Alignment-free prediction of mycobacterial DNA promoters based on pseudo-folding lattice network or star-graph topological indices

Abstract

The importance of the promoter sequences in the function regulation of several important mycobacterial pathogens creates the necessity to design simple and fast theoretical models that can predict them. This work proposes two DNA promoter QSAR models based on pseudo-folding lattice network (LN) and star-graphs (SG) topological indices. In addition, a comparative study with the previous RNA electrostatic parameters of thermodynamically-driven secondary structure folding representations has been carried out. The best model of this work was obtained with only two LN stochastic electrostatic potentials and it is characterized by accuracy, selectivity and specificity of 90.87%, 82.96% and 92.95%, respectively. In addition, we pointed out the SG result dependence on the DNA sequence codification and we proposed a QSAR model based on codons and only three SG spectral moments.

1. Introduction

Protein synthesis promoter sequences play an important role in the function regulation of several important mycobacterial pathogens (Levine and Tjian, 2003; Wyrick and Young, 2002). In this sense, the prediction of the mycobacterial promoter sequences (Mps) could be interesting for the future discovery of new anti-mycobacterial drug targets or for the study of protein metabolism. Mycobacteria have a low transcription rate and a low RNA content per DNA unit. Thus, the transcription and translation signals in Mycobacteria may be different from those in other bacteria such as Esccherichia coli. The large variations among the characterized mycobacterial promoters suggest that the consensus sequences are not representative of these promoters. Consequently, a number of conflicting opinions regarding the presence and characteristics of consensus promoter sequences in the Mycobacteria have been presented in the literature (Mulder et al., 1997). Therefore, understanding the factors responsible for the low level of transcription and the possible mechanisms of regulation of gene expression in Mycobacteria, involves the examination of the mycobacterial promoter structure and the promoter transcription machinery, including chemical information about the involved RNA molecules (Arnvig et al., 2005; Harshey and Ramakrishnan, 1977). Efforts have been made to develop statistical algorithms for the sequence analysis and motif prediction by searching for homologous regions or by comparing the sequence information with a consensus sequence (O’Neill and Chiafari, 1989). Wide variations existing within individual promoter sequences are primarily responsible for the unsatisfactory results yielded by the promoter-site-searching algorithms that in essence perform statistical analysis (Mulligan and McClure, 1986; Mulligan et al., 1984). Therefore, it can be inferred that the recognition of Mps requires a powerful technique capable of unravelling those hidden patterns in the promoter regions, which are difficult to identify directly by sequence alignment.

The bioinformatics methods, based on sequence alignment, may fail in general in cases of low sequence homology between the database query and the template sequences. The lack of function annotation (defined biological function) of the sequences used as template for function prediction constitutes another weakness of the alignment approaches (Dobson and Doig, 2005; Dobson et al., 2004, Dobson et al., 2005). In addition, Chou demonstrated that the 3D structures developed, based on homology modelling are very sensitive to the sequence alignment of the query protein with the structure-known protein (Chou, 2004). A group of researchers shows the growing importance of machine learning methods for predicting protein functional class, independently of sequence similarity (Han et al., 2006). These methods often use the input the 1D sequence numerical parameters, specifically defined to seek sequence–function relationships. For instance, the so-called pseudo amino acid composition approach (Chou, 2001a, Chou, 2005), based on 1D sequence coupling numbers, has been widely used to predict subcellular localization, enzyme family class, structural class, as well as other attributes of proteins based on their sequence similarity (Caballero et al., 2006; Chou and Shen, 2006; Du et al., 2008). Alternatively, the molecular indices that are classically used for small molecules (Aguero-Chapin et al., 2006; Liao and Wang, 2004a, Liao and Wang, 2004b, Liao and Wang, 2004c; Liao and Ding, 2005; Liao et al., 2005, Liao et al., 2006; Liu et al., 2002; Nandy, 1994, Nandy, 1996; Nandy and Basak, 2000; Randic and Vracko, 2000; Randic and Balaban, 2003; Randic and Zupan, 2004; Randic et al., 2000; Song and Tang, 2005; Woodcock et al., 1992; Zupan and Randic, 2005) have been adapted to describe the protein sequences. On the other hand, many authors have introduced 2D or higher dimension representations of sequences prior to the calculation of numerical parameters. This is an important step in order to uncover useful higher-order information not encoded by 1D sequence parameters (Randic, 2004). An example of the 2D representations is the graphs used for proteins and DNA sequences. For instance, the spectral-like and zigzag representations have been used in order to suggest an algorithm to enclose long strings of building blocks (like four DNA bases, 20 natural amino acids, or all 64 possible base triplets) (Aguero-Chapin et al., 2006). The use of the graphic approaches to study biological systems can provide useful insights, as indicated by many previous studies on a series of important biological topics, such as enzyme-catalyzed reactions (Andraos, 2008; Chou, 1981, Chou, 1989; Chou and Forsen, 1980; Chou and Liu, 1981; Chou et al., 1979; Cornish-Bowden, 1979; King and Altman, 1956; Kuzmic et al., 1992; Myers and Palmer, 1985; Zhou and Deng, 1984), protein folding kinetics (Chou, 1990), inhibition kinetics of processive nucleic acid polymerases and nucleases (Althaus et al., 1993a, Althaus et al., 1993b, Althaus et al., 1993c; Chou et al., 1994), analysis of codon usage (Chou and Zhang, 1992; Zhang and Chou, 1993, Zhang and Chou, 1994), analysis of DNA sequence (Qi et al., 2007). Moreover, graphical methods have been introduced for the quantitative structure-activity relationships (QSAR) study (Gonzalez-Diaz et al., 2006c, González-Díaz et al., 2007b; Prado-Prado et al., 2008) as well as used to deal with complicated network systems (Diao et al., 2007; González-Díaz et al., 2007a., González-Díaz et al., 2008). Recently, the “cellular automaton image” (Wolfram, 1984, Wolfram, 2002) has also been applied to study hepatitis B viral infections (Xiao et al., 2006a), HBV virus gene missense mutation (Xiao et al., 2005b), and visual analysis of SARS-CoV (Gao et al., 2006; Wang et al., 2005), as well as to represent complicated biological sequences (Xiao et al., 2005a) and help to identify protein attributes (Xiao and Chou, 2007; Xiao et al., 2006b).

