## 3 Results and Discussion
As depicted in Figure 3a, mutual information was plotted together with a contact map to unravel the long-distance coupled motions of the allosteric network (see magenta dots). However, the only significant correspondence in distant C\(\alpha\) fluctuations was detected between a few residues located in ICL3's midpoint and the distant extracellular parts of TM6, extracellular loop 3 (ECL3), and TM7, which incorporate critical orthosteric binding site residues such as _Phe289_, _Asn293_, and _Asn312_. It is obvious that mutual information between backbone atomic fluctuation was mostly shared by neighboring residues either close in sequence or space. For all MI maps in Figure 3, red dots represent MI values greater than 0.5, and green dots represent MI between 0.25 and 0.5. Any MI less than 0.25 was not displayed. For clarity, the contact map was illustrated in Figure 3a only. Figure 2: Residue-pair cross-correlation map for 3 \(\upmu\)s MD trajectory. Only \(C_{ij}>+0.4\) and \(C_{ij}<-0.4\) are represented. Magenta dots represent the contact map (threshold distance of 6 Å). Figure 3: _Cont._  
The residue-pair MI was next determined for backbone torsion angles \(\phi\) and \(\psi\) in each residue. Despite low MI values observed almost everywhere in the receptor (highest MI = 2.59), the relatively high correspondence between ICL3 and ICL2 regions is noticeable (See Figure 3b). Conformational degrees of freedom were mostly dominated by torsion angles in loop regions; thus, the correspondence of loop regions was anticipated. However, the amount of such correspondence appeared to be limited to a few loop regions when only backbone torsion angles were incorporated. Hence, the next attempt was to combine the information of backbone torsion angles with that of the first side-chain torsion angle, \(\chi_{1}\). Maximum MI was slightly increased to 3.72 from 2.59, and the highest MI values were still observed between ICL3 and ICL2 with increased intensity, as depicted in Figure 3c. Additionally, both ICL3 and ICL2 started to share information with the intracellular part of TM7, and moreover, two distant extracellular loop regions, ECL2 and ECL3, displayed some noticeable correspondence with each other. The total effect of rotational degrees of freedom on MI values can only be disclosed when all possible side-chain torsion angles \((\chi_{i},\ i=1,2,3,4)\) were considered together with backbone torsion angles. As illustrated in Figure 3d,e, the increasing trend in MI values between ICL3 and ICL2 was noticeable. Maximum MI reached a value of 6.55. In addition, ICL3 started to share information with the majority of the receptor, including mainly loop regions such as ICL1, ICL2 at the intracellular part, two ends of the extracellular loop ECL2, the entire ECL3, and also the intracellular part of TM7 with its adjacent tail H8. Overall, it is clear that mutual correspondence driven by torsional degrees of freedom mainly existed between loop regions. It is important to identify residue types most often involved in sharing mutual information, especially among distal ones, as they might point to potential allosteric hub regions. Residue pairs were categorized based on the degree of separation of two residues in the primary sequence, as proximal if 1-4 positions apart and otherwise distal. As illustrated in Figure 4, the highest amount of MI was shared among polar residues, which incorporated _Arg_ predominantly. Moreover, two bulky residues _Phe_ and _Tyr_, with the highest MI values among hydrophobic residues, also paired with polar _Arg_ to a large extent. On the other hand, no significant correspondence was observed among hydrophobic residues. The dom  
Figure 3: Residue-pair mutual information (MI) plotted with contact map (threshold: 6.0 Å) for (**a**) backbone \(C_{a}\), (**b**) backbone torsion angles (\(\phi\), \(\psi\)) (**c**) backbone torsion and first side-chain torsion angles (\(\phi\), \(\psi\), \(\chi_{1}\)), (**d**) backbone torsion and all side-chain torsion angles (\(\phi\), \(\psi\), \(\chi_{i}\), \(i=1,2,3,4\)).