On the Thermodynamics and Kinetics of Hydrophobic Interactions at

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On the Thermodynamics and Kinetics of Hydrophobic Interactions at Interfaces Srivathsan Vembanur,† Amish J. Patel,‡ Sapna Sarupria,§ and Shekhar Garde*,† †

The Howard P. Isermann Department of Chemical and Biological Engineering and The Center for Biotechnology and Interdisciplinary Studies, Rensselaer Polytechnic Institute, Troy, New York 12180, United States ‡ Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States § Department of Chemical and Biomolecular Engineering, Clemson University, Clemson, South Carolina 29634, United States S Supporting Information *

ABSTRACT: We have studied how primitive hydrophobic interactions between two or more small nonpolar solutes are affected by the presence of surfaces. We show that the desolvation barriers present in the potential of mean force between the solutes in bulk water are significantly reduced near an extended hydrophobic surface. Correspondingly, the kinetics of hydrophobic contact formation and breakage are faster near a hydrophobic surface than near a hydrophilic surface or in the bulk. We propose that the reduction in the desolvation barrier is a consequence of the fact that water near extended hydrophobic surfaces is akin to that at a liquid− vapor interface and is easily displaced. We support this proposal with three independent observations. First, when small hydrophobic solutes are brought near a hydrophobic surface, they induce local dewetting, thereby facilitating the reduction of desolvation barriers. Second, our results and those of Patel et al. (Proc. Natl. Acad. Sci. U.S.A. 2011, 108, 17678−17683) show that, whereas the association of small solutes in bulk water is driven by entropy, that near hydrophobic surfaces is driven by enthalpy, suggesting that the physics of interface deformation is important. Third, moving water away from its vapor−liquid coexistence, by applying hydrostatic pressure, leads to recovery of bulklike signatures (e.g., the presence of a desolvation barrier and an entropic driving force) in the association of solutes. These observations for simple solutes also translate to end-to-end contact formation in a model peptide with hydrophobic end groups, for which lowering of the desolvation barrier and acceleration of contact formation are observed near a hydrophobic surface. Our results suggest that extended hydrophobic surfaces, such as air−water or hydrocarbon−water surfaces, could serve as excellent platforms for catalyzing hydrophobically driven assembly.

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INTRODUCTION Water drives the assembly of hydrophobic solutes in solution.1,2 As hydrated solute molecules approach each other, water molecules in the intervening space need to be removed before solute−solute contacts are established. The reversible work for the removal of this intervening water appears as a desolvation barrier in the potential of mean force (PMF) between the solute molecules. Desolvation barriers are observed in a range of processes, from simple association of a pair of small hydrophobic solute molecules3,4 and formation of larger clusters,5 to folding of hydrophobic polymers6 and proteins.7 The presence of a barrier is also reflected in the metastability of water confined between hydrophobic plates or membranes8−11 and the corresponding hysteresis along the association− dissociation coordinate. The ease with which water between the solutes can be removed depends on the nature of the solutes themselves (e.g., solute length scale, shape, solute−water attractions).12−15 In addition, the environment or the context in which the association occurs is expected to significantly influence both the desolvation barrier and the strength of association (e.g., confinement, thermodynamic conditions).16,17 To this end, © 2013 American Chemical Society

aqueous interfaces provide well-defined inhomogeneous environments in which to study hydrophobically driven assembly. Recent studies have shown that water near extended hydrophobic surfaces displays enhanced density fluctuations, higher susceptibility (e.g., compressibility),18−21 and other features that are indicative of water sitting at the edge of a liquid-to-vapor phase transition.2,12,22 How are water-mediated interactions between hydrophobic solutes affected by the presence of an interface in their vicinity? In this work, we report the effects of extended hydrophobic and hydrophilic surfaces on the association of hydrophobic solutes at the pair and many-body levels using extensive molecular dynamics (MD) simulations. We quantify the PMFs for the association of hydrophobic solutes near surfaces of selfassembled monolayers (SAMs) presenting −CH3 (hydrophobic; θcontact ≈ 110°) and −OH (hydrophilic; θcontact ≈ 0°)23 headgroups. We show that, whereas the association at hydrophilic interfaces is similar to that in bulk water, with Received: May 22, 2013 Revised: July 31, 2013 Published: August 1, 2013 10261