In this work, we are proposing a comparative study of the Mycobacterial DNA promoter prediction using pseudo-folding lattice network (LN) and star-graph (SG) topological indices. The first group of indices contains the mean stochastic electrostatic potential (^LN ξ_k), Markov spectral moments (^LN π_k) and Markov entropies (^LN θ_k) of a Markov model (MM) associated to a 2D network that numerically characterize DNA sequences and build a QSAR model to predict Mps. The lattice-like representations (also called maps or graphs) for Mps and control group sequences (Cgs) were derived (González-Díaz et al., 2003a, González-Díaz et al., 2006a, González-Díaz et al., 2005c; González-Díaz, 2007d). The ξ_k, π_k and θ_k values of several types of graphs/networks have been the base for different QSAR studies of DNA/RNA and protein sequences (Du et al., 2007a, Du et al., 2007b; Garcia-Garcia et al., 2004; Marrero-Ponce et al., 2004a, Marrero-Ponce et al., 2004b, Marrero-Ponce et al., 2005b; Meneses-Marcel et al., 2005; Santana et al., 2006). The second group of TIs is derived from the SG representations (Harary, 1969). We subsequently developed a classifier to connect Mps information (represented by the ξ_k, π_k, θ_k and SG TIs values) with the prediction of Cgs as Mps. The Linear Discriminant Analysis (LDA) was selected as a simple but powerful technique (González-Díaz et al., 2006b; González-Díaz, 2003a).

2. Materials and methods

2.1. Pseudo-folding LN

The first MM, also called MARCH-INSIDE, was used to codify the information of 135 Mps (González-Díaz et al., 2005a, González-Díaz et al., 2006a, González-Díaz et al., 2007d) and 511 random Cgs (see Table S.1 in the Supplementary material). Our methodology considers as states of the Markov Chain (MC) any atom, nucleotide or amino acid depending on the class of molecule to be described (González-Díaz et al., 2005e, González-Díaz et al., 2003c). Therefore, MM deals with the calculation of the probabilities (^k p_ij) where the charge distribution of nucleotide moves from any nucleotide in the vicinity i at time t ₀ to another nucleotide j along the protein backbone in discrete time periods, until a stationary state is achieved (Yuan, 1999). As seen from the discussion above, we selected ^LN ξ_{k ,} ^LN π_k and ^LN θ_k based on the utility of non-stochastic (González-Díaz and Uriarte, 2005; González-Díaz et al., 2005d; Ramos de Armas et al., 2004) and stochastic parameters (Randic and Vracko, 2000). Many researchers have demonstrated the possibility of predicting RNA from sequences (Aguero-Chapin et al., 2006) and we used 2D graphs to encode information about Mps sequences (Estrada, 2000, Estrada, 2002; Estrada and González-Díaz, 2003; González-Díaz et al., 2005b; González and Moldes del Carmen Teran, 2004; González et al., 2005, González et al., 2006; Vilar et al., 2005, Vilar et al., 2006). This RNA 2D graphical representation is similar to those previously reported for DNA (Jacchieri, 2000; Nandy, 1994, Nandy, 1996) using four different nucleotides. The construction of the 2D lattice graph corresponding to the Mps of the gene Alpha in Mycobacterum bovis (BCG) is shown in Table 1 and Fig. 1 . Each nucleotide in the sequence is placed in a Cartesian 2D space starting with the first monomer at the (0, 0) coordinates. The coordinates of the successive nucleotide are calculated according to with the following rules:

• (a)Increase by +1 the abscissa axis coordinate for thymine (rightwards-step) or
• (b)Decrease by –1 the abscissa axis coordinate for cytosine (leftwards-step) or
• (c)Increase by +1 the ordinate axis coordinate for adenine (upwards-step) or
• (d)Decrease by –1 the ordinate axis coordinate for guanine (downwards-step).

Table 1.

LN construction rules for the Mps of the gene Alpha in Mycobacterum bovis (BCG)

DNA LN
c1g2a3c4t5t6t7c8g9c10c11c12g13a14a15t16c17g18a19c20
a21t22t23t24g25g26c27c28t29c30c31a32c33a34c35a36c37g38g39t40
a41t42g43t44t45c46t47g48g49c50c51c52g53a54g55c56a57c58a59c60
g61a62c63g64a65
n	Nucleotide	x	y
1	c1a3t5g25	0	0
2	g2c10g26	0	−1
3	c4t16	−1	0
4	t6c8	1	0
5	t7	2	0
6	g9	1	−1
7	c11c27t29	−1	-1
8	c12a14g18c28c30g48	−2	−1
9	g13g49	−2	−2
10	a15c17a19t45t47	−2	0
11	c20a32t44c46	−3	0
12	a21	−3	1
13	t22	−2	1
14	t23	−1	1
15	t24	0	1
16	c31	−3	−1
17	c33g43	−4	0
18	a34t42	−4	1
19	c35a41	−5	1
20	a36	−5	2
21	c37	−6	2
22	g38	−6	1
23	g39	−6	0
24	t40	−5	0
25	c50	−3	−2
26	c51	−4	−2
27	c52a54	−5	−2
28	g53g55	−5	−3
29	c56	−6	−3
30	a57	−6	−2
31	c58	−7	−2
32	a59	−7	−1
33	c60a62	−8	−1
34	g61	−8	−2
35	c63a65	−9	−1
36	g64	−9	−2

Fig. 1.

LN for the Mps of the gene Alpha in Mycobacterum bovis (BCG).

In the next step, we assigned a stochastic matrix ¹ Π to each graph. The elements of ¹ Π are the probabilities ¹ p_ij of reaching a node n_i with the charge Q_i moving through a walk of length of k=1 from another node n_j with charge Q_j (Aguero-Chapin et al., 2006):

$$p_{\mathit{ij}}={\displaystyle \frac{{\displaystyle \frac{Q_j}{d_{j0}}}}{{\sum }_{m=l}^n{\alpha }_{\mathit{il}}·{\displaystyle \frac{Q_j}{d_{l0}}}}}={\displaystyle \frac{{\varphi }_j}{{\sum }_{m=l}^n{\alpha }_{\mathit{il}}·{\varphi }_l}}$$ (1)

$$p_j={\displaystyle \frac{{\displaystyle \frac{Q_j}{d_{j0}}}}{{\sum }_{m=l}^n{\displaystyle \frac{Q_j}{d_{l0}}}}}={\displaystyle \frac{{\varphi }_j}{{\sum }_{m=l}^n{\varphi }_l}}$$ (2)

where α_ij equals to 1 if the nodes n_i and n_j are adjacent in the graph or equal to 0 otherwise; Q_j is equal to the sum of the electrostatic charges of all nucleotides placed at this node. Note that the number of nodes (n) in the graph is equal to the number of rows and columns in ¹ Π but it may be equal or even smaller than the number of DNA bases in the sequence. It then becomes straightforward to calculate different types of invariant parameters for ¹ Π in order to numerically characterize the DNA sequence. In this work we calculated the following invariants:

$${}^{\mathrm{LN}}{\pi }_k=\sum _{i=j}^n{}^k{p}_{\mathit{ij}}$$ (3)

$${}^{\mathrm{LN}}{\xi }_k=\sum _{i=j}^n{}^k{p}_j·{\varphi }_j$$ (4)

