## 2 Methods  
A protein sequence can be viewed as a string of 20 amino acids. Without loss of generality, by the numerical indices 1,2,... 20, we represent the 20 native amino acids according to the alphabetical order of their single-letter codes: A,C,D,E,F,G,H,I,K,L,M,N,P,Q,R,S,T,V,W and Y. Then the frequencies of appearance of the 20 amino acids in a protein sequence are often used to construct a vector  
\[[\text{f}_{1},\text{f}_{2},\ldots,\text{f}_{20}]\]  
This is the conventional amino acid composition. The advantage of such a vector representation is that it is easy in statistical treatment, but it cannot reflect the effect regarding sequence order and property. In what follows, we will take this effect into account through a set of elements in addition to the 20 components. Hydrophobicity, isoelectric point (pI), and relative distance (RD) are three important physicochemical properties of the 20 native amino acids. Here RD can be viewed as an integration of the information on three side chain properties: composition; polarity; and molecular volume--where composition is defined as the atomic weight ratio of hetero (noncarbon) elements in end groups or rings to carbons in the side chain (for details, see [47]). Listed in Table 1 are the original numerical values for hydrophobicity, pI and RD. As can be seen from Table 1, the values of \(P_{1}^{0}\) (Hydrophobicity) is in the range \([-2.53\)-\(1.38]\), and the values of \(P_{2}^{0}\) (isoelectric point) are in the range of 2.97-10.76, while \(P_{3}^{0}\) (relative distance) varies between 1469 and 3355. Therefore, the normalization of these values is needed. Here we normalize them by the formulary below:  
\[p_{n}^{\prime}(AA_{i})=P_{n}^{0}(AA_{i})-\underset{j=1,\ldots,20}{\text{min}} \left\{P_{n}^{0}(AA_{j})\right\},\]  
\[P_{n}^{*}(AA_{i})=\frac{p_{n}^{\prime}(AA_{i})}{\underset{j=1,\ldots,20}{\text{ max}}\left\{p_{n}^{\prime}(AA_{j})\right\}}.\ i=1,2,\ldots,\ 20,\ n=1,2,3. \tag{1}\]  
Clearly, the normalized values for properties of the 20 native amino acids are in the interval [0,1]. The corresponding values are listed in Table 2. The last row in this table gives the average values. \begin{table}
\begin{tabular}{c c c c} \hline
**Amino Acid (AA)** & **Hydrophobicity 1  (\(P_{1}^{0}\))** & **p1  (\(P_{2}^{0}\))** & **RD [\(P_{3}^{0}\))** \\ \hline A & 0.62 & 6.02 & 1889 \\ C & 0.29 & 5.02 & 3355 \\ D & \(-0.90\) & 2.97 & 2209 \\ E & \(-0.74\) & 3.22 & 1812 \\ F & 1.19 & 5.48 & 1916 \\ G & 0.48 & 5.97 & 2078 \\ H & \(-0.40\) & 7.59 & 1507 \\ I & 1.38 & 6.02 & 1765 \\ K & \(-1.50\) & 9.74 & 1797 \\ L & 1.06 & 5.98 & 1822 \\ M & 0.64 & 5.75 & 1689 \\ N & \(-0.78\) & 5.42 & 1943 \\ P & 0.12 & 6.30 & 1720 \\ Q & \(-0.85\) & 5.65 & 1538 \\ R & \(-2.53\) & 10.76 & 1697 \\ S & \(-0.18\) & 5.68 & 2000 \\ T & \(-0.05\) & 6.53 & 1469 \\ V & 1.08 & 5.97 & 1680 \\ W & 0.81 & 5.89 & 2317 \\ Y & 0.26 & 5.66 & 1787 \\ \hline \end{tabular}
\end{table} 
Table 1: The original numerical values for the properties of the 20 native amino acids. For each amino acid (AA), we associate it with a triple (\(t\)(1), \(t\)(2), \(t\)(3)), where  
\[t\left(n\right)= \begin{cases}+1&\text{if }P_{n}^{*}(AA)\geq\overline{P_{n}}\\ -1&\text{otherwise}\end{cases} \left(n=1,2,3\right) \tag{2}\]  
All the amino acids with a same triple form a group. In this way, the 20 native amino acids can be classified into 6 groups:  
\(\text{G}_{\text{I}}=\{\text{A, Y, V, M, L, I}\}\),  
\(\text{G}_{\text{II}}=\{\text{C, W, G, F}\}\),  
\(\text{G}_{\text{III}}=\{\text{D, S, N}\}\),  
\(\text{G}_{\text{IV}}=\{\text{E, Q}\}\),  
\(\text{G}_{\text{V}}=\{\text{H, T, R, K}\}\),  
\(\text{G}_{\text{VI}}=\{\text{P}\}\). For each group, the first amino acid is selected to be the representative. That is, A, C, D, E, H and P are used to stand for the six groups, respectively. The value of the property of a group is defined as the average value of the property of amino acids belonging to the group. Listed in Table 3 are the corresponding values of the six groups.