How Interfaces Affect Hydrophobically Driven Polymer Folding - The

Dec 4, 2008 - Srivathsan Vembanur , Vasudevan Venkateshwaran , and Shekhar Garde. Langmuir 2014 30 (16), 4654-4661. Abstract | Full Text HTML | PDF ...
0 downloads 4 Views 2MB Size
Download PDF
J. Phys. Chem. B 2009, 113, 4093–4101

4093

How Interfaces Affect Hydrophobically Driven Polymer Folding† Sumanth N. Jamadagni, Rahul Godawat, Jonathan S. Dordick, and Shekhar Garde* The Howard P. Isermann Department of Chemical & Biological Engineering and Center for Biotechnology and Interdisciplinary Studies, Rensselaer Polytechnic Institute, Troy, New York 12180 ReceiVed: July 23, 2008; ReVised Manuscript ReceiVed: September 23, 2008

Studies of folding-unfolding of hydrophobic polymers in water provide an excellent starting point to probe manybody hydrophobic interactions in the context of realistic self-assembly processes. Such studies in bulk water have highlighted the similarities between thermodynamics of polymer collapse and of protein folding, and emphasized the role of hydrationswater structure, density, and fluctuationssin the folding kinetics. Hydrophobic polymers are interfacially activesthat is, they prefer locations at aqueous interfaces relative to bulk watersconsistent with their low solubility. How does the presence of a hydrophobic solid surface or an essentially hydrophobic vapor-water interface affect the structural, thermodynamic, and kinetic aspects of polymer folding? Using extensive molecular dynamics simulations, we show that the large hydrophobic driving force for polymer collapse in bulk water is reduced at a solid alkane-water interface and further reduced at a vapor-water interface. As a result, at the solid-water interface, folded structures are marginally stable, whereas the vapor-liquid interface unfolds polymers completely. Structural sampling is also significantly affected by the interface. For example, at the solid-water interface, polymer conformations are quasi-2dimensional, with folded states being pancake-like structures. At the vapor-water interface, the hydrophobic polymer is significantly excluded from the water phase and freely samples a broad range of compact to extended structures. Interestingly, although the driving force for folding is considerably lower, kinetics of folding are faster at both interfaces, highlighting the role of enhanced water fluctuations and dynamics at a hydrophobic interface. I. Introduction The understanding of hydrophobic effects has emerged as a multidimensional challenge for theoretical, computational, and experimental studies. The direct connection of hydrophobicity as a driver for many self-assembly processes in water1-4 has motivated studies in the space of temperature5-9 and pressure10-12 and in the presence of additives, such as salts,13-16 cosolvents, and osmolytes.17-21 At a fundamental level, the lengthscale dependence of hydrophobic effects22-25 as well as the characterization of manybody effects in hydrophobicity15,26-28 (beyond the pair potentials of mean force between two molecular hydrophobes in water) are also of great interest. To this end, hydrophobic polymers have been proposed as excellent model systems to investigate manybody effects relevant to folding-unfolding of more realistic solutes with conformational flexibility.29 Recent simulation studies30 highlight a remarkable similarity of the temperature-dependent parabolic profiles for the free energy of unfolding of such polymers with proteins. Salt and osmolyte effects on the thermodynamic stability of collapsed polymers15,18 also shed light on the molecular mechanism of the action of additives on hydrophobically driven assembly. The role of different lengthscales at play in collapse processes, the corresponding area-volume dependencies, and the differences in the nature of hydration of the polymer interface (compared to that of a single monomer) have also been discussed.30 Although interfaces of water with vapor, liquids, and solids have been the subject of theoretical and experimental attention †

Part of the special section “Aqueous Solutions and Their Interfaces”. * Author to whom correspondence should be addressed. Electronic mail: [email protected], http://www.rpi.edu/∼gardes.

for long,31-35 most studies of hydrophobicity have focused on characterizing them in bulk aqueous solutions. How does the presence of a vapor-liquid or a solid-liquid interface affect hydrophobic interactions? Hydrophobic solutes are poorly soluble in water and will be driven out of solution to an interface, especially if the interface is also hydrophobic. Excess chemical potentials of molecular hydrophobic solutes show a minimum at a water-octane interface36 indicating their affinity for the interface. Experimental data as well as calculations based on simulations and scaled particle theory indicate preferential adsorption of simple alkanes at a water vapor-liquid interface.37 However, the solutes considered in these studies have been mostly rigid, with no conformational degrees of freedom. Since many biological processes and biotechnology applications occur at the interface of materials, it is crucial to understand hydrophobicity in interfacial environments. The stability and structure of proteins and macromolecules on solid supports such as carbon nanotubes,38 silica nanoparticles,39 at chromatographic surfaces,40 and on membrane surfaces41,42 are areas of active research. Experimental43 and lattice model based simulation studies44,45 indicate that proteins unfold and aggregate at hydrophobic air-water and oil-water interfaces. Given that hydrophobic interactions play an important role in folding, its quantification in various interfacial environments will provide a framework for interpretation of those experimental results. Here, we report results of extensive molecular dynamics (MD) simulations of folding-unfolding of a model hydrophobic polymer in bulk water and at two interfaces: a hydrophobic solid alkane-water interface and the vapor-liquid interface of water. In bulk water, Athawale et al.30 show that this polymer folds rapidly into an ensemble of compact globular states, driven

10.1021/jp806528m CCC: $40.75  2009 American Chemical Society Published on Web 12/04/2008

4094 J. Phys. Chem. B, Vol. 113, No. 13, 2009

Jamadagni et al.

