacyGGSbaBcqGGUbGBcqWGAbGwdaWgaaqaaiab=v7aLbqabaGaeiikaGIaemyCaeNaeiykaKcabaGagiiBaWMaeiOBa4Mae8xTdugaaaaa@48CD@

and the generalized fractal dimensions 4344 of the measure are defined as

D ( q ) = τ ( q ) q − 1 , f o r q ≠ 1 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeWaaaqaaiabdseaejabcIcaOiabdghaXjabcMcaPiabg2da9KqbaoaalaaabaacciGae8hXdqNaeiikaGIaemyCaeNaeiykaKcabaGaemyCaeNaeyOeI0IaeGymaedaaOGaeiilaWcabaGaemOzayMaem4Ba8MaemOCaihabaGaemyCaeNaeyiyIKRaeGymaeJaeiilaWcaaaaa@4448@

and

D ( q ) = lim ⁡ ε → 0 Z 1 , ε ln ⁡ ε , f o r q = 1 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeWaaaqaaiabdseaejabcIcaOiabdghaXjabcMcaPiabg2da9maaxababaGagiiBaWMaeiyAaKMaeiyBa0galeaaiiGacqWF1oqzcqGHsgIRcqaIWaamaeqaaKqbaoaalaaabaGaemOwaO1aaSbaaeaacqaIXaqmcqGGSaalcqWF1oqzaeqaaaqaaiGbcYgaSjabc6gaUjab=v7aLbaakiabcYcaSaqaaiabdAgaMjabd+gaVjabdkhaYbqaaiabdghaXjabg2da9iabigdaXiabcYcaSaaaaaa@4D6E@

where Z1,ε=∑μ(B)≠0μ(B)ln⁡μ(B) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOwaO1aaSbaaSqaaiabigdaXiabcYcaSGGaciab=v7aLbqabaGccqGH9aqpdaaeqbqaaiab=X7aTjabcIcaOiabdkeacjabcMcaPiGbcYgaSjabc6gaUjab=X7aTjabcIcaOiabdkeacjabcMcaPaWcbaGae8hVd0MaeiikaGIaemOqaiKaeiykaKIaeyiyIKRaeGimaadabeqdcqGHris5aaaa@46B8@. The generalized fractal dimensions are numerically estimated through a linear regression of ln Z_ε (q)/(q - 1) against ln ε for q ≠ 1, and similarly through a linear regression of Z_{1, ε} against ln ε for q = 1 254245. D(1) is called the information dimension and D(2) the correlation dimension 4344.

The concept of phase transitions in multifractal spectra was introduced in the study of logistic maps, Julia sets, and other simple systems. Evidence of a phase transition was found in the multifractal spectrum of diffusion-limited aggregation 46. By following the thermodynamic formulation of multifractal measures, Canessa 47 derived an expression for the analogous specific heat as

C q ≡ − ∂ 2 τ ( q ) ∂ q 2 ≈ 2 τ ( q ) − τ ( q + 1 ) − τ ( q − 1 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaem4qam0aaSbaaSqaaiabdghaXbqabaGccqGHHjIUcqGHsisljuaGdaWcaaqaaiabgkGi2oaaCaaabeqaaiabikdaYaaaiiGacqWFepaDcqGGOaakcqWGXbqCcqGGPaqkaeaacqGHciITcqWGXbqCdaahaaqabeaacqaIYaGmaaaaaOGaeyisISRaeGOmaiJae8hXdqNaeiikaGIaemyCaeNaeiykaKIaeyOeI0Iae8hXdqNaeiikaGIaemyCaeNaey4kaSIaeGymaeJaeiykaKIaeyOeI0Iae8hXdqNaeiikaGIaemyCaeNaeyOeI0IaeGymaeJaeiykaKIaeiOla4caaa@551E@

He showed that the form of C_q resembles a classical phase transition at a critical point for financial time series.

