## 2 Materials and Methods  
**System Preparation. ** The active state of human \(\beta_{2}\)AR in a complex with a Gs complex and bound to agonist BI-167107 with a PDB id of 3 SN6 [28] was used as an initial state conformation for MD simulations. Prior to the runs, T4 Lysozyme, nanobody, and the agonist were removed, the missing extracellular (_Ala176-His178_) and intracellular (_Phe239-Phe265_) residues were completed via a MODELLER homology modeling tool [29], and the mutations T96M, T98M and E187N, which were used as linkers in crystal structure formation were reverted to their original state via the _mutate_ plugin of the VMD visualization tool [30]. The system was then embedded into a palmitoyloleoyl-phosphatidylcholine (POPC) lipid bilayer using VMD's _membrane_ plugin tool [30], solvated with TIP3P water molecules, and later ionized with 160 Na\({}^{+2}\) and 154 Cl\({}^{-1}\) counter ions for neutralization, which is necessary for the Particle-Mesh Ewald summation method. The system with the dimensions of \(125\times 125\times 165\) A was prepared with a total of 228,299 atoms, of which 54,707 were water molecules. The CHARMM36 forcefield was used to describe the interaction potential of protein and lipids [31]. Periodic boundary conditions were applied in an isothermal, isobaric NPT ensemble with a constant temperature of 310 K and a constant pressure of 1 bar. Temperature and pressure were controlled by the Langevin thermostat and Langevin piston barostat, respectively [32]. The equations of motion were integrated with a 2 fs time step, and the SHAKE algorithm was used to constrain covalent bonds involving non-water hydrogen bonds with a non-bonded cutoff value of 12 A. Two independent 1.5 us long MD runs were performed via the NAMD v2.13 software tool. Each run was initiated with three steps of initial energy minimizations under flexible cell conditions, including (i) the melting of lipid tails when the rest of the atoms were fixed, (ii) minimization and equilibration when protein was constrained but lipid, water, and ion atoms were released, and (iii) minimization and equilibration with the release of all atoms, which was then followed by equilibrium and production runs under constant area according to the membrane proteins' simulation protocol of NAMD [33]. The lipid bilayer in the system was continuously monitored in the minimization and equilibration steps until reaching 63.69 A\({}^{2}\) area per lipid ratio, which was in the range of the experimentally reported value of 64.3 \(\pm\) 1.3 A\({}^{2}\)[34; 35]. **Dynamic Cross-Correlation. ** Correlations between atomic fluctuations from average positions of two residues \(i\) and \(j\) were calculated using the following equation:  
\[C_{i,j}=C\big{(}\Delta R_{i},\Delta R_{j}\big{)}=\frac{\big{<}\Delta R_{i}(t) \cdot\Delta R_{j}(t)\big{>}}{\sqrt{\big{<}(\Delta R_{i})^{2}\big{>}\big{<}( \Delta R_{j})^{2}\big{>}}} \tag{1}\]  
The time average of the dot product of \(\Delta R_{i}(t)\) and \(\Delta R_{j}(t)\) was taken and normalized. \(\Delta R_{i}(t)\) and \(\Delta R_{j}(t)\) represent the atomic fluctuations of \(\alpha\)-Carbons only. If \(C_{ij}=1\), then the fluctuations of atoms \(i\) and \(j\) are perfectly correlated (fluctuates in the same direction), if \(C_{ij}=-1\), then the fluctuations of atoms \(i\) and \(j\) are perfectly anticorrelated (fluctuates in opposite directions), and if \(C_{ij}=0\), then the atoms \(i\) and \(j\) fluctuate independently. Contact map generation. The cutoff distance (\(R_{c}\)) for heavy atoms (C, N, O, S) was taken as 6 A, below which the atoms were considered to be in contact. The incorporation of all heavy atoms provides a more accurate representation of the contact profile than that of \(\alpha\)-Carbons only. The formula used for contact map calculation was defined as:  
\[M_{i,j}=\left\{ \begin{array}{ll}1,&if\ \delta_{i,j}\ \leq\ R_{c}\\ 0,&Otherwise\end{array} \right.