TABLE 1: Lennard-Jones Parameters and Partial Chargesa atom

σ, nm

, kJ/mol

q, e

OW HW monomer CH3 CH2

0.3169 0 0.4400 0.4009 0.4009

0.6502 0 0.8500 0.9497 0.3891

-0.8476 +0.4238 0 0 0

a OW and HW refer to water oxygen and hydrogens, respectively. The bond length rOH ) 0.1 nm, and the angle θHOH ) 109.5°.46 CH3 and CH2 are united atom groups in the solid alkane surface.47

strongly by the hydration component of the free energy. By resolving the free energy into its components, we show that the hydration free energy driving force is significantly reduced at interfaces compared to that in bulk water leading to the destabilization of folded states. The tendency of polymer monomers (which are hydrophobic) to be interfacially active localizes them to a relatively narrow region at the interface. This makes the folding-unfolding roughly 2-dimensional, and correspondingly, the ensemble of structures (especially of the folded state) sampled by the polymer distinctly different from those in the bulk. Interestingly, changes in the magnitude of driving force for folding, coupled with a different structural and dynamic environment at an interface results in faster foldingunfolding kinetics. Collectively, our study provides a framework for understanding thermodynamic, structural, and kinetic aspects of conformationally flexible molecules at various interfaces, especially when their assembly is hydrophobically driven. II. Methods We performed extensive molecular dynamics simulations to study thermodynamic, structural, and kinetic aspects of conformational transitions of a hydrophobic 25-mer (denoted as CG-25, LJ by Athawale et al.30) in bulk water and at vapor-liquid and solid-liquid interfaces. The linear polymer is a freely jointed chain of 25 monomer Lennard-Jones solutes (see Table 1 for parameters) with harmonic bond potentials [Ubond ) 1/2kb(l - l0)2, where kb ) 60 702 kJ/(mol nm2) and l0 ) 0.25 nm]. In the extended state, the polymer is ∼5.1 nm long and roughly represents a coarse-grained version of a freely jointed C50 alkane, with each monomer of the polymer representing an ethanelike unit. Water molecules were represented explicitly using the extended simple point charge (SPC/E) model46 (see Table 1). To study the polymer in bulk water, it was solvated in a cubic box of length ∼5 nm (≈4100 water molecules or more) and simulated in the NPT ensemble at 1 atm and different temperatures ranging from 280 to 360 K in steps of 20 K. Simulations using a larger box gave similar results. To study the polymer at a vapor-liquid (VL) interface of water, we used a slab of water with X-Y cross-sectional area of 6 nm × 6 nm and thickness (along the z-axis) of 4 nm simulated in the NVT ensemble. The z-dimension of the simulation box was 12 nm. If the polymer is initially placed in the bulk region (near z ) 0), it diffuses to the interface over a nanosecond time scale and then remains primarily at the interface. Alternatively, simulations can be started by placing the polymer just above the liquid surface and then allowing it to relax. To generate the solid-liquid (SL) interface a solid surface with cross-sectional area of 6.2 nm × 6.0 nm was created using 270 hexagonally packed linear 11-mer alkane chains with an interchain spacing of 4 Å (see Figure 1). This arrangement

Figure 1. Hydrophobic 25-mer (cyan) at a solid alkane-water interface (gray). The 3D periodic box is shown. For clarity, some water molecules are shown in wireframe representation (white, hydrogens; red, oxygens). The dimensions of the box are 6.2 nm × 6.0 nm × 6.1 nm.

approximates an alkane crystal with the chain axis parallel to the z-axis of the simulation box. Position constraints (4000 pN/ nm) were applied on the central (i.e., sixth) atom of each alkane chain to prevent deformation of the solid surface. The alkanes were represented using the united atom representation, which includes nonbonded Lennard-Jones interactions (see Table 1) as well as bond-length, angle, and dihedral interactions as given in the work of Mondello et al.47 An approximately 5 nm thick slab of water, containing one polymer molecule was then placed on the solid surface and the system equilibrated with an anisotropic barostat such that the normal pressure, pz, is 1 atm. The SL system with 3D periodic boundary conditions, is shown in Figure 1. Similar to that in the vapor-liquid interface system, the polymer placed in the bulk diffuses to the interface and remains primarily at the interface. Umbrella sampling was performed along the radius of gyration, Rg, reaction coordinate to calculate the conformational free energy in the bulk and at the two interfaces (refer to the work of Athawale et al.30 for details). A total of 20 separate windows (each 2 ns long) were used covering Rg ) 0.45-1.4 nm, separated by 0.05 nm. Simulations were performed at five different temperatures (280, 300, 320, 340, and 360 K). The temperature and pressure were maintained using a Berendsen thermostat and barostat,48 respectively in all the simulations. Electrostatics were handled using the Particle Mesh Ewald algorithm.49 All simulations were performed using the molecular dynamics package, GROMACS,50,51 modified to perform umbrella sampling on Rg. The SHAKE algorithm was used to constrain bonds in water molecules. Configurations were stored every picosecond for further analysis. To study the kinetics of hydrophobic collapse, a set of 100 additional simulations were carried out for bulk as well as interfacial systems (at 300 K) each starting from different extended structures with no restraining umbrella potential. To

Hydrophobically Driven Polymer Folding

Figure 2. Potential of mean force, W(Rg), for conformational sampling of the polymer in bulk water and at solid alkane-water and vapor-water interfaces. W(Rg) is reported with reference to its value at Rg ) 1.4 nm, at which polymer conformations are fully extended.

obtain different extended starting structures, configurations were taken every 20 ps from the umbrella simulation window of Rg ) 1.40 nm in each system. III. Results and Discussion A. Thermodynamics and Structure of the Polymer in Bulk Water and Interfacial Environments. A flexible polymer can sample an ensemble of structures in solution with the probability of observing a given conformation depending on its free energy. To characterize conformational sampling, we selected the radius of gyration, Rg, as a reasonable onedimensional reaction coordinate, which allows us to distinguish between compact collapsed (or folded) states (having small Rg values) and extended ones having larger Rg values. Other choices of reaction coordinates, such as the end-to-end distance, exposed surface area, or number of intrapolymer contacts exist. As shown below, the choice of Rg as a reaction coordinate suffices for the present purposes. Figure 2 shows the potential of mean force (PMF) along Rg of the polymer in bulk water and at the two interfaces. In bulk water, hydrophobic interactions drive the collapse of the polymer into compact states (Rg ≈ 0.55 nm) which are favored by a large free energy difference of about 27 kJ/mol relative to extended states, as shown previously.30 The PMF reduces monotonically as the polymer folds from most extended to compact states, and we do not observe a barrier along this 1D projection. For a larger (and purely repulsive) polymer in a coarse-grained model of water, ten-Wolde and Chandler find that formation of a sufficiently large vapor-liquid like interface is the rate limiting step for folding. Miller et al.52 have used the string method53 to compute the minimum free energy pathway for hydrophobic collapse and confirm the importance of lengthscale-dependent hydration in kinetics of polymer collapse. As discussed by Goel et al.,54 the size of the polymer as well as the strength of polymer-water interactions can affect the local water density substantially. We discuss this aspect further in the context of kinetics of folding later. How does the presence of an interface change the conformational free energy profile and structure of the polymer? The PMFs for conformational sampling of the polymer at SL and

