Supporting Information for
Combine Umbrella Sampling with Integrated Tempering Method for Efficient and
Accurate Calculation of Free Energy Changes of Complex Energy Surface
Mingjun Yang,1 Lijiang Yang,2 Yiqin Gao,2 and Hao Hu1,a)
1
Department of Chemistry, The University of Hong Kong, Pokfulam Road, Hong Kong, China
2
College of Chemistry and Molecular Engineering, Peking University, Beijing, 100871, China
a)
To whom correspondence should be addressed: haohu@hku.hk
1
Table SI. Window locations (ω, in degree) and corresponding force constants (k_f, in
kcal/mol/rad2) in US simulations of Ace-Nme systema
For ω:
ω
0
9
18
27
36
45
54
63
72
k_f
70
70
70
70
70
70
70
70
70
ω
81
90
94
96
99
102
104
106
108
k_f
70
70
70
140
90
70
70
70
160
ω
110
112
117
126
135
144
153
162
171
k_f
70
160
70
70
70
70
70
70
70
ω
180
-9
-18
-27
-36
-45
-54
-63
-72
k_f
70
70
70
70
70
70
70
70
70
ω
-81
-90
-94
-96
-99
-102
-104
-106
-108
k_f
70
120
160
120
160
120
160
120
120
ω
-110
-112
-117
-126
-135
-144
-153
-162
-171
k_f
120
120
120
70
70
70
70
70
70
ω
99
108
117
126
135
144
153
162
171
k_f
90
90
90
90
90
90
90
90
ω
-99
-108
-117
-126
-135
-144
-153
-162
k_f
90
90
90
90
90
90
90
90
ω
180
k_f
90
For η:
-171
a
The same set of window locations and force constants along ω were used in 1-D US, 1-D ITSUS, and 2-D US simulations
2
Figure S1. PMF profile for the rotation of dihedral C1-C2-C3-C4 in butane. The error bars
were computed from eight MD trajectories of 32 ns each. For US and ITS-US, 40 evenly
distributed sampling windows were used along the dihedral angle C1-C2-C3-C4 with a uniform
force constant of 45.0 kcal/mol/rad2 for all windows. For ITS, 60 different temperatures in the
range of 273 ~ 450 K were employed. The same set of optimized nk  values were used in both
ITS and ITS-US production simulations.
3
Figure S2. PMF computed from 150ps 1-D ITS-US simulations with different nk values. (A) 1-D
PMF along ω of Ace-Nme system. Black: reference from 2-D US simulations; red: 1-D ITS-US
with nk values optimized using the original potential; green, 1-D ITS-US with nk values
optimized using the original potential with a harmonic restraint at =0; blue, 1-D ITS-US with
nk values optimized using the original potential with a harmonic restraint at =45; yellow, 1-D
ITS-US with nk values optimized using the original potential with a harmonic restraint at =90;
purple, 1-D ITS-US with nk values optimized for each umbrella window using the biased
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potential. The 2-D PMF maps along ω and  were generated from 1-D ITS-US simulations with
nk values optimized using (B) the original potential; (C) the original potential with a harmonic
restraint at =0; (D) the original potential with a harmonic restraint at =45; (E) the original
potential with a harmonic restraint at =90; (F) corresponding biased potential for each
window. Note for clarity, the Y-axis is defined as  in 2-D PMF maps. =- when >0 and
=+ when <0.
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Figure S3. PMF along ω of Ace-Nme system. The total 1500 ps 1-D ITS-US simulations at each
window were evenly divided into ten 150 ps segments and the PMF was calculated for each
segment.
6
Figure S4. Comparison of sampling efficiency between standalone ITS and ITS-US simulations
of Ace-Nme system. All these simulations were carried out using CHARMM force field. The
simulation length for 2-D US and 1-D ITS-US was 600 ps per window. The simulation length for
standalone ITS was 8 × 96 ns. All simulation parameters were the same to these used in
SCCDFTB simulations. The error bar was computed from eight independent ITS simulations and
the PMF of ITS was computed from the average probability distribution.
7
Figure S5. Comparison of sampling efficiency between standalone ITS and ITS-US simulations
of Ace-Nme system. All these simulations were carried out using CHARMM force field. (a) 2-D
PMF maps from 2-D US simulations. (b) 2-D PMF maps from 8 × 96 ns ITS simulations. (c) 2D PMF maps from 1-D ITS-US simulations. Note for clarity, the Y-axis is defined as  in 2-D
PMF maps. =- when >0 and =+ when <0. The unit for both axes is degree.
8