## 4 Conclusions  
Two independent 1.5 \(\upmu\)s long MD simulations were conducted on the apo form of the active state of human \(\beta_{2}\)AR in a complex with a Gs protein. Throughout both trajectories, the active state of the receptor was well preserved with the characteristic tilt in transmembrane helix 6 and ICL3 towards the lipid bilayer to give Gs full access to the binding cavity at the intracellular part. On the extracellular part, since no ligand was attached at the orthosteric site, a minor expansion was observed because of the slightly distancing motion of TM5 from TM3. However, this slight conformational shift at the extracellular part did not cause any allosteric interference in the intracellular region. Distant regions fluctuating in the correspondence are critical as they might point to potential sites along the allosteric pathway. In this study, we attempted to use several metrics for that purpose. First, residue-pair cross-correlations were calculated for \(\alpha\)-Carbon atomic fluctuations from average positions. Distant and correlated regions were mainly observed within the last three transmembrane helices (TM5, TM6, and TM7), including the longest loop region ICL3 and the small extension H8 adjacent to TM7. Moreover, TM6 and TM7 fluctuated in opposite directions with TM1. As cross-correlation ignores the correlated motions in orthogonal directions, the mutual information metric was next used to identify all possible distant sites in correspondence that might be critical for allosteric signaling. First, only \(\alpha\)-Carbon atomic fluctuations were considered. However, not much correspondence was detected in the receptor except between ICL3 and the distant extracellular parts of TM6 and TM7. The next step was to replace \(\alpha\)-Carbon fluctuations with rotameric states of backbone torsion angles \(\phi\) and \(\psi\) in each residue when formulating the mutual correspondence. A considerable change was observed in the profile where ICL3's rotameric states fluctuated in concert and with respect to ICL2. Incorporating side-chain torsion angles further increased the mutual information transferred between ICL3 and ICL2. In addition, ICL3 started to share information with all the other loop regions, including some limited portions of transmembrane helices, TM3, TM6, and TM7. When mutual information was further decomposed based on types of residue pairs, polar ones, especially _Arg_, were identified as the dominating group sharing the highest correspondence with other polar residues. Hydrophobic residues shared the least amount of mutual information, except _Tyr_ and _Phe_, which paired with polar _Arg_. The lowest amount of MI was observed among hydrophobic residues. The dominating feature of polar residues was attributed to their higher abundance in loop regions where the highest mutual information was detected. However, despite its low abundance in loop regions, hydrophobic _Tyr_ with two side-chain torsion angles had average mutual information of 0.11, which was comparable with that of other polar residues with two side-chain torsion angles. Transfer entropy, which is another metric in information theory, relates two states at different times. If the state of residue \(j\) in the future time can be predicted knowing the state of residue \(i\) at the present time, then two sites might communicate with each other as part of an allosteric signaling network. First, transfer entropy was determined for the information about positional fluctuations (C\(\alpha\)). Different profiles were observed in two independent MD runs. In one run, ICL3, ICL2, TM3, and TM4 moderately appeared as entropy donor sites, whereas the intracellular part of TM1 was detected as a dominant acceptor site. In the second run, ICL3, together with the extracellular part of TM7, appeared as two dominant entropy donor sites, whereas no major acceptor site was detected. Clearly, there is no unique communication pathway for backbone C\(\alpha\) displacements.