Extended Surfaces Modulate Hydrophobic Interactions of Neighboring Solutes

Extended surfaces modulate hydrophobic interactions of neighboring solutes

Amish J. Patela, Patrick Varillyb, Sumanth N. Jamadagnia,c, Hari Acharyaa, Shekhar Gardea,1, and David Chandlerb,1

aHoward P. Isermann Department of Chemical and Biological Engineering, and Center for Biotechnology and Interdisciplinary Studies, Rensselaer Polytechnic Institute, Troy, NY 12180; bDepartment of Chemistry, University of California, Berkeley, CA 94720; and cProcter and Gamble Company, Beckett Ridge Technical Center, West Chester, OH 45069

Edited by B. J. Berne, Columbia University, New York, NY, and approved September 1, 2011 (received for review July 5, 2011)

Interfaces are a most common motif in complex systems. To under- phobic surfaces will bind to and catalyze unfolding of proteins, stand how the presence of interfaces affects hydrophobic phenom- which we predict is relevant in the formation of amyloids and the ena, we use molecular simulations and theory to study hydration function of chaperonins. of solutes at interfaces. The solutes range in size from subnanometer to a few nanometers. The interfaces are self-assembled Models monolayers with a range of chemistries, from hydrophilic to hydro- Molecular Simulations. We simulate the solid–water interfaces of phobic. We show that the driving force for assembly in the vicinity self-assembled monolayers (SAMs) of surfactants (Fig. 1A) with of a hydrophobic surface is weaker than that in bulk water and a range of head-group chemistries, from hydrophobic (−CH3)to decreases with increasing temperature, in contrast to that in the hydrophilic (−OH) (8, 10). To study the size dependence of bulk. We explain these distinct features in terms of an interplay hydration at interfaces, we selected cuboid shaped (L × L × W) between interfacial fluctuations and excluded volume effects—the cavities, with thickness W ¼ 0.3 nm, and side L, varying from physics encoded in Lum–Chandler–Weeks theory [Lum K, Chandler small values comparable to the size of a water molecule to as D, Weeks JD (1999) J Phys Chem B 103:4570–4577]. Our results sug- large as 10 times that size. Thicker volumes will show qualitatively gest a catalytic role for hydrophobic interfaces in the unfolding of similar behavior, but will gradually sample the “bulk” region BIOPHYSICS AND proteins, for example, in the interior of chaperonins and in amyloid away from the interface. We model the monolayers and water formation. with reasonably realistic force fields. But to focus specifically on COMPUTATIONAL BIOLOGY solvent-induced hydrophobic interactions, we consider only idea- binding ∣ hydrophobicity ∣ thermodynamics lized solutes—cavities that simply expel water from the volume they occupy. Solute–solvent and solute–solute interactions be- ydrophobic effects are ubiquitous and often the most signif- yond those of excluded volume forces can further enrich our find- icant forces of self-assembly and stability of nanoscale struc- ings (5, 11, 12).