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simulation entailed 0.5 ns of equilibration, followed by a 5-nslong production run. More details are provided at the end of this section. We also calculated PMFs for hydrophobic cluster formation in bulk water and near −CH3 and −OH SAM surfaces. Specifically, we studied the association of 13 hydrophobic solutes in bulk water arranged in a hexagonal close-packed (hcp) lattice (i.e., one solute at the center surrounded by 12 nearest neighbors). We used the lattice spacing, r, as the order parameter and calculated the free energy (i.e., the PMF) along r for r values ranging from 1.25 nm (all solutes separately hydrated) to 0.3 nm (all solutes in contact) and smaller. To understand how thermodynamics of such cluster formation is affected by the presence of an interface, we focused on assembly in planar (two-dimensional) geometries. Specifically, we calculated the PMF for the association of seven solutes (a central solute surrounded by six neighbors in a symmetric hexagonal configuration) with the lattice spacing, r, as an order parameter. Calculations of the PMF were done using the Bennett acceptance ratio method,27 with 44 simulations (each 2 ns long) with r varying from 1.25 to 0.25 nm (in steps of 0.025 nm) used in the analysis. Simulations of a Peptide. To quantify the hydrophobically driven contact formation in a model biological system, we studied the behavior of the peptide Cys-Ala-Gly-Gln-Trp. The peptide was represented in atomic detail using the AMBER-94 force field.28 To ensure effective sampling of the conformational space of the peptide, we followed the approach of Yeh and Hummer.29 We initially performed a high-temperature (700 K) simulation of the peptide solvated by 527 SPC/E water molecules in a three-dimensional periodic box (2.5 × 2.5 × 2.5 nm3) for 500 ps. We selected 20 equally spaced configurations from this high-temperature run as starting configurations for 20 additional NPT simulations (P = 1 bar, T = 300 K), which were equilibrated for 1 ns and run for 20 ns each during the production phase, thus leading to 400 ns of total simulation data for further analysis. A similar process was followed to study the behavior of the peptide near −CH3 and −OH SAM surfaces. The peptide does not adsorb on the hydrophilic −OH surface by itself; therefore, the peptide was placed near the −OH surface, and an additional 9-3 LennardJones potential (ρ = 35.0 nm−3, σ = 0.40 nm, ε = 1.50 kJ/mol) was applied to hold it in the nanoscopic region near the interface. Kinetics. The thermodynamic and structural analysis performed as described in the preceding subsections (and discussed later) highlights the differences in desolvation barriers in bulk and interfacial environments that are expected to alter the kinetics of contact formation/breakage. To estimate the rates of contact formation and breakage for hydrophobic solutes, we performed 40 additional simulations of SAM−water systems containing two hydrophobic solutes (not inserted as in the Widom calculations, but included explicitly), each 20 ns long, with configurations stored every 0.1 ps. In SAM−water systems, the two solutes were restricted to remain in a plane near the interface (z = 0.23 nm near the −CH3 surface and z = 0.25 nm near the −OH surface), thus allowing an estimate of contact formation/breakage in the vicinity of the interfaces to be obtained efficiently. To obtain estimates of contact formation/breakage in the peptide system, we used the 400 ns of simulation data collected as described above in bulk water and near interfaces.

significant desolvation barriers being present, the association near hydrophobic surfaces occurs with much lower (or zero) desolvation barriers under ambient conditions. Correspondingly, the kinetics of hydrophobic contact formation and breakage are faster near the −CH3 surface compared to those at the −OH surface. The reduction of desolvation barriers can be understood using the perspective developed by Chandler and co-workers.2,12,22,24 Under ambient conditions, water is near coexistence with its vapor phase, and the proximity to a hydrophobic surface pushes it even closer to that phase transition. Thus, water near an extended hydrophobic surface is akin to that near a water−vapor interface.12 The association of hydrophobic solutes at such an interface, then, should simply deform that interface continuously, facilitating the removal of water between solutes and the concomitant lowering of the desolvation barrier. To test this idea, we performed simulations under thermodynamic conditions that move water away from phase coexistence, namely, at high hydrostatic pressures. At high pressures, bulklike features, including significant desolvation barriers in the association of solutes, reappear in the PMF between the hydrophobic solutes, even near a hydrophobic surface. Finally, we show that the preceding observations for model hydrophobic solutes translate to more complex and realistic systems. We observed both a reduction in the desolvation barrier and faster kinetics in contact formation between the hydrophobic end groups of a peptide molecule near a hydrophobic surface. Collectively, our results support the idea that hydrophobic surfaces could serve as catalysts for hydrophobically driven (planar) assembly in their vicinity. Aqueous interfaces are ubiquitous in biological systems, as well as in numerous technological applications. Our results highlight the need to understand how interfaces affect the thermodynamic, structural, and kinetic aspects of fundamental watermediated interactions.

