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# Necessity of capillary modes in a minimal model of nanoscale hydrophobic solvation

Edited by Shekhar Garde, Rensselaer Polytechnic Institute, Troy, New York, and accepted by the Editorial Board January 31, 2016 (received for review July 11, 2015)

## Significance

Hydrophobic effects, which play a crucial role in many chemical and biological processes, originate in the statistics of microscopic density fluctuations in liquid water. Chandler has established the foundation for a simple and unified understanding of these effects, by identifying essential aspects of water’s intermolecular structure while accounting for its proximity to phase coexistence. Here, we show that coarse-grained models based on this perspective, when constructed to include the statistics of capillary waves at interfaces, can achieve remarkable agreement with results of atomistically detailed simulations. Highly efficient and lacking adjustable parameters, such models hold promise as powerful tools for studying multiscale problems in hydrophobic solvation.

## Abstract

Modern theories of the hydrophobic effect highlight its dependence on length scale, emphasizing the importance of interfaces in the vicinity of sizable hydrophobes. We recently showed that a faithful treatment of such nanoscale interfaces requires careful attention to the statistics of capillary waves, with significant quantitative implications for the calculation of solvation thermodynamics. Here, we show that a coarse-grained lattice model like that of Chandler [Chandler D (2005) *Nature* 437(7059):640–647], when informed by this understanding, can capture a broad range of hydrophobic behaviors with striking accuracy. Specifically, we calculate probability distributions for microscopic density fluctuations that agree very well with results of atomistic simulations, even many SDs from the mean and even for probe volumes in highly heterogeneous environments. This accuracy is achieved without adjustment of free parameters, because the model is fully specified by well-known properties of liquid water. As examples of its utility, we compute the free-energy profile for a solute crossing the air–water interface, as well as the thermodynamic cost of evacuating the space between extended nanoscale surfaces. These calculations suggest that a highly reduced model for aqueous solvation can enable efficient multiscale modeling of spatial organization driven by hydrophobic and interfacial forces.

Hydrophobic forces play a crucial role in biological self-assembly, protein folding, ion channel gating, and lipid membrane dynamics (1⇓⇓⇓⇓⇓⇓⇓–9). The origin and strength of these forces are well understood at extreme length scales, based on the recognition that accommodating an ideal volume-excluding hydrophobe in water carries the same thermodynamic cost as evacuating solvent from the corresponding volume. On the scale of a small molecule like methane, density fluctuations that enable such evacuation are Gaussian-distributed to a very good approximation, even far from the mean (10). Linear response theories, such as Pratt–Chandler theory, can thus be quite accurate for assessing solvation of individual small hydrophobic species.

Solvation at much larger scales is, by contrast, dominated by water’s proximity to liquid–vapor coexistence. Hydrophobic forces involving extended substrates are shaped by the physics of interfaces and are quantified by macroscopic parameters like surface tension. In between these extremes, a rich variety of hydrophobic effects results from the combined importance of nearby phase coexistence and details of intermolecular structure. Capturing this interplay, for instance, near a biological macromolecule, presents a significant challenge for theory.

Lum–Chandler–Weeks (LCW) theory represents the modern understanding of hydrophobic solvation, providing a conceptual and mathematical framework to couple interfacial forces with the short-wavelength density fluctuations that determine solvation of small molecules (10, 11). The theory’s physical perspective has inspired the development of coarse-grained lattice models, whose applications have revealed interesting and general mechanisms for the role of water in hydrophobic self-assembly processes (2, 12, 13). Primitive versions of these models, however, long appeared unable to achieve close quantitative agreement with atomistically detailed simulations (e.g., for the probabilities of extreme number density fluctuations in nanometer-scale probe volumes). This shortcoming motivated the construction of more elaborate models, which include interactions between nonadjacent lattice sites (13) and/or explicit coupling between density fluctuations at short and long wavelengths (12). Quantitative accuracy improved as a result of these additions, but close agreement with detailed simulations remained elusive, despite the expanded set of adjustable parameters.