H CHEMISTRY tures in liquid matter, from phenomena as simple as micelle formation to those as complex as protein folding and aggregation Theoretical Model. To rationalize the simulation results and obtain (1, 2). These effects depend importantly on length scale (3–5). additional physical insights, we developed a model based on Water molecules near small hydrophobic solutes do not sacrifice Lum–Chandler–Weeks (LCW) theory (4). LCW theory incorpo- hydrogen bonds, but have fewer ways in which to form them, lead- rates the interplay between the small length scale Gaussian den- ing to a large negative entropy of hydration. In contrast, hydrogen sity fluctuations and the physics of interface formation relevant bonds are broken in the hydration of large solutes, resulting in at larger length scales, and captures the length scale dependence an enthalpic penalty. For hydrophobic solutes in bulk water at of hydrophobic hydration in bulk water. Near hydrophobic sur- standard conditions, the cross-over from one regime to the other faces, LCW theory predicts the existence of a soft liquid– occurs at around 1 nm (3–6) and marks a change in the scaling of vapor-like interface, which has been confirmed by simulations the solvation free energy from being linear with solute volume to (7–9). being linear with exposed surface area. In bulk water, this cross- We model this liquid–vapor-like interface near a hydrophobic over provides a framework for understanding the assembly of surface, as an elastic membrane (Fig. 1B), whose energetics are small species into a large aggregate. governed by its interfacial tension and the attractive interactions Typical biological systems contain a high density of interfaces, with the surface. The free energy of cavity hydration, μex,is including those of membranes and proteins, spanning the entire related to the probability of spontaneously emptying out a cavity- spectrum from hydrophilic to hydrophobic. Whereas water near shaped volume, V. Such emptying can be conceptualized as a hydrophilic surfaces is bulk-like in many respects, water near two-step process in which interfacial fluctuations of the mem- hydrophobic surfaces is different, akin to that near a liquid–vapor brane can empty out a large fraction of V in the first step, with interface (3–5, 7–9). Here, we consider how these interfaces alter the remaining volume v emptied out via a density fluctuation hydrophobic effects. Specifically, to shed light on the thermo- (Fig. 1C). When v is small, the probability that it contains N dynamics of hydration at, binding to, and assembly at interfaces, waters is well approximated by a Gaussian (8, 13, 14). The cost we study solutes with a range of sizes at various self-assembled of emptying v can then be obtained from the average and the monolayer interfaces over a range of temperatures using molecular simulations and theory. Author contributions: A.J.P., S.G., and D.C. designed research; A.J.P., P.V., S.N.J., and H.A. Our principal results are that, although the hydration thermo- performed research; A.J.P., P.V., and S.N.J. contributed new reagents/analytic tools; A.J.P., dynamics of hydrophobic solutes at hydrophilic surfaces is similar P.V., S.N.J., and H.A. analyzed data; and A.J.P., P.V., S.G., and D.C. wrote the paper. to that in bulk, changing from entropic to enthalpic with increas- The authors declare no conflict of interest. ing solute size, it is enthalpic for solutes of all length scales near This article is a PNAS Direct Submission. hydrophobic surfaces. Further, the driving force for hydrophobi- 1To whom correspondence may be addressed. E-mail: [email protected] or chandler@ cally driven assembly in the vicinity of hydrophobic surfaces is berkeley.edu. weaker than that in bulk and decreases with increasing tempera- This article contains supporting information online at www.pnas.org/lookup/suppl/ ture, in contrast to that in bulk. These results suggest that hydro- doi:10.1073/pnas.1110703108/-/DCSupplemental.

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Fig. 1. Size-dependent hydrophobic hydration at interfaces. (A) A schematic of a cuboidal cavity (green) at the SAM–water interface. The SAM head groups (black and white), alkane tails (gray), and water (red and white, partially cut out for clarity) are shown. (B) A typical configuration of the model membrane, color coded by its distance from the model surface (gray). (C) Important volumes in estimating the free energy, μex, of emptying the probe volume V (green) using the theoretical model. The region above the membrane is the volume B (blue), and the intersection of V and B is v (dark green). (D) Length-scale dependence of the cavity hydration free energy per unit area, μex∕A, in bulk water and at interfaces, at T ¼ 300 K, obtained from molecular dynamics simulations. (E) Theoretical model estimates of μex∕A, near surfaces with different attraction strengths, η.(F) Connecting the microscopic binding free energy of a cavity to an interface, to the macroscopic surface wettability. The cos θ values were obtained from molecular dynamics simulations of a water droplet on SAM γ surfaces (10). Lines are predictions using Eq. 1 with size-dependent LV taken from D.