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METHODS Our simulation studies comprised calculations of PMFs for pair and many-body hydrophobic association in bulk and interfacial environments and of end-to-end contact formation in a peptide. Details of the simulations and methods are given in the following subsections. MD Simulations of Hydrophobic Hydration and Interactions. The model hydrophobic solutes considered in our study are spherical neutral Lennard-Jonesiums with solute− solute parameters σ = 0.26 nm and ε = 1.234 kJ/mol. Water molecules are represented explicitly using the SPC/E25 water model. We used the Widom test-particle-insertion method26 to calculate the excess chemical potential of a single hydrophobic solute in bulk water and at various locations in the system as a function of the distance from −CH3 and −OH SAMs. We also calculated the potential of mean force between two hydrophobic solutes in bulk water and in planes parallel to the SAM surfaces also using Widom test-particle insertion of two-solute dumbells. To obtain enthalpy and entropy contributions to hydrophobic hydration and interactions in bulk water and near an interface, we performed these calculations over the range of temperatures from 275 to 350 K in steps of 25 K at P = 1 bar. To quantify the effects of hydrostatic pressure on hydrophobicity in bulk water and near the −CH3 SAM surface, we also performed the same calculations at 1, 500, 1000, and 1500 bar over the same temperature range (275−350 K). Each 10262

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Simulation Details. All simulations were performed using GROMACS 3.3.330 with three-dimensional periodic boundary conditions. The Berendsen thermostat and barostat were used to maintain a constant temperature and pressure.31 An anisotropic pressure coupling scheme was used to maintain pressure in systems containing SAMs. The cross section of the SAMs was roughly 3.5 nm × 3.5 nm (see Godawat et al.19 for further details). The Leapfrog algorithm with a time step of 2 fs was used to solve the equations of motion. Electrostatic interactions were calculated using the particle mesh Ewald (PME) method with a real-space cutoff of 1.2 nm. LennardJones interactions were truncated with a plain cutoff of 1.0 nm. Configurations were stored every 0.5 ps for further analysis.

reflects the work needed to remove water molecules between the two solutes and is referred to as the desolvation barrier. We calculated W2(r) in planes parallel to the SAM surfaces located at distances of z ≥ 0.2 nm (see Figure 1A,B). For z < 0.2 nm, the hydrophobic solutes begin to overlap with the atoms of SAM head groups. Details of the water and SAM density profiles shown in Figure 1A,B have been discussed previously.19,20 As pointed out in that work, we observed that water molecules layer near both interfaces; the depletion width is small (less than the size of a water molecule) for both surfaces, but it is larger for the −CH3 surface than for the −OH surface.19 Figure 1D shows that W2(r) near the −OH surface is similar to that in bulk water, with all of the key featurespresence of the CM, SSM, and BARR, along with their locations and heightsbeing unaffected. In contrast, W2(r) shows striking differences near the −CH3 surface (see Figure 1E). Whereas the depth of the CM remains approximately the same, the height of the desolvation barrier decreases significantly as one approaches the −CH3 surface and is negligible next to the −CH3 surface (z = 2.0 Å). Correspondingly, the SSM becomes less stable and is very shallow next to the −CH3 surface. Figure 1F shows the value of the PMF at the desolvation barrier as a function of the distance of the hydrophobic pair from the SAM surfaces and highlights the minimal effect of the −OH surface in contrast to the dramatic effect of the −CH3 surface on hydrophobic association. Simulations by Vaitheeswaran et al.33,34 also showed a destabilization of solvent-separated states (or lowering of the barrier) in the PMFs between hydrophobic solutes adsorbed on the surfaces of water droplets and the inner walls of water-filled carbon nanotubes. Because the hydrophobic interfaces in their work were curved, one would need to renormalize their results using appropriate normalization volumes before the results can be directly compared to each other. Nevertheless, we note that the results are qualitatively similar. Do the effects observed at the pair level translate to the assembly of many hydrophobic solutes? To explore this question, we studied the association of hydrophobic solutes into clusters in bulk water and near interfaces. Figure 2A shows the PMF, W13(r), for the assembly of 13 hydrophobic solutes into a hexagonal close-packed cluster-like configuration along the order parameter, r (lattice distance), as discussed in the Methods section. The key features of W13(r) are qualitatively similar to those of W2(r), with clearly defined CM, SSM, and BARR configurations. As expected, the values of excess free energy involved in the assembly of 13 solutes are considerably larger than those for two solutes. The CM and SSM are stabilized by ∼100 and ∼20 kJ/mol, respectively, relative to the dispersed state. Correspondingly, the desolvation barrier, W13(rBARR) − W13(rSSM), is also large, ∼40 kJ/mol. A similar assembly of 14 slightly larger (methane-like) hydrophobic solutes into a face-centered cubic lattice configuration was studied previously by Rank and Baker.5 The depths of the CM and SSM and the height of the desolvation barrier in their calculations were comparable (if not identical) to those in Figure 2A. To quantify how interfaces affect the hydrophobically driven assembly of more than two solutes, we studied the association of seven hydrophobic solutes in a planar hexagonal closepacked geometry (one solute surrounded by six others, as discussed in the Methods section). The PMFs for association, W7(r), along the lattice separation order parameter, r, for such

