DNA sequence representation by trianders and determinative degree of nucleotides

DNA sequence representation by trianders and determinative degree of nucleotides 1 first-author 2 †E-mail:Steven.A.Duplij@univer.kharkov.ua †E-mail:Steven.A.Duplij@univer.kharkov.ua Steven.A.Duplij@univer.kharkov.ua Institute of Molecular Biology and Genetics, Kiev 03143, Ukraine Institute of Molecular Biology and Genetics, Kiev 03143, Ukraine Theory Group, Nuclear Physics Laboratory, Kharkov National University, Kharkov 61077, Ukraine Theory Group, Nuclear Physics Laboratory, Kharkov National University, Kharkov 61077, Ukraine A new version of DNA walks, where nucleotides are regarded unequal in their contribution to a walk is introduced, which allows us to study thoroughly the “fine structure” of nucleotide sequences. The approach is based on the assumption that nucleotides have an inner abstract characteristic, the determinative degree, which reflects genetic code phenomenological properties and is adjusted to nucleotides physical properties. We consider each codon position independently, which gives three separate walks characterized by different angles and lengths, and that such an object is called triander which reflects the “strength” of branch. A general method for identifying DNA sequence “by triander” which can be treated as a unique “genogram” (or “gene passport”) is proposed. The two- and three-dimensional trianders are considered. The difference of sequences fine structure in genes and the intergenic space is shown. A clear triplet signal in coding sequences was found which is absent in the intergenic space and is independent from the sequence length. This paper presents the topological classification of trianders which can allow us to provide a detailed working out signatures of functionally different genomic regions. DNA walk Triander Determinative degree Analysis DNA sequences Dystrophin Nucleotide INTRODUCTION 1 Genomic DNA sequence analysis using wide range of statistical methods (Torney et al., 1999 1999 ; Bulmer, 1987 1987 ; Luo et al., 1998 1998 ; Nieselt-Struwe, 1997 1997 ; Fickett et al., 1992 1992 ; Buldyrev et al., 1998 1998 ; Azbel, 1995 1995 ) and various symmetry investigations (Findley et al., 1982 1982 ; Hornos and Hornos, 1993 1993 ; Bashford et al., 1997 1997 ; Bhry et al., 1998 1998 ; Forger and Sachse, 1998 1998 ; Frappat et al., 1998 1998 ) is an extremely important tool for extracting hidden information on the dynamic process of evolution, especially after the availability of fully sequenced genomes (Nakamura et al., 2000 2000 ). One of the most promising approaches is the DNA walks method (Hamori, 1985 1985 ; Gates, 1985 1985 ; Berthelsen et al., 1992 1992 ) first introduced by Azbel ( 1973 1973 ; 1995 1995 ) or genomic landscapes (Lobry, 1996 1996 ) based on mapping of a sequence into one-, two-, or multidimensional metric space according to various specific rules. In brief, while drawing a DNA walk, the corresponding mappings assign a direction/unit vector to each nucleotide, to dinucleotide or to purine (pyrimidine). The resulting broken lines endow visual presentation to a formal sequence of 4 symbols, where inhomogeneous regions, fluctuations, patches etc. (Bernardi et al., 1985 1985 ) are immediately seen. A modification of the DNA walks method deals with each position in codons independently, which gives three separate broken lines characterized by different angles and lengths (Cebrat and Dudek, 1998 1998 ), where also addition and subtraction of DNA walks are also considered (Kowalczuk et al., 2001a 2001a ). Here we introduce a new version of DNA walks, where all 4 nucleotides are regarded unequal in the sense that their contribution to a walk differs not only by direction, but also by module. It follows from the assumption in (Duplij and Duplij, 2000 2000 ) that nucleotides