variance of the number of waters in v, which are evaluated by next to the SAM surfaces are captured well by this model, parti- assuming that water density responds linearly to surface–water cularly for the hydrophobic surfaces (with η around one), where adhesive interactions. the potential UðrÞ closely mimics the effect of the real SAM We tune the strength of the model surface–water attraction, on the adjacent water, and the agreement between theory and UðrÞ, using a parameter η, where η ≈ 1 corresponds to a hydro- simulation is nearly quantitative. For the more hydrophilic SAMs, phobic −CH3 SAM-like surface, with higher values representing the comparison is qualitative, because the simple form for UðrÞ increasingly hydrophilic surfaces. The representation of hydro- does not represent dipolar interactions well. philic surfaces in our theoretical model lacks the specific details Fig. 1D also indicates that μex becomes favorable (smaller) of hydrogen bonding interactions (e.g., between the hydrophilic with increasing surface hydrophobicity. The difference in μex at − η Δμex μex − μex OH SAM surface and water), so comparisons between high- an interface and in the bulk, ¼ int bulk, quantifies the model surfaces and hydrophilic SAM surfaces in simulations hydration contribution to the experimentally measurable free are qualitative in nature. Equations that put the above model energy of binding of solutes to interfaces. Because the solvation on a quantitative footing are given in Appendix and the details of large solutes is governed by the physics of interface formation, of its exact implementation are included in SI Text. both in bulk and at the SAM surfaces, we can approximate Δμex γ − γ − γ 2 ¼ Acð SV SL LVÞ, where Ac ¼ L is the cross-sectional Size-Dependent Hydrophobic Hydration at, and Binding to Interfaces. area, γ is the surface tension, and subscripts SV, SL, and LV, Fig. 1D shows the excess free energy, μex, to solvate a cuboidal indicate solid–vapor, solid–liquid, and liquid–vapor interfaces, cavity at temperature T ¼ 300 K, divided by its surface area respectively. Using Young’s equation, γ ¼ γ þ γ cos θ,we 2 SV SL LV (A ¼ 2L þ 4LW). The quantity μex∕A can be thought of as an rewrite effective surface tension of the cavity–water interface. In bulk Δμex − γ 1 − θ [1] water, this value shows a gradual cross-over with increasing L, ¼ Ac LVð cos Þ; as expected (4, 6). Fig. 1D also shows the length-scale dependence of μex∕A for solvating cavities in interfacial environments. where θ is the water droplet contact angle on the solid surface. Near the hydrophilic OH-terminated SAM, the behavior is simi- Although Eq. 1 is strictly valid only for macroscopic cavities, lar to that in bulk water. However, with increasing hydrophobicity it can be applied to sufficiently large microscopic cavities with ex γ~ of the interface, the size dependence of μ ∕A becomes less pro- a length-scale dependent surface tension, LV, which can be ap- − μex ∕ F nounced and is essentially absent near the CH3 surface, suggest- proximated by bulk A. Indeed, lines in Fig. 1 predicted using ing that hydration at hydrophobic surfaces is governed by Eq. 1 are in excellent agreement with simulation data and indi- interfacial physics at all length scales. cate that the strength of binding increases with surface hydropho- Fig. 1E shows the analogous solvation free energies predicted bicity, as well as with solute size. Hydrophobicity of flat surfaces using the theoretical model. The essential features of solvation is frequently characterized at the macroscale and even using

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Fig. 2. Temperature dependence of μex in bulk water and at SAM–water interfaces for large (L ¼ 3.0 nm) cavities (A and B) and for small (L ¼ 0.5 nm, L ¼ 0.75 nm) cavities (C and D), obtained from simulations (A, C) and from the model (B, D) of Eq. 5.

