Generalized lattice graphs for 2D-visualization of biological information.

Several graph representations have been introduced for different data in theoretical biology. For instance, complex networks based on Graph theory are used to represent the structure and/or dynamics of different large biological systems such as protein-protein interaction networks. In addition, Randic, Liao, Nandy, Basak, and many others developed some special types of graph-based representations. This special type of graph includes geometrical constrains to node positioning in space and adopts final geometrical shapes that resemble lattice-like patterns. Lattice networks have been used to visually depict DNA and protein sequences but they are very flexible. However, despite the proved efficacy of new lattice-like graph/networks to represent diverse systems, most works focus on only one specific type of biological data. This work proposes a generalized type of lattice and illustrates how to use it in order to represent and compare biological data from different sources. We exemplify the following cases: protein sequence; mass spectra (MS) of protein peptide mass fingerprints (PMF); molecular dynamic trajectory (MDTs) from structural studies; mRNA microarray data; single nucleotide polymorphisms (SNPs); 1D or 2D-Electrophoresis study of protein polymorphisms and protein-research patent and/or copyright information. We used data available from public sources for some examples but for other, we used experimental results reported herein for the first time. This work may break new ground for the application of Graph theory in theoretical biology and other areas of biomedical sciences.

Introduction

Several graph representations have been introduced for different data in theoretical biology. For instance, complex networks based on Graph theory are used to represent the structure and/or dynamics of different large biological systems such as protein–protein interaction networks. Complex networks are made up of nodes and edges/arcs (node–node connections or links). Drugs, genes, RNAs, proteins, organisms, brain cortex regions, diseases, patients or environmental systems may play the role of nodes. In general, the edges represent similarity/dissimilarity relationships between the nodes. In complex networks, both nodes and edges are placed generally in space without any geometrical constrains; nodes do not need spatial coordinates and edges have not a specific length or shape (Barabasi and Oltvai, 2004; Boccaletti et al., 2006; Estrada, 2006). In addition, Randic, Nandy, Basak, Liao, and many others developed some special types of graph-based representations. This special type of graph includes geometrical constrains to node positioning in space and sometimes adopts final geometrical shapes that resemble lattice-like patterns (Chen et al., 2009; Huang et al., 2009; Liao, 2005; Liao and Wang, 2004; Liao and Ding, 2005; Liao et al., 2005; Liao et al., 2006, Liao et al., 2006; Liao et al., 2009; Novic and Randic, 2008; Randic, 2006; Randic and Balaban, 2003; Randic et al., 2007; Randic et al., 2008; Randic et al., 2009; Randič, 2002 ; Randic′ et al., 2005; Zhang et al., 2009).

Using graphical approaches to study biological problems can provide an intuitive picture or useful insights in order to support the analysis of complicated relations within these systems, as demonstrated by many previous studies on a series of important biological topics, such as enzyme-catalyzed reactions (Andraos, 2008; Cornish-Bowden, 1979; Chou, 1980; Chou, 1981; Chou, 1989; Chou and Forsen, 1980; Chou and Liu, 1981; Chou et al., 1979; King and Altman, 1956; Kuzmic et al., 1992; Myers and Palmer, 1985; Zhou and Deng, 1984), protein folding kinetics and folding rates (Chou, 1990; Chou and Shen, 2009; Shen et al., 2009), inhibition kinetics of processive nucleic acid polymerases and nucleases (Althaus et al., 1993a; Althaus et al., 1993b; Althaus et al., 1993c; Althaus et al., 1996; Althaus et al., 1994a; Althaus et al., 1994b; Chou et al., 1994), analysis of codon usage (Chou and Zhang, 1992; Zhang and Chou, 1993; Zhang and Chou, 1994), base frequencies in the anti-sense strands (Chou et al., 1996), analysis of DNA sequence (Qi et al., 2007). Moreover, graphical methods have been introduced for a QSAR study (González-Díaz et al., 2006; González-Díaz et al., 2007b; Prado-Prado et al., 2008) and they have also been used to deal with complicated network systems (Diao et al., 2007; Gonzalez-Diaz et al., 2008b; González-Díaz et al., 2007a). 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 in representing complicated biological sequences (Xiao et al., 2005a) and helping to identify various protein attributes (Xiao and Chou, 2007; Xiao et al., 2009; Xiao et al., 2006b). In this study, we attempted to propose a different 2D graphical representation for some relevant areas.

In recent reviews, we have discussed the applications of these ones and other graphs in Proteomics and other Biomedical Sciences (Gonzalez-Diaz, 2008; Gonzalez-Diaz et al., 2008a; González-Díaz et al., 2008). However, despite the proved efficacy of new lattice-like graph/networks to represent diverse systems, most works focus on only one specific type of biological data. This work proposes a generalized type of lattice and illustrates how to use it in order to represent and compare biological data from different sources. Specifically, we extend the method from protein sequence to mass spectra (MS) of peptide mass fingerprints (PMF), molecular dynamic (MD) results from protein structural studies, mRNA microarray data, single nucleotide polymorphisms (SNPs), 1D or 2D-Electrophoresis (2DE) study of protein polymorphisms and protein-research patent and/or copyright information.