have an inner abstract characteristic–the determinative degree (Duplij et al., 2000 2000 ) reflecting phenomenological properties of genetic code and is adjusted to nucleotides physical properties. GENETIC CODE REDUNDANCY, DOUBLET MATRIX INNER STRUCTURE AND DETERMINATIVE DEGREE 1 As well-known, the genetic code is a highly organized system (Yčac, 1969 1969 ) and has several general properties: triplet character, uniqueness, non-overlapping, comma less, redundancy (degeneracy), which means that most amino acids can be specified by more than one codon (Lewin, 1983 1983 ; Stent and Kalindar, 1981 1981 ). From 64 possible codons one can extract 16 families each defined by the first two nucleotides. Let we denote a triplet (5′-1-2-3-3′) by XYZ. Then the codon sense can be fully determined by the first two nucleotides X and Y independent of the third Z. There are 8 unmixed families (all 4 codons of which encode the same amino acid), and 8 mixed families for which several patterns of assignment exist, in 6 of the latter the pyrimidine codons (Z=C, T) determine one amino acid, and the purine codons (Z=A, G) determine other amino acids or termination signals (in one family). It was found that two thirds of all DNA bases are identical for all organisms in the first two nucleotides in a triplet, and that the variability of DNA composition is given by the third base (Singer and Berg, 1991 1991 ; Lewin, 1983 1983 ). All 16 doublets XY can be presented as the canonical matrix matrix (Rumer, 1968 1968 ) This ordering is called the rhombic code (Karasev and Sorokin, 1997 1997 ; Karasev, 1976 1976 ). They are grouped together in 2 octets distinguished by the ability of amino acid determination: 8 doublets CC, AC, GC, CU, GU, UC, CG, GG determine amino acid independently of third base (upper part in the rhombic code), and so they can be called strong, and other 8 doublets AA, AU, UU, CA, GA, UG, AG, UA (lower part in the rhombic code) for which third base determines content of codons can be called weak ones (Rumer, 1968 1968 ; Ratner, 1985 1985 ). The strong set of doublets has the following (Ratner, 1985 1985 ) relative content: n (C): n (G): n (U): n (A)=7:5:3:1 while the weak set has the reverse content: n (C): n (G): n (U): n (A)=1:3:5:7 Note that there is only one A in the strong octet, and one C in weak octet, and that all 4 doublets with Y=C completely determine amino acid, but only 2 doublets with Y=G completely determine it, while doublets with Y=A never determine amino acid. Thus, 4 nucleotides can be arranged in descending order C, G, U, A by their determinative ability (strength) (Rumer, 1969 1969 ; 2000 2000 ). We introduce a numerical characterization of the empirical strength–determinative degree of nucleotide d x in the following way $$\begin{gathered}{\text{ Pyrimidine Purine Pyrimidine Purine}} \\{\text{ C G T/U A}} \\{\text{ }}d_{\text{C}} {\text{ = 4 }}d_{\text{G}} {\text{ = 3 }}d_{{\text{T/U}}} {\text{ = 2 }}d_{\text{A}} {\text{ = 1}} \\{\text{Very strong Strong Weak Very weak}} \\{\text{ Completely In 2 cases In 2 cases Never}} \\\end{gathered}$$ which allows us change qualitative to quantitative description of genetic code structure (Duplij and Duplij, 2000 2000 ; 2001 2001 ) We use the notation T/U, because genetic code is read from mRNA, and so we will not differentiate their determinative ability (strength) in what follows. Let us present four bases Eq.( 1 1 ) as the vector-column $${\mathbf{V}} = \left( {\begin{array}{*{20}c}{{\mathbf{V}}_1 } \\{{\mathbf{V}}_2 } \\{{\mathbf{V}}_3 } \\{{\mathbf{V}}_4 } \\\end{array} } \right) = \left( {\begin{array}{*{20}c}{C^{\left( 4 \right)} } \\{G^{\left( 3 \right)} } \\{T^{\left( 2 \right)} } \\{A^{\left( 1 \right)} } \\\end{array} } \right)$$ and the corresponding vector-row V T =(C (4) G (3) T (2) A (1) ) where the upper index for nucleotide denotes determinative degree. We make the exterior product of vector-column Eq.