molecular simulations (15) by measuring the contact angle of water, μex increases with temperature and yields an entropy of a water droplet placed on the surface. Such an approach is not hydration of roughly −25 J · mol−1 ·K−1, characteristic of small feasible for protein surfaces, where topography and chemistry length scale hydrophobic hydration. This negative value is consis- change over subnanometer length scales. Our demonstration of tent with those calculated for spherical solutes of a similar volume the connection between the macroscopic contact angle and the (18). With increasing hydrophobicity, the slope of the μex vs. T microscopic binding free energy suggests a molecular approach curve decreases and becomes negative, indicating a positive en- for characterizing the hydrophobicity of nanoscale surfaces tropy of cavity formation near sufficiently hydrophobic surfaces. (16, 17), for which the contact angle is ill-defined, but the binding Near the most hydrophobic surface (−CH3), the entropy of free energy can be readily measured. hydration of this small cavity is þ30 J · mol−1 ·K−1. Fig. 2D shows that the same behavior is recovered by the Temperature Dependence of Hydration at Interfaces. The differences theoretical model, though the correspondence is clearest at a 0 75 between cavity hydration at interfaces and in bulk are highlighted slightly larger cavity size (L ¼ . nm). Near hydrophilic model BIOPHYSICS AND T μex

most clearly in the dependence of , which characterizes the surfaces, the interface is pulled close to the surface by a strong COMPUTATIONAL BIOLOGY entropic and enthalpic contributions to the free energy. For small attraction, so it is costly to deform it. As a result, small cavities are solutes in bulk, the entropy of hydration is known to be large emptied through bulk-like spontaneous density fluctuations that and negative (18, 19), which reflects the reduced configurational result in a negative entropy of hydration. In contrast, near a space available to the surrounding water molecules. In contrast, hydrophobic surface, the interface is easy to deform, which pro- for large solutes, the entropy of hydration is expected to be vides an additional mechanism for creating cavities. In fact, this positive, consistent with the temperature dependence of the mechanism dominates near sufficiently hydrophobic surfaces, CHEMISTRY liquid–vapor surface tension (20). Fig. 2A shows that μex of large and because the surface tension of water decreases with increas- cuboidal cavities (L ¼ 3 nm) in bulk water indeed decreases with ing temperature, so does μex. Hence, even small cavities have a increasing temperature, although the corresponding hydration positive entropy of hydration near hydrophobic surfaces. The entropy per unit surface area (25 J · mol−1 ·K−1 ·nm−2) is lower continuous spectrum of negative to positive solvation entropies than that expected from the temperature derivative of surface observed in Fig. 2 C and D is thus revealed to be a direct con- tension of water (about 90 J · mol−1 ·K−1 ·nm−2; ref. 20). We sequence of the balance between bulk-like water density fluctua- note that solvation entropies calculated with the Simple Point tions and liquid–vapor-like interfacial fluctuations. Charge Extended (SPC/E) model of water are known to be smal- Fig. 3A shows that, near the hydrophobic CH3-terminated ler than experimental values by about 20% (21). Additionally, SAM, cavity hydration entropies per unit area, Sex∕A, are positive −1 −1 −2 the cavity–water surface tension and its temperature derivative and roughly constant (about 30 J · mol ·K ·nm ) over a for these nanoscopic cavities are expected to be smaller than the broad range of cavity sizes. In contrast, in bulk water, Sex∕A de- corresponding macroscopic values (9). pends on L, and changes from large negative to positive values Fig. 2A also shows that, for large cuboidal cavities (L ¼ 3 nm), with increasing L. The general trend is that, as the physics of μex decreases with increasing temperature not only in bulk water interface formation becomes important (either in the vicinity and near the hydrophilic (−OH) surface, but also near the hydro- of a hydrophobic surface or for increasing cavity sizes), Sex∕A phobic (−CH3) surface, indicating a positive entropy of cavity approaches a positive limiting value. This trend is already well formation. Thus, in all three systems, the thermodynamics of hydration of large cavities is governed by interfacial physics. Although the values of μexðL ¼ 3 nmÞ at 300 K are rather large AB (573 kJ∕mol in bulk water, 568 kJ∕mol at the −OH interface, and 179 kJ∕mol at the −CH3 interface), their variation with temperature, shown in Fig. 2A, is similar in bulk and at interfaces. Fig. 2B shows that this same phenomenology is captured nearly quantitatively by the theoretical model. In the model, the cavity hydration free energies have large but athermal contributions from attractions between water and the model surfaces. The main temperature-dependent contribution to μex is the cost to deform the liquid–vapor-like interface near the surface to accommodate the large cavity. Because the necessary deformation is similar, regardless of the hydrophobicity of the surface, the variation of Fig. 3. Length-scale dependence of the excess solvation entropy per unit μex with temperature is similar as well. surface area for (A) cavities in bulk water and at the −CH3 and −OH SAM– ex Fig. 2C shows the temperature dependence of μ for small water interfaces, and (B) cavities in the model of Eq. 5 near surfaces of cavities (L ¼ 0.5 nm) in bulk and at SAM–water surfaces. In bulk different attraction strengths, η.