Methods

Generalized lattice graphs

Let there be a set of n elements or signals of a biological system, each one identified by a label s, and arranged in the form of a sequence of objects or numeric series. For instance, the one-letter code for all bases in a DNA sequence, amino acids in a protein sequence, gene in a chromosome, signals in a mass spectrum, values from the microarray data results, etc. First, we arrange all these elements n as a vector s=[s 1, s 2, s 3, s,…,s]. Next, we assign to each element a one or more up to m properties or weights ( w) arranged also as vectors: =[1 w 1, 1 w 2, 1 w 3,…, 1 w j,…, 1 w n]; =[2 w 1, 2 w 2, 2 w 3,…, 2 w,…, 2 w]; … =[ w 1, w 2, w 3,…, w,…, w];… and =[ w 1, w 2, w 3,…, w,…, w]. For instance; given an MS we can consider as elements a the n signals in the MS and we can assign at least two weights to each signal a: 1) the mass/charge ratio (m/z) of s and 2) the intensity I of s. Consequently, we have two weight vectors: =[(m/z)1, (m/z)2, (m/z)3,…, (m/z),…, (m/z)] and =[I 1, I 2, I 3,…,I,…, I]. In addition, we can regroup all the elements or signals of the biological system (s) into one or more classes (q) if they obey certain sets of conditions C q. These are usually simple or even composed logical conditions and we assign one letter symbol to all the elements of the same class. For instance, we can label as A, T, G, or C each nucleotide in a sequence if it belongs to the class of Adenine, Thymine, Guanine, or Cytosine. Another example is that we can label as H or L each signal s in an MS if the respective intensity value I is Higher (H) or Lower than the average of all intensities in the MS. Given all these starting facts, we deal here with the following question. How could we graphically visualize, in a simple way, all the information related to systems (sequences or numeric series), elements or signals, weights or properties, sets of conditions and classes if we have one or, even more complicated, up to ith systems or sequences altogether? Our method assigns each element/signal of one sequence as a point with the Cartesian coordinates r 2=(x, y) in a 2D Euclidean space. To this end, we star with the first node (it is not necessarily a data point) at the centre of the system placed at r 2=(0, 0) coordinate. The coordinates of the successive data points are calculated as follows in a similar manner to those for DNA spaces (Randic, 2004) but extended to multiple weight and condition sets for these weights or properties w of the elements or signals s.

Increases in +1 the y axe if w obey the set of conditions C 1 (upwards-step) or:

Increases in +1 the x axe if w obey the set of conditions C 2 but not C 1 (leftwards-step) or:

Decreases in −1 the y axe if w obey the set of conditions C 3 but not C 1 nor C 2 (rightwards-step) or:

Decreases in −1 the x axe otherwise (downwards-step).

Once we have placed the first sequence or system using the following rules we can superpose over it the remnant q sequences. It allows us the display of large databases in a simple 2D picture. We can use colour-scales highlighting systems or sequences with a given property. For instance, use different colours for enzymes of different classes or for MS signals of the blood samples of healthy vs. cancer patients. This type of visual graphs may be interpreted as 2D overlapping or alignment maps. As follows, we give here below some examples to illustrate the high versatility of this approach.

Results and discussion

Classic lattices for protein and peptide sequences

Several authors have used pseudo-folding lattice hydrophobicity-polarity (HP) models to simulate polymer folding by optimizing the lattice structure and resembling the real folding (Berger and Leighton, 1998). However, we can choose notably simpler polymer chain pseudo-folding rules to avoid optimization procedures and speed up notably the construction of the lattice. In this sense, useful graph representations of DNA, RNA and/or protein sequences have been introduced by Gates (1986), Nandy (1996a), Leong and Morgenthaler (1995), Randic et al. (2001) based on 2D coordinate systems. We call these graph representations as polymer sequence pseudo-folding lattice networks because they look like lattice structures and in fact, we force a sequence to fold in a way that does not necessarily occur in nature. In this regard, a novel 2D lattice representation for protein sequence similar to the one proposed by Nandy for DNA sequences was introduced by our group in the study of protein sequences (Nandy, 1996b; Nandy, 2003; Roy et al., 1998). In this 2D graph, each of the four amino acid groups is assigned to each axis direction according to the physicochemical nature of the amino acids (non-polar and non-charged, polar but non-charged, positively charged, or negatively charged) (Aguero-Chapin et al., 2006). These four classes characterize the physicochemical nature of the amino acids as: polar, non-polar, acid, or basic. Classification as positively or negatively charged prevails over polar/non-polar classification in such a way that the four classes do not overlap each other. In mathematical terms, it means that in this example, we used the vector s=[s 0, s 1, s 2,…, s,…,s] to list the labels for the n amino acids s, which are the elements of the system (protein sequence). Here we also used two vectors of weights to characterize numerically the s. The first 1 w=[q 0, q 1, q 2,…, q,…,q] lists the electrostatic charge of each one of the n amino acids in the sequence of the protein or peptide. The second vector 2 w=[μ 1, μ 2, μ 3,…, μ,…, μ] lists the dipolar moments of each amino acid. We also used herein the sets of conditions C 1, C 2, and C 3 that consist of two logic order operations. First, we place the node of the initial amino acid s 0 at the coordinates (0, 0) in a Cartesian 2D space. The coordinates of the successive amino acids are calculated as follows in a similar manner to that for DNA spaces:

The reader must note that the new representation is very similar to the ones previously reported for DNA but it contains a protein sequence of 20 amino acid types instead a DNA sequence of 4 base types. The key of the method we propose to overcome the above-mentioned 10D-space bottleneck is the previous grouping of the 20 natural amino acids into only four groups. As an illustrative example, we have under study a protein that belongs to the family of dyneins. This protein has the accession number LmjF25.0980 in the public database GenDB related to the Sanger institute (http://www.genedb.org/genedb/) and it is expressed by the parasite Leishmania major. Leishmania spp. is required intracellular protozoa that exist in two forms, a promastigote form (elongated cells with a long flagellum) and an amastigote one (ovoid cells that have a very short flagellum). The flagellum is responsible for the motility of trypanosomatids and for their early interaction with the hosts, either by adhering to the insect digestive tract, or by initiating the contact with mammalian cells. Trypanosomatids depend on this adhesion to survive and differentiate. This surface organelle plays a key role in Leishmania motility and sensory reception, and it is essential for parasite migration, invasion and persistence in host tissues. In this regard, some authors have applied lattice representations to study dyneins (Dea-Ayuela et al., 2008). Due to both the high interest of dyneins for the mechanism involved in protein–protein interaction or binding process, some authors have proposed experimental studies of peptide sets found in these proteins (Lajoix et al., 2004). Fig. 1 (A) illustrates the isolated and overlapped lattice graphs only for the first two peptides found in this protein presented in Table 1 . In this table we give the sequence of these peptides and other relevant information (see also next section). We used the first peptide (P01) with the sequences “vlmntlrdir” as example, where the vectors are s=[v 0, l 1, m 2, n 3, t 4, l 5, r 6, d 7, i 8, r 9], 1 w=[00, 01, 02, 03, 04, 05, 16, −17, 08, 19] and 2 w=[00, 01, 02, 13, 04, 05, 16, 17, 08, 19]. The vector 1 w is based on amino acid net charges and the vector 2 w is based on the discrete dipole moments (see Table S1 from the Supplementary material for more details).