The singularities of a measure are characterized by the Lipschitz-Hölder exponent α(q) 44, which is related to τ (q) by

α ( q ) = d d q τ ( q ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGae8xSdeMaeiikaGIaemyCaeNaeiykaKIaeyypa0tcfa4aaSaaaeaacqWGKbazaeaacqWGKbazcqWGXbqCaaGccqWFepaDcqGGOaakcqWGXbqCcqGGPaqkcqGGUaGlaaa@3C60@

Substitution of Eq. (15) into Eq. (19) yields

α ( q ) = lim ⁡ ε → 0 ∑ μ ( B ) ≠ 0 [ μ ( B ) ] q ln ⁡ μ ( B ) Z ε ( q ) ln ⁡ ε . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGae8xSdeMaeiikaGIaemyCaeNaeiykaKIaeyypa0ZaaCbeaeaacyGGSbaBcqGGPbqAcqGGTbqBaSqaaiab=v7aLjabgkziUkabicdaWaqabaqcfa4aaSaaaeaadaaeqbqaaiabcUfaBjab=X7aTjabcIcaOiabdkeacjabcMcaPiabc2faDnaaCaaabeqaaiabdghaXbaacyGGSbaBcqGGUbGBcqWF8oqBcqGGOaakcqWGcbGqcqGGPaqkaeaacqWF8oqBcqGGOaakcqWGcbGqcqGGPaqkcqGHGjsUcqaIWaamaeqacqGHris5aaqaaiabdQfaAnaaBaaabaGae8xTdugabeaacqGGOaakcqWGXbqCcqGGPaqkcyGGSbaBcqGGUbGBcqWF1oqzaaGaeiOla4caaa@5FA8@

Again, the exponent α(q) can be estimated through a linear regression of {∑μ(B)≠0[μ(B)]qln⁡μ(B)}/Zε(q) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaei4EaS3aaabuaeaacqGGBbWwiiGacqWF8oqBcqGGOaakcqWGcbGqcqGGPaqkcqGGDbqxdaahaaWcbeqaaiabdghaXbaakiGbcYgaSjabc6gaUjab=X7aTjabcIcaOiabdkeacjabcMcaPaWcbaGae8hVd0MaeiikaGIaemOqaiKaeiykaKIaeyiyIKRaeGimaadabeqdcqGHris5aOGaeiyFa0Naei4la8IaemOwaO1aaSbaaSqaaiab=v7aLbqabaGccqGGOaakcqWGXbqCcqGGPaqkaaa@4F11@ against ln ε. The multifractal spectrum f (α) versus α can be calculated according to a relationship known as Legendre transformation 44:

f ( α ) = min ⁡ q { q α ( q ) − τ ( q ) } . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOzayMaeiikaGccciGae8xSdeMaeiykaKIaeyypa0ZaaCbeaeaacyGGTbqBcqGGPbqAcqGGUbGBaSqaaiabdghaXbqabaGccqGG7bWEcqWGXbqCcqWFXoqycqGGOaakcqWGXbqCcqGGPaqkcqGHsislcqWFepaDcqGGOaakcqWGXbqCcqGGPaqkcqGG9bqFcqGGUaGlaaa@4774@

We first construct a measure for the numeric sequences according to Eq. (11), then analyze the measure with the above multifractal method. The D(q), C_q, α(q) and f (α) curves for one of the promoter, exon and intron sequences are shown in Figure 2. We select 5 parameters from MFA to distinguish between promoter and non-promoter sequences: D(2), C₁, C_max (the maximum value of C_q), Δα = α_max - α_min and Δf = f (α_max) - f (α_min).

Analogous multifractal analysis is similar to multiaffinity analysis which is a useful method in many fields. It was recently proposed in 9. We denote a time series as X(t), t = 1, 2, ⋯, N. First, the time series is integrated as

y ′ q ( k ) = ∑ t = 1 k ( X ( t ) − X a v e ) q , ( q ∈ ℤ + , k = 1 , 2 , ⋯ , N ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiqbdMha5zaafaWaaSbaaSqaaiabdghaXbqabaGccqGGOaakcqWGRbWAcqGGPaqkcqGH9aqpdaaeWbqaaiabcIcaOiabdIfayjabcIcaOiabdsha0jabcMcaPiabgkHiTiabdIfaynaaBaaaleaacqWGHbqycqWG2bGDcqWGLbqzaeqaaOGaeiykaKYaaWbaaSqabeaacqWGXbqCaaaabaGaemiDaqNaeyypa0JaeGymaedabaGaem4AaSganiabggHiLdGccqGGSaalaeaacqGGOaakcqWGXbqCcqGHiiIZtuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab=rsiAnaaBaaaleaacqGHRaWkaeqaaOGaeiilaWIaem4AaSMaeyypa0JaeGymaeJaeiilaWIaeGOmaiJaeiilaWIaeS47IWKaeiilaWIaemOta4KaeiykaKcaaaaa@64C2@