Patel et al. PNAS Early Edition ∣ 3of6 Downloaded by guest on September 28, 2021 known in the context of solvating hard spheres of various dia- the solvation free energy increases with increasing temperature. meters in liquid water and other bulk solvents (22, 23). Here, When several small hydrophobes come together, water instead however, we provide theoretical tools and results needed to hydrates the aggregate by surrounding it with a liquid–vapor-like establish how these trends play out in contexts of neighboring interface. The corresponding solvation free energy scales as the surfaces. The length scale at which entropy crosses zero, LS, surface area and decreases with increasing temperature. can serve as a thermodynamic cross-over length. In bulk water, Thus, the driving force for assembly of n small solutes (each of ≈ 1 8 0 2 ex∕ μex LS . . nm. The behavior of S A is qualitatively similar surface area A1, volume v1, and solvation free energy of 1;bulk) − ≈ 1 3 0 4 at the OH surface, with LS . . nm. Although the nu- into a large aggregate (with surface area An and volume nv1)in merical value of LS may depend on the shape of the cavity and bulk water is well approximated by on solute–water attractions for nonidealized hydrophobes, the Δμex γ − μex −1∕3 − 1 μex [2] trend in entropy should not. bulk ¼ bulkAn n 1;bulk ¼½Cn n 1;bulk; Fig. 3B shows that our implementation of LCW ideas recovers ∼ γ 2∕3∕μex γ many of the observed trends, with solvation entropy being every- where C ð bulkv1 1;bulkÞ and bulk is a curvature-corrected where positive for the smallest attraction strength η and a ther- effective surface tension (top curve of Fig. 1D). As the surface modynamic cross-over length of just under 1 nm emerging for the tension decreases with increasing temperature, so does the free more hydrophilic model surfaces, similar to that in bulk water. energy to hydrate nanometer-sized aggregates (Fig. 2A). How- Nevertheless, the agreement between Fig. 3 A and B is somewhat ever, the free energy to individually hydrate the small solutes qualitative, mostly as a result of the crude form of UðrÞ used to increases with temperature (Fig. 2C), resulting in a larger driving model hydrophilic surfaces. force for assembly. Conversely, although the driving force for as- Δμex sembly, bulk, is large and negative (favorable) at ambient con- Thermodynamics of Binding to, and Assembly at, Hydrophobic Sur- ditions, it decreases in magnitude with decreasing temperature faces. In the preceding sections, we have examined the hydration (upper portion of Fig. 4) and can even change sign at a sufficiently behavior of single, isolated, idealized cavities near flat surfaces low temperature. When adapted to particular systems, Eq. 2 can, and in the bulk. We now consider the consequences of our ob- with remarkable accuracy, explain complex solvation phenomena servations on hydrophobically driven binding and assembly, sum- like the temperature-dependent aggregation behavior of micelles marized schematically in Fig. 4. (24) and the cold denaturation of proteins (2). Fig. 4 indicates that, although the binding of both small and In the presence of a hydrophobic surface, on the other hand, large solutes (or aggregates) to hydrophobic surfaces is highly we have found that interfacial physics dominates at all length favorable, their thermodynamic signatures are different. Binding scales (Figs. 2 and 3). As a result, the driving force for assembly Δμex 2 of small solutes is entropic and becomes more favorable with at interfaces, int, does not scale as in Eq. , but is instead given increasing temperature, whereas binding of large solutes is en- by thalpic and depends only weakly on temperature. For example, Δμex γ − ∼ −1∕2 − 1 μex [3] for the L ¼ 3 nm solute, the binding free energy is 386 kJ∕mol int ¼ intðAn nA1Þ ½n n 1;int; at 280 K and increases to 397 kJ∕mol at 320 K. γ Fig. 4 also highlights the differences in the thermodynamics where int is the effective surface tension at the interface (the D E γ of hydrophobically driven assembly at interfaces and in bulk, lower curves of Fig. 1 and ). Because int decreases with in- inferred from our length-scale dependence studies. In bulk, the creasing temperature (Fig. 2), so does the hydration contribution solvation of many small, isolated hydrophobes scales as their ex- to the driving force for assembly at a hydrophobic surface, in con- cluded volume. Accommodating small species inside the existing trast to that in bulk. hydrogen-bonding network of water imposes an entropic cost, so The free energy barrier between disperse and assembled states is also expected to be very different in bulk and near hydrophobic surfaces. In bulk, the dispersed state has no liquid–vapor-like interface, whereas the assembled state does. The transition state is a critical nucleus of hydrophobic particles that nucleates the liquid–vapor-like interface. The nucleation barrier can be high and dominates the kinetics of hydrophobic collapse of idealized hydrophobic polymers (25–27) and plates (28). In contrast, at ambient conditions, we expect aggregation near hydrophobic surfaces to be nearly barrierless because an existing liquid–vapor-like interface is deformed continuously between the disperse and assembled states. Finally, and most importantly, we find that, for large aggregates, the driving force of assembly is weaker near interfaces than ex in bulk. In the limit of large n, the terms nμ1 dominate both at interfaces and in bulk (Eqs. 2 and 3), and the results in Fig. 1D μex μex show that 1;int < 1;bulk. The nontrivial behavior of the driving forces and barriers to assembly at interfaces should be relevant in biological systems where hydrophobicity plays an important role. Experiments have shown that hydrophobic surfaces bind and facilitate the unfolding Fig. 4. Schematic illustrating the thermodynamics of binding and assembly. of proteins, including those that form amyloids (29–31). Our re- The points represent free energies of solvating small objects individually sults shed light on these phenomena and suggest that large hydro- (Left) and in the assembled state (Right), in bulk (Top) and at a hydrophobic T T phobic surfaces may generically serve as catalysts for unfolding interface (Bottom), at a lower (blue, L) and a higher (red, H) temperature proteins (32), via solvent-mediated interactions. Indeed, simula- near ambient conditions. Assembly: The driving force for assembly at hydrophobic interfaces is smaller than that in bulk, and is enthalpic, decreasing tions show that the binding of model hydrophobic polymers to with increasing temperature, unlike in bulk. Binding: The driving force for hydrophobic surfaces is accompanied by a conformational rear- binding small objects to a hydrophobic surface increases with temperature, rangement from globular to pancake-like structures (33). Such so it is entropic, whereas for large objects, it is enthalpic. conformations can further assemble into secondary structures,

