Supporting Information for Kirkwood-Buff Integrals for Aqueous Urea Solutions Based upon the Quantum Chemical Electrostatic Potential and Interaction Energies Shuntaro Chiba, 1* Tadaomi Furuta, 2 and Seishi Shimizu 3 1 Education Academy of Computational Life Sciences, Tokyo Institute of Technology, J3-141 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8501 Japan 2 School of Life Science and Technology, Tokyo Institute of Technology, B-62 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8501, Japan. 3 York Structural Biology Laboratory, Department of Chemistry, University of York, Heslington, York YO10 5DD, United Kingdom Contents S1. Experimental thermodynamic quantities and derivatives of molar fraction activity... S2 S2. Parameters for the urea molecule... S4 S3. Radial distribution functions and Kirkwood-Buff Integrals... S7 S4. Thermodynamic quantities calculated with AMOEBA without polarizable effect... S27 References... S28 S1
S1. Experimental thermodynamic quantities and derivatives of molar fraction activity It is necessary to calculate derivatives of the molar fraction activity to derive KBIs. We assumed that G m E could be expressed using the Redlich-Kister polynomial 1 as Chitra et al. 2 did to derive KBIs of urea. We expressed G m E up to the 2 nd order Redlich-Kister polynomial as a function of urea mole fraction, x i : G m E = x i (1 x i )(A + B(1 2x i ) + C(1 2x i ) 2 ), (S1) where A, B, and C are fitting parameters. Eq. (7) and the condition, ln f c (0) = 0, yields the following relationship: ln f c = 1 k B T [ 2(A + 3B + 5C)x c + (A + 9B + 29C)x c 2 4(B + 8C)x c 3 + 12Cx c 4 ]. (S2) By fitting Eq. S2 to experimental values 3 using NonlinearModelFit function in Mathematica version 10.0, 4 we obtained 16.91 kj mol -1, -11.64 kj mol -1, and 4.50 kj mol -1 for A, B, and C with enough small p-values of 10-10 order. The adjusted R 2 of the experimental and fitted ln f c was 0.999987. We also tried the 3 rd and 4 th order Redlich-Kister polynomials for the fitting. Although the correlation between experimental and fitted ln f c improved, p-values of fitted parameters deteriorated to 10-3 and 10-1 order, respectively, indicating overfitting. Hence, we confirmed the 2 nd order form was appropriate. The molar fraction activity thus obtained and other experimental quantities are given in Table S1. S2
Table S1. Experimental thermodynamic quantities and corresponding KB integrals. c a d a mol L -1 g ml -1 V m,u a cm 3 mol -1 V m,w a cm 3 mol -1 κ T a GPa -1 f uu a Fitted a G m E b kj mol -1 G uu b cm 3 mol -1 G uw b cm 3 mol -1 G ww b cm 3 mol -1 N uu b N uw b N wu b N ww b 0.00 0.9970 44.22 18.07 0.446 0.000-16.99-0.94 0.51 1.0051 44.36 18.07 0.436-0.031 0.091-8.83-44.07-16.36 0.00-2.40-0.02-0.89 1.01 1.0130 44.49 18.07 0.427-0.056 0.181-13.63-44.83-15.71-0.01-2.36-0.04-0.83 2.06 1.0293 44.74 18.06 0.406-0.096 0.378-23.77-45.76-14.14-0.05-2.28-0.10-0.70 3.15 1.0460 44.97 18.05 0.386-0.119 0.591-33.51-45.74-12.37-0.10-2.19-0.14-0.59 6.20 1.0910 45.47 17.99 0.334-0.122 1.253-50.94-42.18-7.39-0.33-1.66-0.27-0.29 7.83 1.1143 45.65 17.96 0.312-0.132 1.656-52.91-40.44-4.33-0.41-1.46-0.31-0.16 a) The original Experimental values were obtained from the literature: mole concentration, c; density, d; 5 partial molar volume, V m,i ; 6 isothermal compressibility, κ T. 7 Data for each value were fitted to a polynomial as a function of concentration. f uu was obtained by fitting experimental activity coefficients to a model 3 (see section 2.1 KBIs and thermodynamic quantities). b) Based on experimental values, the mole excess Gibbs energy, G E m, and KB integrals, G ij were calculated. The excess coordination numbers, N ij were calculated via N ij = ρ j G ij. S3