J. Phys. Chem. B, Vol. 113, No. 13, 2009 4095 VL interfaces shown in Figure 2 are different from that in bulk water both qualitatively and quantitatively. At the SL interface, the free energy minimum is stabilized by only 20 kJ/mol and is shifted to higher Rg (≈0.70 nm). The minimum is also not as sharply defined as in bulk water but shows a broad basin in the range of 0.70 < Rg < 0.90 nm. The nature of the PMF profile is dramatically different at the VL interface. When compared to that in bulk water, the VL interface destabilizes the compact states in favor of more extended states. Overall, the PMF is essentially flat with a very shallow and broad minimum (≈6 kJ/mol in magnitude) over the Rg range from 0.6 nm to roughly 1 nm. There are no apparent barriers for conformational sampling for Rg > 0.60 nm, and in absence of the umbrella potential, a broad sampling of this region is indeed observed. Thus, PMF profiles indicate that the polymer folds into an ensemble of compact folded states in bulk water, and into less stable semicompact states at the SL interface, and freely samples a broad range of conformations at the VL interface with no dominant conformational basin. As shown later, in vacuum, the intramolecular attractions lead to folding of the polymer at room temperature into compact states. Thus, while bulk water and vacuum environments are “bad solvents” for the solvation of this polymer, the SL interface acts as a better solvent, whereas the VL interface acts like a “good solvent” allowing broad sampling of a range of conformations. Polymer structures at the PMF basin are different in bulk water and at interfaces. In bulk water, the polymer collapses into an ensemble of compact globular structures (Figure 3a). Although the internal packing within these structure is not perfect, they are compact, have a relatively high number of intrapolymer contacts and minimize the exposure to solvent. Interfaces offer an inhomogeneous environment with a rapidly varying solvent density in the z direction. As discussed previously,36,37,55 individual monomers (or molecular hydrophobic solutes) are interfacially active and reside preferentially at the interface (i.e., adsorb). As a result, the polymer, comprising hydrophobic monomers is also interfacially active, and remains mostly in the plane of the interface and samples quasi-2-dimensional conformations. The degree of pinning of the polymer to the interface as well as whether the quasi-2dimensional conformations are folded (pancake-like) or unfolded are different at the SL and VL interface as discussed below. Figures 3b and c show representative pictures of the polymer in the free energy basin at the SL interface. The side views (bottom panels) show that each monomer of the polymer is in direct contact with the solid surface, and the top views (top panels) show that these conformations are compact, pancakelike (S-shaped or spiral) structures that attempt to maximize the intrapolymer contacts. Similar type of pancake-like structures of polymers near a solid surface in a bad solvent have been observed in a recent simulation study.56 At the VL interface (Figure 3d), consistent with the PMF profile, the polymer samples a broad range of conformations from compact to fully extended ones. Although these conformations are approximately planar, the pinning of each monomer (and therefore, of the polymer) to the VL surface is much softer than at the SL interface. Frequently, the polymer is partially lifted from the VL surface (into the vapor phase) leading to only a partial hydration (below 50% of that in bulk) for many conformations. The differences in pinning of the polymer to the interface and the quasi-2-dimensional nature of the conformations is better characterized by the average monomer density profiles along the z direction (see Figure 4.) To locate the interface, density

4096 J. Phys. Chem. B, Vol. 113, No. 13, 2009

Jamadagni et al.

Figure 3. Top views (top panels) and side views (bottom panels) of representative structures of the hydrophobic polymer in bulk water (panel a), at the SL interface (panels b and c), and at the VL interface (panel d). The polymer is shown in a spacefill representation (cyan), water molecules in stick representation (red, oxygen; white, hydrogens), and the solid surface in gray. We note that the zoom level is different in different panels. The size of the polymer monomer or that of a water molecule serves as a scale bar.

Figure 4. Density profiles of various solution components along the axis normal to the interface. Interface location, defined roughly as the point at which water density (blue) is half of that in the bulk, is at z ) 0. The density of polymer monomers (red) is shown for two umbrella simulation windows, Rg ) 0.75 nm (broader) and 1.2 nm (sharper) (top panel), and at for Rg ) 0.75 in the bottom panel. The density profile for the crystalline solid surface (scaled by a factor of 3) is magenta. In both panels, a density profile for polymer monomers from the folded state ensemble in the bulk (Rg ) 0.55 nm) projected onto the Z-axis is shown for comparison (green). Both the localization of the polymer and its relatively flat (quasi-2-dimensional) conformations at the interface is apparent from these profiles.

At the VL interface, the polymer is largely excluded from the liquid phase; the peak of the monomer density distribution is in the region of low water density of about 100 kg/m3. The balance between favorable LJ interactions of the polymer with water and unfavorable free energy of solvating a large hydrophobic solute (cavity creation work) is so maintained that the polymer neither enters the liquid phase nor does it detach and go into the vapor phase completely, but stays at the interface. The softness of pinning of the polymer to the VL interface is evident in the monomer density profile in the top panel of Figure 2 shown from umbrella runs for two different Rg values. Although the density profile is more well defined compared to that in bulk water, it is not as sharply defined as at the SL interface. This is consistent with partial lifting and dehydration of the polymer at a much softer VL interface that is characterized by larger density fluctuations compared to that at a SL interface. B. Decomposition of PMF into Hydration and Other Contributions. The PMF for conformational sampling of the polymer in bulk water includes contributions from intrapolymer, polymer-water, and water-water (or hydration) terms. When the polymer is present at the solid-liquid interface, polymersurface and water-surface interactions make additional contributions to the PMF. Resolving the overall free energy into its contributions provides a route toward understanding the role and importance of different interactions in the thermodynamics of polymer collapse at interfaces and in bulk water. We follow the decomposition of the PMF proposed by Athawale et al.,30 the rationale for which is discussed in detail in their paper. Thus,

W(Rg) ) Wvac(Rg) + 〈Upw(Rg)〉 + Whyd(Rg) profiles of water and the solid surface (in case of SL interface) are also shown in Figure 4 for reference. The crystallinity of the solid material itself is evident in the peaks of carbon density in the bottom panel. Water molecules display typical layering in z direction with a well defined first hydration layer near the SL interface. The monomer density profile of the polymer at the SL interface shows a well defined single peak (for all Rg > 0.65) with the base-width of about 3 Å, indicating relatively tight pinning of individual monomers of the polymer to the solid interface. We note that the size of each monomer is slightly larger than that of a water molecule, explaining the peak location being slightly to the right of the first peak of water density. In contrast, a similar monomer density profile for the compact states in bulk water (shown for reference by the green line) is significantly broader consistent with the approximately spherical or globular nature of those conformations.