$${}^{\mathrm{LN}}{\theta }_k=-\sum _{i=j}^n{}^k{p}_j·\log ({}^k{p}_j)$$ (5)

where ^LN π_k are the Markov spectral moments and indicate that we sum all the values in the main diagonal of the matrices ^LN π_k=tr(^k Π)=tr[(¹ Π)^k] (tr is the trace operator), ^LN ξ_k are the mean values of electrostatic potentials and ^LN θ_k are the Markov entropies (González-Díaz et al., 2007e). All calculations of the ^LN ξ_{k ,} ^LN π_k and ^LN θ_k values for the DNA sequences of both groups (Mps and Cgs) were carried out with our in-house software MARCH-INSIDE, version 2.0^® (González-Díaz et al., 2007e), including sequence representation.

2.2. SG topological indices

Each DNA sequence is a real network where the nucleotides are the vertices/nodes, connected in a specific sequence by the phosphodiester bonds. SG is an abstract representation of the real network that has a dummy non-nucleotide centre and a number of “rays” equal with the nucleotide types. In the case of DNA, we can consider two codifications: the nucleotide code (as in the case of the amino acid protein sequences) and the DNA codons (the final incomplete codons are ignored). In the first codification, there are only four branches (“rays”) of the star corresponding to the four types of nucleotides: adenosine (a), thymidine (t), cytidine (c) and guanosine (g). Using the codons, the DNA sequences are virtually translated into amino acid sequences that generate 21 branches, 20 standard amino acids and an extra X non-amino acid corresponding to the STOP DNA codons (Griffiths et al., 1999). Even if the promoters are not naturally translated in proteins, the second codification is useful for a comparison with the protein SG calculations. The same DNA/protein can be represented by different forms associated to distinct distance matrices (Randic et al., 2007). Standard SG were constructed for each DNA promoter: each nucleotide/vertex holds the position in the original sequence and the branches are labelled by the standard letters of the nucleotides (a, t, c and g). If the initial connectivity in the DNA sequence is included, the graph is embedded. In order to qualitatively evaluate the graphs, it is necessary to transform the graphical representation into correspondent connectivity matrix, distance matrix and degree matrix. In the case of embedded graph, the matrices of the connectivity in the sequence and in the SG are combined. These matrices and the normalized ones are the base of the calculation of the topological indices.

For a visual comparison of the lattice and SG representations, the same promoter sequence from Table 1 was used to generate a standard SG based on codons that are virtually translated to amino acids (see Table 2 and Fig. 2 ).

Table 2.

SG codifications for the virtually translated Mps of the gene Alpha in Mycobacterum bovis (BCG)

DNA codon SG
DNA nucleotide sequence	c1g2a3c4t5t6t7c8g9c10c11c12g13a14a15t16c17g18a19c20a21
	t22t23t24g25g26c27c28t29c30c31a32c33a34c35a36c37g38g39
	t40a41t42g43t44t45c46t47g48g49c50c51c52g53a54g55c56a57c58a59c60g61a62c63
DNA codons sequence	cga1ctt2tcg3ccc4gaa5tcg6aca7
	ttt8ggc9ctc10cac11aca12cgg13
	tat14gtt15ctg16gcc17cga18gca19cac20gac21
Virtually translated amino acid sequence	R1L2S3P4E5S6T7F8G9L10H11T12R13Y14V15L16A17R18A19H20D21

Fig. 2.

SG for the Mps of the gene Alpha in Mycobacterum bovis (BCG).

The SG topological indices are obtained with the in-house sequence to star networks (S2SNet) python application. This tool can transform any character string in SG topological indices. Our recent works (Munteanu et al., 2008a, Munteanu et al., 2008b) proved the potential of S2SNet in protein QSAR models. The calculations presented in this work are characterized by embedded (E) and non-embedded (nE) TIs, non-weights, Markov normalization and power of matrices/indices (n) up to 5. The result file contains the following embedded (super index “e”) or non-embedded TIs (Todeschini and Consonni, 2002):

Shannon Entropy of the n powered Markov matrices (^SG θ_n):

$${}^{\mathrm{SG}}{\theta }_n^{(e)}=-\sum _i{p}_i*\log (p_i)$$ (6)

where p_i are the n_i elements of the p vector, resulted from the matrix multiplication of the powered Markov normalized matrix (n_i×n_i) and a vector (n_i×1) with each element equal to 1/n_i;

The trace of the n connectivity matrices (^SG π_n):

$${}^{\mathrm{SG}}{\pi }_n^{(e)}=-\sum _i(M^n{)}_{\mathit{ii}}$$ (7)

where n=0−power limit, ^SG M=SG connectivity matrix (i*i dimension); ii=ith diagonal element;

