Driving Force for the Association of Hydrophobic Peptides: The

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Driving Force for the Association of Hydrophobic Peptides: The Importance of Electrostatic Interactions in Coarse-Grained Water Models Zhe Wu, Qiang Cui,* and Arun Yethiraj* Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, 1101 University Avenue, Madison, Wisconsin 53706, United States ABSTRACT: The hydrophobic effect plays a central role in many biological processes, including protein folding and aggregation. The hydrophobic interaction between solutes, such as helical peptides, is believed to be of entropic origin and driven by the increase in the entropy of water due to association. In this work, we examine the association between peptides in water using several coarse-grained (CG) models, such as MARTINI (MAR), polarizable MARTINI (POL), and big multipole water (BMW) models, where four atomistic water molecules are grouped into a single CG unit. All models predict that a pair of helical peptides (Ala20 and Leu20) has favorable association free energy. The BMW model is the only model, however, in which this association is entropy-driven, as has been previously established with atomistic simulations and experiments. The MAR and POL models, where the CG water particles do not have a quadrupole moment, predict an enthalpy-driven association, with a negligible entropy change upon association. Similarly, the association of two rigid cylinders in water is found to be enthalpy-driven when the water is described with the CG model of Shinoda et al. that includes a soft-core nonelectrostatic interaction, while BMW predicts an entropy-driven association. These results emphasize the importance of electrostatic interactions in water for the qualitative features of the thermodynamics of biophysical systems. SECTION: Statistical Mechanics, Thermodynamics, Medium Effects

W

ater is an important and ubiquitous component in biological systems. A concept of central importance in biology is the hydrophobic effect, which causes nonpolar solutes to associate in water. Many fundamental processes, such as protein folding or binding, are driven in part by this water-mediated interaction between moieties.1
4 For most small (bio)molecular solutes, the hydrophobic effect is believed to be entropic in nature,5,6 that is, the thermodynamic driving force for the association between small hydrophobic objects is the associated increase in the entropy of water. For example, computer simulations of the association of R-helices of peptides composed of hydrophobic residues (alanine and leucine) show that the total association free energy has a large favorable entropic contribution and a small enthalpic barrier,7 consistent with experimental8
10 and theoretical studies.11 Computer simulations have become an important tool for the theoretical study of biological systems. The length scales and time scales of interest are quite large, however, and atomistic simulations are prohibitively intensive for many applications. Therefore, there has been considerable interest in developing coarse-grained (CG) models for biomolecules and water, where, for example, several water molecules are lumped into a single CG unit. In this work, we study the association of hydrophobic peptides using various CG water models and demonstrate the importance of electrostatic interactions on the qualitative features, that is, entropic or enthalpic, of the thermodynamics of hydrophobic aggregation. r 2011 American Chemical Society

A number of CG water models have been recently introduced. The use of uncharged CG units to represent water molecules is computationally appealing because long-ranged electrostatic interactions do not have to be calculated, for example, using the Ewald sum, thus resulting in a significant savings in computer time. A popular model is the MARTINI (MAR)12 model where four water molecules are grouped into a single unit and two such units interact via a Lennard-Jones potential. Other similarly uncharged CG water models have also been proposed,13
18 and it has been suggested that for a water model to reproduce the density and diffusion anomalies, the potential has to be a function of temperature and density.15 CG water models with explicit electrostatic interactions have also been developed. Yesylevsky et al.19 have introduced a polarizable MARTINI (POL) model where each unit contains two point charges and dipole fluctuation; another model similar in spirit is the two-site CG model of Riniker and van Gunsteren.20 Recently, we introduced the big multipole water (BMW) model,21 which mimics both the dipole and quadrupole moment tensors of atomistic four-water clusters using a three-site topology. The Wat Four model22 is featured with four point charges and also has quadrupole moments; although the model was constructed to map 5 water molecules into one CG unit, the calculated properties Received: May 17, 2011 Accepted: July 5, 2011 Published: July 05, 2011 1794