(1)

where Wvac is the PMF of the polymer in vacuum and is calculated by performing umbrella simulations of the polymer in vacuum. 〈Upw(Rg)〉 is the ensemble averaged polymer-water interaction energy and Whyd(Rg) is the solvent (i.e., water) contribution to the PMF and quantifies the driving force provided by solvent-solvent interactions as well as solvent entropy to the conformational free energy of the polymer in solution. At the SL interface, motivated by the decomposition by Athawale et al., we propose the following separation.

W(Rg) ) Wvac-sur(Rg) + 〈Upw(Rg)〉 + 〈Usw(Rg)〉 + Whyd(Rg) (2)

Hydrophobically Driven Polymer Folding

Figure 5. Hydration and other contributions to the PMF, W(Rg), at 300 K obtained from bulk and interfacial simulations. (a) Average polymer-water interaction energy, 〈(Upw)〉. (b) PMF in vacuum, Wvac(Rg). (c) Average polymer-surface, 〈Ups〉, and surface water, 〈Usw〉 interaction energies. (d) Hydration contribution to the PMF, Whyd. Unless otherwise indicated, colors correspond to specific systems: bulk (red), solid-water (blue), vapor-water (magenta).

where Wvac-sur(Rg) is the PMF of the polymer on the solid surface in the absence of solvent and was calculated using appropriate umbrella simulations of that system, and 〈Usw(Rg)〉 is the additional ensemble averaged surface water interaction energy that contributes to the overall PMF. Figure 5 shows different contributions in the above decompositions for polymer in bulk water as well as at VL and SL interfaces. Because we are only interested in relative changes, as in Figure 2, all curves are shifted vertically to make the corresponding value at Rg ) 1.4 nm equal to zero. Collectively, these profiles suggest that the hydration contribution is an important factor in determining different conformational preferences of the polymer in bulk and at interfaces. Below, we first describe the behavior of the polymer in vacuum, followed by the effects of attractive interactions with the medium, leading to the above conclusion. In absence of solvent, Wvac determines the conformational preferences of the polymer. Because the monomer-monomer interactions in this freely jointed chain are attractive, at ambient temperatures, the polymer preferentially folds into compact structures with low Rg as indicated by the minimum in Wvac in Figure 5b. At the SL interface, in absence of solvent, the Wvacsur is essentially flat, except below Rg values of 0.8 nm, where it becomes unfavorable. In vacuum, the attractive polymersurface interactions pull the polymer toward the surface and the polymer primarily samples quasi-2D-structures. As Rg is reduced further, the smallest Rg structures of the polymer are globular, which means that at least some monomers of the polymer must detach partly from the interface. This leads to the loss of polymer-surface interactions and increases Wvac-sur values in the low Rg regime. A similar mechanism is also at play when water is present, as shown in Figure 5c, where ensemble averaged polymer-surface interaction is almost flat and then becomes increasingly unfavorable with decreasing Rg values. However, when water is present, the partially detached monomers create additional space for water to interact with the surface, leading to an opposite behavior

J. Phys. Chem. B, Vol. 113, No. 13, 2009 4097 for the average water-surface interaction energy. Thus, at a SL interface, changes in polymer-surface and water-surface interactions approximately cancel each other. This behavior is not surprising, given that both the solid surface and liquid water are high density condensed phases with roughly similar strength of van der Waals interactions with the polymer. Such is not the case at a VL interface, where the densities of adjoining water and vapor phases are significantly different. The average polymer-water interactions in three different systems referenced to Rg ) 1.4 nm state are shown in Figure 5a. As expected, the most extended states of the polymer have the highest water exposure and, therefore, the most favorable interaction with water. As the polymer folds, this exposure is reduced and intrapolymer interactions replace polymer-water ones. The average polymer-water interaction energy therefore becomes relatively unfavorable. This effect is most prominent in bulk water. At the SL and VL interfaces, the polymer-surface is already partially dehydrated, due to the presence of the interface, resulting in a smaller increase in 〈Upw〉 upon folding. Interestingly, although the magnitude of change in 〈Upw〉 for SL and VL interfaces is similar at the end point, the intermediate behavior is different due to different conformational details (such as the extent of pinning and quasi-2D nature) at the two interfaces. Figure 5d shows the resulting hydration contributions to polymer folding in bulk water and at the two interfaces referenced to its value at Rg ) 1.4 nm. In bulk water, the hydration contribution to folding is large, consistent with the calculations of Athawale et al., and drives the folding of the polymer into compact states. As shown by Athawale et al. and recently by Goel et al.,54 one advantage of the separation of PMF in this manner is that the Whyd contribution is relatively insensitive to polymer-water interactions. That is, Whyd quantifies the “cavity” or “hydrophobic” potential of mean force for conformational changes of the polymer. Indeed, if the polymerwater interactions were purely repulsive (as is the case for WCA polymer considered by Athawale et al.), the 〈Upw〉 term would make a smaller contribution, while Whyd would remain the same, leading to a large stabilization of the folded states. In the present case, the attractive polymer-water interactions mitigate this effect leading to the overall PMF profile in Figure 2. At the VL interface, the hydration contribution, which quantifies the hydrophobic driving force, again favors folding but its magnitude is smaller, about 20% of that in the bulk. This is consistent with the observation that the area of the polymer in contact with water is rather small when it is at the VL interface. The small driving force for folding results in the overall flat PMF profile (conformational) at a VL interface (Figure 2). At the SL interface, the behavior is intermediate to that in bulk water and at a VL interface, except for the increase of Whyd observed for Rg values below 0.7 nm. This increase might appear surprising at first. However, as Rg is reduced below 0.7 nm, the polymer can no longer sustain flat pancake-like conformations in contact with the surface. As the polymer compacts into more globular structures, at least some of the monomers are lifted from the solid surface, exposing buried surfaces. This increased exposure of total hydrophobic surface area of the polymer and the surface is unfavorable and contributes to the increase of Whyd in that region. Although the detailed decomposition of the PMF discussed above is useful, it is instructive to focus on the contributions of Whyd and “everything else”, (W - Whyd), which includes vacuum as well as polymer-medium interaction energies and is shown

4098 J. Phys. Chem. B, Vol. 113, No. 13, 2009

Jamadagni et al.

