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J. Phys. Chem. 1996, 100, 5616-5619
Testing the Modified Hydration-Shell Hydrogen-Bond Model of Hydrophobic Effects Using Molecular Dynamics Simulation Keith E. Laidig and Valerie Daggett* Department of Medicinal Chemistry, UniVersity of Washington, Box 357610, Seattle, Washington 98195-7610 ReceiVed: February 20, 1996X
Molecular dynamics simulations are used to test the central hypothesis of Muller’s “modified hydration-shell hydrogen-bond” model for the molecular explanation of the hydrophobic effect in order to provide an objective assessment of the model that would be difficult to achieve experimentally. The fraction of broken waterwater hydrogen bonds within the first solvation shell around simple hydrocarbons is compared with that found within the bulk for “infinitely dilute” aqueous simulations of n-butane, n-hexane, n-octane, and benzene. Simulations confirm Muller’s hypothesis that a larger fraction of the hydrogen bonds within the first solvation shell are broken in comparison to the bulk solution. Simulations are also used to test Muller’s extension of this model to aqueous-urea solutions, which predict further decreases in the fraction of hydrogen bonds intact in the hydration shell. This predicted decrease is observed in the simulation.
Despite the large number of studies and discussions about the molecular basis of the hydrophobic effect, there is little consensus about it origins.1 The prevailing “structure making” model explains the loss in entropy of transfer from pure liquids to aqueous solutions for hydrocarbons as increased “order” within the aqueous solution via the formation of water aggregates or “icebergs”2 in the vicinity of the hydrophobic solute. While this model has gained wide acceptance, there are a number of conceptual problems that remain to be addressed.3-5 Muller recently proposed a “modified hydration-shell, hydrogen-bond model” (HSHB) to explain the solubility behavior of hydrophobic solutes in aqueous solutions over a range of temperatures.3,4 The model has been used to explain the effects of urea upon the solubility of hydrophobic solutes,6 to discount the destabilizing effect of hydrophobic hydration upon protein native structures,7 and to investigate the relationship between the convergence temperatures T*h and T*s in protein unfolding.8 This model avoids a number of the conundrums associated with the structure making and breaking model of solvation3,4,6 and provides a hypothesis which can be tested using molecular dynamics. Our study provides an objective assessment of this model which would be difficult to achieve experimentally. The HSHB model is based upon the difference in the hydrogen-bond formation within the hydration shell of water surrounding the hydrophobic solute and the bulk. In a dilute solution, the formation and breaking of hydrogen bonds are treated as a pseudo-chemical equilibrium:
H-bond (intact) h H-bond (broken) with equilibrium constants defined as
Khs ) exp(-∆Hhs/RT + ∆Shs/R) Kb ) exp(-∆Hb/RT + ∆Sb/R) where the subscript hs refers to the hydration shell and b refers to the bulk, and the fraction of hydrogen bonds broken given as
fhs ) Khs/(1 + Khs) fb ) Kb/(1 + Kb)
Consideration of the experimental results then requires ∆Hhs * Corresponding author. Phone: (206) 685-7420. FAX: (206) 6853252. E-mail:
[email protected]. X Abstract published in AdVance ACS Abstracts, March 15, 1996.
0022-3654/96/20100-5616$12.00/0
TABLE 1: Parameters for the F3C Water Modela bO-H (Å) θH-O-H (deg) Kb (kcal mol-1 Å-2) Kθ (kcal mol-1 rad-2) qO (e) qH (e) rOO (Å) OO (kcal mol-1) AOO (kcal Å12 mol-1) BOO (kcal Å6 mol-1) rHH (Å) HH (kcal mol-1) AHH (kcal Å12 mol-1) BHH (kcal Å6 mol-1)
1.000 109.47 250 60 -0.820 0.410 3.5532 0.1848 748 407 -743.8 0.9100 0.0101 0.0028 0.0106
ab O-H is the equilibrium bond length, Kb is the bond stretching force constant, θH-O-H is the equilibrium H-O-H bond angle, Kθ is the bond stretching force constant, qX is the atomic charge of X, rX-X is the distance of minimum energy, XX is the depth of the potential well at rX-X, with AXX and BXX the repulsive and attractive coefficients of the X-X Lennard-Jones, 12-6 intermolecular potential intermolecular potential, respectively. The O-H parameters are determined from the OO and HH parameters by taking the square root of the product; POH ) (POOPHH)1/2.