( 2 2 ) and vector-row Eq.( 3 3 ) as follows (Duplij and Duplij, 2000 2000 ; Duplij et al., 2000 2000 ): $${\mathbf{M}} = {\mathbf{V}} \times {\mathbf{V}}^{\text{T}} = \left( {\begin{array}{*{20}c}{C^{(4)} {\text{C}}^{(4)} } & {{\text{C}}^{(4)} {\text{G}}^{\left( 3 \right)} } & {{\text{C}}^{(4)} {\text{T}}^{\left( 2 \right)} } & {{\text{C}}^{(4)} {\text{A}}^{\left( 1 \right)} } \\{{\text{G}}^{\left( 3 \right)} {\text{C}}^{(4)} } & {{\text{G}}^{\left( 3 \right)} {\text{G}}^{\left( 3 \right)} } & {{\text{G}}^{\left( 3 \right)} {\text{T}}^{\left( 2 \right)} } & {{\text{G}}^{\left( 3 \right)} {\text{A}}^{\left( 1 \right)} } \\{{\text{T}}^{\left( 2 \right)} {\text{C}}^{(4)} } & {{\text{T}}^{\left( 2 \right)} {\text{G}}^{\left( 3 \right)} } & {{\text{T}}^{\left( 2 \right)} {\text{T}}^{\left( 2 \right)} } & {{\text{T}}^{\left( 2 \right)} {\text{A}}^{\left( 1 \right)} } \\{{\text{A}}^{\left( 1 \right)} {\text{C}}^{(4)} } & {{\text{A}}^{\left( 1 \right)} {\text{G}}^{\left( 3 \right)} } & {{\text{A}}^{\left( 1 \right)} {\text{T}}^{\left( 2 \right)} } & {{\text{A}}^{\left( 1 \right)} {\text{A}}^{\left( 1 \right)} } \\\end{array} } \right)$$ It is remarkable that the matrix M in Eq.( 4 4 ) fully coincides with the canonical matrix of doublets in the rhombic code, if and only if the vector V has the determinative degree order C, G, U, A in Eq.( 1 1 ). Although there are 4!=24 possibilities to place 4 bases in row, but all others except one presented in Eq.( 1 1 ) do not reflect the phenomenological properties of the genetic code. It follows that the intuitive rhombic code and genetic vocabulary (Rumer, 1968 1968 ; Karasev and Sorokin, 1997 1997 ; Karasev, 1976 1976 ) have their own inner abstract structure uniquely defined by the exterior product of special vectors in Eq.( 4 4 ). This ordering is also adjusted to the schemes (Sukhodolec, 1985 1985 ; Maslov, 1981 1981 ), and also (partially) adjusted to the half time of nucleotide substitution under mutational pressure (Kowalczuk et al., 2001b 2001b ) and the nucleotides information weights (Dudek et al., 2002 2002 ). Indeed these facts allows us to introduce the determinative degree, as an abstract variable being a numerical measure of nucleotide difference in ability to determine sense of codon (Duplij and Duplij, 2000 2000 ; Duplij et al., 2000 2000 ) An analogous model for the triplet genetic code can be constructed using triple exterior product in the same way (Duplij and Duplij, 2000 2000 ). We dispose the doublet matrix M on the XY plane and multiply it on the vector-column V Eq.( 2 2 ) disposed along Z axis, i.e. we construct the triple exterior product K = V × M Thus we obtain three-dimensional matrix over the set of all triplets, and, since each codon (except three terminal ones) corresponds to an amino acid, that can be treated as a cubic matrix model of the genetic code (Duplij and Duplij, 2000 2000 ). DETERMINATIVE DEGREE AND NUCLEOTIDE PROPERTIES 1 The connection bulk DNA structure and various properties of nucleotides were studied in (Zheltovsky et al., 1989 1989 ; Govorun et al., 1992 1992 ). It is well-known that by chemical structure the 4 nitrous bases can be divided into: 1. purine (A, G) and pyrimidine (C, T); 2. having amino (A, C) group and (G, T) keto group; 3. making 3 (strong) hydrogen bonds (C, G) and 2 (weak) hydrogen bonds (A, T). They give rise to 3 symmetry transformations: 1. Purine-pyrimidine symmetry $$\left( \begin{gathered}{\text{T}}^{\left( 2 \right)} \\{\text{A}}^{\left( 1 \right)} \\{\text{C}}^{(4)} \\{\text{G}}^{\left( 3 \right)} \\\end{gathered} \right) = \left( {\begin{array}{*{20}c}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\\end{array} } \right)\left( \begin{gathered}{\text{C}}^{\left( 4 \right)} \\{\text{G}}^{\left( 3 \right)} \\{\text{T}}^{\left( 