Free Energy Calculation

The potential of mean force (PMF), also known as the free energy landscape, shows how the free energy in a system changes as a function of some specific reaction coordinates; in this work, we use the TM3–TM6 distance on the cytoplasmic side of the protein and the connector region’s RMSD vs the inactive state as the two reaction coordinates. The free energy landscape is calculated from the simulation trajectories using the following equation: −k_BT ln(P_{i, j}) where k_B is the Boltzmann constant, T is the temperature, and P_{i, j} is the percent of the protein population located at bin i, j for all trajectories; our free energy landscapes were generated using 30 bins in both x and y directions. The free energy landscape includes the entire trajectory data set, with the exception of the first 3000 frames for the agonist-bound simulation and 1230 frames for the antagonist-bound simulation to allow the protein conformations to properly orient themselves away from their initial homology models. These particular ranges of frames were skipped for the following reasons: for the antagonist after frame 1230 (or frame 123 in other plots), the Leu62-Pro517 distance (Fig. 1(b)) and NPxxY RMSDs (Fig. 1(c)) both stabilize at their respective values for the majority of the simulation; for the agonist before frame 3000 (or frame 300 in the other plots), the connector region RMSDs (Figure 3(d)) stabilize, and the Leu62–Pro517 distance (Figure 3(b)) and the seven transmembrane region RMSDs (data not shown) both have low values that are never encountered again during these frames. Note that the above equation uses the total population to calculate P_{i, j}, thus in Fig. 6 there is a slight bias towards the antagonist-bound simulation over the agonist-bound simulation, from which 25670 and 16500 frames were used to generate the graph, respectively. This, however, did not have a significant impact as the free energy landscapes obtained from each of the antagonist- and agonist-bound simulations (see Suppl. Fig. S2) are still clearly present on the combined free energy landscape.

Data from antagonist-bound conformations after 538ns aMD runs. Shown are the TM3–TM6 distance (a), Leu62-Pro517 distance (b), the root mean squared deviation (RMSD) of the NPxxY region (N516-A521) in TM7 from the inactive conformation (c), and the RMSD of the connector region (I108-F476) from the inactive conformation (d).

Data from agonist-bound trajectories after 390ns aMD runs. Shown are the TM3–TM6 distance (a), Leu62-Pro517 distance (b), the root mean squared deviation (RMSD) of the NPxxY region (N516-A521) in TM7 from the inactive conformation (c), and the RMSD of the connector region (I108-F476) from the inactive conformation (d).

Combined free energy landscape of both the agonist and antagonist simulations. Four possible different energy wells are shown: Active, Inactive, and two Intermediate states (Inter 1 and Inter 2). The two parameters used were TM3–TM6 distance, and the connector region RMSD compared to the inactive state.

In aMD simulations, transition barriers between low-energy states are decreased, causing higher populations of the systems at these energetically less energetically-favorable states to be present in these simulations. Theoretically, the free energy landscape of aMD simulations can be reweighted using the Boltzmann factor of the boost potential applied to each trajectory frame: e^ΔV(r)/k_BT. However, high boost potentials needed to be applied to the simulations, likely due to the large size of the system, causing overflow errors to be encountered when calculating the weights. In addition, aMD simulations performed on another GPCR, M2 muscarinic, observed that large energetic noise was encountered when attempting to reweigh, causing large fluctuations in the calculation of the free energy.⁹ In this paper, Miao et al. further demonstrate that the unweighted PMF profiles obtained from aMD simulations match well to PMF profiles obtained from conventional MD simulations. Thus, in this study we present our unweighted free energy landscapes.