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RESULTS AND DISCUSSION How Interfaces Affect Hydrophobic Interactions at Pair and Many-Body Levels. Figure 1 captures the effects of

Figure 1. (A,B) Normalized density profiles of the heavy atoms in the (A) −CH3 and (B) −OH SAM systems (gray, SAM atoms; black, water oxygens), along the z direction calculated in bins of 0.01 nm. The location at which the SAM heavy-atom density is equal to 1% of its bulk density defines the z = 0 reference. (C) Snapshot of the solutes at the −CH3 SAM−water interface (gray, SAM heavy atoms; white, SAM hydrogen atoms; cyan, solutes; red and white wireframe, water molecules). (D,E) Solute−solute PMF, W2(r), calculated in planes parallel to the (D) −CH3 and (E) −OH surfaces. The PMFs are colorcoded consistent with the locations of the planes (vertical dashed lines) in panels A and B. The approximate locations are also indicated by the labels next to the curves. The PMF in bulk water is shown in black. The PMFs are shifted vertically by 2 kJ/mol for visual clarity. (F) Barrier height, W2(0.45 nm), as a function of distance from the −CH3 (red) and −OH (blue) surfaces.

the −OH and −CH3 interfaces on hydrophobic interactions at the pair level. The PMF, W2(r), between two hydrophobic solutes in bulk water (black curves in Figure 1D,E) displays a contact minimum (CM) near 0.3 nm, a solvent-separated minimum (SSM) at 0.6 nm, and a desolvation barrier (BARR) located between the two minima. The hydration of a small hydrophobic solute in water is known to be dominated by an unfavorable (i.e., large negative) entropy, which is frequently interpreted in terms of the ordering of water molecules around the solute. When the solutes associate into a contact pair, some “ordered” water molecules are released, making the contact minimum in free energy favored by entropy. In contrast, the solvent-separated minimum, in which a layer of solvation water molecules separates the two solutes, has been shown to be stabilized by enthalpy.32 The barrier between the two minima 10263

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terized this behavior of water as “sitting at the edge” of a dewetting transition: Water under ambient conditions is close to coexistence with its vapor and is driven even closer to that transition near an extended hydrophobic surface. Further unfavorable perturbation can make the liquid metastable and trigger dewetting. Because the presence of the hydrophobic solutes at the interface can be thought of as providing that additional perturbation, we examined the water density near the solutes in the proximity of interfaces. Structural Origins. Figure 3 shows the in-plane solute−water radial distribution function (rdf) calculated in 0.1-nm-thick