2 \right)} \\{\text{A}}^{\left( 1 \right)} \\\end{gathered} \right) = {\mathbf{R}}_{{\text{pur}}} {\mathbf{V}}$$ 2. Amino-keto symmetry $$\left( {\begin{array}{*{20}c}{{\text{A}}^{\left( 1 \right)} } \\{{\text{T}}^{\left( 2 \right)} } \\{{\text{G}}^{\left( 3 \right)} } \\{{\text{C}}^{\left( 4 \right)} } \\\end{array} } \right) = \left( {\begin{array}{*{20}c}0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \\\end{array} } \right)\left( {\begin{array}{*{20}c}{{\text{C}}^{\left( 4 \right)} } \\{{\text{G}}^{\left( 3 \right)} } \\{{\text{T}}^{\left( 2 \right)} } \\{{\text{A}}^{\left( 1 \right)} } \\\end{array} } \right) = {\mathbf{R}}_{{\text{amino}}} {\mathbf{V}}$$ 3. Complementary symmetry (leaving invariant the double helix) $$\left( {\begin{array}{*{20}c}{{\text{G}}^{\left( 3 \right)} } \\{{\text{C}}^{\left( 4 \right)} } \\{{\text{A}}^{\left( 1 \right)} } \\{{\text{T}}^{\left( 2 \right)} } \\\end{array} } \right) = \left( {\begin{array}{*{20}c}0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 \\\end{array} } \right)\left( {\begin{array}{*{20}c}{{\text{C}}^{\left( 4 \right)} } \\{{\text{G}}^{\left( 3 \right)} } \\{{\text{T}}^{\left( 2 \right)} } \\{{\text{A}}^{\left( 1 \right)} } \\\end{array} } \right) = {\mathbf{R}}_{{\text{compl}}} {\mathbf{V}}$$ where the even (because determinant is +1) permutation matrices R pur , R amino , R compl satisfy R pur R amino R compl = I and two of them, e.g. R pur , R compl , together with the identity matrix I form the dihedral group D 2 which is the symmetry group of the dihedron, or regular double-pyramid, with vertices on the unit-sphere (Ziegler, 1995 1995 ). Another representation of this group by 3×3 rotational matrices is called a DNA group (Zhang, 1997 1997 ). The difference in the number of hydrogen bonds causes different interaction with its complementary nucleotide: each strong nucleotide (C and G) has 3 bonds and the energy of C-G interaction is −10.05 kJ/mol, and each weak nucleotide (T and A) has only 2 bonds with the energy of A-T interaction being −5.02 kJ/mol (Lewin, 1983 1983 ). Therefore each base has its own properties and so dividing them into only 2 groups is not sufficient. We then can search whether the ordering Eq.( 1 1 ) is adjusted to some physical properties of nucleotides. First we observe that the dipole moment of bases is proportional to the determinative degree as shown in Fig. 1 1 . Then we see that the weight of hydration sites for bases is also proportional to the determinative degree Fig. 2 2 . We can conclude that the determinative degree reflects not only redundancy of genetic code in the third position, but also connected with some energy properties of the bases themselves. TRIANDERS AND THEIR CHARACTERISTICS 1 It can be assumed that the phenomenological properties of genetic code and inequality of bases (reflected in Eq.( 1 1 )) will become apparent in real nucleotide sequences. Here we use the introduced determinative degree to build a new kind of sequence analysis based on some special modification of DNA walks method (Berthelsen et al., 1992 1992 ; Azbel, 1973 1973 ; Lobry, 1996 1996 ; Cebrat and Dudek, 1998 1998 ). TRIANDER CONSTRUCTION 1 We embed a nucleotide sequence into the two-dimensional determinative degree space (DD plane) in the following way. The axis assignment corresponds to the value of nucleotide determinative degree as Axis x : {A}=(−1, 0); {T}=(+2, 0); Axis y : {G}=(0, −3); {C}=(0, +4) Moving along a sequence produces a walk in the determinative degree space which we call a determinative degree walk. In general, a current point on DD plane after i steps is determined by the coordinates $$x_i^{{\text{DD}}} & = d_{\text{T}} n_{\text{T}} (i) - d_{\text{A}} n_{\text{A}} (i) \\y_i^{{\text{DD}}} & = d_{\text{C}} n_{\text{C}} (i) - d_{\text{G}} n_{\text{G}} (i) \\$$ where n X ( i ) is cumulative quantity of nucleotide X after I steps and d X is the determinative degree of nucleotide X. The standard DNA walks (Cebrat and Dudek, 1998 1998 ) (genome landscapes (Azbel, 1973 1973 )) have all d X =1 in Eq.