The prominence of interfacial physics in this understanding of hydrophobic effects suggests that the quantitative success of a coarse-grained model hinges on its ability to accurately capture the natural shape fluctuations of a liquid–vapor interface (1, 14⇓–16). We recently showed that doing so with LCW-inspired lattice models requires closer attention to the statistical mechanics of capillary waves than was previously paid (14, 17). These fluctuations are pronounced in molecular simulations but present in lattice models only for sufficiently weak coupling *ε* between lattice sites [i.e., only at temperatures above the roughening transition *ε* and the macroscopic surface tension γ is nontrivial. This previously unrecognized connection, which is essential for faithfully representing the spectrum of capillary waves, yields a lattice model parameterization that is substantially different from that in previous work (17), and has thus presented, to our knowledge, the first primitive LCW-inspired lattice model that fully respects the statistical mechanics of capillary waves.

Here, we put the lattice model of ref. 17 to a number of exacting tests, which probe the models’ ability to accurately describe density fluctuations on the nanometer scales relevant to protein biophysics. Despite its coarseness and lack of adjustable parameters, this model achieves remarkably close agreement with atomistic simulations, even in scenarios with strong spatial heterogeneity. These tests assess the importance of details we have omitted, such as explicit coupling between short- and long-wavelength density fields.

A careful analysis of our results underscores the interplay of length scales accomplished by the coarse-grained model, highlights the importance of capillary fluctuations, and emphasizes the special environment for solvation presented by extended interfaces. Our ultimate conclusion is that diverse hydrophobic phenomena can be captured quantitatively at a coarse-grained level, with minimal attention to atomic-scale intermolecular structure. It is sufficient to capture the correct physics at extreme length scales and link them with simple excluded volume constraints. We illustrate the promise of such models as practical tools with an application to the association of hydrophobic plates.

## Coarse-Graining Water

Coarse-grained models motivated by the LCW approach separately account for density fluctuations at small and large length scales. As described by Chandler and coworkers in refs. 12 and 13, long-wavelength variations are represented on a lattice with microscopic resolution on the order of a molecular diameter. In the absence of solutes, walls, or constraints, the corresponding binary occupation variables *i*, are governed by a lattice gas Hamiltonian*μ* is the chemical potential for cell occupation and

Regions that are locally liquid-like

We consider ideal hydrophobic solutes, whose influence on the solvent is to exclude it from a volume *v*. Weak, smoothly varying attractive interactions between solute and solvent amount to a small perturbation in this context. The effect of, for example, dispersion forces can therefore be reasonably addressed using perturbation theory (12, 13).

We model such idealized solutes by imposing a constraint of solvent evacuation: the total density within *v* must vanish,*i* that intersect *v*, and

Gaussian fluctuations in the rapidly varying field *m* ideal volume-excluding solutes (12, 13),*m*-component column vector with elements*i*, and *C* is given by

The last two terms in Eq. **5** improve upon previous lattice-based models, providing a computationally tractable yet quantitatively accurate approximation for solute–solute interactions mediated by Gaussian density fluctuations in the surrounding solvent. Starting from a Gaussian field theory (11), the fluctuations may be derived by applying the constraint in Eq. **4** separately to each solute’s excluded volume. Details of this derivation are included in the *Supporting Information*.

Note that the coarse-grained model defined by Eq. **5** has essentially no free parameters—in light of Eq. **2**, there is very little freedom in the choice of *ε* and *δ* (17). We only require the surface tension of water and the bulk radial distribution function **4**.

Below, we compare calculations based on this coarse-grained model with results of atomistic molecular dynamics (MD) simulations. Molecular simulations were performed using the extended simple point charge (SPC/E) model of water (*Methods*). Several of these calculations scrutinize the extreme wings of probability distributions, which were accessed using standard techniques of umbrella sampling (*Methods*).

## Results and Discussion

We have performed calculations that stringently assess the ability of an LCW-inspired lattice model to capture details of hydrophobic solvation at small-, large-, and intermediate-length scales. We focus on characterizing and comparing the statistics of density fluctuations within microscopic probe volumes, in part because of the fluctuations’ direct relevance to solubility. Specifically, we calculate the probability *N* solvent molecules within a probe volume of size *v*. The probability distribution’s extreme value determines the excess chemical potential

In addition to atomistically detailed MD simulations and our LCW-inspired lattice model, we present results for several less sophisticated models. These simpler descriptions lack one or more of the physical ingredients underlying LCW theory and thus shed light on their relative importance. For example, short-wavelength fluctuations can be straightforwardly neglected by studying the conventional lattice gas described by Eq. **1**, which lacks biases from Gaussian fluctuations in

To assess the importance of capillary fluctuations, we examine a different parameterization of Eq. **1**

Together, these calculations explore the interplay between short- and long-wavelength aspects of hydrophobicity. We find in general that a simple LCW-inspired lattice model can describe with surprising accuracy the statistics of density fluctuations observed in detailed molecular simulations. This success is compromised substantially, in most cases, by omitting the effects of short-wavelength fluctuations and/or capillary waves, suggesting that our coarse-grained model contains a minimum of microscopic detail required to quantitatively capture the solvation and association of nanoscale hydrophobic species.

