Designing Heteropolymers To Fold into Unique Structures via Water

Oct 6, 2010 - Author to whom correspondence should be addressed. Electronic mail: [email protected]; homepage: http://www.rpi.edu/∼gardes. Cite this:J...
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Designing Heteropolymers To Fold into Unique Structures via Water-Mediated Interactions Sumanth N. Jamadagni, Christian Bosoy, 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: May 28, 2010; ReVised Manuscript ReceiVed: July 26, 2010

Hydrophobic homopolymers collapse into globular structures in water driven by hydrophobic interactions. Here we employ extensive molecular dynamics simulations to study the collapse of heteropolymers containing one or two pairs of oppositely charged monomers. We show that charging a pair of monomers can dramatically alter the most stable conformations from compact globular to more open hairpin-like. We systematically explore a subset of the sequence space of one- and two-charge-pair polymers, focusing on the locations of the charge pairs. Conformational stability is governed by a balance of hydrophobic interactions, hydration and interactions of charge groups, water-mediated charged-hydrophobic monomer repulsions, and other factors. As a result, placing charge pairs in the middle, away from the hairpin ends, leads to stable hairpin-like structures. Turning off the monomer-water attractions enhances hydrophobic interactions significantly leading to a collapse into compact globular structures even for two-charge-pair heteropolymers. In contrast, the addition of salt leads to open and extended structures, suggesting that solvation of charged monomer sites by salt ions dominates the salt-induced enhancement of hydrophobic interactions. We also test the ability of a predictive scheme based on the additivity of free energy of contact formation. The success of the scheme for symmetric twocharge-pair sequences and the failure for their flipped versions highlight the complexity of the heteropolymer conformation space and of the design problem. Collectively, our results underscore the ability of tuning watermediated interactions to design stable nonglobular structures in water and present model heteropolymers for further studies in the extended thermodynamic space and in inhomogeneous environments. I. Introduction Coarse-grained lattice or off-lattice models of polymers and proteins have long served as conceptual devices for the understanding of the conformational behavior of these complex molecules.1-6 Recently, several groups have reported interesting simulations of homopolymers (e.g., freely joined chains, alkanes, polyglutamine, etc.) in explicit water to understand conformational equilibria, including hydrophobically driven collapse,7-11 its thermodynamics and dynamics,12,13 and the role of solvent fluctuations and dewetting in the mechanism of collapse.14 Hydrophobic homopolymers collapse into globular states in water without much internal structure. Here we use extensive molecular dynamics (MD) simulations to show that adding a pair of opposite charges on two of the monomers can dramatically shift the conformational equilibrium from globular to hairpin-like states. Our study highlights the richness of this problem by systematically exploring the design space of heteropolymers containing one and two pairs of charges. The equilibrium structure of a heteropolymer is governed by the balance of hydrophobic interactions, hydration and interactions of charge groups, water-mediated repulsions between charged and hydrophobic monomers, and other factors. We explore how the heteropolymer sequence (i.e., locations of the charges) affects the balance of these factors. The results are nonintuitive, with globular, partially globular, or hairpin-like structures being stable depending on the sequence. We also explored whether the behavior of heteropolymers containing two pairs of charges can be predicted using information from those containing one pair. That is, we tested to see if * Author to whom correspondence should be addressed. Electronic mail: [email protected]; homepage: http://www.rpi.edu/∼gardes.

the effects of two pairs of opposite charges are simply additive. The results are again nonintuitive, with the additive or nonadditive nature depending on the order in which the charges are located in the sequence. Adding salt or changing the strength of monomer-water van der Waals interactions provides another handle on the balance of water-mediated hydrophobic and electrostatic interactions, which can be used to tune the resulting equilibrium conformations and their thermodynamic stability. The conformational behavior of heteropolymers in water results from the nature of water-mediated interactions, which will be difficult to reproduce using implicit solvent schemes. Collectively, our results show that marginally stable folded structures of heteropolymers can be designed by tuning water-mediated interactions. Our study also presents specific heteropolymer models for further studies in the extended thermodynamic space and in inhomogeneous interfacial or confined environments. II. Methods The hydrophobic homopolymer on which the present heteropolymer is based has been extensively characterized by our group in water7 and at interfaces.15 That polymer (denoted as CG-25 by Athawale et al.7) is a freely jointed 25-mer of Lennard-Jones (LJ) solutes (see Table 1) with harmonic bond potentials [Ubond ) 1/2kb(l - lo)2, with lo ) 0.25 nm and kb ) 60 702 kJ/(mol nm2)]. To allow for the formation of symmetric hairpin structures, we reduced the number of monomers to 24 and elongated the bond between monomers 12 and 13 from 0.25 to 0.45 nm. We denote this modified polymer as CG-24. One or two charge pairs were placed on the CG-24 polymer to make it a heteropolymer. The heteropolymer was solvated in a periodic cubic box of length ∼5.4 nm containing 5369 water molecules.

