AAC is the frequency of amino acid residues in a protein sequence [50, 51, 52, 53, 54]. The frequencies $f\left(x\right)$of 20 residues can be calculated by

(1)$$f\left(x\right)=\frac{N\left(x\right)}{N}x\in \{\text{𝐴𝐶𝐷𝐸𝐹𝐺𝐻𝐼𝐾𝐿𝑀𝑁𝑃𝑄𝑅𝑆𝑇𝑉𝑊𝑌}\}$$

where $N\left(x\right)$ is the x-th residue in a protein sequence with N residues.

CKSAAP describes the occurrence of amino acid pairs disengaged by any K amino acid (K = 0, 1, 2, 3, 4, 5). It [50] is demarcated as k-spaced residual pairs $Q_{xy}$ which is formulated as

(2)$${\mathrm{Q}}_{\mathrm{xy}}=\frac{{\mathrm{N}}_{\mathrm{xy}}}{\mathrm{N}-\mathrm{k}}\mathit{\hspace{1em}}(\mathrm{k}=0,1,2,3,4,5\text{and}\mathrm{xy}=\text{type of}\mathrm{AA})$$

where $N_{xy}$ is the number of residue pairs and ‘k’ denotes the number of residues. In this work, for saving calculation time, the value of ‘k’ was set to 3 and the dimension of the features is 1600.

It contains the occurrence frequency of amino acids and the correlation of physiochemical properties between two amino acid residues [55]. It comprises of $A{c}_i$ and $A{c}_{\partial i}$ which can be formulated as

(3)${\mathrm{Ac}}_{\mathrm{i}}={\displaystyle \frac{{\mathrm{N}}_{\mathrm{i}}}{1+\omega \times {\sum }_{\mathrm{i}=1}^{20}{\theta }_{\mathrm{i}}}}$ $(\text{here}{\theta }_{\mathrm{i}}={\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{N}-\mathrm{d}}{\left({\mathrm{Q}}_{\mathrm{i}}-{\mathrm{Q}}_{\mathrm{i}+\mathrm{d}}\right)}^2}{{\mathrm{N}}_{\mathrm{Q}}}},(\mathrm{i}=1,2,3\mathrm{\dots },20))$

(4)$$A{c}_{\partial i}=\frac{\omega \times {\theta }_i}{1+\omega \times {\sum }_{i=1}^{20}{\theta }_i},(\text{here},\omega =0.05)$$

where $N_Q$ is properties number and $N_i$ is the i-th frequency of amino acid. $Q_i$ is the i-th physicochemical property value and ${\theta }_i$ is order factor of protein sequence.

CTD describes the composition, transition and distribution of AAC in a protein sequence [56]. Amino acids are separated into three different classes on the basis of their physiochemical properties [18, 57, 58]. It is calculated as follows:

(5)$$C_a=\frac{N_a}{N}\mathit{\hspace{1em}}(a=1,2,3\mathrm{\dots })$$

(6)$$T_b=\frac{N_{b,c}+N_{c,b}}{N-1}\mathit{\hspace{1em}}(b=1,2,3\mathrm{\dots },c\ne b)$$

(7)$${\mathrm{D}}_{\mathrm{b},\mathrm{z}}=\frac{{\mathrm{N}}_{\mathrm{b},\mathrm{z}}}{\mathrm{N}}\mathit{\hspace{1em}}(\mathrm{b}=1,2\mathrm{\dots },\text{and}\mathrm{z}=1,0.15\mathrm{N}\mathrm{\dots },\mathrm{N})$$

where $N_a$ is number of classes, $N_{b,c}$ is the contiguous number of classes b and c and $N_{b,z}$ is the amino acids number which is in the z-th of b-th class.

It is a type of association descriptor and has an extreme resemblance with M-descriptor [59]. The mathematical manifestation is shown as $Q(m)$:

(8)$$\mathrm{Q}(\mathrm{m})=\frac{\mathrm{N}-1}{2\times (\mathrm{N}-\mathrm{m})}\times \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{N}-\mathrm{m}}{\left({\mathrm{P}}_{\mathrm{i}}-{\mathrm{P}}_{\mathrm{i}+\mathrm{m}}\right)}^2}{{\sum }_{\mathrm{i}=1}^{\mathrm{N}}{\left({\mathrm{P}}_{\mathrm{i}}-\mathrm{m}\right)}^2}\mathit{\hspace{1em}}(\mathrm{m}=1,2,\mathrm{\dots },20)$$

where $P_i$ is i-th amino acids property value in the amino-acid index.

It is a kind of autocorrelation function [22] and has a resemblance with Moran-descriptor. The mathematical manifestation is shown as

(9)$$\mathrm{Q}(\mathrm{m})=\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{N}-\mathrm{m}}\left({\mathrm{P}}_{\mathrm{i}}\times {\mathrm{P}}_{\mathrm{i}+\mathrm{m}}\right)}{\mathrm{N}-\mathrm{m}}\mathit{\hspace{1em}}(\mathrm{m}=1,2,3,\mathrm{\dots },20)$$

where $P_i$ is i-th amino acids property value in the amino-acid index.