( 9 9 ), i.e. $$\begin{aligned}x_i^{{\text{standard}}} & {\text{ = }}n_{\text{T}} (i) - n_{\text{A}} (i) \\y_i^{{\text{standard}}} & {\text{ = }}n_{\text{C}} (i) - n_{\text{G}} (i) \\\end{aligned}$$ The one-dimensional (purine/pyrimidine) (PP) DNA walks are defined by only one coordinate, while x is chosen as position, i.e. $$\begin{aligned}x_i^{{\text{pp}}} & {\text{ = }}i \\y_i^{{\text{pp}}} & {\text{ = }}n_{\text{C}} (i) + n_{\text{T}} (i) - n_{\text{A}} (i) - n_{\text{G}} (i) \\\end{aligned}$$ Therefore, while purine/pyrimidine DNA walks manifestly show the purine/pyrimidine imbalance, the standard DNA walks Eq.( 10 10 ) applied for one strand show DNA asymmetry (Wu, 1991 1991 ; Francino and Ochman, 1997 1997 ) (violation of the second parity rule (Sueoka, 1995 1995 )), the determinative degree walk Eq.( 9 9 ) visually shows strength imbalance in one strand. Then we build 3 independent determinative degree walks beginning from 1st nucleotide with step 3 (due to the triplet structure of genetic code). In this way we obtain 3 broken lines (branches) starting from the point of origin, and each of them presents the determinative degree walk through the following nucleotide numbers: 1st branch goes through 1,4,7,10,13,... positions; 2nd branch goes through 2,5,8,11,14,... positions; 3rd branch goes through 3,6,9,12,15,... positions. These 3 branches on the determinative degree plane are called triander. If 1st letter corresponds to the first start codon nucleotide, then the triander branches represent nucleotide sets in three codon positions independently. As distinct from previous versions of DNA walks in which all 4 nucleotides are regarded equivalent in the sense that they give equal by module shifts, in our approach each nucleotide gives contribution different by module (which is taken equal to its determinative degree). So, although we obtain at first sight isomorphic to (Cebrat and Dudek, 1998 1998 ) plot, the trianders show not only quantitative composition and pure statistical laws of symbol strings, but also reflect the connection between nucleotide sequences and inner phenomenological properties of genetic code and physicochemical properties of bases. As an example of triander we will take the dystrophin gene which is the largest gene found in nature, measuring 2.4 millions bases pairs, responsible for Duchenne (DMD) and Becker (BMD) muscular myodystrophy (Yagi et al., 2003 2003 ). The dystrophin RNA is differentially spliced, producing a range of different transcripts, encoding a large set of protein isoforms. Dystrophin is a large, rod-like cytoskeletal protein found at the inner surface of muscle fibers. The triander for the dystrophin gene is shown in Fig. 3 3 . For comparison we also show the triander for a shuffled sequence of the same nucleotide composition. Obviously the ideal triander for uniformly random sequence consists of 3 flowing together lines from the origin having 45 degrees slope. This line also corresponds to the symmetric sequence satisfying the second parity rule (Sueoka, 1995 1995 ): N C = N G , N T = N A . Such lines are presented on all triander plots below for normalization. DETERMINATIVE DEGREE ANGLE 1 An important visual characteristic of a triander is the slope of its branches, we call it determinative degree (DD) angle ( α ), which for a current point ( i ) can be calculated by $$\tan \alpha (i) = \frac{{4n_{\text{C}} (i) - 3n_{\text{G}} (i)}}{{2n_{\text{T}} (i) - n_{\text{A}} (i)}}$$ Evidently, for uniformly random sequence or a symmetric sequence satisfying the parity rule 2 (Sueoka, 1995 1995 ) the angle will be 45 degree (horizontal dashed line of the