## 2 Methods
\begin{table}
\begin{tabular}{c c c c} \hline
**AA** & \(\text{P}_{1}^{*}\) & \(\text{P}_{2}^{*}\) & \(\text{P}_{3}^{*}\) \\ \hline A & 0.8056 & 0.3915 & 0.2227 \\ C & 0.7212 & 0.2632 & 1.0000 \\ D & 0.4169 & 0 & 0.3924 \\ E & 0.4578 & 0.0321 & 0.1819 \\ F & 0.9514 & 0.3222 & 0.2370 \\ G & 0.7698 & 0.3851 & 0.3229 \\ H & 0.5448 & 0.5931 & 0.0201 \\ I & 1.0000 & 0.3915 & 0.1569 \\ K & 0.2634 & 0.8691 & 0.1739 \\ L & 0.9182 & 0.3864 & 0.1872 \\ M & 0.8107 & 0.3569 & 0.1166 \\ N & 0.4476 & 0.3145 & 0.2513 \\ P & 0.6777 & 0.4275 & 0.1331 \\ Q & 0.4297 & 0.3440 & 0.0366 \\ R & 0 & 1.0000 & 0.1209 \\ S & 0.6010 & 0.3479 & 0.2815 \\ T & 0.6343 & 0.4570 & 0 \\ V & 0.9233 & 0.3851 & 0.1119 \\ W & 0.8542 & 0.3748 & 0.4496 \\ Y & 0.7136 & 0.3453 & 0.1686 \\ \hline \(\overline{P_{n}}\) & 0.6471 & 0.3994 & 0.2283 \\ \hline \end{tabular}
\end{table} 
Table 2: The normalized values for the properties of the 20 native amino acids. \begin{table}
\begin{tabular}{c c c c c} \hline
**Group** & **Representative** & \(\text{P}_{1}\) & \(\text{P}_{2}\) & \(\text{P}_{3}\) \\ \hline \(\text{G}_{\text{I}}\) & A & 0.8619 & 0.3761 & 0.1607 \\ \(\text{G}_{\text{II}}\) & C & 0.8242 & 0.3363 & 0.5024 \\ \(\text{G}_{\text{III}}\) & D & 0.4885 & 0.2208 & 0.3084 \\ \(\text{G}_{\text{IV}}\) & E & 0.4437 & 0.1881 & 0.1092 \\ \(\text{G}_{\text{V}}\) & H & 0.3606 & 0.7298 & 0.0787 \\ \(\text{G}_{\text{VI}}\) & P & 0.6777 & 0.4275 & 0.1331 \\ \hline \end{tabular}
\end{table} 
Table 3: The values for properties of the six groups. At the same time, a protein primary sequence can be reduced into a six-letter sequence by replacing each element in the protein sequence with its representative letter. Suppose \(S=S_{1}S_{2}\ldots S_{L}\) is a given six-letter sequence, we inspect it by stepping one element at a time. For the step \(k\) (\(k=1,2,\ldots,\ L\)), a 3-D space point \(q_{k}=(x_{k},y_{k},z_{k})\) can be constructed as follows:  
\[(x_{k},y_{k},z_{k})=(x_{k-1},y_{k-1},z_{k-1})+(P_{1}\left(S_{k}\right),P_{2} \left(S_{k}\right),P_{3}\left(S_{k}\right)), \tag{3}\]  
where \((x_{0},y_{0},z_{0})=(0,0,0)\). When \(k\) runs from 1 to \(L\), we get \(L\) points \(q_{1},q_{2},\ldots,\ q_{L}\). Connecting these points one by one sequentially with straight lines, a three-dimensional curve can be drawn. One can further associate the graph with some structural matrices. Here we adopt the \(L\)/\(L\) matrix and denote it by \(M\), whose (\(i\),\(j\))-entry is defined as follows:  
\[m_{ij}= \begin{cases}\frac{d(i,j)}{d(i+1)+d(i+1,j+2)+\ldots+d(j-1,j)}&\text{if $i<j$}\\ 0&\text{if $i=j$}\\ m_{ji}&\text{if $i>j$}\end{cases} , \tag{4}\]  
where \(d(i,j)\) is the Euclidean distance between points \(q_{i}\) and \(q_{j}\). It is not difficult to see that \(\lim_{t\rightarrow+\infty}\)\({}^{t}\)\(M\) is a (0,1) matrix; here \({}^{t}\)\(M\) stands for the product of Hadmannmard multiplication of the matrix \(M\) by itself \(t\)-times. In this paper, we call the limit matrix as a generalized adjacency matrix (GAM) generated by points \(q_{1},q_{2},\ldots,\ q_{L}\), and denote it by \(M_{G}\). Obviously, \(\left[M_{G}\right]_{ij}=1\) if and only if \(q_{i}\) and \(q_{j}\) lie on a straight line in the graph. As mentioned above, once a symmetric matrix is given, one can calculate its ALE-index by the following formula:  
\[\chi=\frac{1}{2}\left(\frac{1}{L}\ ||\cdot||_{m1}+\sqrt{\frac{L-1}{L}}\ ||\cdot||_{F}\right), \tag{5}\]  
where \(L\) is the order of the matrix, \(||\cdot||_{m1}\) and \(||\cdot||_{F}\) are the \(m1\)- and \(F\)-norms of a matrix, respectively. In order to reduce variations caused by comparison of matrices with different sizes, we consider instead of \(\chi\left(M_{G}\right)\) a normalized ALE-index \(\chi^{\prime}\left(M_{G}\right)=\frac{\chi\left(M_{G}\right)}{\sqrt{6L}}\). In addition, following the similar procedures in capturing the sequence-order information of a protein [33; 34], for the six-letter sequence \(S=S_{1}S_{2}\cdots S_{L}\), we extract a set of new order-correlated factors as defined below:  
\[