Figure 7. (a) Free energy of unfolding ∆Gu of the polymer as a function of temperature in bulk water and at the two interfaces. A cutoff of Rg ) 0.75 nm was used to define folded and unfolded states in the bulk and 0.90 nm at the interfaces. Points are simulation data and lines are fits using eq 4. Data at the LV interface are essentially flat, and therefore, were not fit. The dashed line guides the eye. (b) and (c): ∆Hu (dashed lines) and T∆Su (solid black lines) obtained from the fits also shown in bulk water and at the SL interface respectively.

Figure 6. PMF, W(Rg), and the hydration and nonhydration contributions to the PMF (in units of kJ/mol) at 300 K for the polymer in bulk water and at the two interfaces obtained using eqs 1 and 2.

in Figure 6. In bulk water, both Whyd as well as nonhydration contributions are large, but work in opposite directions. The Whyd contribution dominates and favors folding. At the SL interface, different contributions to the nonhydration part mostly cancel each other and the overall PMF is essentially dominated by the hydration contribution. In contrast, the hydration contribution at the VL interface is smaller in magnitude and the polymer-water attractions dominate, especially, in the low Rg region, favoring unfolding of the polymer (relative to that in bulk water). C. Temperature Dependence of the Free Energy of Polymer Unfolding. The potential of mean force, W(Rg), shown in Figure 2 characterizes conformation-dependent free energy over the entire Rg range. That PMF can be integrated to obtain the Gibbs energy of unfolding in the context of a two-state model for folding-unfolding as,

( )

-∆Gu exp ) kbT

∫RR ∫RR

max

cut cut

min

exp(-W(Rg)/kbT) dRg

(3) exp(-W(Rg)/kbT) dRg

where Rcut defines the boundary between “folded” (Rg < Rcut) and “unfolded” (Rg > Rcut) ensembles, as done previously by Athawale et al.30 Rcut may be easy to identify if the PMF profile shows two distinct minima, which is not the case here. Clearly,

the numerical value of ∆Gu depends on the choice of Rcut. In addition, because the folded state ensemble is structurally different in bulk water and at interfaces, a single choice of Rcut is not suitable. One important requirement is that the basin of the PMF for folded states should be included in the region with Rg < Rcut. With this in mind, we select Rcut equal to 0.75 nm in bulk water and 0.90 nm at interfaces. Increasing or decreasing these values changes the numerical value of ∆Gu but not appreciably its temperature dependence. As noted before, there is no clear thermodynamically stable folded state at the VL interface because the polymer samples folded or compact as well as extended states almost uniformly. However, Rcut ) 0.90 nm is still helpful in differentiating compact and unfolded states at that interface. Figure 7a shows ∆Gu for the polymer in bulk water and at interfaces as a function of temperature. Fits to those data using ref ∆Gu(T) ) ∆Href u + ∆Cp(T - Tref) - T(∆Su + ∆Cp ln(T/Tref)) (4)

with Tref ) 300 K in bulk and at the SL interface are also shown. In bulk and at SL interfaces, the ∆Gu profile is roughly parabolic in nature, similar to that for unfolding of globular proteins in water or for transfer of hydrophobic solutes to water. The stability of folded states in bulk water is 10-12 kJ/mol, proteinlike and decreases significantly at the SL interface. At the VL interface, ∆Gu is close to zero and essentially independent of temperature (and we therefore did not fit the data to eq 4). The curvature of these profiles characterizes the change in heat capacity, ∆Cp, upon unfolding, which is similar in magnitude, ≈0.72 and 0.68 kJ/(mol K) for bulk and SL interfacial systems, respectively, and close to zero for the VL system (resulting from the flat ∆Gu(T) profile). For proteins, the ∆Cp value in aqueous solution is generally positive, and is proportional to the change in hydrophobic area of the protein exposed to solvent upon unfolding. In light of this, the smaller value of ∆Cp for unfolding at the VL interface appears reasonable, although the origin of similarity between values in the bulk and at the SL interface is