y q ( k ) = ∑ t = 1 k | X ( t ) − X a v e | q , ( q ≠ 0 , k = 1 , 2 , ⋯ , N ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiabdMha5naaBaaaleaacqWGXbqCaeqaaOGaeiikaGIaem4AaSMaeiykaKIaeyypa0ZaaabCaeaacqGG8baFcqWGybawcqGGOaakcqWG0baDcqGGPaqkcqGHsislcqWGybawdaWgaaWcbaGaemyyaeMaemODayNaemyzaugabeaakiabcYha8naaCaaaleqabaGaemyCaehaaaqaaiabdsha0jabg2da9iabigdaXaqaaiabdUgaRbqdcqGHris5aOGaeiilaWcabaGaeiikaGIaemyCaeNaeyiyIKRaeGimaaJaeiilaWIaem4AaSMaeyypa0JaeGymaeJaeiilaWIaeGOmaiJaeiilaWIaeS47IWKaeiilaWIaemOta4KaeiykaKcaaaaa@5B57@

where X_ave is the average over the whole time period and k ∈ [1, N]. Then two quantities M_q (L) and M′q(L) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmyta0KbauaadaWgaaWcbaGaemyCaehabeaakiabcIcaOiabdYeamjabcMcaPaaa@3176@ are defined as

M ′ q ( L ) = [ 〈 | y ′ ( j ) − y ′ ( j + L ) | 〉 j ] 1 q , ( q ∈ ℤ + ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiqbd2eanzaafaWaaSbaaSqaaiabdghaXbqabaGccqGGOaakcqWGmbatcqGGPaqkcqGH9aqpcqGGBbWwcqGHPms4cqGG8baFcuWG5bqEgaqbaiabcIcaOiabdQgaQjabcMcaPiabgkHiTiqbdMha5zaafaGaeiikaGIaemOAaOMaey4kaSIaemitaWKaeiykaKIaeiiFaWNaeyOkJe=aaSbaaSqaaiabdQgaQbqabaGccqGGDbqxdaahaaWcbeqcfayaamaalaaabaGaeGymaedabaGaemyCaehaaaaakiabcYcaSaqaaiabcIcaOiabdghaXjabgIGioprr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae8hjHO1aaSbaaSqaaiabgUcaRaqabaGccqGGPaqkaaaaaa@5E18@

M q ( L ) = [ 〈 | y ( j ) − y ( j + L ) | 〉 j ] 1 q , ( q ≠ 0 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiabd2eannaaBaaaleaacqWGXbqCaeqaaOGaeiikaGIaemitaWKaeiykaKIaeyypa0Jaei4waSLaeyykJeUaeiiFaWNaemyEaKNaeiikaGIaemOAaOMaeiykaKIaeyOeI0IaemyEaKNaeiikaGIaemOAaOMaey4kaSIaemitaWKaeiykaKIaeiiFaWNaeyOkJe=aaSbaaSqaaiabdQgaQbqabaGccqGGDbqxdaahaaWcbeqcfayaamaalaaabaGaeGymaedabaGaemyCaehaaaaakiabcYcaSaqaaiabcIcaOiabdghaXjabgcMi5kabicdaWiabcMcaPaaaaaa@5347@

where 〈〉_j denotes the average over j, j = 1, 2, ⋯, N – L; L typically varies from 1 to N₁ in which the linear fit is good. From the ln L vs ln M_q (L) and ln L vs ln M′q(L) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmyta0KbauaadaWgaaWcbaGaemyCaehabeaakiabcIcaOiabdYeamjabcMcaPaaa@3176@ planes, one can determine the relations:

M ′ q ( L ) ∝ L h ′ ( q ) f o r q ∈ ℤ + , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeWaaaqaaiqbd2eanzaafaWaaSbaaSqaaiabdghaXbqabaGccqGGOaakcqWGmbatcqGGPaqkcqGHDisTcqWGmbatdaahaaWcbeqaaiqbdIgaOzaafaGaeiikaGIaemyCaeNaeiykaKcaaaGcbaGaemOzayMaem4Ba8MaemOCaihabaGaemyCaeNaeyicI48efv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFKeIwdaWgaaWcbaGaey4kaScabeaakiabcYcaSaaaaaa@4D03@