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The Journal of Physical Chemistry Letters (e.g., density) suggested that one CG unit effectively reflects 11 water molecules. Other more detailed models, such as the soft sticky dipole model23,24 and the GB-EMP model,25 are closer in character to atomistic models and offer little computational advantage over them. The purpose of this study is to investigate the qualitative features of the thermodynamics of peptide association in CG models of water. There are several reasons that this is important. While the savings in computer time is definitely useful, the utility of the models is limited if the physical insight obtained is not reliable. A worry in this regard is the entropy of the water in CG models. For example, grouping several molecules into a single site results in a loss of translational entropy of the molecules. It is also not clear how removing electrostatic interactions affects the orientational entropy of the molecules. While the CG interaction parameters are chosen to reproduce experimental values for bulk properties and solute
water interactions can be chosen to reproduce solubility and/or transfer free energies, it is not clear if the CG models are reliable for more subtle issues related to the thermodynamics of the hydrophobic effect. In this work, we study the thermodynamics of peptide association and consider three CG water models, MAR, POL, and BMW. In all three models, four water molecules are mapped into one CG unit; the POL and BMW models are based on the MAR model in this respect. These models have increasing levels of electrostatic complexity. The MAR model has no electrostatic interactions between the CG beads. In the POL model, each unit contains two point charges connected by springs to a neutral center particle; therefore, it has a dipole moment, is flexible (and thus polarizable), but has no quadrupole moment. The BMW model has two positively charged and one negatively charged sites and is constructed to reproduce the dipole and quadrupole moment of four-molecule clusters in atomistic water simulations. The BMW also has a softer nonelectrostatic repulsion compared to the POL and MAR models. (See the Abstract graphic; the CG beads are colored according to charge: tan for charge-neutral, green for charge-positive, and blue for charge-negative. The boundary for the MAR and POL beads is smooth, while that for the BMW bead is fuzzy, reflecting the softer nonpolar potential in the BMW model.) We calculate the potentials of mean force (PMFs) between two alanine and two leucine 20-mer helices at different temperatures (298 and 318 K) and use the results to compute the entropic and enthalpic contributions to the free energy of association. The BMW model is more accurate than the MAR and POL models for the PMF between the helices when compared to results from published atomistic simulations using the OPLS force field.7 Figures 1 and 2 depict the PMF between Ala20 and Leu20 helices, respectively, for the three models. With all models, an attractive PMF is obtained at short distances. Only steric dewetting in the helix dimerization is found, with a single layer of CG water between the helices, similar to what is seen in atomistic simulations.7 The MAR and POL models overestimate the depth of the association minimum in Ala20, that is, MAR gives
118 kJ/ mol and POL gives
101 kJ/mol compared to the atomistic value of
60 kJ/mol. The BMW model, which gives a value of
43 kJ/mol, is much closer to the atomistic result for this quantity. For Leu 20 , the BMW model predicts an attractive depth of
123 kJ/mol, which is substantially stronger than that for Ala20 but smaller in magnitude than the atomistic result of
165 kJ/ mol. The MAR and POL models, however, predict a weaker

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Figure 1. The potential of mean force (PMF, black) and the corresponding enthalpy (red) and entropy (
ST, green) for the association of two Ala20 helices as functions of the interhelical distance calculated with the big multipole water (BMW), MARTINI (MAR), and polarizable MARTINI (POL) models.