### Statistics of Density Fluctuations in Bulk Water.

We first examine density fluctuations in the simplest aqueous environment (i.e., bulk liquid water). It has been well established by MD simulations that for small probe volumes ^{3}) in this homogeneous setting, *v*, low-density fluctuations are strongly biased by the small chemical potential difference between macroscopic liquid and vapor. *B*, the model of Eq. **5**, when parameterized with attention to capillary fluctuations, does an excellent job reproducing distributions obtained from atomistic MD simulations for nanometer-scale cubic probe volumes, over a very wide range of *N*.

Reversibly decreasing *N* in atomistic simulations from its average value induces formation of a small, roughly cubic cavity that grows to span *v* as *N* water molecules in bulk is *C*) unsurprisingly fails to capture the Gaussian character of *v*, erring by more than 70

Fig. 1*C* also shows results obtained from simulations of the lattice gas with **9**, accurately describing the low-density slope, but not the scale or peak behavior, of

The parameterization of Eq. **1** that we have advocated, which does capture capillary fluctuations at the liquid–vapor interface, significantly improves agreement between atomistic simulations and the conventional lattice gas. Plotted in Fig. 1*C*, the

The success achieved by the full coarse-grained model of Eq. **5** is thus not a transparent consequence of the limiting behaviors motivating its form. Instead, a subtle cooperation of interfacial fluctuations and thermodynamics, together with Gaussian density statistics at the molecular scale, underlies the equation’s accurate prediction for *N*.

LCW-inspired models based on the **5** accurately predicts *N*, where interfacial fluctuations figure prominently (Fig. S1). Ref. 13 outlines two strategies to address this shortcoming. Smearing out discrete interfaces with a numerical interpolation scheme improves predictions substantially but still fails to achieve quantitative accuracy in the extreme tail of

### Density Fluctuations at a Liquid–Vapor Interface.

The interplay of physical factors determining hydrophobic solvation can resolve much differently in spatially heterogeneous environments. To explore basic effects of such nonuniformity, we have examined microscopic density fluctuations at the interface between air and water. Specifically, we consider a cubic probe volume that straddles the plane of a macroscopic phase boundary (i.e., the Gibbs dividing surface between liquid and vapor). The overall shape of

The LCW-inspired lattice model of Eq. **5** again matches atomistic simulation results very well, both near the mean of

Although the shape of *B*. In contrast to our bulk liquid results, neglecting short-wavelength density fluctuations in this case effects only a modest suppression of extreme low-density excursions; the shape and scale of

The long-wavelength density component thus dominates the response of the LCW-inspired model in this spatially heterogeneous scenario, highlighting the key importance of capillary fluctuations at the air–water interface. Evacuation of a probe volume can be inexpensively achieved near a preexisting interface by simply deforming the probe’s shape. Refs. 2 and 21 have also pointed to interfacial deformation as a mechanism for extreme density fluctuations near ideal hydrophobic surfaces and hydrophobic biological molecules.

The statistics of finer scale density variations have nonnegligible quantitative impact on the predictions of Eq. **1** (e.g., reducing the cavitation free energy by roughly 5

### Association of a Hydrophobic Solute with the Interface.

The occupation statistics discussed above hint at thermodynamic driving forces that govern solvation of nonpolar species in interfacial environments. To make this connection explicit, we have computed the excess chemical potential *z* from the air–water interface (with *R*. The estimate clearly neglects the role of capillary fluctuations, which we have argued can be the dominant mode of solvent response in such scenarios.

Atomistic simulations show that Eq. **10** poorly approximates the free-energy profile for an ideally hydrophobic nanometer-scale solute. As shown in Fig. 3 for *z* well beyond the solute’s radius, with **10**.