S2. Parameters for the urea molecule The following parameters were used to calculate the Kirkwood-Buff integrals and related thermodynamic quantities. The vdw parameters were not optimized in the following parameters. The units for energy and length are given in kcal mol -1 and angstrom, respectively. Note that the unit for out-of-plane bend was set to 0.02191418. This made urea structure almost plane. Atom name: Type Class Symbol Description Atomic Mass Valence 401 401 C nonplanarurea 6 12.011 3 404 404 O nonplanarurea 8 15.999 1 402 402 N nonplanarurea 7 14.007 3 403 403 H nonplanarurea 1 1.008 1 Bond Stretching Parameters: Atom Classes K(S) Length 401 402 482.000 1.3927 401 404 601.800 1.2146 402 403 542.000 1.0092 Angle Bending Parameters: Atom Classes K(B) Angle 402 401 402 60.000 112.564 402 401 404 60.000 123.718 401 402 403 35.000 111.723 403 402 403 22.300 113.274 Stretch-Bend Parameters: Atom Classes K(SB)-1 K(SB)-2 402 401 402 18.700 18.700 401 402 403 4.300 7.200 Out-of-Plane Bend Parameters: Atom Classes K(OPB) 401 402 0 0 12.949 402 401 0 0 107.910 403 402 0 0 5.755 404 401 0 0 46.833 S4
Torsional Parameters: Atom Classes 1-Fold 2-Fold 3-Fold 4-Fold 5-Fold 6-Fold 402 401 402 403 0.00 0 1.20 180 0.80 0 0.00 0 0.00 0 0.00 0 404 401 402 403 0.00 0-0.66 180-0.36 0 0.00 0 0.00 0 0.00 0 Additional Pi-Orbital Torsion Parameters : Atom Class 2-Fold 401 402 6.850 van der Waals Parameters : Atom Class Size Epsilon Reduction 401 3.8200 0.1060 0.000 402 3.7100 0.1100 0.000 403 2.5900 0.0220 0.900 404 3.3000 0.1120 0.000 Atomic Multipole Parameters : Atom Type Coordinate Frame Definition Multipole Moments 401 402 402 0 Bisector 0.99454 0.00000 0.00000-0.28471 0.25909 0.00000-1.50666 0.00000 0.00000 1.24757 402 401 403 0 Z-then-X -0.36496 0.00000 0.00000 0.74107-0.45969 0.00000-0.45969 0.00000 0.00000 0.91938 403 402 401 0 Z-then-X 0.12066-0.24861 0.00000-0.42123 0.04233 0.00000 0.26188-0.28294 0.00000-0.30421 404 401 402 0 Z-then-X -0.74728 0.00000 0.00000 0.16268-0.51965 0.00000-0.17330 0.00000 0.00000 0.69295 S5
Atomic Dipole Polarizability Parameters : Atom Type Alpha Damp Group Atom Types 401 1.334 0.390 404 402 404 0.837 0.390 401 402 1.073 0.390 401 403 403 0.496 0.390 402 S6
S3. Radial distribution functions and Kirkwood-Buff Integrals Using sampling structures obtained from molecular dynamics simulation, radial distribution functions (RDF) and Kirkwood-Buff integrals (KBI) were calculated. In Figure S1-S4, the RDFs and KBIs for AMOEBA, GAFF, optimized AMOEBA, and KBFF for 0.51 7.83 mol L -1 are shown. The KBIs calculated by averaging the values of KBIs from the range of 0.95 1.3 nm are tabulated in Table S2 with statistical uncertainties calculated from individual production runs. The solution density, d, the partial molar volume, V m,i, and the excess coordination number, N ij were calculated using the calculated KBIs using Eqs (3), (5)-(7). The molar excess Gibbs E energy of the mixture, G m was calculated with the same procedure as outlined in Section S1. These thermodynamic quantities for AMOEBA, GAFF, optimized AMOEBA, and KBFF are tabulated in Table S3-S6. Comparing these parameters given by AMOEBA and the vdw-optimized AMOEBA, improvement of the latter can be observed in Figure S5 in terms of almost all the thermodynamic quantities. The exceptions were partial molar volume and isothermal compressibility, whose deviation from experimental values increased slightly. S7