Figure 2. The PMF (black) and the corresponding enthalpy (red) and entropy (
ST, green) for the association of two Leu20 helices as functions of the interhelical distance calculated with the BMW, MAR, and POL models.

association between Leu20 than Ala20, which is in contrast with atomistic and BMW results. It is also an unphysical result because Leu20 is more hydrophobic than Ala20. Alanine has a single methane group as the side chain, and each residue in the CG scheme is modeled with a single backbone bead. Leucine has an i-butane side chain and in the CG scheme is modeled as a backbone bead plus one hydrophobic side-chain bead. It is therefore surprising that MAR and POL models predict that the more hydrophobic helices have a weaker attraction in the PMF. The most significant difference between the models is that the BMW model predicts that the hydrophobic association is entropic in nature, and the other models predict that it is enthalpic. With the BMW model, the PMF is different for the two temperatures, with a slightly stronger attraction at the higher temperature. 1795

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The Journal of Physical Chemistry Letters With the MAR and POL models, the PMF is insensitive to temperature. When the temperature derivative is used to decompose the PMF into entropic and enthalpic contributions, the BMW model predicts a strongly attractive entropic contribution and a weakly repulsive enthalpic contribution. Therefore, in the BMW model, the association of the helices is entropy-driven, similar to atomistic simulations.7 The magnitude of the entropic and enthalpic contributions also agrees fairly well with those from atomistic simulations. For the Ala20 helices, the BMW model predicts an entropy minimum of
50 kJ/mol and an enthalpic maximum of 35 kJ/mol, which can be compared to the atomistic (OPLS) simulation values of
72 and 56 kJ/mol, respectively. For the Leu20 helices, the BMW model also agrees qualitatively with atomistic results in terms of enthalpic and entropic components of the PMF for most interhelical distances. At interhelical separation shorter than 1.2 nm, however, the BMW model predicts unfavorable enthalpic contributions, while atomistic simulations7 found favorable enthalpic contributions; such discrepancy is likely due to the lower resolution of the BMW model, which limits its quantitative accuracy in describing the solvent structure and interaction in the presence of strong perturbation introduced by two adjacent hydrophobic helices. In the MAR and POL models, enthalpy is the sole driving force for hydrophobic association; as shown in Figures 1 and 2, the entropic component for association is nearly zero or slightly repulsive with these models. Therefore, the BMW model is in qualitative agreement with experiment and atomistic simulations,7
10 and the MAR and POL models are not. It is of interest to elucidate what aspect of the water model results in such a qualitative difference in the thermodynamics of association. To do so, we would like to understand the effect of (1) structure of the solutes, (2) the degree of hydrophobicity of the solutes, (3) the softness in the nonelectrostatic interaction in the water model (the BMW uses a soft core while the others use a Lennard-Jones interaction), (4) bulk thermodynamic properties of water, and (5) specific mapping of four water molecules into one CG site. To differentiate these different effects, we study the association PMF of rigid, water-repulsive CG cylinders (to address points 1 and 2) in water. We also study the association PMF in a fourth water model (SDK) proposed by Shinoda and coworkers,26 which has a 3 to 1 mapping of atomistic to CG water and also includes a soft-core 12
4 nonelectrostatic interaction (to address points 3 and 5). All of these models have different values for the bulk thermodynamic properties of water. Figure 3 compares the association PMF of cylinders in these four water models. Only the BMW model gives entropy-driven hydrophobic association of the cylinders. The entropic effect does not correlate with the predicted thermodynamic properties or the softness of the nonelectrostatic interaction. We conclude that including a reasonable description of the electrostatic interactions is important in CG models of water. In particular, the quadrupole moment plays an important role in the entropy of the water. A dramatic observation from this study is the qualitatively different results obtained for the hydrophobic association of helices. With the quadrupole moment included, as is done with the BMW model, this association is entropy-driven, consistent with experiment and atomistic simulations. With the quadrupole moment absent, as in the MAR and POL models, the association is enthalpy-driven. We therefore suggest that for problems involving protein folding, binding, or self-assembly, a water model that incorporates the quadrupole moment is indispensable.

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Figure 3. The PMF (black) and the corresponding enthalpy (red) and entropy (
ST, green) for the association of two water-repulsive cylinders as functions of the interhelical distance calculated with the BMW, MAR, POL, and Shinoda’s water (SDK) models.