Figure 2. (A) PMF between 13 solutes confined to a threedimensional hexagonal close-packed (hcp) lattice in bulk water, along the lattice spacing order parameter, r. Snapshots of the system in the CM, BARR, and SSM states are shown (cyan, solutes; red, water oxygens; white, water hydrogens). Water molecules close to the solutes (3.5ρbulk) in the space between the solutes (slightly off-center, i.e., above and below the line joining the solute centers). Near the −CH3 surface, however, the hydration shells are less well-defined, as might be expected from the single-solute hydration data in Figure 3B. Importantly, the density of water between the solutes, is significantly reduced (∼1.5ρbulk), consistent with the observed destabilization of the SSM and the lowering (or absence) of the BARR in the assembly of hydrophobic solutes near hydrophobic surfaces. Thermodynamic Origins. The differences in the hydration of hydrophobic solutes near a hydrophobic interface are also highlighted by the thermodynamic data in Figure 5. The hydration of small hydrophobic solutes in water is dominated by a large negative entropy,38 with the free energy of hydration increasing with temperature under ambient conditions.39 This is indeed true for the hydration of a hydrophobic solute in bulk water, as well as near the −OH SAM (see Figure 5). Near the −CH3 SAM, however, the excess chemical potential is approximately insensitive to temperature, suggesting that its hydration is dominated by enthalpy. This result is consistent with calculations of Patel et al.,24 who found that the sizedependent crossover in hydrophobic hydration is absent near hydrophobic surfaces. Patel et al. argued that solvating a hydrophobic solute at an extended −CH3 interface can be viewed simply as the deformation of the already present vapor− liquid-like interface. Thus, near an extended hydrophobic surface, the physics of interface deformation is expected to govern hydrophobic solvation at all length scales, including small ones. Panels C−E of Figure 5 show the temperature dependence of the PMF, W2(r), between two hydrophobic solutes in bulk water and near the −OH and −CH3 SAMs, respectively. The temperature dependence of the free energy at the contact

minimum, W2(rCM), in the three cases is shown in Figure 5B. As expected, in bulk water and near the −OH SAM, the free energy of association [W2(rCM)] decreases (stronger association) as temperature increases, indicating that the association is dominated by favorable (i.e., positive) entropy. Near the −CH3 SAM, the free energy of the CM configuration is only weakly dependent on temperature, suggesting a negligible entropy contribution and a dominant enthalpy contribution to association thermodynamics. Such an enthalpy-dominated association near the −CH3 surface is consistent with thermodynamics of interface formation (or deformation) governing the association, as anticipated from the work of Patel et al.24 Moving Away from the Edge. The structural and thermodynamic data presented in the preceding subsections strongly suggest that the disappearance of desolvation barriers at the −CH3 SAM−water interface is due to the interface being soft and deformable, akin to a vapor−liquid interface. If we destroy this vapor−liquid-like interface by moving the system away from coexistence, do we recover characteristics of “bulklike” solvation and association near a hydrophobic surface? Specifically, do desolvation barriers reappear as water is moved away from coexistence with its vapor? An effective way to move away from vapor−liquid coexistence is to apply hydrostatic pressure. Figure 6 shows how increasing the hydrostatic pressure affects the structural and thermodynamic aspects of hydration and the interactions between hydrophobic solutes near the −CH3 SAM. With increasing pressure, the first hydration shell in the solute−water rdf becomes well-defined, with the first peak height increasing to above 2 at a pressure of 1500 bar. Correspondingly, the solute excess chemical potential becomes temperature-dependent, bulklike, and monotonically increasing with temperature over the range studied here, as shown in the inset of Figure 6A. Similarly, at higher pressures, the PMF, W2(r), recovers bulklike features, including the desolvation barrier and the solvent10265

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Figure 7. (A) Schematic defining the three states for a pair of hydrophobic solutes: state 1, contact; state 2, solvent-separated; state 3, dispersed. (B,D) Time-series data for distance between the solutes near the −CH3 (red) and −OH (blue) surfaces from sample trajectories, where panel D shows a small portion of the trajectory where transitions between the three states are clearly visible. (C) PMF between hydrophobic solutes, W2(r), near the −CH3 (red) and −OH (blue) SAMs. Regions of the PMF corresponding to states 1 and 3 are shown with a gray background, and state 2 is shown with a white background. Panel D has the same background colors. In panel C, the vertical dotted lines mark the cutoffs used in the kinetics analysis following the method of Buchete and Hummer.40

crossing the r1 boundary (Figure 7C), and true 1 → 2 transitions are characterized by trajectories starting in state 1 and crossing the r21 boundary. Similarly, the r3 and r23 boundaries are used to identify the 2 → 3 and 3 → 2 transitions, respectively. However, there are no direct 1 → 3 or 3 → 1 transitions. The mean first passage times (MFPTs) for transitions from state i to state j are reported in Table 1. The reciprocal of the

Figure 6. (A) Solute−water rdf, gSW(r), calculated in a slab of 0.1 nm thickness near the −CH3 surface at four different pressures: 1 bar (red), 500 bar (magenta), 1 kbar (green), and 1.5 kbar (blue). The rdfs are shifted horizontally by 0.1 nm for visual clarity. The inset shows the change in excess chemical potential of the solute with increasing temperature at 1 bar and 1.5 kbar. (B) W2(r) calculated in a plane near the −CH3 surface at four different pressures. The curves are shifted vertically for visual clarity. (C) Contact minimum free energy, W2(0.3 nm), as a function of temperature, relative to its value at 275 K. The color code is the same as in panels A and B.

