Theory of Adaptive Optimization for Umbrella Sampling

We present a theory of adaptive optimization for umbrella sampling. With the analytical bias force constant obtained from the constrained thermodynamic length along the reaction coordinate, the windows are distributed to optimize the overlap between neighbors. Combining with the replica exchange method, we propose a method of adaptive window exchange umbrella sampling. The efficiency gain in sampling by the present method originates from the optimal window distribution, in which windows are concentrated to the region where the free energy is steep, as well as consequently improved random walk.

a The number following ± sign is the asymptotic standard error. b Although τ is larger, DKL is smaller than that from WEUSMD (see Figure S1). c data from the fit to b exp(−t/τ )

S1.2. The number of y-barrier crossing N cross
Here, we provide the average number of y-barrier crossing N cross for all the tested parameter sets. For each parameter set, the number of y-barrier crossing was calculated from the trajectory of each replica for each set of initial configurations and then averaged, from which N cross and its standard error were estimated. The data are shown in Figure S2 and summarized in Table S2.
The average number of y-barrier crossing N cross . For each N , the left, center, and right bars are the data calculated fromUSMD, WEUSMD, and aWEUSMD, respectively. The green boxes represent the number of y-barrier crossing from y > 0 to y < 0 and the red boxes represent those in the opposite direction.
The error bars are the standard error of N cross . As H y increases, the improvement becomes less obvious.

S1.3. The random walk of replicas in WEUSMD and aWEUSMD
Here, we provide the data representing the quality of random walk of replicas along window space, which is shown by the fraction of replicas that have visited the lowest-index window most recently, f (i), in eq 19 in the main text. For each parameter set, the fraction was calculated for each set of initial configurations and then averaged, from which f (i) and its standard error were estimated (see Figure S3). The improvement in the quality of random walk is more apparent for smaller number of windows. As N increases, the improvement becomes less obvious.

S2. ADDITIONAL DATA FOR THE GPA-TM ASSEMBLY
S2.1. The average RMSD R a (r HH , Ω) for IS1 and IS2 Here, we provide the average RMSD of sampled conformations, R a (r HH , Ω), from WEUSMD and aWEUSMD for IS1 and IS2. The RMSD with respect to the NMR structure was calculated for each conformation at (r HH ,Ω) and then averaged over the simulation time to obtain R a (r HH , Ω).
As shown in Figure S4, aWEUSMD was able to sample wider conformations along Ω, especially, at r HH < 9Å. The sampled region by aWEUSMD for IS1 includes the NMR structure-like conformations, which supports the improved sampling efficiency by aWEUSMD over WEUSMD. R a (r HH , Ω), calculated from the results of (C) WEUSMD and (D) aWEUSMD for the IS2. In RMSD calculations, C α and C β atoms were used and then averaged over the simulation time.

S2.2. Data from WEUSMD and aWEUSMD for IS3
Here, we present the results from WEUSMD and aWEUSMD starting with the initial configurations with the left-handed helix-dimer interfaces (IS3). The PMFs obtained from the results of WEUSMD and aWEUSMD differ within the error bar ( Figure S5A). The average relaxation time τ of window parameters from IS3 aWEUSMD are about 38 ns and 3.8 ns for the bias force constants and window centers, respectively, which are consistent with those obtained from the IS1 and IS2 aWEUSMD. As shown in Figure S6, the sampling power of WEUSMD and aWEUSMD were comparable for IS3, which is different from the results of aWEUSMD for IS1 and IS2. This suggests that WEUSMD starting with the initial configurations with parallel helix-dimer interfaces can be a good choice for the TM assembly of unknown interfaces. aWEUSMD for IS3. The average RMSD of sampled conformations with respect to the NMR structure, R a (r HH , Ω), calculated from the results of (C) WEUSMD and (D) aWEUSMD for IS3.