The BMW is computationally more intensive (by about a factor of 6) than the MAR model, but the qualitative difference in prediction makes it a physically more appealing choice. These conclusions are not restricted to the MAR or POL models. Any nonelectrostatic model of water13,14 is expected to have the same qualitative drawbacks as the MAR model unless the interaction potential is a function of temperature and density. These results could have significance in understanding a large class of processes. If association free energy at a single temperature is of interest, it is possible that a carefully parametrized CG model without sophisticated electrostatics is adequate.27 If the models are expected to be robust over a range of temperatures and/or pressures,5 then getting the driving force correctly is important; only the BMW predicts a temperature-dependent PMF in this study. A useful way of parametrizing and testing models is to compare to experimental data for the phase behavior of lipid/peptide mixtures, and these experiments are not always done at room temperature. Another important area is in the competitive binding of different molecules to a substrate where the subtle balance between enthalpic and entropic effects could play a role.28 We therefore conclude that a robust model of CG water should include a reasonable representation of electrostatic effects.

’ COMPUTATIONAL METHOD PMF calculations are performed in GROMACS 4.0.7.29 in the NPxyPzT ensemble. The mapping of the amino acids is the same in all CG models, and the secondary structures are constrained by a dihedral angle force constant for all three CG models, and the interaction parameters are obtained specifically from each force field so that the solvation free energy for each particle type agrees with experiments.19,30,31 The interaction parameters are different in the three force fields because the different interactions between water molecules necessitates a reparameterization of all of the other parameters. 1796

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The Journal of Physical Chemistry Letters

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(force constant of 8000 kJ/(mol 3 nm2)), each window for 120/ 160/200/300 ns with BMW/POL/MAR/SDK models. The water
water interaction in the SDK model is a 12
4 potential, which is very similar to modified BMH potential used in BMW. For the region between 0.43 and 1 nm and with parameters from the published work,26 good fitting correlation up to 0.999 can be achieved with modified BMH with ε = 3.71 kJ/mol, rm = 0.5 nm, f = 3.977, and a = 6.1514 (figure not shown). The relatively large f (close to 6) and f/a (close to 1) indicate softness around the potential minimum.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] (Q.C.); [email protected] (A.Y.). Figure 4. The repulsive potential between the cylinder and water (solid) and the original interaction curve (dash) taken from the MAR force field (ε = 2.0 kJ/mol and σ = 0.47 nm).12

The helices are constrained in a box of water molecules in a manner identical to the atomistic simulations.7 The starting conformations are generated by simulated annealing with each model, and the two-helix structures adopt a coiled antiparallel geometry with a slightly different interhelix angle as the optimal position with different models. For a given model, the same interhelix angle is kept in different simulations and different temperatures. In the PMF simulation, backbone particles on one helix are restrained in three dimensions, and the other has restraints on only two dimensions (x,y), with the side chains unrestrained (for poly-leucine). The helical conformation and the angle between the helices are fixed during the simulation, and no rotation or translation in other direction is allowed. The only remaining degree of freedom is the distance between the two helices, and this is chosen as the coordinate r for the PMF calculation. The PMF is obtained using umbrella sampling with 115 windows evenly between a 0.72 and 3 nm interhelix distance (force constant of 8000 kJ/(mol 3 nm2)). At each temperature, each window is simulated for five independent 120/160/200 ns runs for the BMW/POL/MAR model to obtain the average PMF and statistic errors. In order to obtain the entropy and enthalpy contributions to the total association free energy, the PMF calculations are performed for two different temperature (298 and 318 K). The entropy is calculated from the thermodynamic definition, S =
(∂G/∂T)N,P, where G is the Gibbs free energy, with a finite difference approximation for the derivative. For each distance r SðrÞ ¼