4of6 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1110703108 Patel et al. Downloaded by guest on September 28, 2021 such as β-sheets (30–32, 34), and we predict that the solvent contribution to this assembly at the hydrophobic surface will be governed by interfacial physics. This behavior implies that manipulating the liquid-vapor surface tension, either by changing the temperature or by adding salts or cosolutes, will allow one to manipulate the driving force for assembly. We further speculate that the catalysis of unfolding by hydrophobic surfaces may play a role in chaperonin function (35). The interior walls of chaperonins in the open conformation are hydrophobic and can bind misfolded proteins, whereupon their unfolding is catalyzed (36, 37). Subsequent ATP-driven conformational changes render the chaperonin walls hydrophilic (35, 36). As a result, the unfolded protein is released from the wall, as the free energy for a hydrophobe to bind to a hydrophilic surface is much lower than that to bind to a hydrophobic one (Fig. 1D). Our results also provide insights into the interactions between biomolecules and nonbiological hydrophobic surfaces, such as those of graphite and of certain metals, which have been shown – to bind and unfold proteins (38, 39). Such interactions are of Fig. 5. Power spectrum of the instantaneous liquid vapor interface at T ¼ 300 K. A liquid–vapor interface was simulated using a 24 × 24 × 3 nm3 interest in diverse applications including nanotoxicology (40) and 3 slab of SPC/E water in a periodic box of size 24 × 24 × 9 nm and the instan- ~ biofouling (39). taneous interface configuration, hðx;yÞ, and its Fourier transform, hðkÞ, were Collectively, our findings highlight that the magnitude and evaluated as in ref. 53. The power spectrum of our simulated instantaneous temperature dependences of the driving forces for assembly near interface is in good agreement with the capillary-wave theory prediction ~ 2 2 hydrophobic surfaces are different from that in bulk and near (hjhðkÞj i ∼ 1∕βγk ) for wavevectors smaller than approximately 2π∕9 Å. For hydrophilic surfaces. Experimental measurements of the thermo- larger wavevectors, the power spectrum is sensitive to molecular detail (i.e., the coarse-graining length, ξ, used to define the intrinsic interface), dynamics of protein folding have been performed primarily in −1 as expected (54). Fitting the ξ ¼ 2.0 Å data in the range 0.01 Å < k < bulk water (41). Although many experiments have probed how −1 0.3 Å yields γ ¼ 62.0 0.5 mJ∕m2, in reasonable agreement with the