M q ( L ) ∝ L h ( q ) f o r q ≠ 0. MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeWaaaqaaiabd2eannaaBaaaleaacqWGXbqCaeqaaOGaeiikaGIaemitaWKaeiykaKIaeyyhIuRaemitaW0aaWbaaSqabeaacqWGObaAcqGGOaakcqWGXbqCcqGGPaqkaaaakeaacqWGMbGzcqWGVbWBcqWGYbGCaeaacqWGXbqCcqGHGjsUcqaIWaamcqGGUaGlaaaaaa@4242@

Linear regressions of ln M′q(L) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmyta0KbauaadaWgaaWcbaGaemyCaehabeaakiabcIcaOiabdYeamjabcMcaPaaa@3176@ and ln M_q (L) against ln L will yield the exponents h' (q) and h(q) respectively.

The exponent h(q) has a nonlinear dependence on q. When q = 1, the methods are just those reported in Refs. 4849 and these methods are used to study the length sequences from the complete genomes by Yu et al. 49. M'(L) may be assessed to determine long-range correlation 50. From Ref. 49, the linear fit to get the exponent h(1) is better than that to get the exponent h'(1). Our numerical results show that the exponents h(q) are more robust than the exponents h'(q), so we suggest to use the exponents h(q). We have used h(q) in clustering the structure of large proteins and it turns out to be a useful method 9.

Figure 3 gives an example in applying the AMFA to the free energy sequence of a promoter sequence. It shows a good linear relationship between ln M(L) and ln(L). For different values of q, we get the exponents h(q) from linear regressions of ln M(L) against ln (L) according to Eq. (27). The exponent spectrum h(q) of the promoter sequence is shown in the right panel of Figure 3. We extract four parameters from AMFA: h(-2), h(-1), h(1) and h(2).

The concept of the Z curve representation of a DNA sequence was first proposed by Zhang and Zhang 51, and was used to distinguish coding and noncoding DNA sequences 5253. A new system based on ZC, Z CURVE 1.0, for finding protein-coding genes in bacterial and archaeal genomes has been proposed 10. Recently, another new self-training system based on the ZC method, ZCURVE_V 11, for recognizing protein-coding genes in viral and phage genomes was reported.

In this paper, we apply the ZC method in distinguishing promoter and non-promoter sequences. For convenience, we give a brief description of the methods in Refs. 10 and 11. The frequencies of bases A, C, G and T occurring in a promoter/non-promoter sequence with bases at positions 1, 4, 7, ⋯; 2, 5, 8, ⋯; 3, 6, 9, ⋯, are denoted by a₁, c₁, g₁, t₁; a₂, c₂, g₂, t₂; a₃, c₃, g₃, t₃, respectively. They are in fact the frequencies of bases at the first, second and third codon positions, which can be called codon-position-dependent frequencies of mononucleotides. Based on the ZC 54, a_i, c_i, g_i, t_i for each i can be used to construct three coordinates, denoted by x_i, y_i and z_i according to the Z transform 54:

{ x i = ( a i + g i ) − ( c i + t i ) , y i = ( a i + c i ) − ( g i + t i ) , z i = ( a i + t i ) − ( g i + c i ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7150@

where x_i, y_i, z_i ∈ [-1, 1], i = 1, 2, 3.

We can use the above 9 parameters in the promoter/non-promoter problem. We can also consider the codon-position-independent frequencies of single bases, which results in the following three coordinates:

{ x = ( a + g ) − ( c + t ) , y = ( a + c ) − ( g + t ) , z = ( a + t ) − ( g + c ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWaaiqaaeaafaqaaeWabaaabaGaemiEaGNaeyypa0JaeiikaGIaemyyaeMaey4kaSIaem4zaCMaeiykaKIaeyOeI0IaeiikaGIaem4yamMaey4kaSIaemiDaqNaeiykaKIaeiilaWcabaGaemyEaKNaeyypa0JaeiikaGIaemyyaeMaey4kaSIaem4yamMaeiykaKIaeyOeI0IaeiikaGIaem4zaCMaey4kaSIaemiDaqNaeiykaKIaeiilaWcabaGaemOEaONaeyypa0JaeiikaGIaemyyaeMaey4kaSIaemiDaqNaeiykaKIaeyOeI0IaeiikaGIaem4zaCMaey4kaSIaem4yamMaeiykaKIaeiilaWcaaaGaay5Eaaaaaa@59D1@

where x, y, z ∈ [-1, 1], a, c, g and t are the frequencies for the bases A, C, G and T in a promoter/non-promoter sequence, respectively.