ΔGðr, T þ ΔTÞ
ΔGðr, TÞ ΔT

ð1Þ

where ΔG(r) is the difference between G(r) and its value at infinite separation. The enthalpy is obtained from a subtraction of the entropy from ΔG(r). For the PMF calculation of cylinders, the same restraints are applied as those in the case of helices; therefore, the only degree of freedom is the distance between the two cylinders. The cylinder
water interaction is the same with all four water models, with a repulsive potential to mimic absolute “hydrophobicity” (shown in Figure 4); the cylinders, which are 1.2 nm in diameter and 4.2 nm in length, do not interact with each other. Simulations are performed at temperatures of 298 and 318 K with umbrella sampling with 95 windows evenly between a 1.12 and 3 nm intercylinder distance

’ ACKNOWLEDGMENT We thank Prof. P. Tieleman for helpful discussions regarding the atomistic PMF results of ref 7 and Dr. R. Devane for discussions on Shinoda’s CG water model. The research has been supported by the National Science Foundation (CHE0957285 to Q.C. and CHE-0717569 to A.Y.). Computational resources at the Centre for High Throughput Computing (CHTC) at UW—Madison are acknowledged. ’ REFERENCES (1) Makhatadze, G. I.; Privalov, P. L. Adv. Protein Chem. 1995, 47, 307–425. (2) Baldwin, R. L. J. Mol. Biol. 2007, 371, 283–301. (3) Levy, Y.; Onuchic, J. N. Annu. Rev. Biophys. Biomol. Struct. 2006, 35, 389–415. (4) Berne, B. J.; Weeks, J. D.; Zhou, R. H. Annu. Rev. Phys. Chem. 2009, 60, 85–103. (5) Hummer, G.; Garde, S.; Garcia, A. E.; Paulaitis, M. E.; Pratt, L. R. J. Phys. Chem. B 1998, 102, 10469–10482. (6) Ashbaugh, H. S.; Pratt, L. R. Rev. Mod. Phys. 2006, 78, 159–178. (7) MacCallum, J. L.; Moghaddam, M. S.; Chan, H. S.; Tieleman, D. P. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 6206–6210. (8) Segawa, S. I. Biopolymers 1984, 23, 2473–2488. (9) Schindler, T.; Schmid, F. X. Biochemistry 1996, 35, 16833–16842. (10) Scalley, M. L.; Baker, D. Proc. Natl. Acad. Sci. U.S.A. 1997, 94, 10636–10640. (11) Lui, Z.; Chan, H. S. J. Mol. Biol. 2005, 349, 872–889. (12) Marrink, S. J.; de Vries, A. H.; Mark, A. E. J. Phys. Chem. B 2004, 108, 750–760. (13) Shelley, J. C.; Shelley, M. Y.; Reeder, R. C.; Bandyopadhyay, S.; Klein, M. L. J. Phys. Chem. B 2001, 105, 4464–4470. (14) Ayton, V. S.; Noid, W. G.; Voth, G. A. MRS Bull. 2007, 32, 929–934. (15) Chaimovich, A.; Shell, M. S. Phys. Chem. Chem. Phys. 2009, 11, 1901–1915. (16) Egorovj, S. A. Hydrophobic Interactions with Coarse-Grained Model for Water. Submitted, arXiv: 1105.0222v1. (17) Yan, Z. Y.; Buldyrev, S. V.; Giovambattista, N.; Debenedetti, P. G.; Stanley, H. E. Phys. Rev. E 2006, 73, 051204. (18) Head-Gordon, T. Chem. Phys. Lett. 1994, 227, 215–220. (19) Yesylevskyy, S. O.; Sch€afer, L. V.; Sengupta, D.; Marrink, S. J. PLoS Comput. Biol. 2010, 6, e1000810. (20) Riniker, S.; van Gunsteren, W. F. J. Chem. Phys. 2011, 134, 084110. (21) Wu, Z.; Cui, Q.; Yethiraj, A. J. Phys. Chem. B 2010, 114, 10524– 10529. (22) Darre, L.; Machado, M. R.; Dans, P. D.; Herrera, F. E.; Pantano, S. J. Chem. Theory Comput. 2010, 6, 3793–3807. 1797

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