Harary number (H):

$$H^{(e)}=\sum _{i<j}(m_{\mathit{ij}}/d_{\mathit{ij}})$$ (8)

where d_ij are the elements of the distance matrix and m_ij are the elements of the M connectivity matrix;

Wiener index (W):

$$W^{(e)}=\sum _{i<j}{d}_{\mathit{ij}}$$ (9)

Gutman topological index (S ₆):

$$S_6^{(e)}=\sum _{\mathit{ij}}\mathit{de}{g}_i*\mathit{de}{g}_j/\mathit{de}{g}_{\mathit{ij}}$$ (10)

where deg_i are the elements of the degree matrix;

Schultz topological index (non-trivial part) (S):

$$S^{(e)}=\sum _{i<j}(\mathit{de}{g}_i+\mathit{de}{g}_j)*d$$ (11)

Balaban distance connectivity index (J):

$$J^{(e)}=(\mathit{edges}-\mathit{nodes}+2)*\sum _{i<j}{m}_{\mathit{ij}}*\mathrm{sqrt}\left(\sum _k{d}_{\mathit{ik}}*\sum _k{d}_{\mathit{kj}}\right)$$ (12)

where nodes+1=AA numbers/node number in the SG+origin, ∑_k d_ik is the node distance degree;

Kier–Hall connectivity indices (ⁿ X):

$${}^0{X}^{(e)}=\sum _{i}1/\mathrm{sqrt}(\mathit{de}{g}_i)$$ (13)

$${}^0{X}^{(e)}=\sum _{i<j<k}{m}_{\mathit{ij}}*m_{\mathit{jk}}/\mathrm{sqrt}(\mathit{de}{g}_i*\mathit{de}{g}_j*\mathit{de}{g}_k)$$ (14)

$${}^3{X}^{(e)}=\sum _{i<j<k<m}{m}_{\mathit{ij}}*m_{\mathit{jk}}*m_{\mathit{km}}/\mathrm{sqrt}(\mathit{de}{g}_i*\mathit{de}{g}_j*\mathit{de}{g}_k*\mathit{de}{g}_m)$$ (15)

$${}^4{X}^{(e)}=\sum _{i<j<k<m<0}{m}_{\mathit{ij}}*m_{\mathit{jk}}*m_{\mathit{km}}*m_{\mathit{mo}}/\mathrm{sqrt}(\mathit{de}{g}_i*\mathit{de}{g}_j*\mathit{de}{g}_k*\mathit{de}{g}_m*\mathit{de}{g}_o)$$ (16)

$${}^5{X}^{(e)}=\sum _{i<j<k<m<o<q}{m}_{\mathit{ij}}*m_{\mathit{jk}}*m_{\mathit{km}}*m_{\mathit{mo}}*m_{\mathit{oq}}/\mathrm{sqrt}(\mathit{de}{g}_i*\mathit{de}{g}_j*\mathit{de}{g}_k*\mathit{de}{g}_m*\mathit{de}{g}_o*\mathit{de}{g}_q)$$ (17)

Randic connectivity index (¹ XR):

$${}^{1}X{R}^{(e)}=\sum _{i<j}{m}_{\mathit{ij}}/\mathrm{sqrt}(\mathit{de}{g}_i*\mathit{de}{g}_j)$$ (18)

The embedded and non-embedded SG TIs are used to construct a DNA promoter classification model using the LDA statistical methods.

2.3. Linear discriminant analysis

LDA forward stepwise analysis from STATISTICA (StatSoft.Inc., 2002) was carried out for a variable selection to build up the model (Garcia-Garcia et al., 2004; Kutner et al., 2005; Marrero-Ponce et al., 2004a, Marrero-Ponce et al., 2004b, Marrero-Ponce et al., 2005b; Meneses-Marcel et al., 2005; Santana et al., 2006). In order to decide whether a DNA sequence is classified as a mycobacterial promoter (Prom) or not (nProm), we added an extra dummy variable named Prom/nProm (binary values of 1/−1 for LN and 1/0 for SG) and a cross-validation variable (CV). The best cross-validation methods in practice are the independent dataset test, the subsampling test and the jackknife test (Chou and Zhang, 1995). The jackknife test has been increasingly used by investigators to examine the accuracy of various predictors (Chen and Li, 2007; Chou and Shen, 2007a, Chou and Shen, 2008; Diao et al., 2007; Ding et al., 2007; Lin, 2008; Xiao and Chou, 2007). In the actual work, the independent data test is used by splitting the data at random in a training series (train, 75%) used for model construction and a prediction one (val, 25%) for model validation (the CV column is filled by repeating 3 train and 1 val). All the variables included in the models were standardized in order to bring them onto the same scale. Subsequently, standardized linear discriminant equations that allow comparison of their coefficients were obtained (Chiti et al., 2003; Pawar et al., 2005).