Hydrophobically Driven Polymer Folding not entirely clear. It may be that instead of physical waterexposed area, the relative magnitude of the hydration contribution is more important in determining ∆Cp values, i.e., at 300 K, conformational sampling in bulk and at the SL interface is dominated by Whyd, whereas at VL interface, it is not. To our knowledge, data on thermodynamics of protein folding-unfolding equilibria in interfacial environments are not available. ∆Hu and ∆Su obtained by taking temperature derivatives of the ∆Gu profiles are shown in Figure 7b and c. Temperatures TH and TS at which enthalpy and entropy of unfolding go to zero indicate maxima in the values of ∆Gu/kT and ∆Gu, respectively, and have been characterized for many globular proteins, as well as for hydrophobic transfer processes. For both the bulk and SL systems, we find TS > TH, with TH in the range of 310-320 K. We do not further interpret the fine differences between the bulk and interfacial results as they will be dependent on the choice of Rcut as well as statistical uncertainties in the data used to fit the thermodynamics. These results on the thermodynamic properties on folding-unfolding of polymers in the bulk and at interfaces should motivate experimental studies focused on characterizing thermodynamics of selfassembly at interfaces. IV. Kinetics In bulk water, the collapsed states of the polymer are favored by a significant free energy driving force. Correspondingly, in simulations started with extended states, without an umbrella potential, the polymer collapses rapidly into compact globular folded states. A similar observation is made at the SL interface that the folded states are flat and pancake-like. In contrast, at the VL interface, as shown before, the PMF for conformational sampling is essentially flat, and correspondingly, the polymer samples a broad range of conformations in a simulation without the constraining umbrella potential. We are interested in characterizing the kinetics of conformational transitions. To this end, we performed one hundred simulations of polymer folding in bulk water as well as in interfacial systems, all starting with the polymer in the extended state (Rg ≈ 1.40 nm), and the overall system in microscopically different starting configurations (see Methods). Figure 8 shows time variation of Rg in bulk water and in the two interfacial systems. Five selected trajectories (out of 100) as well as the average of all 100 simulations are shown in each plate. To characterize the heterogeneity of folding times, we also calculated standard deviations of Rg at each time point. For all systems, the dynamics of folding or conformational transitions are relatively fast, in the subnanosecond time scale regime. The differences are, however, interesting. In bulk water, although the collapse occurs over a subnanosecond time scale, there is significant heterogeneity in the evolution of Rg in individual trajectories. A fast collapse is observed in some trajectories, whereas in others Rg changes in steps, with significant fluctuations up and down as a function of time. The folding of a hydrophobic polymer in water is accompanied by a corresponding reduction in solvent exposure. Naturally, the solvent must get out of the way as the polymer is driven into more compact states.57 The dynamics of solvent molecules and of local density fluctuations will depend on polymer solvent interactions.29,52 Depending on the local structural arrangements of water molecules around the polymer in a given configuration (the effects of which are integrated out in the one-dimensional projection of the PMF in Figure 2), it may be easier or harder for water to “get out of way” in bulk water. This role of structure and fluctuations of the vicinal

J. Phys. Chem. B, Vol. 113, No. 13, 2009 4099

Figure 8. Folding kinetics from independent trajectories. Panels a (bulk), b (SL interface), and c (VL interface) show the time evolution of Rg in selected trajectories (out of 100 each) from simulations all starting with the polymer with Rg ≈ 1.4 nm. Average of all one hundred trajectories in bulk and SL systems is shown by a bold red and blue line, respectively, with dotted black lines showing one standard deviation of the observed Rg at a given time point. No averages or standard deviation are shown in the VL system, but one trajectory is shown by a bold black line for visual clarity. Panel d shows average time evolution in bulk and at a SL interface again for purposes of max comparison; R*g ) (Rg - Rmin - Rmin g )/(Rg g ). The inset shows raw Rg(t) data.

solvent molecules is reflected in the heterogeneity of timescales for folding of this hydrophobic polymer in bulk water. Panel b of Figure 8 shows the kinetic data for polymer folding at the SL interface. Folding at the SL interface is faster compared to that in bulk although the driving force for folding is smaller than that in bulk water. In almost all trajectories at the SL interface, the polymer folds into compact pancake-like structures (in the basin with Rg < 0.80 nm) within about 300 ps. Further, different trajectories are relatively homogeneous, and correspondingly, the folding time distribution is narrower compared to that in bulk water. Why are the collapse kinetics at a SL interface different (faster and more homogeneous) compared to that in bulk water? One might argue that because the pancakelike folded states lie in the free energy basin with Rg values approximately between 0.7 and 0.9 nm, the polymer has to travel a shorter distance in the conformation space at the SL interface compared to that in bulk, in which the folded states are near Rg ≈ 0.55 nm. However, the direct comparison of the evolution of normalized averages in the bulk and at SL interface in Figure 8d reveals that conformational changes occur at a faster rate (∼3 times that in bulk) at the SL interface, indicating additional factors at play. As shown earlier, the first of these factors is simply that the polymer conformations at the SL interface are quasi-2dimensional, be they extended or collapsed. As the extended conformers fold at the SL surface, the surface acts as a catalyst or a guide, maintaining a quasi-2D structure of the polymer, and reducing the conformational space the polymer explores as it folds. The quasi-2D conformations have the solid surface on one side and thus are only partially hydrated (∼50%). Also, the hydrophobic nature of the surface is known to enhance local density fluctuations of the vicinal water. Collectively, the partial dehydration and higher solvent density fluctuations represent the second factor which may aid faster folding of the polymer.

4100 J. Phys. Chem. B, Vol. 113, No. 13, 2009

Jamadagni et al. curves can be scaled to approximately overlap with each other. The scaling factors are 1 for the bulk, 1.4 for the SL interface, and 7.5 for the VL interface, again consistent with the relative order of kinetics observed above. V. Conclusions

Figure 9. Diffusivity, D(Rg), characterizing conformational dynamics in bulk water and at interfaces. The inset shows the diffusivity data (in units of 10-2 Å2/ps), where values at VL and SL interfaces are scaled by factors of 7.5 and 1.4, respectively.

Indeed, the dynamics of water molecules for motions parallel to a similar solid hydrophobic surface as well as that in the interior of a nanotube58 are shown to be faster than that in the bulk. It will be interesting to see if translational diffusion of the polymer along the surface is also enhanced due to these factors compared to that in the bulk medium. At the VL interface, the PMF profile (see Figure 2) is shallow and essentially flat. Neither folded nor unfolded states are stable. As discussed earlier, the polymer structures are only partially hydrated, limiting the role of further dehydration or dewetting processes in the kinetics of folding-unfolding for this hydrophobic polymer. As a result, as shown in Figure 8, the polymer samples a broad range of conformations over fast (∼100 ps) timescales with no significant barriers between these conformational states. As the polymer samples different conformations, the onedimensional reaction coordinate, Rg, can be viewed as performing a random walk with some drift. The diffusion as well as drift are local in that they depend on the value of Rg, as discussed by Socci et al.59 or by Hummer.60 Calculating the diffusivity, D(Rg), of a reaction coordinate over the conformational space provides an alternate quantitative method to understand the relative kinetics of polymer conformational equilibria. Following Hummer, we obtain D(Rg) using data from umbrella simulations using,

τ)

∫0∞ 〈δRg(0)δRg(t)〉 dt 〈δRg(0)2〉

D(Rg) )

〈δRg(0)2〉 τ

(5)