> ∆Hb and ∆Shs > ∆Sb so that at every temperature Khs > Kb and fhs > fb. Although the hydration shell water-water hydrogen bonds are enthalpically stronger, a larger fraction of them is broken. Muller has formulated a set of parameters (∆Hhs ) 2.56 kcal mol-1, ∆Hb ) 2.34 kcal mol-1, ∆Shs ) 6.54 cal mol-1 K-1, ∆Sb ) 5.16 cal mol-1 K-1)3,4 that successfully predict hydrophobic hydration effects in aqueous solutions over a large range of temperatures. We performed a series of molecular dynamics simulations of “infinitely dilute” aqueous solutions of n-butane, n-hexane, n-octane, and benzene in order to test the central hypothesis of the HSHB model, that a larger fraction of the hydrogen bonds within the solvation shell are broken in comparison to the bulk. A simple, three-centered water model (F3C) is used as it reproduces the thermodynamic, structural, and dynamic characteristics of water over a wide range of temperatures and the parameters are listed in Table 1.9 Molecular dynamics was performed using ENCAD,10 with standard protocols11 and potential functions9,12 for the hydrocarbons and water. Simulations were done over the temperature range 273-523 K, with the density of the solvent set to the experimental values13 (T e 423 K) or to values extrapolated from experiment using a rational function (T > 423 K).13 After setting the density, the © 1996 American Chemical Society
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J. Phys. Chem., Vol. 100, No. 14, 1996 5617
Figure 1. Fractions of hydrogen bonds broken per water molecule in the bulk and hydration shell for four hydrocarbons in pure water as a function of the simulation temperature. The solid lines represent the hydration-shell fraction and the dashed lines represent the bulk fraction.
Figure 2. Representative structure from the octane simulation sliced through the box to show the hydration shell and some bulk waters. The thin, dashed lines between water molecules denote hydrogen bonds.
box volume was held constant and periodic boundary conditions were employed. A 0.2 fs (1 fs ) 10-15 s) time step was used for the full simulation of 20 ps (1 × 105 steps) for each solution. Analyses were performed on the last 500 structures of the simulation (e.g., the second 10 ps of the run). Tests of different simulation times, time steps, box sizes, and demonstration that the velocity distribution was Gaussian over the time period sampled indicate that the large number of molecular dynamics steps and the very short time step result in rapid convergence
of the solvation hydrogen bonding properties addressed here (data not presented). Further simulation of n-butane in water, at 298 K for 100 ps (5 × 105 steps), showed no change in the results obtained (data not presented). The number of water molecules produced a layer of at least 15 Å from the solute, yielding 1373, 1501, 1634, and 1518 water molecules for the n-butane, n-hexane, n-octane, and benzene simulations, respectively. Both the hydrogen bonds within the solvation shell around the hydrophobic solute and within the bulk were defined
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Figure 3. Graphical depiction of the angular distribution of water around solute. The tables list the number of water molecules found at various distances from the solute heavy atoms and their angular distribution (the angle between the H-O-H angle bisector and the O-solute vector) as percentages. The distributions about water in bulk water and about butane in the infinitely dilute butane solution are given. These angular distributions are depicted in approximate contour diagrams above each table.
as having intermolecular O-H‚‚‚distances of e2.6 Å and O-H‚‚‚O angles within 35° of linearity, with the hydration shell defined as those molecules e4.5 Å of the solute. Use of different O-H‚‚‚O distances and angles resulted in different absolute numbers of hydrogen bonds, but the relative numbers of hydrogen bonds remained the same. The fraction of hydrogen bonds broken in the hydration shell and the bulk are shown in Figure 1. All of the straight-chain hydrocarbons behave in a similar manner, with fhs being quite close to that of bulk at lower temperatures then increasing more rapidly as the temperature is increased. For example, at 273 K fb ) 0.119 ((0.002) for n-hexane and fhs is 0.139 ((0.007), while at 523 K fb ) 0.397 ((0.003) and fhs is 0.479 ((0.008). Benzene behaves differently at lower temperatures, having fhs values greater than for the simple hydrocarbons until 323 K. This is the temperature range over which these molecules reach their minimum solubility14 and the simplicity of Muller’s model may not account for the ability of aromatic rings to act as hydrogen-bond acceptors.15 However, as predicted by Muller’s model,3,4 fhs is greater than fb over the entire temperature range for all of the simulations, with the difference between the two fractions being statistically significant. Muller suggested3,4 that 0.15 is a reasonable value for fb at 273 K, based upon the model values of ∆Hhs and ∆Hb, which agreed with the value based upon sublimation energies and enthalpies of fusion for ice16 and a value determined from IR studies.17 Given the qualitative nature of Muller’s values,4 our values of fb (0.12-0.13) appear to be reasonable. In addition, both the model4 and the simulations (Figure 1) predict that both fb and fhs increase with temperature and that the difference (fhs - fb) also increases. The orientation of water molecules in the first hydration shell changes from the bulk upon introduction of the hydrocarbon, pulling away from the solute slightly and “flattening out” to maximize hydrogen bonding with neighboring water molecules (Figure 2). Thus, the introduction of a nonpolar solute restricts the angular distribution of the water molecules in and near the hydration shell in comparison to bulk. For example, in the hydration shell around the solute, neighboring waters cluster with an angle of 120° from the H-O-H angle bisector to the