In the case of LN, the general QSAR formula is the following:

$${}^{\mathrm{LN}}\mathrm{Mps}‐\mathrm{score}=a_0+\sum _{k=0}^5{b}_k\times {}^{\mathrm{LN}}{\pi }_k+\sum _{k=0}^5{c}_k\times {}^{\mathrm{LN}}{\theta }_k+\sum _{k=0}^5{d}_k\times {}^{\mathrm{LN}}{\xi }_k$$ (19)

where ^LNMps-score is the continue score value for the DNA mycobacterial promoter classification corresponding to the lattice representation, ^LN π_k are Markov spectral moments (traces), ^LN θ_k are the Markov entropies, ^LN ξ_k the mean stochastic electrostatic potential, b_k, c_k, d_k are the coefficients of the previous indices and a ₀ is the independent term. A similar formula defines the SG QSAR model in Eq. (20).

$${}^{\mathrm{SG}}\mathrm{Mps}‐\mathrm{score}=e_0+\sum _{k=0}^5{f}_k\times {}^{\mathrm{SG}}{\pi }_k+\sum _{k=0}^5{f}_k^e\times {}^{\mathrm{SG}}{\pi }_k^e+\sum _{k=0}^5{g}_k\times {}^{\mathrm{SG}}{\theta }_k+\sum _{k=0}^5{g}_k^e\times {}^{\mathrm{SG}}{\theta }_k^e+\sum _{k=0}^{10}{e}_k\times {\mathrm{TI}}_k+\sum _{k=0}^{10}{e}_k\times {\mathrm{TI}}_k^e$$ (20)

where ^SGMps-score is the continue score value for the DNA mycobacterial promoter classification corresponding to the SG representation, ^SG π_k ^e/^SG π_k and ^SG θ_k ^e/^SG θ_k are embedded/non-embedded traces (Markov spectral moments) and the Shannon entropies, TI_k ^e/TI_k are the other 22 standard SG embedded and non-embedded TIs (H, W, S ₆, S, J, ⁰ X, ^{2 − 5} X, ¹ XR, H ^e, W ^e, S ₆ ^e, S ^e, J ^e, ⁰ X ^e, ^{2 − 5} X ^e, ¹ XR ^e), f_k ^e/f_k, g_k ^e/g_k and e_k ^e/e_k are the TIs coefficients and e ₀ is the independent term. Accuracy, specificity, sensitivity, F, Wilk's (λ) statistic (λ=0 perfect discrimination, being 0<λ<1) were examined in order to assess the discriminatory power of the model.

3. Results and discussion

Many different parameters can be used to encode RNA sequence information and further assign or predict the function or physical properties (González-Díaz and Uriarte, 2005). The present approach involves the calculation of different sequence parameters, which can be applied to different types of molecular graphs (Aguero-Chapin et al., 2006), including DNA, RNA and proteins (Di Francesco, 1999; González-Díaz et al., 2005c). MM has been applied successfully to Genomics and Proteomics and represents an important tool to analyse biological sequence data. In particular, MM has been used for protein folding recognition (Chou, 2001b) and for prediction of protein signal sequences (Chou and Shen, 2007b; Van Waterbeemd, 1995). This work compared two models based on different TIs including π_k and θ_k values of the stochastic matrices ¹ Π(LN) and ¹ Π(SG) (^SGM) associated with LN and SG, ^LN ξ_k parameters of ¹ Π(LN) as well as classic TIs for ¹ Π(SG). These parameters describe the distribution of the nucleotides of the DNA sequence in the above graphs/networks. This calculation was carried out for two groups of DNA sequences, one made up of Mps and the other formed by Cgs. In addition, previous results of the RNA secondary structure (2S) QSAR are compared.