interfaces affect protein folding, structure, and function (29, 42), BIOPHYSICS AND experimental value of 72 mJ∕m2 and some simulated values of the SPC/E to the best of our knowledge, there are no temperature-dependent 2 surface tension (e.g., 63.6 1.5 mJ∕m ; ref. 55), but not others (e.g., COMPUTATIONAL BIOLOGY thermodynamic measurements of self-assembly at interfaces. 52.9 mJ∕m2; ref. 54). Extensions of atomic force microscope experiments of pulling a hydrophobic polymer attached to a surface (43) have the potential contain modes with wavevectors below 2π∕9 Å. At any instant to quantify the thermodynamics of hydrophobic interactions in in time, part of V can be empty due to an interfacial fluctuation. interfacial environments described here. We hope that our results The number of waters in the remaining volume, v, fluctuates, and will motivate such measurements. we denote by PvðNÞ the probability that v contains N waters. We CHEMISTRY thus estimate the free energy for emptying V completely to be Appendix Z Simulation Details. Our simulation setup and force fields are μex − D −1 −βH½hðx;yÞ 0 [5] similar to those described in refs. 8 and 10. Simulations were per- ðVÞ¼ kBT ln hZ e Pvð Þ; formed in the canonical (NVT) ensemble with a periodic box (7 × 7 × 9 nm) that has a buffering liquid–vapor interface at the where Z ¼ ∫ Dh expf−βH½hðx;yÞg is the partition function of top of the box, for reasons explained in ref. 9. It has been shown the membrane. The volume v depends on the interfacial config- that free energies obtained in the above ensemble are indistin- uration hðx;yÞ—i.e., v ¼ v½hðx;yÞ. guishable from those obtained in the isothermal-isobaric ensem- It is known that PvðNÞ is well approximated by a Gaussian ble at a pressure of 1 bar (44). We have chosen the SPC/E model when v is small (8, 13, 14). If water were far from liquid–vapor of water (45) because it adequately captures experimentally coexistence, then PvðNÞ would also be close to Gaussian for arbi- known features of water, such as surface tension, compressibility, trarily large v. The fact that water at ambient conditions is near and local tetrahedral order, which play important roles in the liquid–vapor coexistence, and that there is a liquid–vapor-like in- hydrophobic effect (5). Electrostatic interactions were calculated terface near the SAM, is captured by the additional interfacial using the particle mesh Ewald method (46), and bonds in water energy factor Z−1 expf−βH½hðx;yÞg in Eq. 5. The net result is that were constrained using SHAKE (47). Solvation free energies the thermal average of Eq. 5 is dominated by interface configura- were calculated using test particle insertions (48) for smaller tions where v is small, so that even at ambient conditions, we can cavities (L < 1 nm) and the indirect umbrella sampling method approximate (9, 44) for larger cavities. ≈ 2πσ −1∕2 − − 2∕2σ PvðNÞ ð vÞ exp½ ðN hNivÞ v; Theoretical Model. We model the liquid–vapor-like interface near hydrophobic surfaces as a periodic elastic membrane, z ¼ hðx;yÞ, where hNiv is the average number of waters in v and σ δ 2 with an associated Hamiltonian, H½hðx;yÞ: v ¼hð NÞ iv is the variance. We estimate these by noting that Z Z the solvent density responds linearly to the attractive potential, γ ∞ r C ∇ 2 ρ r [4] Uð Þ, in the volume occupied by the water, B, depicted in Fig. 1 H½hðx;yÞ ¼ j hðx;yÞj þ ℓUð Þ : (13, 49, 50). Hence, x;y 2 z h x;y ¼ ð Þ Z Z Z Z γ – ≈ ρ − χ r r0 β r0 σ ≈ χ r r0 Here, is the experimental liquid vapor surface tension of water, hNiv ℓv ð ; Þ Uð Þ; v ð ; Þ; rϵ r0ϵ rϵ r0ϵ ρℓ is the bulk water density, and UðrÞ is the interaction potential v B v v 0 0 2 0 between the model surface and a water molecule at position where χðr;r Þ¼ρℓδðr − r Þþρℓ½gðjr − r jÞ − 1: r ¼ðx;y;zÞ. The square-gradient term in Eq. 4 accurately captures the energetics of interfacial capillary waves only for wavelengths Here, gðrÞ is the oxygen–oxygen radial distribution function of larger than atomic dimensions (Fig. 5), so we restrict hðx;yÞ to water (51).