Table 1. Mean First Passage Times (in Picoseconds) for Transitions between the Three States Defined in Figure 7A, near the −CH3 and −OH Surfacesa

separated minimum (Figure 6B). Also, the free energy at the contact minimum decreases with temperature, indicating that entropy drives the association at higher pressure even near the −CH3 surface (Figure 6C). At about 1500 bar, the entropy of hydrophobic association near the −CH3 surface becomes comparable to that in bulk water under ambient conditions. Kinetics of Hydrophobic Association at Interfaces. From the lowering of the desolvation barriers in association near a hydrophobic surface, one would expect to find a difference in the kinetics of hydrophobic contact formation/ breakage near hydrophobic and hydrophilic interfaces. To explore this connection, we classified the configurations of a pair of hydrophobic solutes into three coarse-grained states, namely, state 1 with r < 0.45 nm (contact), state 2 with 0.45 < r < 0.77 nm (solvent-separated), and state 3 with r > 0.77 nm (dispersed) (see Figure 7), and then analyzed the kinetics of transitions between these states. To identify “true” transitions from one state to another, we followed the transition-pathbased method proposed by Buchete and Hummer,40 which discounts quick ballistic transitions from near the boundaries. In the Buchete and Hummer approach, true 2 → 1 transitions are characterized by trajectories starting from state 2 and

transition

near −OH

near −CH3

→ → → →

5.9 (0.09) 14.3 (0.28) 7.2 (0.09) 58.4 (1.19)

2.6 (0.02) 3.7 (0.03) 3.5 (0.03) 26.1 (0.31)

1 2 2 3 a

2 1 3 2

Errors in estimating the times are shown in parentheses.

MFPT for a given transition characterizes the rate for that transition. Given that the analysis was performed for a collection of very long trajectories (∼800 ns of total data near each surface) and that there were no nonergodicity issues in the sampling of the states, we confirmed that the MPFT data were also consistent with equilibrium concentrations obtained from the integration of solute−solute PMFs (see the Supporting Information). Near the −OH surface, 1 → 2, 2 → 1, and 2 → 3 transitions occur over a time scale of roughly 10 ps, whereas the 3 → 2 transition occurs over a significantly longer time scale. This is expected because the state 3 is a dispersed state and includes all configurations with solute−solute separations larger than 0.77 nm. Naturally, the sampling of state 3 and of 3 → 2 transitions 10266

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shown that solvent degrees of freedom will likely need to be considered in defining such a reaction coordinate.43,44 End-to-End Contact Formation for a Peptide. Do the results presented so far for small hydrophobic solutes translate to more complex biological molecules? The model peptide studied previously by Yeh and Hummer,29 Cys-Ala-Gly-GlnTrp, is ideal for addressing this question because it contains two hydrophobic groups at the ends. Their simulation study was motivated by the experimental work of Lapidus et al.,45 which focused on the loop-closure kinetics in biomolecules. Yeh and Hummer reported the end-to-end PMF as well as contact formation kinetics of the peptide in bulk water using molecular dynamics simulations. There is a difference between their peptide and what we have simulated here: Their peptide was capped with neutral CH3−CO− and −NH−CH3 groups, whereas we used an uncapped version with amino and carboxyl end groups. We simulated the peptide in bulk water as well as at the two interfaces as discussed in the Methods section. In each system, we used 400 ns of simulation data for the analysis presented here. Figure 9E shows the PMF between the ends of the peptide, Wend‑to‑end(r), where r is the distance between the sulfur atom of cysteine and the carbon atom in the tryptophan ring that is closest to the sulfur atom. This PMF is in excellent agreement with that reported by Yeh and Hummer for a capped version of this peptide.29 We note that the PMF includes the water-mediated contribution as well as others, such as the peptide chain entropy and intrapeptide and peptide− surface interactions, and is therefore quantitatively different from that between two monomeric hydrophobic solutes. Nevertheless, as for simple solutes, both in bulk water and at the −OH surface, the PMF shows at least two distinct minima, corresponding to a contact minimum and a solvent-separated state, with a desolvation barrier at a separation of r ≈ 0.5 nm. Importantly, as for simple hydrophobic solutes, the height of the desolvation barrier in the end-to-end PMF of the peptide decreases significantly near the −CH3 surface. Because a number of factors, such as electrostatics, can contribute to the peptide end-to-end PMF, an understanding of how hydrophobic interfaces affect the other interactions in their vicinity is needed to fully understand the quantitative differences between the PMFs, especially the significant stabilization of the contact state at −CH3 surface. However, the broader observation of a reduction in the desolvation barrier for the peptide near an extended hydrophobic surface is consistent with the similar observation for the primitive hydrophobic interaction between small spherical solutes. To perform the kinetic analysis, we divided the peptide configurations into two states: state 1 defined by r < 0.52 nm (closed/contact) and state 2 defined by r > 0.52 nm (open/ extended) (see Figure 9A). Following Buchete and Hummer,40 we defined appropriate boundaries (see Figure 9E) to identify transitions between the two states. In bulk water and near the −OH surface, the MFPT for contact formation is approximately 0.4 ns, whereas near the −CH3 surface, the MPFT drops to about 0.1 ns, suggesting a significant acceleration of contact formation, consistent with lowering of the desolvation barrier. The faster end-to-end contact formation near the −CH3 surface is also evident from the rapid decay of the survival probability, S2(t), of the peptide in the extended state, as shown in Figure 9F.