Our LCW-inspired lattice model, by contrast, faithfully captures these features of

### Density Fluctuations Between Ideal Hydrophobic Plates.

The calculations described so far demonstrate that a simple numerical implementation of the LCW perspective can accurately describe hydrophobic effects involving both microscopic structural response and macroscopic bistability, featuring interfaces that may be preexisting or emergent. This success encourages use of such a model to address more complicated and specific situations that arise in modern biophysics and materials science [e.g., water flow in nanotubes (25), gating of transmembrane ion channels (26), or the development of tertiary and quaternary protein structure (27)]. In each of these phenomena, large-scale atomistic simulations have revealed intriguing functional roles for hydrophobic response as solute configurations rearrange. Toward these frontier applications, we consider as a final example aqueous density fluctuations in a confined hydrophobic environment.

When confined at the nanometer scale or below, water can exhibit physical properties markedly distinct from those of the bulk liquid (28). Interactions with the containing boundaries become a critical consideration, with hydrophilic and hydrophobic walls generating very different structural motifs and susceptibilities. Hydrophobic walls are known to greatly enhance density fluctuations, so that even weak external fields can induce nanoscale drying (3). By poising water near a highly cooperative transition, such constraints can thus be used to engineer switching under minor perturbation, which in turn can sharply modulate functional behaviors like transport and self-assembly (1).

Using atomistic and coarse-grained simulations, we examined a model confinement scenario featuring two ideally hydrophobic parallel plates *D*, immersed in liquid water, as depicted in Fig. 4*A*. As in our other examples, we focus on occupation statistics of a probe volume, here comprising the space between the two plates (a volume of size *B*) and *C*). For comparison, we also show the corresponding probability distributions for a probe volume of the same size and shape placed in homogeneous bulk water, obtained from atomistic simulations.

The presence of these hydrophobic plates is not sufficient to induce drying for either value of *D* considered. The plates are not sufficiently large to induce a vapor layer in their vicinity. The average density between plates is in fact greater than in bulk water, because of the tendency of molecules in dense liquids to pack tightly against hard walls. This modest elevation of

The hydrophobic plates have a much stronger impact on the tails of *T* per molecule would be sufficient to induce drying. This profound impact of hydrophobic confinement on susceptibility to weak perturbations, despite negligible influence on typical fluctuations, has been emphasized in previous work and explored in detail in the context of protein complex formation (2, 27, 28).

These behaviors too are well described by the LCW-inspired lattice model. The slope of

As in the case of a probe volume of comparable size in bulk solvent (Fig. 1), the conventional lattice gas model (with *N*¿, whose weight is nonnegligible. Increasing plate size should facilitate the formation of extended interfaces, eventually making capillary fluctuations the only relevant source of large deviations in *N* (as in the case of the preexisting interface of Fig. 2).

## Conclusions

Hydrophobic effects drive the formation of diverse assemblies in biological and materials systems. Our results suggest that the microscopic basis of these effects is thoroughly described by the physical perspective put forth by Lum, Chandler, and Weeks (31). Statistics of extreme fluctuations that determine solvation thermodynamics can be captured with quantitative accuracy in a lattice model based on the LCW perspective. Doing so, however, requires careful attention to the softness of air–water interfaces, a property lacking in many previous models.

The lattice model defined by Eq. **5** appears to be truly minimal for this purpose. Omitting any of the models’ contributions degrades the close agreement with atomistic simulations that we have demonstrated. Moreover, the model’s parameters are highly constrained by basic experimental observations, namely surface tension, molecular pair correlations, and the spectrum of long-wavelength capillary waves.

Notably absent in our model are several ingredients introduced in previous studies to improve agreement with detailed molecular simulations. We have not introduced explicit coupling between the rapidly and slowly varying components of the density field [i.e., between the lattice variables **5**. Direct interactions among the lattice variables **1** are of course sufficient to stabilize interfaces, the primary and essential role of unbalanced forces in the mean field theories of refs. 31 and 32. Rather, Eq. **5** neglects the specific source of unbalanced attraction attributable to short-wavelength structure, a coupling whose form and strength are not transparent for an associated liquid like water (32). Nor does our omission imply a lack of coupling between **5** permit short-wavelength density fluctuations only in regions that are liquid-like

The lattice gas on which our model is built includes only nearest neighbor interactions. As emphasized in ref. 13, the ground state of this model possesses unphysical degeneracies in the shape of closed, convex interfaces. These degeneracies can be removed by introducing lattice interactions between nonneighboring cells. In a lattice gas below its roughening transition temperature, the impact of these additional interactions is dramatic, because the interactions suppress the only affordable mode of interfacial shape variation. Our studies of lattice gases above the roughening transition temperature suggest that such degeneracies are much less important in the presence of natural capillary fluctuations.