Patel et al. PNAS Early Edition ∣ 5of6 Downloaded by guest on September 28, 2021 – r r – The surface water interaction is modeled by a potential, Uð Þ, Utailð Þ, similarly captures the alkane tail water interaction, mod- that closely mimics the attractive potential exerted by the −CH3 eled as a uniform half-space of OPLS/UA CH2 LJ interaction ρ ζ SAM on water: sites of volume density tail at a distance below the head groups. The parameters R0, ζ, μ , and ρ are dictated by the geometry r r η r r head tail Uð Þ¼Uwallð Þþ Uheadð ÞþUtailð Þ: of the SAM (see SI Text for details). r The first term, Uwallð Þ, is a sharply repulsive potential in the ACKNOWLEDGMENTS. The authors thank Steve Granick, Bruce Berne, and region z < R0 that captures the hard-core exclusion of a plane Frank Stillinger for providing helpful comments on an earlier draft. A.J.P. and of head groups at z ¼ 0 with hard-sphere radius R0. The second P.V. were supported by National Institutes of Health Grant R01-GM078102- r η – 04. S.G. gratefully acknowledges partial financial support of the National term, Uheadð Þ, is scaled by , and captures the head-group water Science Foundation (CBET-0933169, CBET-1134341, NSF-CBET-0967937) interaction, modeled as a plane of CH3 Lennard–Jones (LJ) in- 0 μ grants. D.C. was supported by the Director, Office of Science, Office of Basic teraction sites at z ¼ with an area density of head. The sites are Energy Sciences, Materials Sciences and Engineering Division and Chemical parametrized with the Optimized Potential for Liquid Simula- Sciences, Geosciences, and Biosciences Division of the US Department of tions United Atom (OPLS/UA) force field (52). The final term, Energy under Contract DE-AC02-05CH11231.

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