plots below), and so the difference from this value will say be about nontrivial ordering. The plots of current values of α for the dystrophin gene and for a shuffled sequence of the same nucleotide composition are presented in Fig. 4 4 . We stress that trianders show not only quantitative composition, but allow us to find local motives in a more clear way, because different modules for nucleotides lead to less number of superposition and selfintersections. Also trianders more accurately reflect the tendency of the sequence as a whole similarly to DNA walks. Thus triander can be treated as a picture, genome passport or genogram of a given sequence. If we remember that third base in codon has maximal redundancy, then 3rd branch of a triander gets a definite physical sense. Let us assume that the determinative degree is an additive variable (which can be made as first approximation at least (Duplij and Duplij, 2000 2000 )), then 3rd branch can show the current strength of the sequence, that is the bulk ability to determine sense of codon. In this scheme the other two branches can be treated as 3rd branch with shifted ORF (Open Reading Frame). EUCLIDEAN AND MANHATTAN DISTANCES 1 As the measure of the sequence strength we can choose length of the radius-vector from the origin to the current point of the triander, i.e. the Euclidean distance $$D_{\text{E}} (i){\text{ = }}\sqrt {(4n_{\text{C}} (i) - 3n_{\text{G}} (i))^2 + (2n_{\text{T}} (i) - n_{\text{A}} (i))^2 }$$ We can also use the Manhattan distance (also known as rectilinear distance, and it can be treated as the distance that would be travelled to get from one data point to the other if a grid-like path is followed (a car driving in a city laid out in square blocks, like Manhattan)) $$D_{\text{M}} (i){\text{ = }}\left| {4n_{\text{C}} (i) - 3n_{\text{G}} (i)} \right| + \left| {2n_{\text{T}} (i) - n_{\text{A}} (i)} \right|$$ which is the distance between two points measured along axes at right angles (Skiena, 1990 1990 ). In case of symmetric sequence (equal number of all nucleotides) at the step i the Euclidean and Manhattan distances are $$D_{\text{E}} (i) = i/\sqrt 2$$ and D M ( i )= i /2 (which is shown by dashed lines in Fig. 5 5 and Fig. 6 6 . VISUALIZATION OF THE GENETIC CODE TRIPLET NATURE 1 Now we make sure that the triplet character of the genetic code can be seen directly from sequences representation by trianders. As an example we take gene of Homo Sapiens Che-1 mRNA. We consider additionally analogs of trianders with different phases=4,5,7. The result is presented in Fig. 7 7 showing that only the case phase=3 provide nontrivial ordering leading to definite branches, that is we have clear visual presentation of the strong triplet signal. In such a way one could search for higher phase statistical correlations and possible structures, if any, in nucleotide sequences. TRANSFORMATIONS OF TRIANDERS 1 Here we illustrate how the symmetry transformations influence the triander. As an example we take the Homo Sapiens dystrophin in Fig. 3 3 , and the result of various symmetry transformations Eq.( 6 6 )~Eq.( 8 8 ) and reversing the sequence is shown on Fig. 8 8 . The original non-transformed triander is shown in Fig. 3 3 . We observe that the reverse triander is very similar to the original one in Fig. 3 3 . THREE-DIMENSIONAL TRIANDERS 1 The previously constructed two-dimensional trianders have disadvantage in form, because it is not clear, where in the sequence a given point is. To improve this we introduce three-dimensional trianders which are defined by the formula $$\begin{aligned}x_i^{3{\text{D}}} & {\text{ = }}d_{\text{T}} n_{\text{T}} (i) - d_{\text{A}} n_{\text{A}} (i) \\y_i^{3{\text{D}}} & {\text{ = }}d_{\text{C}} n_{\text{C}} (i) - d_{\text{G}} n_{\text{G}} (i) \\z_i^{3{\text{D}}} & {\text{ = }}i \\\end{aligned}$$ which can be treated as mixing of one-dimensional and two-dimensional cases with taking into account the determinative degree. Then, any on the DD space structure can be definitely visually localized using vertical axis. In Fig. 9 9 we show examples of chaotic and ordered trianders. All the graphs start from one point, the origin, and have different length (which can be simply calculated from Eq.