# Novel Graphical Representation and Numerical Characterization of DNA Sequences
## 2 Methods
### Numerical Characterization of DNA Sequences  
The graphical representations not only offer the visual inspection of data, helping in recognizing major differences among DNA sequences, but also provide with the numerical characterization that facilitates quantitative comparisons of DNA sequences. One way to arrive at the numerical characterization of a DNA sequence is to convert its graphical representation into some structural matrices, and use matrix invariants, e.g., the leading eigenvalues, as descriptors of the DNA sequence [8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 31; 32]. It is expected that effective invariants will emerge and enable to uniquely characterize the sequences considered. However, the difficulties associated with computing various parameters for very large matrices that are natural for long sequences have restricted the numerical characterizations, for instance, leading eigenvalues and the like [17; 24]. The search for novel descriptors may be an endless project. The art is in finding useful descriptors, and those that have plausible structural interpretation, at least within the model considered [8]. In this section, we bypass the difficulty of calculating the invariants like the leading eigenvalue and propose a novel descriptor to numerically characterize a DNA sequence. As described above, the pattern, including shape and trend, of curves for the 18 DNA sequences provides useful information in an efficient way. This inspires us to numerically characterize a DNA sequence with an idea of \"piecewise function\" as below. For a given 3-D graphical representation with \(n\) vertices, by the order in which these vertices appear in the curve, we partition it into \(K\) parts, each of which is called a cell. All the cells contain \(m=\left\lfloor\frac{n}{K}\right\rfloor\) vertices except the last one. For the \(i\)-th cell, \(i\) = 1,2,...,\(K\), the geometric center \(U_{i}=(x_{i},y_{i},z_{i})\) is viewed as its respective. Then we have  
\[U_{i-1}^{\rightarrow}U_{i}=(x_{i}-x_{i-1},y_{i}-y_{i-1},z_{i}-z_{i-1}) \tag{3}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Species** & **AC (GenBank)** & **Location** \\ \hline
1 & _Human_ & U01317 & join(62187\_62278,62409\_62631,63482\_63610) \\
2 & _Homo_ & APO07546 & join(180\_271,402\_624,1475\_1603) \\
3 & _Gorilla_ & X61109 & join(4538\_4630,4761\_4982,5833\_\(\sim\)5881) \\
4 & _Chimpantzee_ & X02345 & join(4189\_4293,4412\_4633,5484\_\(\sim\)5532) \\
5 & _Lemur_ & M15734 & join(154\_245,376\_598,1467\_1595) \\
6 & _CebusaPella_ & AY279115 & join(946\_1037,1168\_1390,2218\_2346) \\
7 & _LagethrixLogotricha_ & AY279114 & join(952\_1043,1174\_1396,2227\_3255) \\
8 & _Bovine_ & X00376 & join(278\_363,492\_714,1613\_1741) \\
9 & _Gaat_ & M15387 & join(279\_364,493\_715,1621\_1749) \\
10 & _Sheep_ & DQ352470 & join(238\_323,452\_674,1580\_1708) \\
11 & _Mouflon_ & DQ352468 & join(238\_323,452\_674,1578\_1706) \\
12 & _European hare_ & Y00347 & join(1485\_1576,1703\_1925,2492\_2620) \\
13 & _Rabbit_ & V00882 & join(277\_368,495\_717,1291\_1419) \\
14 & _Mouse_ & V00722 & join(275\_367,484\_705,1334\_1462) \\
15 & _Rat_ & X06701 & join(310\_401,517\_739,1377\_\(\sim\)1505) \\
16 & _Opossum_ & J03643 & join(467\_558,672\_894,2360\_2488) \\
17 & _Gallus_ & V00409 & join(465\_5556,649\_871,1682\_1810) \\
18 & _Muscopy duck_ & X15739 & join(291\_382,495\_717,1742\_1870) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The CDS (Coding DNA Sequence) of \(\beta\)-globn gene of 18 species. where \(U_{0}=(0,0,0)\). It is not difficult to find that \(U_{i-1}^{\rightarrow}U_{i}\) reflects a certain \"growing trend\" of these cells. For convenience, we call \(U_{i-1}^{\rightarrow}U_{i}\) the trend-point. On the basis of the \(K\) trend-points, a DNA sequence can be characterized by a \(3K\)-dimensional vector \(V_{tp}\):  
\[ \begin{array}{ll}V_{tp}=&(x_{1}-x_{0},x_{2}-x_{1},\cdots,x_{k}-x_{k-1},\\ &y_{1}-y_{0},y_{2}-y_{1},\cdots,y_{k}-y_{k-1},\\ &z_{1}-z_{0},z_{2}-z_{1},\cdots,z_{k}-z_{k-1})\end{array}