Such additional couplings may be needed to further improve quantitative predictions or to address more complicated scenarios. These goals may also require attention to molecular details that have not yet been incorporated into LCW-inspired models, for instance, concerning the geometry of hydrogen bonds, coordination statistics, or the specific form of interaction potentials. Pratt has made significant advances toward understanding and quantifying the role of these effects in hydrophobicity (29). Incorporating these advances into a lattice-based model poses significant challenges.

In addition to helping establish a conceptual foundation for complex hydrophobic phenomena, our studies advance the more pragmatic goal of faithfully simulating systems that comprise very large numbers of water molecules. This challenge limits, for example, the scale of biomolecular problems that can be examined by simulation without reducing the description of solvent fluctuations to a gross caricature. For the systems we have discussed, our coarse-grained approach reduces the computational cost of representing explicit solvent fluctuations by more than two orders of magnitude (relative to atomistic simulations), while preserving microscopic realism with surprising accuracy. This advantage should become even more significant for very large systems, whose computational burden scales exactly linearly in Eq. **5**. The model we have described could thus enable the study of problems that currently lie outside of the reach of atomistically detailed simulations.

## Methods

All atomistic simulations included 6,912 rigid water molecules, interacting through the SPC/E potential (34), in a periodically replicated simulation cell with dimensions 75 Å

Coarse-grained simulations were performed with an in-house software package that is available upon request. These systems comprised *Supporting Information*. Although complicated, these expressions significantly reduce computation time for solutes that move continuously in space.

## Integrating Out Gaussian Fluctuations in the Presence of Multiple Ideal Hydrophobes

The presence of a solvent-excluding volume *v* introduces a constraint on the total solvent density field,**S1**, and **S2** by a single constraint on the average density in *v*, as in Eq. **4**. The resulting expression for **4** introduces spurious correlations between solutes that never vanish. If *v* is the union of *m* nonintersecting regions,**4** separately to each solute’s excluded volume). The infinite product of delta functions in Eq. **S2** then becomes a product of *m* delta functions,**S2** becomes**5**. For two solutes *α* and *β*, the off-diagonal matrix element **S3** reduces to a simple sum of uncoupled terms,

## Exact Expression for the Overlap Between the Lattice and Spherical Solute

The Hamiltonian given in Eq. **5** requires that we compute

To efficiently compute the overlap between cells of the lattice and spherical objects, we derived an exact formula using ordinary calculus. There are eight distinct cases to consider, each with several subcases. The calculation is a tedious, but straightforward, exercise in enumerating the cases. There are two main, equivalent scenarios. First, if the cubic region lies entirely within the solute, then the overlap is simply

Consider a sphere of radius *R* centered at the origin. We will work through the case that *δ* is the coarse-graining length (the side length of the cubes that form the lattice). The overlap between the sphere and a cubic lattice cell can always be decomposed into contributions from each of the eight octants of the sphere. Without loss of generality, we will select the first octant

With this particular case fully specified, we now carry out the integral

This expression is valid when **S4**.

The integral specifying the overlap can be calculated analytically. We have not been able to simplify the expression into a convenient form, but the analytical result can easily be used within computer code.

The simplified expression for Eq. **S4** is

## Acknowledgments

S.V. and P.L.G. were supported by the US Department of Energy, Office of Basic Energy Sciences, through the Chemical Sciences Division (CSD) of the Lawrence Berkeley National Laboratory (LBNL), under Contract DE-AC02-05CH11231. S.V. was supported by The University of Chicago for later stages of this project. G.R. was supported by a National Science Foundation (NSF) Graduate Research Fellowship. A.H. was supported by NIH Training Grant T32GM008295. We acknowledge computational resources obtained under NSF Award CHE-1048789. Computing resources of the Midway-Research Computing Center computing cluster at The University of Chicago are also acknowledged.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: svaikunt{at}uchicago.edu.

Author contributions: S.V., G.R., A.H., and P.L.G. designed research; S.V., G.R., A.H., and P.L.G. performed research; S.V., G.R., A.H., and P.L.G. contributed new reagents/analytic tools; S.V., G.R., A.H., and P.L.G. analyzed data; and S.V., G.R., A.H., and P.L.G. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. S.G. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1513659113/-/DCSupplemental.

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