Figure 4. Complex phase convergence problems can arise due to insufficient sampling of protein and water degrees of freedom, e.g., a sidechain rotamer and intra-barstar contact for A42T and an inter-chain contact and neighboring waters for R87A.

(A) Residual complex phase free energy difference (ΔG) time series for AREX simulations of A42T (number of states = 24 and simulation time =10 ns/replica), where the residual ΔG is computed as $\Delta \text{G}\left(t\right)-\Delta \text{G}(t=10\text{ns})$. Blue curve represents the time series for the AREX simulation without restraints and green curve represents the time series for the AREX simulation with heavy atoms restraints (force constant = 50 kcal/moIŲ). Shaded regions represent ± two standard deviations, which were computed by bootstrapping the decorrelated reduced potential matrices 200 times. (B) Time series for $\partial U/\partial \lambda$ (left y-axis, purple) and ${\chi }_1$ angle for residue T42 (right y-axis, gray) for a representative replica (replica 4) of the A42T complex phase AREX simulation (number of states = 24, simulation time = 50 ns/replica). PCC indicates Pearson correlation coefficient and $g$ indicates statistical inefficiency, which is proportional to the correlation time. $g=0.1\text{ns}$ indicates very thorough sampling (because the sampling interval is 0.1 ns) and large values of $g$ indicate poor sampling. (C) Time series for $\partial U/\partial \lambda$ (left y-axis, purple) and T42-E76 distance (right y-axis, gray) for a representative replica (replica 4) of the A42T complex phase AREX simulation (number of states = 24, simulation time = 50 ns/replica). (D) Same as (A), but for R87A instead of A42T (number of states = 36 and simulation time = 10 ns/replica) and using a force constant of 75 kcal/molŲ instead of 50 kcal/molŲ. (E) Time series for $\partial U/\partial \lambda$ (left y-axis, purple) and R87-D39 distance (right y-axis, gray) for a representative replica (replica 25) of the R87A complex phase AREX simulation (number of states = 36, simulation time = 50 ns/replica). (F) Time series for $\partial U/\partial \lambda$ (left y-axis, purple) and number of waters within 5 Å of A87 (right y-axis, gray) for a representative replica (replica 25) of the R87A complex phase AREX simulation (number of states = 36, simulation time =50 ns/replica).