3.1. Results for DNA LN indices

In the first study of the DNA LN representations, the best QSAR equation that classifies a novel sequence as Mps or not is the following (Table 3 ):

$${}^{\mathrm{LN}}\mathrm{Mps}‐\mathrm{score}=-1.2-4.1\times {}^{\mathrm{LN}}{\xi }_1+2.1\times {}^{\mathrm{LN}}{\xi }_5$$ (21)

The statistical parameters of this equation were Wilk's statistic (λ=0.95) and the error level (p-level<0.001). This discriminant function misclassified only 36 cases out of 511 Cgs used, reaching a high level of accuracy (90.87%). More specifically, the model classified correctly 112/135 (82.9%) of Mps and 475/511 (92.9%) of the control group. Conversely, the remaining four descriptors ^LN ξ _0, ^LN ξ _2, ^LN ξ ₃ and ^LN ξ ₄ do not have a significant relationship with the Mps characteristic. The use of only six molecular descriptors to model a data set of 585 sequences prevents us by large from chance correlation. In physical terms, the above results confirm other studies about the relationship between the electrostatic potential of the DNA molecule and its biological activity. However, in this case, not all the electrostatic interactions affect the activity in the same way. Finally, long-term electrostatic interaction potentials (^LN ξ _0, ^LN ξ _2, ^LN ξ ₃ and ^LN ξ ₄) do not correlate with the Mps activity. The detailed results of the forward stepwise analysis are given in Table 3.

Table 3.

Summary of the LDA results for DNA LN and SG models vs. RNA 2S folding representations

TI	Ac (%)	Se (%)	Sp (%)	Final TIs	Vars.	λ	F	p	Ref.
Primary structure of DNA nucleotide & LN
LNθk	78.33	72.59	79.84	LNθ0	1	0.74	230.5	0.0001	a
LNπk	81.73	78.52	82.58	LNπ0, LNπ1, LNπ5,	3	0.89	76.3	0.0001	a
LNξk	90.87	82.96	92.95	LNξ1, LNξ5	2	0.82	142.1	0.0001	a
Pool	92.88	75.56	97.46	LNθ0, LNπ0, LNξ1, LNξ5	4	0.83	130.8	0.0001	a
Primary structure of DNA nucleotide sequences & SG
SGθk	66.25	81.48	62.23	SGθ1e, SGθ4e	2	0.78	69.62	0.001	a
SGπk	71.21	85.19	67.51	SGπ0e, SGπ2e, SGπ5e	3	0.76	49.54	0.001	a
TIk	75.39	68.15	77.30	W, Je, 0Xe	3	0.73	58.19	0.001	a
Pool	81.58	68.15	85.13	SGπ5e, H, 1XRe	3	0.67	79.94	0.001	a
Primary structure of DNA codon sequences & SG
SGθk	70.43	76.30	68.88	SGθ0, SGθ1, SGθ4e	3	0.75	52.31	0.001	a
SGπk	74.77	82.96	72.60	SGπ4, SGπ4e, SGπ5e	3	0.74	56.37	0.001	a
TIk	76.16	59.26	80.63	S, 0X, 1XRe	3	0.72	60.98	0.001	a
Pool	80.80	74.81	82.39	SGθ0, SGθ4e, SGπ4e, SGπ5e, W	5	0.67	47.04	0.001	a
RNA electrostatic parameter of thermodynamically-driven 2S folding
2Sθk	97.60	93.30	100.00	2Sθ0	1	0.34	724.47	0.001	b
2Sπk	93.83	83.70	98.89	2Sπ0, 2Sπ2	2	0.44	515.03	0.05	c
2Sξk	96.58	85.19	100.00	2Sξ0, 2Sξ1	2	0.41	38.8	0.001	d

Note: the terms Ac, Se, and Sp mean accuracy, sensitivity and specificity, and measure the ratio of the number of total, Mps, or Cgs sequences correctly classified by the model with respect to the real classification; Vars.=no of variables in the QSAR equations; SG=star-graph; LN=lattice network; 2S=secondary structure; super index “e” represents the embedded calculations; references (Ref.) are a: this work, b: (González-Díaz et al., 2007c), c: (González-Díaz et al., 2005a) and d: (González-Díaz et al., 2006a).

Analyzing the above equations, it is important to highlight that, the combination of a negative contribution of ^LN ξ ₁ and a positive contribution of ^LN ξ ₅ in Eq. (21) point to a pseudo-folding rule for the biological activity. A validation procedure was subsequently performed in order to assess the model predictability. This validation was carried out with an external series of Mps and randomized Cgs. The present model showed an accuracy of 90.87%, which is similar in comparison to results obtained by other researchers when using the LDA method in QSAR studies (González-Díaz et al., 2007f). These results are also consistent with many others that we have recently reviewed in-depth and published as a review article where we used different network-like indices in small-sized, nucleic acid, and protein QSAR (González-Díaz et al., 2005d, González-Díaz et al., 2007d, González-Díaz et al., 2007f; Marrero-Ponce et al., 2005a, Marrero-Ponce et al., 2005b; Van Waterbeemd, 1995).