Fig. 1

Sequence vs. MDT and MS lattice graphs for peptides found on PMF of proteins.

Table 1

Some information for peptides used to construct sequence, MDT, or PMF-MS lattices.

Peptide	Sequence	AAs	E0	E1	E100	ACCR	(m/z)j
P01	vlmntlrdir	10	635.4	−175.49	−22.38	0.48	1246.67
P02	dqelhfsefk	10	264.1	−15.71	107.65	0.46	1279.71
P03	hgimvvgpamcgk	13	17719.8	67.89	211.26	0.47	1356.70
P04	hwqeimkvsgr	11	10779.3	−34.40	133.33	0.47	1370.71
P05	qvmeylchfr	10	456.1	−75.29	75.14	0.47	1382.71
P06	mdsanglidalsger	15	714.9	−84.51	65.21	0.48	1564.81
P07	mnpkaitapqmfgr	14	15383.8	−42.80	137.60	0.47	1593.84
P08	mmytiaryyptr	12	16704.0	−116.06	62.84	0.47	1597.85
P09	lratmnadgqmlpr	14	14499.2	−145.61	26.08	0.48	1605.85
P10	ldfsslfiptadsvr	15	1325865.3	−80.48	98.25	0.47	1667.86
P11	lvrhgimvvgpamcgk	16	18520.6	18.45	197.03	0.48	1740.96
P12	eavahdaaivahgeaeakk	19	1343.8	13.43	222.83	0.47	1917.03
P13	qvvemsqvydlskpgvr	17	15611.8	−124.14	84.37	0.47	1935.04
P14	qvvemsqvydlskpgvrr	18	15565.5	−184.71	56.38	0.48	2091.14
P15	ylqsldtyfdvlyssnlqr	19	1532.4	−184.84	73.61	0.47	2325.15
P16	aqskpwetitdavtllrvwk	20	43367.0	−104.21	167.46	0.47	2342.16
P17	ldfsslfiptadsvrlhylak	21	1.4×10−7	−62.93	193.21	0.47	2393.28
P18	iwvtsephnsvpigllqmsikltneppqgik	31	1.5×10−7	−66.01	298.05	0.47	3442.90

Lattices for MD outcomes

The 3D structure of the L. major dynein protein sequence represented in the previous example is unknown; which is the case for many other proteins nowadays. In this sense, and taking into consideration the issues discussed in the previous section, the study of the 3D structure of its component peptides is of major interest. Since the advent of MD in bioscience with the study carried out by McCammon et al. on the dynamics of the bovine pancreatic trypsin inhibitor, MD has become the by the foremost a well-established computational technique to investigate the 3D structure and function of peptides and proteins (Karplus and McCammon, 2002; McCammon et al., 1977). Consequently, MD studies of peptides of the template protein used in the previous example are also interesting. In general, the analysis of the MD-Trajectories (MDTs) resulting from the integration of the motion equations in MD remains, however, the greatest challenge and requires a great deal of insight, experience, and effort. In a recent and very important work, Hamacher (2007) has proposed a new, theoretical sound, and versatile analysis procedure that provides scientists with a semi-quantitative tool to compare various scenarios of their respective simulations. In this regard, we extended the lattice representations of proteins to allow easy comparison of MDTs. In Fig. 2 we illustrate an example that consists of the superposition or 2D-alignment of 18 lattices derived from 100-steps MDTs results. Each MDT was obtained after a Monte Carlo study of 18 peptides found on the PMF of a very important parasite protein. In Table 1 we summarize some details on the MD study used here as example. The key of the method we propose is the regrouping into four classes the energy values E obtained from different steps (s) of one MD trajectory obtained in peptide structure study with the Monte Carlo method. These four classes characterize the deviation of the energy value E from the average energy of the same MDT at different steps (MD-average); or the deviation from average energy values in the same step for other MDTs (step-average).

Fig. 2

2D/1D Electrophoresis experiments reported in this work and examples of lattices.

First, we place the values of energy for a MDT in a Cartesian 2D space starting with s 0 at the coordinates (0, 0). In this example, we used the vector s=[s 0, s 1, s 2,…, s,…,s 100] to list the labels for optimization steps s in the MDT, which are the signals or elements of the system. Herein, we also used three vectors of weights to numerically characterize s. The first 1 w=[E 0, E 1, E 2,…, E,…, E 100] =101 lists the energy values for each s in the MDT numeric sequence (of one peptide). The second vector 2 w=[ E 1, E 2, E 3,…, E,…, E 9] lists the average of E for each one of the ith MDT for the ith peptides. The third vector 3 w=[ E 0, E 1, E 2,…, E, …, E 100] lists the average of E for each one of the jth steps of all MDT for all peptides. We also used the sets of conditions C 1, C 2, and C 3 that consist of two logical order operations. These operations perform the comparison (“>” or “<”) with respect to the average values MDT-average and step-average:

In Fig. 1(B), we depict the 2D alignment for MD results obtained after the optimization of the structure of 18 peptides and successive Monte Carlo search of different conformations. Remarkably, the MDT lattice obtained for peptide P03 notably deviates from the other peptides while the graph for P15 lies in the middle of the rest of peptides. It may indicate that this type of lattice is useful to differentiate visually peptides with high initial energy after MD geometry optimization (E 1) and not optimal MDT from the rest of peptides (see Table 1). The reader may note the differences between sequence and MD lattice graphs; which indicates that both types of graphs may be used as complementary information visualization techniques.