In addition to the frequencies of codon-position-dependent mononucleotide, we also consider the frequencies of phase-specific dinucleotides. We denote the frequencies of the 16 dinucleotides AA, AC, ⋯, and TT occurring at the codon positions 1–2 and 2–3 of a promoter or non-promoter sequence by p₁₂(AA), p₁₂(AC), ⋯, p₁₂(T T); p₂₃(AA), p₂₃(AC), ⋯, and p₂₃(T T), respectively. Using the Z transform 54, the following 24 coordinates can be defined:

{ x k X = ( p k ( X A ) + p k ( X G ) ) − ( p k ( X C ) + p k ( X T ) ) , y k X = ( p k ( X A ) + p k ( X C ) ) − ( p k ( X G ) + p k ( X T ) ) , z k X = ( p k ( X A ) + p k ( X T ) ) − ( p k ( X G ) + p k ( X C ) ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@A62A@

where xkX,ykX,zkX∈[−1,1] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemiEaG3aa0baaSqaaiabdUgaRbqaaiabdIfaybaakiabcYcaSiabdMha5naaDaaaleaacqWGRbWAaeaacqWGybawaaGccqGGSaalcqWG6bGEdaqhaaWcbaGaem4AaSgabaGaemiwaGfaaOGaeyicI4Saei4waSLaeyOeI0IaeGymaeJaeiilaWIaeGymaeJaeiyxa0faaa@4222@p_k (XY) = n_k(XY)/[n_k (XA) + n_k (XC) + n_k(XG) + n_k (XT)], n_k(XY) are the occurring times of dinucleotides XY, X, Y = A, C, G, T, k = 12, 23.

We can also consider the frequencies of phase-specific dinucleotides and the frequencies of phase-independent dinucleotides. For this purpose, a sliding window with size 2nt is used and moved forward one base each time to count the number of times of the occurring dinucleotides. With this method, 12 new coordinates can be defined:

{ x X = ( p ( X A ) + p ( X G ) ) − ( p ( X C ) + p ( X T ) ) , y X = ( p ( X A ) + p ( X C ) ) − ( p ( X G ) + p ( X T ) ) , z X = ( p ( X A ) + p ( X T ) ) − ( p ( X G ) + p ( X C ) ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8F11@

where x^X, y^X, z^X ∈ [-1, 1], p(XY) = n(XY)/[n(XA) + n(XC) + n(XG) + n(XT)], n(XY) are the occurring times of dinucleotides XY, X, Y = A, C, G, T.

Gao and Zhang 32 compared various algorithms for recognizing short coding sequences of human genes and they defined 48 quantities, which were the frequencies of phase-dependent tri-nucleotides. In Ref. 32, Gao and Zhang used a sliding window with size 3nt and the window was moved forward three bases each time to count the frequencies for the 64 tri-nucleotides. Now we move forward the sliding window with size 3nt one base each time. The definition for the 48 coordinates is

{ x X Y = ( p ( X Y A ) + p ( X Y G ) ) − ( p ( X Y C ) + p ( X Y T ) ) , y X Y = ( p ( X Y A ) + p ( X Y C ) ) − ( p ( X Y G ) + p ( X Y T ) ) , z X Y = ( p ( X Y A ) + p ( X Y T ) ) − ( p ( X Y G ) + p ( X Y C ) ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@A186@

where x^XY, y^XY, z^XY ∈ [-1, 1], p(XYZ) = n(XYZ)/[n(XYA) + n(XYC) + n(XYG) + n(XYT)], n(XY Z) are the occurring times of trinucleotides XYZ, X, Y, Z = A, C, G, T. The difference between Ref. 32 and here is in the calculation of n(XYZ); the present method can be regarded as a phase-independent method.

Dubchak et al. 12 proposed a method for predicting protein folding classes based on a global protein chain description. The protein-chain descriptor includes overall composition, transition, and distribution of amino acid attributes. Similar methods have also been used in Refs. 55565758. In this paper, we propose the global descriptor of promoter/non-promoter sequences.

