# Novel Graphical Representation and Numerical Characterization of DNA Sequences
## 2 Methods  
### The 3-D Graphical Representation  
As we know, the four nucleic acid bases A, G, C, and T can be classified into three categories:  
\[\text{R}\,=\,\{\text{A},\text{G}\}/\text{Y}\,=\,\{\text{C},\text{T}\};\,\text{ M}\,=\,\{\text{A},\text{C}\}/\text{K}\,=\,\{\text{G},\text{T}\};\,\text{W}\,=\,\{ \text{A},\text{T}\}/\text{S}\,=\,\{\text{G},\text{C}\}.\]  
In fact, these groups are just all of the non-repetition 2-combinations of set {A,G,C,T}. If repetition is allowed, in other words, if we consider multiset \(\{\infty\cdot\text{A},\infty\cdot\text{G},\infty\cdot\text{C},\infty\cdot \text{T}\}\) instead of the set {A,G,C,T}, then the number of 2-combinations equals 10 (see Table 1). Let \(V\) be a regular tetrahedron whose center is at the origin \(O=(0,0,0)\). \(V_{1}\) = (+1,+1,+1), \(V_{2}\) = (\(-1,-1,+1\)), \(V_{3}\) = (+1,\(-1,-1\)), and \(V_{4}\) = (\(-1,+1,-1\)) are its four vertices. To each of the vertices we assign one of the four nucleic acid bases A, C, G and T. Moreover, to the midpoint of the line segment AC we assign M, and K to the midpoint of the line segment GT, R to that of the line segment AG, Y to that of the line segment CT, W to that of the line segment AT, and S to that of the line segment CG. We thus obtain ten fixed directions: \(\overrightarrow{OA},\overrightarrow{OC},\overrightarrow{OG}, \overrightarrow{OT},\overrightarrow{OM},\overrightarrow{OK}, \overrightarrow{OY},\overrightarrow{OS}\), based on which we can derive ten unit vectors:  
\[r_{A}=\frac{1}{||\overrightarrow{OA}||}\cdot\overrightarrow{OA},\,r_{C}= \frac{1}{||\overrightarrow{OC}||}\cdot\overrightarrow{OC},\ldots,\,r_{S}= \frac{1}{||\overrightarrow{OS}||}\cdot\overrightarrow{OS} \tag{1}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Base** & **A** & **G** & **C** & **T** \\ \hline A & {A,A} & {A,G} & {A,C} & {A,T} \\ G & - & {G,G} & {G,C} & {G,T} \\ C & - & - & {C,C} & {C,T} \\ T & - & - & - & {T,T} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The 2-combinations of multiset \(\{\infty\cdot\text{A},\infty\cdot\text{G},\infty\cdot\text{C},\infty\cdot \text{T}\}\). Obviously, the ten unit vectors are ten points on a unit sphere. An idea arises naturally: each of the ten 2-combinations can be associated with one of the ten unit vectors. In detail, we have  
\[ \begin{array}{c}\{A,A\}\gets r_{A},\ \{A,G\}\gets r_{R},\ \{A,C\}\gets r_{M},\{A,T\}\gets r_{W},\\ \{G,G\}\gets r_{G},\{G,C\}\gets r_{S},\{G,T\}\gets r_{K},\\ \{C,C\}\gets r_{C},\ \{C,T\}\gets r_{Y},\ \{T,T\}\gets r_{T}. \end{array}  \tag{2}\]  
To obtain the spatial curve of a DNA sequence, we move a unit length in the direction that the above assignment dictates. Taking sequence segment ATGGTGCACCTGACTCCTGATCTGTA as an example, we inspect it by stepping two nucleotides at a time. Starting from the origin \(O=(0,0,0)\), we move in the direction dictated by the first dinucleotide AT, \(r_{W}\), and arrive at \(P_{1}\), the first point of the 3-D curve. From this point, we move in the direction dictated by the second dinucleotide TG, \(r_{K}\), and arrive at the second point \(P_{2}\). From here we move in the direction dictated by the third dinucleotide GG, \(r_{G}\), and come to the third point \(P_{3}\). Continuation of this process is illustrated in Table 2, and the corresponding 3-D graphical representation is shown in Figure 1.  
As the characterization of a research object, a good visualization representation should allow us to see a pattern that may be difficult or impossible to see when the same data is presented in its original form. In order to provide a direct insight into the local and global characteristics of a DNA sequence, the proposed 3-D curve can be projected on planes (\(x\),\(y\)), (\(y\),\(z\)) or (\(x\),\(z\)), and thus three different 2-D graphical representations will be yielded. Figure 2 shows the projections of 3-D curves of 18 different DNA sequences listed in Table 3.