is expected to be governed by the diffusion of solutes on the surface. The MFPTs for transitions near the −CH3 surface, listed in Table 1, are significantly lower than those for similar transitions near the −OH surface. In particular, the lowering of the MFPT for the 2 → 1 transition by a factor of ∼4 is noteworthy and is qualitatively consistent with the corresponding lowering of the desolvation barrier for association of the solutes. The fact that the MFPT for the 1 → 2 transition is lower than that for the 2 → 1 transition (Table 1) might appear to be in contradiction to the expectation from the barrier heights in the solute−solute PMFs. We note, however, that, for association in a planar geometry, a Jacobian factor of ln(r) is also involved, indicating that the volume of state 2 is larger than that of state 1, thus explaining the observed trend (see the Supporting Information). We note that not only are the contact formation and breakage rates higher near the −CH3 surface, but the 3 → 2 transition, which is predominantly governed by diffusion, is also faster (by a factor of ∼2) near the hydrophobic −CH3 surface. Faster diffusion at the hydrophobic surface is consistent with previous simulations studies showing that water molecules41,42 and polymers35 adsorbed on hydrophobic surfaces diffuse faster than molecules adsorbed on hydrophilic surfaces. An alternate way to quantify kinetics in this system is to measure the survival probability, Si(t), in state i, namely, the probability that the solute pair stays in state i throughout the time interval t, given that it was in that state at time t = 0 (see Figure 8). Consistent with the faster kinetics near the −CH3

Figure 8. Survival probability, Si(t), that the solutes will remain in state i continuously over the time interval t given that they were in that state at t = 0. Si(t) is shown for the three states near the −CH3 (red) and −OH (blue) surfaces.

surface as deduced from the MFPTs, the Si(t) curves decay more rapidly near the −CH3 surface than those near the −OH surface for all states i. Specifically, the faster decay of states 1 and 2 near the −CH3 surface again highlights faster contact formation and breakage events. We note that the observation that state 2 decays faster than state 1 might appear counterintuitive when compared to the MFPT data in Table 1. However, note that state 2 can make two possible transitions (to either state 1 or state 3) and the rates add up. The preceding kinetic analysis using MFPTs and survival probabilities is sufficient to demonstrate the faster kinetics of hydrophobic contact formation/breakage near the −CH3 surface. Connecting the kinetic data quantitatively to the PMFs would require one to obtain the appropriate reaction coordinate for the solvent-mediated interactions. Past work has 10267

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Figure 9. (A) Peptide Cys-Ala-Gly-Gln-Trp in the open (2) and closed (1) states. The hydrophobic ends of the peptide (sulfur atom of cysteine and tryptophan ring) are highlighted using space-filling representation. (B−D) Snapshots of typical structures adopted by the peptide (B) in the bulk and at the (C) −CH3 and (D) −OH interfaces. (E) End-to-end PMF for the peptide in bulk water (black) and at the −CH3 (red) and −OH (blue) surfaces. r is the distance between the sulfur atom of the cysteine end and the closest carbon atom of the tryptophan group. The PMFs are zeroed at r = 1.0 nm. r = 0.52 nm (vertical dashed line) defines the boundary between closed (gray background) and open (white background) states of the peptide. For kinetic analysis, two additional cutoffs, r = 0.42 and r = 0.72 nm, shown by vertical dotted lines, were used to identify transitions between the two states, following the method of Buchete and Hummer.40 (F) Survival probabilities for the open/extended state of the peptide, S2(t).