The global description contains three parts: composition (Comp), transition (Tran) and distribution (Dist). In order to explain the method, we suppose that a sequence consists of only two kinds of letters (A and B). The composition is used to measure the frequency of occurrence of each kind of letters in the sequences. For example, for the sequence: BABBABABBABBAABABABBAAAB-BABABA, there are 14 As and 16 Bs, hence the frequencies for A and B are 100.00 × 14/(14+16) = 46.67, 100.00 × 16/(14+16) = 53.33, respectively. These two numbers represent the first part of the global description, Comp. The second part, Tran, characterizes the percent frequency with which A is followed by B or B is followed by A. For example, for the above sequence, there are 21 transitions of this type, that is, (21/29) × 100.00 = 72.14. The third part of the global description, Dist, measures the chain length within which the first, 25%, 50%, 75% and 100% of certain type of letters is located, respectively. For example, for the above sequence, the first, 25%, 50%, 75% and 100% of Bs are located within the first, 6th, 12th, 20th and 29th nucleotides, respectively. The Dist descriptor for Bs is thus: 1/30 × 100.00 = 3.33, 6/30 × 100.00 = 20.00, 12/30 × 100.00 = 40.00, 20/30 × 100.00 = 66.67 and 29/30 × 100.00 = 96.67. Likewise, the Dist descriptor for As is 6.67, 23.33, 53.33, 73.33 and 100.00. As a result, the global description for the above sequence is (Comp; Tran; Dist) = (46.67, 53.33; 72.14; 6.67, 23.33, 53.33, 73.33, 100.00, 3.33, 20.00, 40.00, 66.67, 96.67). A more detailed description of global description of sequences is given in Refs. 1255565758.

The global description for the promoter/non-promoter sequences can be computed by a similar procedure. As the sequences consist of four types of nucleotides (A, C, G and T), there are 4 parameters for Comp, 6 parameters for Tran and 20 parameters for Dist. Overall, a total of 30 parameters are used to give a global description of a promoter/non-promoter sequence.

The Entropy Density Profile (EDP) model is a global statistical description for a DNA sequence, which employs Shannon's artificial linguistic description for a DNA sequence of finite length like an open reading frame (ORF) 59. Zhu et al. 59 developed a new non-supervised gene prediction algorithm for bacterial and archaeal genomes based on EDP. Here we describe such method briefly. If p_i(i = 1, 2, 3, 4) are the frequencies for the four types of nucleotides of a promoter/non-promoter sequence, then an EDP vector S = {s_i} inferred from {p_i} is used to represent the sequence with an emphasis on the information content, where i is the index of the four kinds of nucleotides. The EDP s_i is defined as 59

s i = − 1 H p i log ⁡ p i , i = 1 , 2 , 3 , 4 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiabdohaZnaaBaaaleaacqWGPbqAaeqaaOGaeyypa0JaeyOeI0scfa4aaSaaaeaacqaIXaqmaeaacqWGibasaaGccqWGWbaCdaWgaaWcbaGaemyAaKgabeaakiGbcYgaSjabc+gaVjabcEgaNjabdchaWnaaBaaaleaacqWGPbqAaeqaaOGaeiilaWcabaGaemyAaKMaeyypa0JaeGymaeJaeiilaWIaeGOmaiJaeiilaWIaeG4mamJaeiilaWIaeGinaqJaeiilaWcaaaaa@4871@

where H=−∑i=14pilog⁡pi MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemisaGKaeyypa0JaeyOeI0YaaabCaeaacqWGWbaCdaWgaaWcbaGaemyAaKgabeaakiGbcYgaSjabc+gaVjabcEgaNjabdchaWnaaBaaaleaacqWGPbqAaeqaaaqaaiabdMgaPjabg2da9iabigdaXaqaaiabisda0aqdcqGHris5aaaa@3F65@ is the Shannon entropy.

It was shown that P=p12+p22+p32+p42 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemiuaaLaeyypa0JaemiCaa3aa0baaSqaaiabigdaXaqaaiabikdaYaaakiabgUcaRiabdchaWnaaDaaaleaacqaIYaGmaeaacqaIYaGmaaGccqGHRaWkcqWGWbaCdaqhaaWcbaGaeG4mamdabaGaeGOmaidaaOGaey4kaSIaemiCaa3aa0baaSqaaiabisda0aqaaiabikdaYaaaaaa@3EB2@ is a useful statistical quantity for analysis of DNA sequences 5460, which was called a nucleotide composition constraint of genomes 61. As a result, we obtain 6 parameters s₁, s₂, s₃, s₄, H and P from EDP.

Overall, combining the above two description systems, we get 36 parameters for the global descriptor of a promoter/non-promoter sequence.