Lattices for MS of peptide mass fingerprints

The study of peptides found on the PMFs of new proteins may become an interesting source to discover new peptides with potential use as drug, in vaccine design, or as disease biomarkers. In particular, toxicity and inefficacy of actual organic drugs against Leishmaniosis justify research projects to find new drugs or drug molecular targets in Leishmania species including L. infantum and L. major, both important pathogens (Chenik et al., 2006; Dea-Ayuela et al., 2008; Roldos et al., 2008; Sarciron et al., 2005). In the two previous examples, we used lattices to study the sequences and MD results of peptides found in a dynein of L. major. In this example, we propose to construct PMF lattices, in analogy to sequences and MDTs lattices. To this end, we use a real experiment as example. We isolate all the peptides found on the PMF of a protein expressed on the parasite L. infantum with 2DE and characterize them with MALDI-TOF MS. After a MASCOT search of similar PMF-MS, we found that this new protein is similar to the protein of L. major studied in the previous examples. In Fig. 2(A), we illustrate the 2DE map experimentally obtained and highlight the position of the spot for the new protein. In Table 1 we give details on the (m/z) values for the peptides found on the PMF of the new protein. We report the experimental study of this protein for the first time but the method used is essentially the same we had used before for other dynein protein. That is why we omit the experimental details in this work and refer to the previous work (Dea-Ayuela et al., 2008).

Next, we report the generation of the 2D lattice graphs for large MS data generated in PMF experiments. The idea of using the graph to study MS is a promising field of research. Bartels proposed for the first time the application of Graph theory to MS for peptide sequencing (Bartels, 1990). The fundamental idea consists in transforming an MS into a graph called the spectrum graph, each peak in the experimental spectrum being represented as a graph node (or several nodes). Directed edges (or arc) connect between two vertices if the mass difference of the two vertices equals the mass of one or several amino acids. “SeqMS” (Fernandez-de-Cossio et al., 1995), “Lutefisk” (Taylor and Johnson, 1997), “Sherenga” (Dancík et al., 1999) and “PepNovo” (Frank and Pevzner, 2005) are the most popular algorithms that make use of spectrum graphs based on the basic idea proposed by Bartels.

In our lattice graph, MS signals are placed in a Cartesian 2D space starting with the first data point at the coordinate r 2=(0, 0). The coordinates of the successive data points are calculated as follows. In this example, we used the vector s=[s 1, s 2, s 3,…, s,…, s 68] to list the labels for the 68 MS signals s of the new protein. We also used four vectors of weights; the first 1 w=[1(m/z)]=[1(m/z)1, 1(m/z)2,…,1(m/z),…,1(m/z)68] lists the mass/charge ratio values for each s. The other three vectors are: 2 w=[2(m/z)1, 2(m/z)2,…, 2(m/z) ≤68]; 3 w=[3(m/z)1, 3(m/z)2,…, 3(m/z),…, 3(m/z) ≤68]; and 4 w=[4(m/z)1, 4(m/z)2,…,4(m/z),…,4(m/z) ≤68]. These vectors list, in an increasing order, the (m/z) values for s also present in the MS of the three most similar template proteins found after the MASCOT search. In order to generalize the procedure, we can refer to the vectors: w=[(m/z)], w=[(m/z)], and w=[(m/z)]. These vectors list the (m/z) values of the three proteins (kth-triad) placed at positions k, k+1, and k+2 in the list of template proteins, found after the MASCOT search, ordered from higher to lower similarity to the query protein. Next, we can use a set of conditions C 1, C 2, and C 3 to align many triads and detect the similarity patterns.

In Fig. 1(C), we depict the alignment of these types of lattice graphs for the query protein vs. 17 triads found with MASCOT in the template database. It is relevant that the method perfectly discriminates the alignment (black colour) with the best triad (more similar proteins), with respect to triads formed by other less similar or dissimilar proteins (grey colour). Last, we can apply alternatively and somehow complementary operations * C 1, * C 2, * C 3 and * C 4 if our aim is the study of the s in the query protein that does not match up with any template protein.

Note that the first lattice graph based on C 1, C 2, C 3 and C 4 plots the MS signals present in both the query protein and at least one of the triad of template proteins selected. Consequently, this graph gives us a visual idea on how similar our query protein is with respect to the known template proteins (like in BLAST). Conversely, the second type of graph based on * C 1, * C 2, * C 3 and * C 4 plots precisely those MS signals that do not match up with MS signals found on the triad of template proteins. In consonance, this graph may give an idea on how dissimilar this protein is and then how useful it may be to decide an investigation of unknown peptides.

Lattices for mass spectra of proteins serum profiles (PSP-MS)