3.2. Results for DNA SG indices

The second study used the SG–QSAR models in order to evaluate the same mycobacterial DNA promoter property (see Table 3). The grouping of the embedded and non-embedded TIs was done similar to the lattice models: the traces (^SG π_k ^e/^SG π_k), the Shannon entropies (^SG θ_k ^e/^SG θ_k), the rest of embedded and non-embedded TIs (H, W, S ₆, S, J, ⁰ X, ^{2 − 5} X, ¹ XR, H ^e, W ^e, S ₆ ^e, S ^e, J ^e, ⁰ X ^e, ^{2 − 5} X ^e, ¹ XR ^e) and all SG TIs (pool). The forward stepwise selection variable method, conjugated with the nE & E TIs of the virtually translated DNA sequences, provides better results for the codon grouping of the nucleotides, with accuracy, sensitivity and specificity greater than 70% for the ^SG π_k ^e/^SG π_k and for the pool (Table 3). Even if the accuracy of the simple nucleotide sequences are up to 81.58% (pool), the selectivity and the specificity have values lower than 70%. The best QSAR model using the SG based on the codon sequences is defined with the ^SG π_k ^e/^SG π_k group of indices in Eq. (22) and is characterized by 74.77% accuracy, 82.96% sensitivity and 72.60% specificity.

$${}^{\mathrm{SG}}\mathrm{Mps}‐\mathrm{score}=-1.9+1.3\times {}^{\mathrm{SG}}{\pi }_4-1.9\times {}^{\mathrm{SG}}{\pi }_4^e-1.2\times {}^{\mathrm{SG}}{\pi }_5^e$$ (22)

Despite the good values of accuracy, sensitivity and specificity (80.80%, 74.81%, 82.39%) for the pool group of TIs (^SG θ ₀, ^SG θ ₄ ^e, ^SG π ₄ ^e, ^SG π ₅ ^e, W), the QSAR model cannot be considered due to the low sensitivity for the CV set (66.67%). Thus, the results based on the traces (spectral moments) are similar in the case of LN and SG representations, maintaining the ^SG π ₅ ^e/^LN π ₅ in the equations.

3.3. Comparison with RNA 2S and other indices

In previous works, we have published QSAR models to predict Mps using RNA electrostatic parameters of thermodynamically-driven 2S folding representations. These models were based on the ^2S θ_k (González-Díaz et al., 2007c), ^2S π_k (González-Díaz et al., 2005a) and ^2S ξ_k (González-Díaz et al., 2006a) values for the ¹ Π(2S) matrix associated to RNA 2S folding representations. In Table 3 we illustrate that the best values of accuracy, sensitivity and specificity of 97.60%, 93.30% and 100% were found for ^2S θ ₀. This TI is present in the QSAR equations for DNA LN/SG and RNA 2S folding representations. All these observations pointed out the importance of the spectral moments, entropies and in the stochastic electrostatic potentials in the DNA/RNA QSAR models. In general, the results for RNA 2S folding representation are better, but they require additional calculations for the optimization of RNA 2S. Therefore, several RNA 2S are possible for the same DNA sequence (theoretically because the promoters have no correspondent RNA) by introducing an indeterminacy in the final model prediction. In Fig. 3 we depict a possible 2S for the RNA sequence corresponding to the DNA sequences used in Fig. 1, Fig. 2 (dG is the free energy). This RNA 2S was obtained with the online DINAMelt server (Markham and Zuker, 2005). The SG TIs that show to not be important for the DNA/RNA models (H, W, S, J) can successfully describe protein QSAR models (Munteanu et al., 2008b). This work pointed out the conclusion that the models based on SG, LN, as well as 2S, which are linear and have few variables, can be compared very favourably in terms of complexity with other models previously reported by Kalate et al. (2003). These authors used a non-linear artificial neural network and a large parameter space.

Fig. 3.

RNA 2S for the Mps of the gene Alpha in Mycobacterum bovis (BCG).

5. Conclusions

The work presents a comparative study of the parameters associated with LN and SG representations in order to predict the mycobacterial DNA promoters. LN QSAR classifier successfully discriminates between Mps and a control group, with values significantly better than the SG-QSAR results based on the DNA codon sequences. In addition, the DNA nucleotide sequences (used for LN) were not able to create a good model based on SG representations. The work promotes the use of the experience accumulated in small-molecules QSAR with spectral moments and other kind of indices (entropies and spectral moments) in new types of DNA QSAR studies, today in the focus of attention of many researchers worldwide.

Appendix A. Supporting Information

Supporting Information A