## 4 Multi-Angle Analysis of RGN Predicted Results
### Multi-Angle Analysis of Tertiary Data
#### 4.2.2 Transformation of Tertiary File into Torsion Angle Data  
In the process of protein structure conformation formation, the torsion angle determines the generation of tertiary structure conformation and ultimately determines the function of the predicted structure. From the perspective of conformational analysis, studying the change of torsion angle is a relatively important content in structural comparison. We propose a method to convert tertiary structure files into torsion angle data.  
Firstly, the protein structure is formed through the extension of the bonds between atoms, the rotation, and the rotation of the torsion angle. The torsion angle is formed by bonds. When a single bond rotates, it will cross with other bonds on adjacent atoms to form a certain angle. When other bonds rotate, they will form an angle with other bonds. Finally, the conformation of the bond formed constitutes the tertiary structure of the protein. The torsion angle is composed of three angles: \(\phi\) angle, \(\psi\) angle, \(\omega\) angle. As shown in Figure 2, which shows the schematic diagram of the protein chain after the RGN predicted structure.  
Secondly, when calculating the torsion angle, we use the method of calculating the dihedral angle. All the atoms of a protein chain are extracted and recorded as [\(P_{1}\), \(P_{2}\),... \(P_{n}\)], \(P_{i}\) is the \(i\) atom in the protein chain, and \(n\) is the number of atoms, where the coordinates of \(P_{i}\) are (\(x_{i}\), \(y_{i}\), \(z_{i}\)). As can be seen from Figure 2, the two peptide faces are composed of 4 atoms so that a torsion angle can be calculated for every four atoms, and the difference between \(\phi\) angle, \(\psi\) angle, \(\omega\) angle is the key to calculate the torsion angle. Therefore, the specific steps are:  
Figure 2: Examples of torsion angles \(\phi\), \(\psi\), \(\omega\) of the protein spatial structure. They represent the rotation angles of N atoms and C\(\alpha\) atoms, C atoms and C\(\alpha\) atoms, and C atoms and N atoms. Since the peptide plane is rigid, the absolute value of the \(\omega\) angle is generally close to 180\({}^{\circ}\) (rarely 10\({}^{\circ}\)) ([PERSON] [33]).  
1. According to the coordinates of the four atoms, the vectors \(v_{1}\), \(v_{2}\), \(v_{3}\) of the three bonds will be calculated.
2. Then, the normal vectors \(n_{1}\) and \(n_{2}\) of the two peptide planes are calculated according to the value of the vector in step (1).
3. The torsion angles of the two peptide planes are calculated according to the normal vectors of the two peptide planes in step (2) and formula: \[\text{angel }=\arccos\left(\frac{<n1,n2>}{|n1||n2|}\right)\] (4) where \(<n1,n2>\) is the inner product of two vectors, \(|n1||n2|\) is the product of two vector modules.  
At this time, the torsion angle of the two peptide planes is calculated. However, in the three-dimensional space, the value range of torsion angle is \([-180^{\circ},180^{\circ}]\), while the value range of torsion angle calculated by the above method is \([0^{\circ},180^{\circ}]\). At this time, it is necessary to use the positive and negative values of \(<v_{1}\times v_{2},v_{3}>\) to determine the direction.