In the previous paragraphs, we have introduced 2D lattice graph representations for DNA/protein sequences, MDT results, and PMF-MS experiments. Now, we report the generation of the 2D lattice graphs for large MS data generated in PSP-MS experiments with blood samples. Blood proteome is continuously changing due to the effect of the drug-induced damage in the affected organ. After the separation of the small peptide fragments from the actual insult, the remaining mixture of peptides retains the specificity of the disease due to the specific biomarker amplification process in a unique tissue microenvironment in the organ where the toxicity occurs (Hu et al., 2006). Therefore, we can use the serum, the saliva, or the urine because they are protein-rich information reservoirs containing blood traces (Hu et al., 2006). In addition, it is well-known the optimal performance in the low mass range demonstrated by the mass spectroscopy (Kantor, 2002; McDonald and Yates, 2002) applied to proteomics by offering the great chance of discovering these early stage composition changes. The main problems in the identification of a single disease-related protein are the following: there are thousands of intact and cleaved proteins in the PSP that require the separation and identification of each protein biomarker and most toxicity biomarkers appear only when significant organ damage has occurred. Thus, the pattern identification in PSP-MS becomes a realistic complementary approach compared with the direct identification of a single marker candidate. Consequently, we can state that PSP-MS may allow detecting disease biomarkers at the first stages. In this regard, the development of new graph representations becomes significant to visually depict interesting similarity/dissimilarity patterns between PSP-MS of different groups of patients. In a recent work, we have introduced novel Randic's spiral network representation of PSP-MS (Cruz-Monteagudo et al., 2008a). Other example is the previous theoretical study of Human Prostate Cancer with new graph representations (Ferino et al., 2008). In previous works, our group has extended for the first time the spiral, star (Cruz-Monteagudo et al., 2008b), and lattice (Petricoin et al., 2004) graphs to represent the PSP-MS with a very high number of intensity (I) signals and wide (m/z) bandwidth. As these types of graphs had been studied before, we do not depict a PSP-MS lattice here, by reasons of space. However, we give as follows a mathematical formalization of this type of graph in order to generalize them and show more possibilities to codify information in PSP-MS experiments. For it, each signal in the MS is placed in a Cartesian 2D space starting with the first data point at the coordinate r 2=(0, 0). The coordinates of the successive data points are calculated by using the following mathematical formalism. Although the binned process reduces efficiently the number of data points, it is still unmanageable for graph generation. Hence, the number of data points in the binned data files was condensed by taking the averaged *(m/z) and * I values for consecutive regions containing a fixed number n * of (m/z) and I data points. The value n * may be changed according to the interest of the research; in particular when we keep n *=1 the number of averaged regions * s is equal to the number of original signals s. In this example, we used the vector s=[s 1, s 2, s 3,…,s,…,s *] to list the labels for the MS signals s of the new protein. We also used two vectors of weights; the first 1 w=[*(m/z)]=[*(m/z)1, *(m/z)2,…, *(m/z),…,*(m/z)* ] list the average mass/charge ratio values for each region * s out of * n altogether. The other vector lists in a similar way the average intensity values 2 w=[* I 1, * I 2,…,* I,…,* I * ]. We also used the sets of conditions C 1, C 2, C 3 and C 4:

Lattices of protein polymorphisms determined by electrophoresis

Different electrophoresis such as: immunofixation electrophoresis, capillary electrophoresis, 2D-gel electrophoresis or 2DE, are used to characterize protein polymorphism in populations (Alper and Johnson, 1969; Hadi et al., 1998; Kanamori-Kataoka and Seto, 2009; Lopez-Galvez et al., 1995). In fact, the amount of protein variation undetected by electrophoresis may be reasonably small and at the protein level, a typical sexually-reproducing organism may be heterozygous at 20 or more percent of the gene loci. Although the evidence is limited, it seems that at the level of the DNA nucleotide sequence every individual is heterozygous at every locus-if introns as well as exons are taken into account (Ayala, 1983). In the present example, we characterized experimentally for the first time the polymorphism for 17 enzymes in three populations of Fasciola hepatica (F. hepatica). The parasite F. hepatica is the causal agent of fasciolosis infection, an important cause of lost productivity in livestock worldwide. Effective control of fasciolosis is difficult, especially in milking cows, which can only be treated during dry periods, a control strategy that has not been evaluated yet. Recently, our group has studied the effect of the type of flukicide treatment on the prevalence and intensity of infection in dairy cattle from Galicia, an area where fasciolosis is endemic and which is also the main milk-producing region in Spain (Mezo et al., 2008). In the present preliminary study, we found that 8 loci out of 17 studied presented polymorphisms expressing up to 3 different isoforms of the enzyme. The polymorphic enzymes were: Aconitate Hydratase or Aconitase (ACO), Adenilate Kynase (AK), Glutamate Oxaloacetate Transaminase (GOT), Hexokinase (HK), Isocitrate Dehydrogenase (IDH), Phosphogluconate Dehydrogenase (PGD),: Phosphoglucomutase 1 (PGM1), and Phosphoglucomutase 2 (PGM2). Considering that: (1) heterozygous organisms are common, (2) each protein has one or even two locus, and (3) each locus may present one out of two or more alleles; we can construct large databases with the information obtained by electrophoresis for individuals in different populations. This situation determines the necessity of the use of computational techniques. Actually, the necessity of the use of computational techniques for phenotypic analysis in adults and eggs of F. hepatica has been recently proposed by Valero et al. (2005).

In this regard, the present type of data is another interesting candidate to be studied with lattice graphs. Consequently, in this example we need 1 lattice graph for each parasite individual with 8 enzymes. Altogether, each of the 6 enzymes is encoded by 1 gene; which presents 2 loci that may express 1 out of 3 possible isoforms of the enzyme. In addition, one of the enzymes PGM is codified by two different gene producing two different proteins, PGM1 and PGM2. We used the vectors a s=[ s 1, s 2, s 3,…, s,…, s 8] and b s=[ s 1, s 2, s 3,…, s,…, s 8] to list the labels for the two possible alleles s and s for two enzyme isoforms A and B codified by a gen. Commonly, we use the magnitude called Retention factor (Rf) to characterize each signal in 1D electrophoresis; which measures the chromatographic displacement distance (electrophoresis in this case) of the band from the point of application. In Fig. 2(B), we illustrate the 1D electrophoresis bands for one individual parasite experimentally characterized in this work. Herein, we used an integer-value scale for Rf=1, 2, or 3 for the bands with lower absolute displacement, second higher displacement, etc. Then, we also used two vectors of weights; the first a w=[ Rf]=[ Rf 1, Rf 2,…, Rf,…, Rf 8] and the second b w=[ Rf]=[ Rf 1, Rf 2,…, Rf,…, Rf 8]. These vectors list the first 1 Rf and the second 2 Rf values of Rf for two isoenzymes codified by the same gene. In general, when 1 Rf≠2 Rf the organism is heterozygote for this character (enzyme) it means that the two loci of the gene codify different alleles. In this case we assign the lower absolute value of retention factor (1 Rf>2 Rf) to the first vector 1 s the enzyme. Otherwise, 1 Rf>2 Rf the two alleles of the gen produce the same enzyme and the organism is homozygote for this character. We can use the following sets of conditions C 1, C 2, C 3, and C 4 to obtain the lattice graph for 1 s or 2 s separately. In particular, C 4 refers to cases when the Rf could not be accurately determined and the genotypic information is not clear; the condition sets are as follows:

However, both alleles are determinant in the polymorphism. Therefore, it is more interesting to generate graphical plots for one individual containing both alleles at the same time. In this regard, we extended our mathematical formalism as follows (see the previous example on SNPs). Let there be, c s=a s∪b s=[ s 1, s 2, s 3,…, s,…, s, s 1, s 2, s 3,…, s,…, s] the vector that list the labels of two possible alleles of n genes and c w=a w∪b w=[1 Rf 1, 1 Rf 2,…, 1 Rf,…,1 Rf 8,2 Rf 1, 2 Rf 2,…,2 Rf,…,2 Rf 8]; we can use it to characterize the polymorphism of one individual in the following manner. We can refer to s and w as genotypic-polymorphism vectors and apply the same rules outlined above. Fig. 3 depicts the c s-alignment of all individuals belonging to different populations using the same above-mentioned rules C 1, C 2, C 3 and C 4.

Fig. 3

Parasite polymorphism lattices for different populations derived with 1D Electrophoresis results.

In any case, as the elements s and s for i=j are haplotypes of the same loci that codify the same enzyme, it is easier to list them successively. In this regard, it is probably easier to use the vectors s c=a s∩b s=[ s 1, s 1, s, s 2, s 3, s 3,…, s, s,…, s, s] and w c=a w∩b w=[1 Rf 1, 2 Rf 1,1 Rf 2, 2 Rf 2,…,1 Rf, 2 Rf,…,1 Rf 8, 2 Rf 8] to list the labels and weights of two possible alleles of n genes. In terms of computational cost, both procedures are equivalent but, with respect to facilitating data input, the s c vectors are more user-friendly. In any case, it is important to note that, in general, the lattice graph is different for c s and s c schemes; which may offer alternative solutions to the same problem. In Fig. 2(B), we also give examples of lattice graphs for one individual parasite using c s=a s∪b s or c s=a s∩b s as alternative schemes. On the other hand, if you are interested not in the characterization of the polymorphism of individuals within a population but in specific enzymes in different individuals you have to invert the previous approach using one vector of labels to list individuals and vectors of weights to characterize a haplotypes of the specific a enzyme in different individuals.

Last, in other type of electrophoresis methods such as 2D electrophoresis (see Fig. 2 for instance) the different proteins are characterized by Mass (Mj) and isoelectric point (pI) instead of only one Rf value (Dea-Ayuela and Bolás-Fernández, 2005). In these cases, we may use one label vector s, two weight vectors 1 w=[M] and 2 w=[pI], and cut-off values δ 1 and δ 2, to define condition sets C 1, C 2, C 3 and C 4 similar to those used for PSP-MS above:

Lattices of single nucleotide polymorphisms (SNPs)

In the previous paragraphs, we have introduced 2D lattice graph representations for DNA/protein sequences, MDT results, and MS outcomes. Now, we report the generation of the 2D lattice graphs for large SNP of schizophrenia patients. The HTR2A and DRD3 genes codify protein receptors for the biogenic amine serotonin (5-HT) and dopamine (DA) neurotransmitter; which are the primary targets of the antipsychotic drugs in the schizophrenia treatment (Meltzer et al., 1989). The silent SNP T102C (rs6313) at HTR2A as well as the non-synonymous SNP Ser9Gly (rs6280) at DRD3 have been extensively analyzed in schizophrenia case-control studies (Abdolmaleky et al., 2004; Jonsson et al., 2003). In this example we aim to study a SNPs database based on the 17bp-long and 32bp-long SNPs of the DRD3 and HTR2A gene respectively, from 260 schizophrenic patients and 354 control subjects (Dominguez et al., 2007). These SNPs are codified with the following haplotypes: 0 if the first allele is homozygous, 1 if heterozygous, 2 if the second allele is homozygous and 3 if it is unknown. As a result, we have a large amount of information contained in raw data of 614 patients with 17 inputs each one making 17×614=10,438 input data points altogether for DRD3 gene or 32×614=19,648 input data points altogether for HTR2A gene. In this type of cases, the use of simple graph methods may be very interesting in order to perform a fast visualization of the large database. In addition, the alignment of superposition in some way of all inputs may unravel hidden patterns of similarity/dissimilarity between all patients. In any case, to the best of our knowledge, no lattice graph has been reported to represent and perform 2D alignment of SNPs in schizophrenia patients.

In this example, the sequences of the SNPs genotype information are transformed into lattice graphs using the following mathematical formalism. To this end, each nucleotide in the 32bp SNPs sequence of one patient was placed as a node in a Cartesian 2D space starting with the first data point at the coordinate r 2=(0, 0). The coordinates of the successive data points are calculated as follows. In this example, we used the vector 1 s=[1 s 1, 1 s 2, 1 s 3,…,1 s j,…,1 s 17] and 2 s=[2 s 1, 2 s 2, 2 s 3,…,2 s j,…,2 s 32] to list the labels for each of the 17 signals s for gene DRD3 or 32 signals for gene HTR2A of one patient. We also used two vectors of weights; the first 1 w=[1 h]=[1 h 1, 1 h 2,…, 1 h,…,1 h 17] and the second 2 w=[2 h]=[2 h 1, 2 h 2,…, 2 h,…, 2 h 32] to list the haplotype types 1 h or 2 h=0, 1, 2, or 3 for each s of DRD3 or HTR2A gene respectively. We also used the following sets of conditions C 1, C 2, C 3, and C 4:

We may apply these conditions to the vectors 1 s and 1 w in order to obtain a 2D alignment of all SNPs for all patients by using gene DRD3 or HTR2A separately. However, it has been admitted that both genes were involved in schizophrenia so the generation of graphical plots for both sets of SNPs is interesting. In this regard, we extended our mathematical formalism as follows. Let there be, s=1 s∪2 s,…,∪ s=[1 s 1, 1 s 2, 1 s 3,…,1 s,…,1 s 1, 2 s 1, 2 s 2, 2 s 3,…,2 s,…,2 s 2,…, s 1, s 2, s 3,…, s,…, s] the vector that lists the labels of all SNPs for n genes with n1, n2,…,nn SNPs and w=1 w∪2 w,…,∪ w=[1 h 1, 1 h 2, 1 h 3,…,1 h,…,1 h 1, 2 h 1, 2 h 2, 2 h 3,…,2 h,…,2 h 2,…, h 1, h 2, h 3,…, h,…, h]; we can consider it as a single list of SNPs for n genes instead of only one. We can refer to s and w as partial or total chromosome vectors if they incorporate all gene in the same chromosome or only some of them. We can refer to s and w as ordered if the order of union of vectors is the same as in the original chromosome. Last, s and w are mixed and/or disordered if they assemble vectors coming from different chromosomes and/or in another order, different from the one specific to natural chromosome order. Last, the vectors s and w generated with all the genes of an organism may be classified as hole-SNPs genome vectors. In this example we can construct the vectors: s=1 s∪2 s=[1 s 1, 1 s 2, 1 s 3,…,1 s,…,1 s 17, 2 s 1, 2 s 2, 2 s 3,…,2 s,…,2 s 32] and w=1 w ∪2 w=[1 h 1, 1 h 2, 1 h 3,…,1 h,…,1 h 17, 1 h 1, 1 h 2, 2 h 3,…,2 h,…,2 h 32] that list the labels and weights for SNPs of a patient and the two genes DRD3 and HTR2A at the same time. Fig. 4 A depicts the 2D-alignment of all these vectors for all patients using the same above-mentioned rules C 1, C 2, C 3 and C 4. We can note that both groups overlap notably in the lattices of schizophrenia patients (in grey), by expanding leftwards to areas not covered by healthy patients, but there are no significant results because this region is the consequence of the unknown allele-type component (1 h=3). The area with interesting overlap differences that can be used to perform further research to find SNPs biomarkers for schizophrenia is the upper part of the lattice that is generated by the homozygous allele (1 h=0). The graph allowed us to depict visually 17×614+32×614=10,438+19,648=30,086 SNPs points for 614 healthy vs. schizophrenia patients in a single 2D graph.

Fig. 4

Examples of lattices for: (A) SNPs of schizophrenia patients, (B) microarray for cancer patients and (C) patents related to protein-research methods.

Lattices for mRNA microarrays

In the previous sections, we have introduced 2D lattice graph representations for DNA/protein sequences, MDT results, MS outcomes, and SNPs. In this example, we introduce 2D lattice graphs for the results obtained in mRNA microarray experiments. Microarrays have been used to find gene expression patterns with special relevance as molecular biomarkers for different diseases including cancer. Specifically, human breast cancer (HBCa) is the most common neoplasia in women since approximately 211,000 women are diagnosed with it annually in the United States. In spite of earlier detection and improved treatment, it remains the second leading cause of cancer-related death in the United States and in other developed countries. The genetic background of patients and the tumor's genetic and epigenetic anomalies create, in combination, molecularly distinct subtypes arising from distinct cell types within the ductal epithelium. This genetic complexity underlies the clinical heterogeneity of HBCa limiting a rational selection of treatment tailored to individual patient/tumor characteristics. In this regard, Modlich et al. (2005) published a very interesting study whose declared goal was to identify gene signatures predictive of response to preoperative systemic chemotherapy (PST) with epirubicin/cyclophosphamide in patients with primary HBCa. The authors obtained pre-treatment needle biopsies from 83 patients with breast cancer and profiled mRNA on Affymetrix HG-U133A arrays. Response ranged from pathologically confirmed complete remission (pCR), to partial remission (PR), to stable or progressive disease, “No Change” (NC). A primary analysis was performed in breast tissue samples from 56 patients and 5 normal healthy individuals as a training cohort for predictive marker identification. The high complexity of this dataset makes these results another interesting candidate to be visually depicted with lattice graphs. In addition, the 2D alignment or superposition of all inputs may unravel hidden patterns of similarity/dissimilarity between all patients. In any case, we have not found a previous report using lattice graphs to represent and/or perform 2D alignment of mRNA microarray results in cancer patients or another disease.

In this example, values of mRNA levels for each patient obtained with Affymetrix HG-U133A arrays are directly transformed into one lattice graph using the following mathematical formalism. To this end, each value for one specific mRNA for one patient is placed as point (node) in a Cartesian 2D space starting with the first data point at the coordinate r 2=(0, 0). We calculated the coordinates of the successive data points as follows. In this example, we used the vector 1 s=[1 s 1, 1 s 2, 1 s 3,…, 1 s,…,1 s] to list the labels s for the different mRNA profiled with the Affymetrix kit. We also used the vector of weights: 1 w=[1 c]=[1 c 1, 1 c 2,…,1 c,…,1 c] to list the numeric value of the level of the mRNA. Last, we used the following sets of conditions C 1, C 2, C 3, and C 4:

where c is the average of 1 c of the same mRNA value for all patients whereas c is the average of 1 c for mRNA value of all gen in a given patient. Fig. 4B depicts the 2D-alignment of all these vectors for a sub-set of patients using the same above-mentioned rules C 1, C 2, C 3 and C 4. In this graph, both NC and PR patients are displayed in black whereas pCR patients are coloured in grey. The lattice shows that in fact, both populations share common areas but NC and PR patients with no positive answer to drug treatment distribute downwards to regions not covered by lattices of healthy patients. In any case, this is only a technical-note illustrative example on how to carry out the construction of mRNA microarrays lattices and we need to perform further research with larger databases in order to draw more convincing conclusions.

Lattices for research trends, copyright and patent protection in biological research

In the previous sections, we have introduced 2D lattice graphs for different molecular experiments. However, the applications in proteome research of these lattices may have further implications. For instance, we can use these graphs to analyze the scientific production and copyright or patent protection of this scientific production. It may help proteome research scientists, development managers, and/or politicians to decide which directions on proteome I+D are promising for further investment in order to introduce final protected products in the market. It may help also to detect relevant communities, groups, and/or research networks in their respective areas of interest. The use of Graph theory to analyze scientific production trends is not new (Malin and Carley, 2007; Rosvall and Bergstrom, 2008). Thus, in this example we report for the first time the construction of 2D lattice graphs with this aim.