nearest solute atom, while in bulk water the angular distribution is roughly equally distributed between 90° and 120° (see Figure 3). Muller has extended his model to address the increased solubility of hydrophobic solutes in urea solutions.6 The cosolvent (urea) sterically reduces the average number of hydration shell hydrogen bonds associated with each solute molecule, as well as the number of water molecules in the shell. Consequently, this reduces the hydration enthalpy, hydration entropy, and the Gibb’s free energy arising from the changes in the water-water hydrogen bonding equilibrium due to introduction of hydrophobic solute. For the urea solution of n-butane, a larger box was created, with the density corrected to that of the final solution,18 containing 3094 water molecules. 104 water molecules were swapped for urea molecules (this proportion of urea was used for comparison to Muller), using a urea model developed within this laboratory19 and simulation of this system was carried out as described. The fraction of broken water-water hydrogen bonds is greater in the hydration shell in the presence of urea than in pure water. For example, at 298 K, fhs ) 0.17 in urea and 0.15 in water. Thus, as predicted by Muller, the fraction of broken hydrogen bonds in the solvation shell is larger in aqueous urea solutions than in water. Also, an important assumption in Muller’s formulation is that the cosolvent (urea) only affects hydrophobic hydration by occupying space in the hydration shell that would otherwise contain water molecules and that the cosolvent has little effect on water hydrogen bonding. In support of this assertion, fb was essentially the same in the water and aqueous urea simulations. Muller’s model provides a relatively simple molecular explanation for hydrophobic hydration with simple hydrocarbons in water and urea-water solutions. While this model does not account for all observations associated with hydrophobic hydration, such as the consequences of charged species, ionic strength, complicated mixed solvent systems, etc., it appears to provide a promising basis for a simple systematic view of this phenomenon. Extensions to more complicated systems such as amino acids and proteins are presently underway.
Letters Acknowledgment is made for partial support of this work from the National Science Foundation (MCB 940 7903), the donors of The Petroleum Research Fund, administered by the American Chemical Society (ACS-PRF 28022-64), and the Young Investigator Program of the Office of Naval Research (N00014-95-1-0484). We thank Dr. D. O. V. Alonso for helpful discussions and comments on the manuscript. References and Notes (1) See the special issue entitled “Thermodynamics of Hydration”, Biophys. J. 1994, 51, (2) Frank, H. S.; Evans, M. W. J. Chem. Phys. 1945, 13, 507. (3) Muller, N. J. Solut. Chem. 1988, 17, 661. (4) Muller, N. Acc. Chem. Res. 1990, 23, 28. (5) Ben-Naim, A. In Water, A ComprehensiVe Treatise; 1973, Franks, F., Ed.; Plenum Press: New York, 1973; Vol. 2. (6) Muller, N. J. Phys. Chem. 1990, 94, 3856. (7) Muller, N. Trends Biochem. Sci. 1992, 17, 459. (8) Baldwin, R. L.; Muller, N. Proc. Nat. Acad. Sci. U.S.A. 1992, 89, 7110. (9) Levitt, M.; Hirshberg, M.; Sharon, R.; Daggett, V. “Calibration and Testing of a Water Model for Simulation of the Molecular Dynamics
J. Phys. Chem., Vol. 100, No. 14, 1996 5619 of Proteins and Nucleic Acids in Solution”, submitted. (10) Levitt, M. ENCADsEnergy Calculations and Dynamics. Molecular Applications Group, Stanford University and Yeda, 1990. (11) Daggett, V.; Levitt, M. J. Mol. Biol. 1992, 223, 1121. (12) Levitt, M.; Hirshberg, M.; Sharon, R.; Daggett, V. Comput. Phys. Commun. 1995, 91, 215. (13) Kell, G. S. J. Chem. Eng. Data 1967, 12, 66. (14) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; Wiley-Interscience: New York, 1980. (15) Levitt, M.; Perutz, M. F. J. Mol. Biol. 1988, 201, 751. (16) Pauling, L. The Nature of the Chemical Bond; Cornell University Press: Ithaca, NY, 1960. (17) Luck, W. A. P. In Water, A ComprehensiVe Treatise; Franks, F., Ed.; Plenum Press: New York, 1973; Vol 2. (18) Timmermans, J. The Physico-chemical Constants of Binary Systems in Concentrated Solutions; Interscience Publishers, Inc.: New York, 1960; Vol. 4. (19) Laidig, K. E.; Daggett, V. Macromolecular Simulations in Mixed Solvents: Development of the Solvent Systems, in preparation. Laidig, K. E.; Kazmirski, S.; Daggett, V.; Macromolecular Simulations in Mixed Solvents: Solvent-Dependent Conformational Properties of Helix A of Hen Egg White Lysozyme, in preparation.
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