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CONCLUSIONS Water plays an important role in mediating the assembly of hydrophobic solutes and in the folding and aggregation of proteins and other macromolecules. Surfaces of solutes or biomolecules are typically wet, and the intervening water needs to be removed before direct contacts can be established in the assembled state. The free energy to remove the hydration water molecules appears as a desolvation barrier in the assembly process. Connections between the desolvation barrier and the kinetics of assembly have been highlighted for the association of two hydrophobic plates,46,47 for the collapse of a hydrophobic polymer in water,6 and for assembly in biological systems.48 The ease with which water between the associating solutes can be removed depends on the chemical and physical context (e.g., topography, curvature) provided by the surfaces of the solutes, as well as the presence of other solutes or inhomogenieties in the system. We showed that the desolvation barriers present in the PMF for the assembly of two or more solutes in bulk water decrease significantly when the assembly occurs near an extended hydrophobic surface. This observation can be understood using the perspective of Patel et al.:22 Water under ambient conditions is close to vapor−liquid coexistence and is pushed even closer to that transition when it is present near an extended hydrophobic surface. Thus, a vapor−liquid-like interface is present at a water−hydrophobic surface, and this interface is continuously deformed during the assembly of hydrophobic solutes at the interface. In other words, the proximity to a phase transition makes it easier to remove water molecules between the solutes when association occurs in the vicinity of the hydrophobic surface, resulting in the destabilization of the SSM state and a reduction in the desolvation barrier. We show that moving water away from its coexistence, for example, by increasing the hydrostatic pressure or by making the surface hydrophilic, leads to recovery of the bulklike signatures, including the desolvation barrier, in the solute−solute PMF. Also, thermodynamic calculations, including resolution of the free energy of solvation into entropic and enthalpic contributions, support the notion that assembly at hydrophobic surfaces is governed by the physics of interface deformation. Our studies of hydrophobically driven association were performed approximately in the “infinite-dilution” limit, that is,

for a few solutes on a relatively large surface. When many solutes adsorb onto the surface, there could be complete monolayer or even multilayer coverage. Even in such cases, the vapor−liquid-like nature of the hydrophobic interface is expected to persist, making our key results still relevant. In the “high-concentration” limit, many-body interactions in semithree-dimensional geometries will also need to be considered, presenting a potentially rich and more complex problem for future studies. Although the solute−solute separation might not accurately represent the reaction coordinate for assembly (even for a pair of simple hydrophobic solutes), we found that lowering of the desolvation barrier in assembly near a hydrophobic surface is accompanied by the acceleration of hydrophobic contact formation and breakage. These observations from studies of simple hydrophobic solutes also appear to translate to end-toend hydrophobic contact formation in a model peptide, for which lowering of the desolvation barrier and acceleration of contact formation are observed near a hydrophobic surface. Our results and those of Patel et al.24 suggest that extended hydrophobic surfaces, such as the vapor−liquid interface of water, water−hydrocarbon liquid−liquid interfaces, or interfaces of water with self-assembled monolayers, could serve as excellent platforms for catalyzing hydrophobically driven selfassembly. These ideas are consistent with the experimental observation of a many-fold increase in the rate of fibril formation of amyloidogenic proteins in the presence of an air− water or Teflon bead−water interface.49 Nanoparticles have also been known to provide sites for nucleation of biological aggregation phenomena.50 Our simulation study focused mainly on how an extended hydrophobic interface affects hydrophobic interactions in its vicinity. A deeper understanding of biological assembly or aggregation mediated by interfaces would also require a systematic study of how other relevant interactions (e.g., electrostatic interactions) are affected by the presence of interfaces.

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ASSOCIATED CONTENT

S Supporting Information *

One section on kinetics showing that the rate equations are satisfied in our calculations, and one section containing the PMFs with the Jacobian of transformation included. This 10268

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material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge financial support from NSF Grants CBET-1159990 and DMR-1207411. We thank Pablo Debenedetti, David Chandler, Hari Acharya, and Vasudevan Venkateshwaran for numerous useful discussions. We also thank Gerhard Hummer for insightful comments and suggestions regarding the kinetic analysis presented in this article.

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