In this example, we get outputs of patent search including the last 500 inputs containing the word protein in the field title from the European Patent Office (EPO) web (http://ep.espacenet.com/). Now, we report the generation of the 2D lattice graphs for this dataset as a sort of illustrative example, more detailed research is expected to be used in other fields. The starting point has coordinates r 2= (0,0) placed at the centre of a Cartesian 2D space. The coordinates of the successive data points were calculated as follows. First, we assign to each patent a vector s=[s 0, s 1, s 2,…,s,…,s 20] that lists the labels for different search terms s (s 0 is the word method+20 additional terms). We also used two weighting vectors; the first 1 w=[1 f]=[1 f 0, 1 f 2,…,1 f,…,1 f 20] lists the frequency of each term s in the 500 patents studied. The other vector: 1 w=[1 δ]=[1 δ 0, 1 δ 2,…,1 δ,…,1 δ 20] lists the values 1 δ=1 if the term s is present in the field of the patent; 1 δ=0 otherwise. We also used the sets of conditions C 1, C 2, C 3 and C 4.

In Fig. 4C we depict the alignment of the 500 protein-research related patents studied here using these rules. For instance, in this case, we can note which areas are common to US, WO, or other patents and which are not, in order to profile patenting strategies in US.

Conclusions

The construction of 2D-generalized lattice graphs constrained into a Cartesian coordinate system is a useful technique for biological data visualization not necessarily limited to DNA sequences. For instance, we demonstrated how to extend it in order to depict protein sequences, SNPs, parasite enzyme poly-morfirms, peptide MD, protein MS, PSP-MS, mRNA microarray outcomes and protein-research patent information. The present results break new ground in applying the Graph theory for knowledge discovery in proteome research as well as other areas of biological sciences.

C1: Increases in +1 the y axe if qj > 0 (upwards-step) or:

C2: Increases in +1 the x axe if qj=0 and μj≠0 (rightwards-step) or:

C3: Decreases in −1 the y axe if qj<0 (downwards-step) or:

C4: Decreases in −1 the x axe otherwise (leftwards-step).

C1: Increases in +1 the y axe; if Ej>avgEj and Ej>avgEi (upwards-step) or:

C2: Increases in +1 the x axe; if Ej>avgEj and Ej<avgEi (rightwards-step) or:

C3: Decreases in −1 the y axe; if Ej<avgEj and Ej<avgEi (downwards-step) or:

C4: Decreases in −1 the x axe; otherwise (leftwards-step).

C1: Increases in +1 the y axe if 1(m/z)j ∈ kw and ∉ k+1w and ∉ k+2w (upwards-step) or:

C2: Increases in +1 the x axe if 1(m/z)j ∉ kw and ∈ k+1w and ∉ k+2w (rightwards-step) or:

C3: Decreases in −1 the y axe if 1(m/z)j ∉ kw and ∉ k+1w nor ∈ k+2w (downwards-step).

C4: Decreases in −1 the x axe if otherwise (leftwards-step).

*C1: Increases in +1 the y axe if 1(m/z)j ∉ 2w (upwards-step) or:

*C2: Increases in +1 the x axe if 1(m/z)j ∉ 2w and ∉3w neither (rightwards-step) or:

*C3: Decreases in −1 the y axe if 1(m/z)j ∉ 2w and ∉ 3w and ∉ 4w neither (downwards-step).

*C4: Decreases in −1 the x axe if otherwise (leftwards-step).

C1: Increases in +1 the y axe if *(m/z)j>0.5 and *Ij>0.5 for *sj (upwards-step) or:

C2: Increases in +1 the x axe if *(m/z)j>0.5 and *Ij<0.5 for *sj (rightwards-step) or:

C3: Decreases in −1 the y axe if *(m/z)j<0.5 and *Ij<0.5 for *sj (downwards-step).

C4: Decreases in −1 the x axe if otherwise (leftwards-step).

C1: Increases in +1 the y axe if kRfj=1 (upwards-step) or:

C2: Increases in +1 the x axe if kRfj=2 (rightwards-step) or:

C3: Decreases in −1 the y axe if kRfj=3 (downwards-step).

C4: Decreases in −1 the x axe otherwise (leftwards-step).

C1: Increases in +1 the y axe if Mj>δ1 and pIj >δ2 (upwards-step) or:

C2: Increases in +1 the x axe if Mj>δ1 and pIj<δ2 (rightwards-step) or:

C3: Decreases in −1 the y axe if Mj<δ1 and pIj<δ2 (downwards-step).

C4: Decreases in −1 the x axe otherwise (leftwards-step).

C1: Increases in +1 the y axe if 1hj=0 (upwards-step) or:

C2: Increases in +1 the x axe if 1hj=1 (rightwards-step) or:

C3: Decreases in −1 the y axe if 1hj=2 (downwards-step).

C4: Decreases in −1 the x axe if otherwise (leftwards-step).

C1: Increases in +1 the y axe; if 1cj > avgcj and 1cj > avgci (upwards-step) or:

C2: Increases in +1 the x axe; if 1cj > avgcj and 1cj<avgci (rightwards-step) or:

C3: Decreases in −1 the y axe; if 1cj<avgcj and 1cj<avgci (downwards-step) or:

C4: Decreases in −1 the x axe; otherwise (leftwards-step),

C1: Increases in +1 the y axe if 1δ0=1 and 1δj=1 and 1fj > *fj (upwards-step) or:

C2: Increases in +1 the x axe if 1δ0=0 and 1δj=1 and 1fj > *fj (rightwards-step) or:

C3: Decreases in −1 the y axe if 1δ0=1 and 1δj=1 and 1fj<*fj (downwards-step).

C4: Decreases in −1 the x axe if otherwise (leftwards-step).