( 15 15 )), characterizing them as a whole. In Fig. 10 10 we show the three-dimensional triander of the Homo Sapiens zinc finger and its shuffled version. TOPOLOGICAL CLASSIFICATION OF TRIANDERS 1 Here we propose the topological classification of trianders by their branches belonging to different quadrants on the DD plane and to the number of intersections and return point. This makes possible studying the fine structure of any length sequences with exactly established functions (genes, intergenic space, repeat regions, etc.) and comparing various loci, as well as searching for homological regions, which can allow us to work out mathematically strong genomic signature formalism (Wu, 2002 2002 ; Bergmann et al., 2002 2002 ). We note that there exist many types of trianders. A triander corresponding to a gene we call a genogram, and a triander corresponding to intergenic space we call a gapgram. If some branches intersect each other we call them intersecting trianders, if a branch intersects itself producing knots, we call it knot triander. Branches with (multiple) return points, we call returned trianders. Thus, the determinative degree walk topologizing in our sense means that we identify trianders having definite structure topological features (knot, intersection, return point) and place them into a special class. So we may hope that such topological classification of trianders can actually help in visually solving the inverse problem: for a given sequence to predict its possible function. Let n X ( i ) be cumulative quantity of nucleotide X after i steps, then the DD plane quadrants are defined by Eq.( 9 9 ), and therefore I: 2 n T ( i )− n A ( i ) > 0; 4 n C ( i )−3 n G ( i ) > 0; II: 2 n T ( i )− n A ( i ) < 0; 4 n C ( i )−3 n G ( i ) > 0; III: 2 n T ( i )− n A ( i ) < 0; 4 n C ( i )−3 n G ( i ) < 0; IV: 2 n T ( i )− n A ( i ) > 0; 4 n C ( i )−3 n G ( i ) < 0. After examination of around 2000 eukaryotic and prokaryotic sequences we found that all trianders can be distinguished into several types. The first type is a chaotic triander with no definite branch structure, other types can be called ordered trianders. To work out the general classification of ordered trianders and description of branches we introduce the notion: Type A-B-C F ( x , y ) (E) where A is quadrant where the 1st branch lies, B is quadrant where the 2nd branch lies, C is quadrant where the 3rd branch lies; E is comprised of triander characteristics as a whole; indices F and ( x , y ) describe properties of corresponding separate branches (also for A and B) which will be explained below. In general, there are 4 3 =64 possible ordered triander types classified by quadrants only. We will identify trianders which differ by permutation, because it corresponds to ORF shift, thus decreasing to 24 types. Nevertheless, observation showed that there exist only 7 triander types: I-I-I, I-I-II, I-I-III, I-I-IV, I-II-III, I-II-IV, I-III-IV. For example, the Type I-I-II includes the Types I-II-I and Types II-I-I, if we shift ORF to 1 and 2, but on figures we present only first of them. If e.g. a branch crosses from I quadrant to II quadrant, we denote that by fraction I/II. For instance, the triander of Homo Sapiens dystrophin gene Fig. 3 3 is of Type II-I-I/IV. The additional qualitative features of triander as a whole observed from sequence examination are E=sharp, flat, parallel For branch properties we have x , y denotes axis to which a branch is