Figure S1 S8
Figure S1 (Continued) S9
Figure S1 (Continued) S10
Figure S1 (Continued) Figure S1. Radial distribution functions (RDFs), enlarged RDFs, and Kirkwood-Buff integrals (KBIs) for urea-urea (uu), urea-water (uw), and water-water (ww) pairs at various urea concentrations calculated using AMOEBA. Green dash and cyan chain lines represent the standard error, respectively. For short r, some figures of the enlarged RDFs lack RDF curves because no data to plot curves in the range of the Y-axis were available. The KBIs were calculated with the DKBI method. S11
Figure S2 S12
Figure S2 (Continued) S13
Figure S2 (Continued) S14
Figure S2 (Continued) Figure S2. Radial distribution functions (RDFs), enlarged RDFs, and Kirkwood-Buff integrals (KBIs) for urea-urea (uu), urea-water (uw), and water-water (ww) pairs at various urea concentrations calculated using GAFF. Green dash and cyan chain lines represent the standard error, respectively. For short r, some figures of the enlarged RDFs lack RDF curves because no data to plot curves in the range of the Y-axis were available. The KBIs were calculated with the DKBI method. S15
Figure S3 S16
Figure S3 (Continued) S17
Figure S3 (Continued) Figure S3. Radial distribution functions (RDFs), enlarged RDFs, and Kirkwood-Buff integrals (KBIs) for urea-urea (uu), urea-water (uw), and water-water (ww) pairs at various urea concentrations calculated using the vdw-optimized AMOEBA. Green dash and cyan chain lines represent the standard error, respectively. For short r, some figures of the enlarged RDFs lack RDF curves because no data to plot curves in the range of the Y-axis were available. The KBIs were calculated with the DKBI method. The RDF and KBI of neat water is the same as those determined with AMOEBA. S18
Figure S4 S19
Figure S4 (Continued) S20
Figure S4 (Continued) S21
Figure S4 (Continued) Figure S4. Radial distribution functions (RDFs), enlarged RDFs, and Kirkwood-Buff integrals (KBIs) for urea-urea (uu), urea-water (uw), and water-water (ww) pairs at various urea concentrations calculated using KBFF for urea. Green dash and cyan chain lines represent the standard error, respectively. For short r, some figures of the enlarged RDFs lack RDF curves because no data to plot curves in the range of the Y-axis were available. The KBIs were calculated with the DKBI method. S22
Table S2. Calculated KB integrals using AMOEBA, GAFF, optimized AMOEBA, and KBFF. c AMOEBA c GAFF c mol L -1 a G uu SE G uw SE G ww SE G uu SE G uw SE G ww SE 0.00-15.9 0.0-15.8 0.0 0.51 81.0 11.9-45.0 0.3-14.6 0.0 227.8 8.6-40.3 0.2-15.3 0.0 1.01 72.7 12.5-47.9 0.6-13.8 0.0 245.6 8.6-46.5 0.3-14.6 0.0 2.06 43.5 7.3-52.0 0.8-11.6 0.1 299.8 5.4-64.8 0.5-11.6 0.0 3.15 37.0 8.6-57.3 1.5-8.4 0.3 330.3 7.5-87.8 1.1-5.7 0.2 6.20-22.8 2.6-53.5 1.1-0.4 0.5 415.1 10.8-188.7 3.7 42.1 1.2 7.83-37.2 2.4-49.7 1.4 3.5 0.9 392.4 11.4-241.5 5.4 91.2 2.6 (Continued) c vdw-optimized AMOEBA c KBFF c mol L -1 a G uu SE G uw SE G ww SE G uu SE G uw SE G ww SE 0.00-15.3 0.0-15.8 0.0 0.51 32.2 10.7-43.5 0.25-14.7 0.0 50.9 9.0-44.0 0.2-15.2 0.0 1.01 41.9 11.8-46.1 0.58-13.9 0.0 33.4 7.0-45.6 0.3-14.4 0.0 2.06 25.5 7.4-49.5 0.77-11.2 0.1 19.5 4.5-48.6 0.5-12.6 0.0 3.15 11.1 6.6-52.3 1.11-9.3 0.2 2.9 2.4-50.1 0.4-10.3 0.1 6.20-27.3 3.1-50.9 1.24-1.6 0.5-38.0 1.0-45.3 0.4-5.0 0.1 7.83-39.1 2.8-47.7 1.56 2.3 0.9-49.2 0.7-40.4 0.4-3.3 0.2 KBIs were calculated using the direct KBI method. The standard errors (SEs) were calculated from the individual production runs (see Section 3.2). a) c stands for mole concentration b) Numbers of urea and water molecules calculated using the experimental mole concentration, c and density, d in Table 1, which were used to build simulation systems. c) All the KB integrals (G ij ) and standard errors (SEs) are given in cm 3 mol -1. S23