(6)

where the autocorrelation function (ACF) for Rg is determined in each umbrella window. Figure 9 shows D(Rg) profiles for the hydrophobic polymer in bulk water and in the two interfacial systems. The overall nature of all these curves is similarsdiffusivity is larger for higher values of Rg and decreases monotonically as the polymer folds into more compact states. Consistent with the kinetic data reported above, we find that the diffusivity values are highest at the VL interface, and comparable at the SL interface and in bulk water, although in the Rg > 0.9 nm, D(Rg)SL > D(Rg)bulk. We empirically find that the three diffusivity

We have performed extensive molecular dynamics simulations to understand the effect of solid-liquid and vapor-liquid interfaces on structural, thermodynamic, and kinetic aspects of folding-unfolding of a hydrophobic polymer relative to that in bulk water. The solid interface we considered is hydrophobic in nature with alkane-like interactions with water and with polymer. In bulk water, the hydrophobic polymer collapses into an ensemble of compact globular states that are thermodynamically stable relative to extended states. Studies of one hundred different trajectories for folding indicate the time scale for folding to be about a nanosecond long for these short and flexible hydrophobic polymers, with a broad distribution of folding times ranging from 100 ps to 1 ns. Simulations of SL and VL interfaces show that the hydrophobic polymer is interfacially activesthat is, thermodynamically, it prefers to be in the interfacial region relative to bulk. At the SL interface, the polymer conformations are quasi-2dimensional, with its compact or folded states being pancakelike spirals or S-shaped conformers. These pancake-like conformations are stable relative to the corresponding extended conformations at the SL interface. That stability is, however, not as high as that of globular structures relative to extended ones in the bulk water phase. From the point of view of overall system free energy, however, the compact pancake-like structures at the interface are most stablesthat is, a folded polymer in bulk water, if left alone, will diffuse to the hydrophobic solid-water interface and rearrange into pancake-like structures spontaneously. In contrast to that at the SL interface, the conformational free energy landscape of the polymer at the VL interface of water is shallow and relatively flat. That is, a broad range of conformations are observed with no clear bias for folded or unfolded states. The transitions between different conformations at the VL interface are fast and occur over a 100 ps time scale for the flexible 25-mer polymer. The faster kinetics of conformational transitions for hydrophobic polymer at interfaces relative to that in bulk indicates the role of vicinal water (structure and dynamics) in kinetics of the folding process. The weaker hydration of the polymer situated at the interface along with enhanced density fluctuations of water at the interface appear to aid faster dynamics of conformational transitions. Our study, although focused on folding-unfolding of a simple model hydrophobic polymer, provides a framework for further studies of solvent-surface-macromolecule interactions specifically focused on understanding the effects of interfacial environments. We note that the polymer model we use is a coarse grained one, without internal angle or dihedral potentials. Simulations that include such intrapolymer potentials for C18 alkanes in water and in other solutions have been reported recently.61 Extensions of those studies to longer polymers and to interfacial environments would provide numerical data on realistic systems of sizes comparable to those reported here. Inclusion of intrapolymer potentials would alter numerical detailsssuch as slowing down of folding kinetics and destabilization of folded states. However, broader aspects of results reported here are robust. Conformational equilibria in interfacial environments is an interesting multidimensional problem. In our study, we consid-

Hydrophobically Driven Polymer Folding ered only a vapor-liquid interface and an atomically flat hydrophobic solid surface. Many new questions can be pursued in the near future. How does the surface curvature affect the conformational behavior of macromolecules in the vicinity? How do solution conditions (e.g., salt or additives, co-solvents) affect the interfacial behavior of macromolecules? How does changing the chemistry of the polymer affect its interfacial thermodynamics, structure and dynamics? To address this, new heteropolymer models have to be developed, characterized in bulk-aqueous solutions and then studied in interfacial environments. What are the effects of surface chemistries from hydrophobic to hydrophilic on the folding structure, thermodynamics, and kinetics of folding-unfolding? Hydrophobic interfaces are ubiquitous in biological systems and in bio and nano technology applications such as molecular separations,40 development of active biofunctional materials (e.g., nanoparticle-enzyme conjugates),38 or in thermal transport.62 Our studies present an excellent starting point for exploration in those directions. Acknowledgment. We gratefully acknowledge partial financial support of NSF (BES and NSEC DMR-0642573) and NIH (GM66712) grants. We also acknowledge generous support of the computational resources through an IBM-SUR grant. We also thank many useful discussions with Prof. Thomas M. Truskett, Gaurav Goel, and Dr. Manoj Athawale. References and Notes (1) Kauzmann, W. AdV. Protein Chem. 1959, 14, 1–63. (2) Tanford, C. J. Am. Chem. Soc. 1962, 84, 4240. (3) Dill, K. A. Biochemistry 1990, 29, 7133–7155. (4) Pratt, L. R. Annu. ReV. Phys. Chem. 2002, 53, 409–436. (5) Guillot, B.; Guissani, Y. J. Chem. Phys. 1993, 99, 8075–8094. (6) Ludemann, S.; Schreiber, H.; Abseher, R.; Steinhauser, O. J. Chem. Phys. 1996, 104, 286–295. (7) Silverstein, K. A. T.; Haymet, A. D. J.; Dill, K. A. J. Am. Chem. Soc. 1998, 120, 3166–3175. (8) Garde, S.; Hummer, G.; Garcia, A. E.; Paulaitis, M. E.; Pratt, L. R. Phys. ReV. Lett. 1996, 77, 4966–4968. (9) Garde, S.; Garcia, A. E.; Pratt, L. R.; Hummer, G. Biophys. Chem. 1999, 78, 21–32. (10) Wallqvist, A. J. Chem. Phys. 1992, 96, 1655–1656. (11) Hummer, G.; Garde, S.; Garcia, A. E.; Paulaitis, M. E.; Pratt, L. R. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 1552–1555. (12) Ghosh, T.; Garcia, A. E.; Garde, S. J. Am. Chem. Soc. 2001, 123, 10997–11003. (13) Smith, P. E. J. Phys. Chem. B 1999, 103, 525–534. (14) Kalra, A.; Tugcu, N.; Cramer, S. M.; Garde, S. J. Phys. Chem. B 2001, 105, 6380–6386. (15) Ghosh, T.; Kalra, A.; Garde, S. J. Phys. Chem. B 2005, 109, 642– 651. (16) Thomas, A. S.; Elcock, A. H. J. Am. Chem. Soc. 2007, 129, 14887– 14898. (17) Graziano, G. Can. J. Chem.-ReV. Can. Chim 2002, 80, 388–400. (18) Athawale, M. V.; Dordick, J. S.; Garde, S. Biophys. J. 2005, 89, 858–866. (19) Rosgen, J.; Pettitt, B. M.; Bolen, D. W. Protein Sci. 2007, 16, 733– 743. (20) Oostenbrink, C.; van Gunsteren, W. F. Phys. Chem. Chem. Phys. 2005, 7, 53–58. (21) Trzesniak, D.; van der Vegt, N. F. A.; van Gunsteren, W. F. Phys. Chem. Chem. Phys. 2004, 6, 697–702. (22) Stillinger, F. H. J. Sol. Chem. 1973, 2, 141–158.

