## 2 Materials and Methods
### Methods
#### 2.2.2 PSSM Process  
The PSSM is a feature that incorporates evolutionary information from protein sequences [28]. This feature has been widely used in feature extraction steps for various protein-related experiments. The PSSM calculated by Blast is usually a matrix of size L * 20 [21], where L represents the length of the protein sequence. However, commonly used machine learning classifiers cannot handle such size-variable data very well. To better extract evolutionary information from protein sequences and adapt to machine learning models, many researchers have proposed PSSM-based features and corresponding calculation methods [29]. In this paper, we used three PSSM-based features to fully obtain the evolutionary information of protein sequences and improve the performance of the classifier.  
DPC-PSSM: Dipeptide composition (DPC) is a method that can calculate the probability of consecutive amino acids in a protein sequence [30]. Unlike some other feature extraction methods, DPC not only extracts amino acid composition information but also contains a part of amino acid sequence information. The evolutionary information alone cannot fully characterize the entire protein. So we try to add the DPC information into the PSSM and generate an 800-dimensional (20 \(\times\) 20) feature [31; 32]. The features can be formulated as:  
\[\text{DPC}-\text{PSSM}=(\text{D}_{1,1},\dots,\text{D}_{1,20},\text{D}_{2,1}, \dots,\text{D}_{2,20},\dots,\text{D}_{20,1},\dots,\text{D}_{20,20})^{\text{T}} \tag{1}\]  
where  
\[\text{D}_{i,j}=\frac{1}{\text{L}-1}\sum_{\text{k}=1}^{\text{L}-1}\text{D}_{ \text{k},i}\times\text{D}_{\text{k}+\text{i},j}(1\leq\text{i},\text{j}\leq 20) \tag{2}\]  
KSB-PSSM: KSB-PSSM provides good feedback on the relationship between disjoint amino acids in protein sequences [33]. It is an extension of DPC-PSSM. In this method, the key parameter k indicates the amino acid distance at the time of calculation. In this paper, KSB-PSSM is calculated by setting k = 3. In this case, the transfer probability between amino acids split by two amino acids is calculated, and a 400-dimensional feature is generated. The calculation is as follows:  
\[\text{KSB}-\text{PSSM}=(\text{K}_{1,1}(\text{k}),\dots,\text{K}_{1,20}(\text{ k}),\text{K}_{2,1}(\text{k}),\dots,\text{K}_{2,20}(\text{k}),\dots,\text{K}_{20,1}( \text{k}),\dots,\text{K}_{20,20}(\text{k}))^{\text{T}} \tag{3}\]and  
\[\mathrm{K_{ij}(k)}=\sum_{t=1}^{L-k}\mathrm{K_{t,i}}\times\mathrm{K_{t+k,j}}(1\leq i,j\leq 20) \tag{4}\]  
S-FPSSM: The S-FPSSM is a 400-dimensional feature derived from the FPSSM by row transformation [34]. In addition, the FPSSM is a matrix filtering the negative values in the original PSSM. The S-FPSSM is calculated as follows:  
\[\mathrm{S_{j}^{(i)}}=\sum_{k=1}^{L}\mathrm{fp_{k,j}}\times\delta_{k,i} \tag{5}\]  
which  
\[\begin{cases}\delta_{k,i}=1\text{ if }\mathrm{r_{k}}=\mathrm{a_{i}}\\ \delta_{k,i}=0\text{ otherwise}\end{cases}(1\leq i,j\leq 20) \tag{6}\]  
In the above equation, \(\mathrm{fp_{k,j}}\) denotes the element in the kth row and jth column of the FPSSM, \(\mathrm{r_{k}}\) denotes the kth amino acid in the original protein sequence, and \(\mathrm{a_{i}}\) denotes the amino acid ranked at position i in the PSSM (20 positions in total).