Table S3. Calculated thermodynamic quantities with AMOEBA. c mol L -1 d a g ml -1 V m,u cm 3 mol -1 V m,w cm 3 mol -1 f uu G m E b kj mol -1 κ T Eq. (4) GPa -1 κ T c GPa -1 N uu N uw N wu N ww 0.00 1.001 17.99 0.000 1.090 0.636-0.850 0.51 1.008 44.81 18.01-0.072 0.185 1.080 0.633 0.041-2.444-0.023-0.794 1.01 1.016 45.10 18.01-0.134 0.377 1.069 0.609 0.073-2.540-0.048-0.732 2.06 1.029 45.67 18.02-0.212 0.781 1.044 0.578 0.089-2.617-0.107-0.585 3.15 1.043 46.18 18.02-0.296 1.217 1.015 0.548 0.116-2.721-0.180-0.400 6.20 1.080 47.00 18.01-0.308 2.577 0.928 0.483-0.140-2.118-0.328-0.017 7.83 1.099 47.32 17.99-0.294 3.401 0.873 0.460-0.286-1.757-0.382 0.125 a) The standard error calculated from the individual production runs was less than 10-4 g ml -1. b) Using calculated KBIs and Eq. S2, the fitting parameters, A (34.61 kj mol -1 ), B (-23.92 kj mol -1 ), and C (9.52 kj mol -1 ), were obtained. The maximum p-value (0.068) of the fitted parameters was higher compared to that obtained when we fitted Eq. S2 to the experimental values, indicating over fitting. c) These values were calculated via the volume fluctuation formula (( V 2 V 2 ) k B T V ). Table S4. Calculated thermodynamic quantities with GAFF. c mol L -1 d a g ml -1 V m,u cm 3 mol -1 V m,w cm 3 mol -1 f uu G m E b kj mol -1 κ T Eq. (4) GPa -1 κ T GPa -1 N uu N uw N wu N ww 0.00 0.998 18.04 0.000 0.902 0.470-0.876 0.51 1.009 37.38 18.05-0.130 0.186 0.903 0.454 0.117-2.189-0.021-0.833 1.01 1.021 37.61 18.05-0.243 0.377 0.901 0.449 0.249-2.479-0.047-0.778 2.06 1.045 38.05 18.04-0.456 0.775 0.889 0.422 0.627-3.310-0.136-0.592 3.15 1.070 38.36 18.01-0.602 1.197 0.873 0.411 1.063-4.273-0.282-0.279 6.20 1.141 38.90 17.92-0.823 2.468 0.793 0.359 2.684-7.887-1.220 1.759 7.83 1.179 39.05 17.86-0.867 3.206 0.727 0.336 3.240-9.168-1.994 3.460 a) The standard error calculated from the individual production runs was less than 10-4 g ml -1. b) Using calculated KBIs and Eq. S2, the fitting parameters, A (30.15 kj mol -1 ), B (-18.57 kj mol -1 ), and C (8.77 kj mol -1 ), were obtained. The maximum p-value of the fitted parameters was 0.016. c) These values were calculated via the volume fluctuation formula (( V 2 V 2 ) k B T V ). S24
Table S5. Calculated thermodynamic quantities with optimized AMOEBA. c mol L -1 d a g ml -1 V m,u cm 3 mol -1 V m,w cm 3 mol -1 f uu G m E b kj mol -1 κ T Eq. (4) GPa -1 κ T GPa -1 N uu N uw N wu N ww 0.00 1.001 17.99 0.000 1.090 0.636-0.850 0.51 1.008 44.46 18.00-0.050 0.050 1.080 0.638 0.016-2.362-0.022-0.796 1.01 1.016 44.82 18.01-0.107 0.102 1.070 0.617 0.042-2.446-0.047-0.737 2.06 1.030 45.17 18.03-0.183 0.208 1.050 0.587 0.053-2.493-0.102-0.600 3.15 1.044 45.71 18.03-0.238 0.318 1.026 0.555 0.035-2.483-0.164-0.443 6.20 1.083 46.54 18.02-0.279 0.639 0.949 0.502-0.168-2.019-0.312-0.063 7.83 1.102 46.84 18.02-0.270 0.820 0.902 0.475-0.302-1.691-0.368 0.080 a) The standard error calculated from the individual production runs was less than 10-4 g ml -1. b) Using calculated KBIs and Eq. S2, the fitting parameters, A (7.65 kj mol -1 ), B (-5.18 kj mol -1 ), and C (3.08 kj mol -1 ), were obtained. The maximum p-value (0.59) of the fitted parameters was higher compared to that obtained when we fitted Eq. S2, indicating over fitting. And G m E are not reliable. c) These values were calculated via the volume fluctuation formula (( V 2 V 2 ) k B T V ). Table S6. Calculated thermodynamic quantities with KBFF. c mol L -1 d a g ml -1 V m,u cm 3 mol -1 V m,w cm 3 mol -1 f uu G m E b kj mol -1 κ T Eq. (4) GPa -1 κ T GPa -1 N uu N uw N wu N ww 0.00 0.998 18.05 0.000 0.907 0.461-0.875 0.51 1.006 44.12 18.05-0.059 0.137 0.899 0.454 0.026-2.384-0.022-0.822 1.01 1.014 44.31 18.06-0.098 0.279 0.894 0.452 0.034-2.414-0.046-0.765 2.06 1.030 44.57 18.06-0.171 0.577 0.875 0.436 0.040-2.447-0.100-0.633 3.15 1.046 44.79 18.06-0.215 0.899 0.848 0.424 0.009-2.387-0.158-0.492 6.20 1.092 45.07 18.04-0.203 1.904 0.729 0.398-0.235-1.811-0.280-0.200 7.83 1.117 45.14 18.02-0.154 2.514 0.641 0.384-0.385-1.453-0.316-0.120 a) The standard error calculated from the individual production runs was less than 5 10-5 g ml -1. b) Using calculated KBIs and Eq. S2, the fitting parameters, A (25.89 kj mol -1 ), B (-18.51 kj mol -1 ), and C (7.57 kj mol -1 ), were obtained. The maximum p-value (0.09) of the fitted parameters was higher compared to that obtained when we fitted Eq. S2, indicating over fitting and that G m E are not reliable. c) These values were calculated via the volume fluctuation formula (( V 2 V 2 ) k B T V ). S25
Figure S5. Comparison of calculated thermodynamic quantities for 0 to 7.83 mol L -1. Solution density, d, partial molar volume, V m,i, activity derivative, f uu, isothermal compressibility, κ T, and excess coordination numbers, N ij, as a function of urea concentration, are plotted against molar concentration of urea. κ T was calculated via Eq. (4) or the volume fluctuation formula (( V 2 V 2 ) k B T V ). 8 The latter is plotted with a thin line marked with an arrow. S26
S4. Thermodynamic quantities calculated with AMOEBA without polarizable effect We estimated how much the polarizable effect of AMOEBA contributed to the improvement of reproducing thermodynamic quantities. We conducted MD simulations for 0.51- and 7.83-mol L -1 solution systems for 1 ns without incorporating polarizability, where interaction energy was calculated without induced dipole moments. The density of each system reached plateau by the first 0.5 ns and the following 0.5-ns trajectory was used to calculate the average system density. The resulting system densities significantly deviated from the densities calculated with the induced dipole and experimental values, i.e., 0.796 ± 0.0005 and 0.922 ± 0.0001 g ml -1 for 0.51- and 7.83-mol L -1 solution systems, respectively. (The experimental densities and calculated densities calculated with the induced dipole are shown in Table S1 and S3, respectively). Lack of the induced dipole would weaken favorable interaction of molecules, resulting in expansion of the system. Figure S2 shows a plot of the AMOEBA energies against the QC energies indicating that the incorporation of the induced dipole corrected underestimated favorable interaction energies. Figure S6. Interaction energy for urea-urea, urea-water, and water-water pairs calculated with AMOEBA is plotted against those calculated with MP2/aug-cc-pVTZ using the BSSE correction. Blue triangle and red circle AMOEBA s correspond to interaction energies calculated without and with induced dipole. S27
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