J. Phys. Chem. B, Vol. 113, No. 13, 2009 4101 (23) Lum, K.; Chandler, D.; Weeks, J. D. J. Phys. Chem. B 1999, 103, 4570–4577. (24) Rajamani, S.; Truskett, T. M.; Garde, S. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 9475–9480. (25) Chandler, D. Nature 2005, 437, 640–647. (26) Rank, J. A.; Baker, D. Protein Sci. 1997, 6, 347–354. (27) Shimizu, S.; Chan, H. S. Proteins 2002, 48, 15–30. (28) Czaplewski, C.; Rodziewicz-Motowidlo, S.; Dabal, M.; Liwo, A.; Ripoll, D. R.; Scheraga, H. A. Biophys. Chem. 2003, 105, 339–359. (29) ten Wolde, P. R.; Chandler, D. Proc. Natl. Acad. Sci. U. S. A. 2002, 99, 6539–6543. (30) Athawale, M. V.; Goel, G.; Ghosh, T.; Truskett, T. M.; Garde, S. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 733–738. (31) Pratt, L. R.; Pohorille, A. Chem. ReV. 2002, 102, 2671–2691. (32) Schwartz, D. K.; Schlossman, M. L.; Kawamoto, E. H.; Kellogg, G. J.; Pershan, P. S.; Ocko, B. M. Phys. ReV. A 1990, 41, 5687–5690. (33) Mitrinovic, D. M.; Tikhonov, A. M.; Li, M.; Huang, Z. Q.; Schlossman, M. L. Phys. ReV. Lett. 2000, 85, 582–585. (34) Giovambattista, N.; Debenedetti, P. G.; Rossky, P. J. J. Phys. Chem. B 2007, 111, 9581–9587. (35) Richmond, G. L. Chem. ReV. 2002, 102, 2693–2724. (36) Patel, H. A.; Nauman, E. B.; Garde, S. J. Chem. Phys. 2003, 119, 9199–9206. (37) Ashbaugh, H. S.; Pethica, B. A. Langmuir 2003, 19, 7638–7645. (38) Karajanagi, S. S.; Vertegel, A. A.; Kane, R. S.; Dordick, J. S. Langmuir 2004, 20, 11594–11599. (39) Asuri, P.; Karajanagi, S. S.; Vertegel, A. A.; Dordick, J. S.; Kane, R. S. J. Nanosci. Nanotechnol. 2007, 7, 1675–1678. (40) Sane, S. U.; Cramer, S. M.; Przybycien, T. M. J. Chromatogr. A 1999, 849, 149–159. (41) Sethuraman, A.; Han, M.; Kane, R. S.; Belfort, G. Langmuir 2004, 20, 7779–7788. (42) Sethuraman, A.; Vedantham, G.; Imoto, T.; Przybycien, T.; Belfort, G. Proteins 2004, 56, 669–678. (43) Pereira, L. G. C.; Theodoly, O.; Blanch, H. W.; Radke, C. J. Langmuir 2003, 19, 2349–2356. (44) Leonhard, K.; Prausnitz, J. M.; Radke, C. J. Langmuir 2006, 22, 3265–3272. (45) Anderson, R. E.; Pande, V. S.; Radke, C. J. J. Chem. Phys. 2000, 112, 9167–9185. (46) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. J. Phys. Chem. 1987, 91, 6269–6271. (47) Mondello, M.; Grest, G. S.; Webb, E. B.; Peczak, P. J. Chem. Phys. 1998, 109, 798–805. (48) Berendsen, H. J. C.; Postma, J. P. M.; Vangunsteren, W. F.; Dinola, A.; Haak, J. R. J. Chem. Phys. 1984, 81, 3684–3690. (49) Essmann, U.; Perera, L.; Berkowitz, M. L.; Darden, T.; Lee, H.; Pedersen, L. G. J. Chem. Phys. 1995, 103, 8577–8593. (50) Lindahl, E.; Hess, B.; van der Spoel, D. J. Mol. Model. 2001, 7, 306–317. (51) Berendsen, H. J. C.; Vanderspoel, D.; Vandrunen, R. Comput. Phys. Commun. 1995, 91, 43–56. (52) Miller, T. F.; Vanden-Eijnden, E.; Chandler, D. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 14559–14564. (53) Maragliano, L.; Fischer, A.; Vanden-Eijnden, E.; Ciccotti, G. J. Chem. Phys. 2006, 125, 024106. (54) Goel, G.; Athawale, M. V.; Garde, S.; Truskett, T. M. J. Phys. Chem. B 2008, 112, 13193–13196. (55) Pohorille, A.; Benjamin, I. J. Chem. Phys. 1991, 94, 5599–5605. (56) Desai, T.; Keblinski, P.; Kumar, S. K. Polymer 2006, 47, 722– 727. (57) Hummer, G. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 14883–14884. (58) Kalra, A.; Garde, S.; Hummer, G. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 10175–10180. (59) Socci, N. D.; Onuchic, J. N.; Wolynes, P. G. J. Chem. Phys. 1996, 104, 5860–5868. (60) Hummer, G. New J. Phys. 2005, 7. (61) Sun, L.; Siepmann, J. I.; Schure, M. R. J. Phys. Chem. B 2006, 110, 10519–10525. (62) Patel, H. A.; Garde, S.; Keblinski, P. Nano Lett. 2005, 5, 2225–2231.

JP806528M