CGTM workflow

Ran 15-μs simulations of inactive and partially active A_2AR.

Voronoi tessellation to determine first-shell lipids, second-shell lipids, etc.

For each configuration, determine the number of each type of lipid in the first shell. For each configuration, project the composition onto discretized states in 5-mol% increments of each lipid type, totaling 100%.

Seeded additional 50-ns simulations in a range of initial states selected to achieve more complete coverage of the state space.

Voronoi tessellation to determine shells of lipids in 50-ns simulations.

Binned sampled configurations (from 50-ns simulations) into states.

Populated CGTM using results from step 6 above (Eqs. 1 and 2 below).

Solved matrix equation to obtain equilibrium probability distribution (Eq. 3 below).

Tested for convergence: if the number of trajectories or length of trajectories is increased, does π^Eq change?

Note that the outline above reflects the historical development of the method. Long simulation trajectories were already available, but the analysis showed that they were not well converged. Additional simulations were therefore run (step 4) to improve convergence. The CGTM method anticipates that no such long simulations are available, in which case one would complete steps 4–9 with initial states selected from an informed ansatz, followed by convergence checks and then running additional simulations to improve until convergence is judged satisfactory.

Two 15-μs trajectories, one of the inactive and one of the partially active receptor, in step 1 of CGTM workflow were run previously (34). The lipid mixture (0.55:0.15:0.30 DPPC:DOPC:Chol) used is in the liquid-liquid coexistence region of the phase diagram.

Lipids were assigned to a solvation shell around A_2AR by a 2D Voronoi tessellation in step 2 of CGTM workflow. The Voronoi tessellation followed the method described in Beaven et al. (37), in which the α-carbon atoms of the receptor represent its structure and lipid locations are defined by the center of mass of each lipid projected onto the z = 0 plane (the membrane plane). Positions of lipids and the receptor were taken from coordinate files of the trajectories, sampled every 240 ps. The area around each lipid was divided by tessellation borders, which represent spatial boundaries. A lipid was considered first shell if its tessellation area shared a border with the receptor; second-shell lipids shared a border with first-shell lipids, and so on. Lipids in the fifth shell were considered “bulk” because this shell was farthest from the receptor and also contained ratios of lipid types nearest to the whole system averages (within 1%). Because site-specific binding was not considered, both leaflets were averaged together. Fig. 1 shows an example of Voronoi boundaries for a single leaflet in a single frame.

Example of Voronoi areas in a single leaflet. Voronoi cells for receptor (gray), DOPC (blue), DPPC (red), and cholesterol (yellow) are shown. Agonist-bound receptor is depicted. To see this figure in color, go online.

Tessellation data were then used to bin lipids into solvation shells in step 3 of CGTM workflow. The lipid composition of the first shell was characterized by the percent of each lipid type rounded to the nearest 5% and constrained to total 100%. For example, a first-shell solvation state could be DPPC/DOPC/cholesterol in a ratio of 0.35/0.45/0.20. Because this binning does not consider the lipids’ spatial distributions, the resulting states are coarse grained.

After binning, 16 states selected to span configuration space were selected for each receptor configuration (inactive and partially active) in step 4 of CGTM workflow. Short 50-ns simulations initialized in these states were used to populate a transition matrix. MD simulation details for these short simulations are described below under MD simulations. Initial configurations were taken from the 15-μs simulations for undersampled states. The more populated states were adequately represented in the original 15-μs simulations; therefore, nonoverlapping segments of the original trajectories were extracted. Thus, of the trajectories used to populate the CGTM, half were new. The half that were extracted from the 15-μs comprise roughly one-sixth of the original 15-μs trajectory.

To compute the equilibrium distribution of states (π^Eq), a CGTM was populated from the 50-ns trajectories in step 7 of CGTM workflow (after Voronoi tessellation and binning into solvation states of the 50-ns trajectories in steps 5 and 6). The procedure is identical to constructing a Markov state model but with coarse graining resulting from the lack of spatial resolution; hence, the name CGTM modeling. Assuming the system can occupy a state s_i existing in s = [s₁, s₂, …, s_N], discrete sampling of the system results in a probability distribution π = [π₁, π₂, …, π_N] in which π_i is the sampled probability of the system occupying s_i. Transition probabilities between states are computed from c_ij of transition events between s_i and s_j:

A CGTM T containing these probabilities describes the evolution of state populations π_i over time:

When there is no net population flow among states, the system is in a steady state (in this case, equilibrium). π^Eq can be computed by solving the matrix equation (step 8 of CGTM workflow):

This is guaranteed to be true if detailed balance is satisfied:

Detailed balance is therefore a sufficient condition for a system to exhibit equilibrium. Detailed balance implies “coarse balance”—balanced flux between every pair of arbitrarily defined set of microstates (38). Although coarse graining configuration space results in loss of the Markov property, the coarse balance property implies the reliability of equilibrium properties estimated using trajectories that sample the equilibrium probability distribution. Further verification of the computed π^Eq is therefore provided by checking the system for detailed balance and showing the robustness of π^Eq as a function of lagtime (time between discrete sampling events).