parallel, F=blurry, loop, smooth, oscillative (horizontal, vertical). Separately we can describe interaction of branches as: 1. Single intersection of A and B is denoted by sign A#B, which gives intersecting triander; 2. Multiple intersection A and B is called braiding and denoted A & B, which gives braiding triander (Fig. 11 11 ). We thoroughly analyzed 150 sequences different by function and evolution level, and for each sequence there were also constructed 100 shuffled sequences having the same nucleotide composition, but not coinciding with the examined one. For every class we show a typical triander of Fig. 12 12 , where the following real sequences are presented: 1. Chaotic triander (Fig. 12a 12a ). Dengue virus type 1 strain FGA/NA d1d, intergenic space: AF226686; 2. Type I-I-I y (Fig. 12b 12b ). Homo Sapiens cytochrome P450, family 2, subfamily F, polypeptide 1 (CYP2F1), mRNA: NM000774; 3. Type II blury -I x -I y (flat) (Fig. 12c 12c ). Homo Sapiens Cbp/p300-interacting transactivator, mRNA: NM006079; 4. Type III oscill -I blury -I oscill (Fig. 12d 12d ). Homo Sapiens collagen, type IX, alpha 2 (COL9A2), mRNA: NM001852; 5. Type IV-I#I (Fig. 12e 12e ). Caenorhabditis elegans immunoglobulin domain-containing protein family member (106 400), mRNA: NM171617; 6. Type III-II-I x (sharp) (Fig. 12f 12f ). Homo Sapiens H1 histone family, member 5 (H1F5), mRNA: NM005322; 7. Type II#I-IV (Fig. 12g 12g ). Homo Sapiens dystrophin (muscular dystrophy, Duchenne and Becker types) (DMD), transcript variant D140ab, mRNA: NM004022; 8. Type III loop -I-IV (Fig. 12h 12h ). Homo sapiens utrophin (homologous to dystrophin) (UTRN), mRNA: NM007124. Further more careful topological classification and analysis of two- and three-dimensional trianders can be made using some of the topological curves methods (Petrovskiy, 1938 1938 ; Rokhlin, 1974 1974 ; Arnold and Oleinik, 1979 1979 ) or the knot theory (Turaev, 1994 1994 ; Kauffman, 1991 1991 ). CONCLUSION 1 We can conclude that the introduced determinative degree DNA walk method confirms the mosaic structure of genome, shows parts with different nucleotide content and strength, and so allows us to find the fine structure of nucleotide sequences. We propose a general method for identification of DNA sequence by triander, which can be treated as a unique genogram, gene passport, etc. The two- and three-dimensional trianders are introduced and their features are studied. The difference of the nucleotide sequences fine structure in genes and the intergenic space is shown. 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In Spectroscopy of Biological Molecules , 1989 . : Bologna Societa Editrice Esculapio , 159 - 172 . . In Spectroscopy of Biological Molecules GM Ziegler . Lectures on Polytopes , 1995 . : Berlin Springer-Verlag , . Lectures on Polytopes We sincerely thank S. Cebrat and M.R. Dudek for their kind hospitality at SmORFland (Inst. Microbiology, Wroclaw) and for sharing their useful DNA work experience with us. Also, we are grateful to A.Y. Berezhnoy, N.A. Chashchin, A.V. Elskaya, A.A. Gusakov, V. Knecht, A.I. Kornelyuk, S.S. Maliuta, G.C. Kourinnoy, P. Mackiewicz, B.V. Novikov, L.A. Livshits, S. Sachse, Y.G. Shckorbatov, O.A. Tretyakov for their very fruitful discussions, and J. Bashford, G. Findley, H. Grubmüller, D.M. Hovorun, P. Jarvis, E. Korotkov, N. Tidjani, D.C. Torney, C. Zhang for kind sending us their interesting papers and useful comments during correspondence. We are thankful to S.M. Donets for language checking and V. Kalashnikov for creation of original special programs for sequence analysis and indispensable assistance in the related GNUPLOT/PERL/C programming. 10.1631/jzus.2005.B0743 2005-04-26 2005-05-30 2005-07-29 2022-01-05 Copyright © 2005, Journal of Zhejiang University Science 2005 Journal of Zhejiang University-SCIENCE B (Biomedicine and Biotechnology) 08 6B