10.1021/jp104924g  2010 American Chemical Society Published on Web 10/06/2010

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TABLE 1: Force Field Parameters Used in MD Simulations molecule

atom type

σ, nm

ε, kJ/mol

q, e

polymera waterb

PC OW HW Na Cl

0.4400 0.3169 0 0.2583 0.4401

0.8500 0.6502 0 0.4185 0.4185

0.00 -0.8476 +0.4238 +1.00 -1.00

ionsc a

From ref 7. b SPC/E model for water, ref 18. c From refs 16 and

17.

To study the effect of salt, simulations were performed in NaCl solution (see Table 1 for parameters). The simulation contained 180 ion pairs and 5009 water molecules, corresponding to a mole fraction of ∼0.033 or to a concentration of 1.76 M. An additional simulation at a mole fraction of ∼0.10 or a concentration of ∼5 M was also performed to study the effect of high salt concentration. To verify that the conformations of the heteropolymer were sampled well, we also performed umbrella sampling on the radius of gyration order parameter as well as replica exchange MD (REMD) simulations for a few select cases. For umbrella sampling simulations, 20 equally spaced windows spanning Rg ) 0.45 nm to Rg ) 1.40 nm were used. A harmonic potential, U ) 1/2k(Rg - Rg*)2 with k ) 7000 kJ/mol/nm2, was used to constrain the radius of gyration in the neighborhood of Rg*. For the REMD simulations, 50 replicas spanning a temperature range of 280-438 K were used. Each replica was simulated for 8 ns for a total simulation time of 400 ns. All MD simulations (excluding umbrella and REMD) were at least 30 ns long, with selected systems simulated for 80 ns. Polymer coordinates were stored every 0.5 ps for analysis. The SPC/E model18 was used to represent water molecules explicitly. Electrostatics were handled using the PME algorithm.19 Temperature (300 K) and pressure (1 atm) were maintained using the Berendsen coupling scheme.20 Simulations were performed using GROMACS.21,22 Clustering. A modified “greedy” algorithm23 was used to identify dominant polymer conformations from a long simulation trajectory. The root-mean-square deviations (rmsd) between all pairs of structures in a given long trajectory were calculated, and all structures whose rmsd was within a cutoff (3 Å) of each other were identified as belonging to a single cluster. This dominant cluster was then removed and the analysis repeated to identify the second most dominant cluster and so on. III. Results and Discussion A. Heteropolymer Design Problem. The bottom panel of Figure 1a shows the potential of mean force (PMF), W(Rg), for CG-24 and CG-25 polymers. The reduction in the number of monomers from 25 to 24 and the elongation of 12-13 bond in CG-24 homopolymer have a negligible effect on conformational preferences. Both polymers collapse into globular states with Rg ≈ 0.55 nm, with W(Rg) ≈ -28 kJ/mol relative to the extended states. We are interested in quantifying the effects of charging one or two pairs of monomers on the conformational preferences of CG-24 polymer (Figure 1b-c). Although the negative (N) and positive (P) charges can be placed on any two monomers, we restrict our attention to a subset of sequences that may potentially fold into symmetric hairpin-like structures in water. Specifically, we focus on sequences in which monomers i and 25 - i are charged. We denote the polymers containing one charge pair by NiP25-i. For polymers with two pairs of charges,

two different versions, NiNjP25-jP25-i or NiPjN25-jP25-i (with the orientation of one of the charge pairs “flipped”), are possible (Figure 1d). Even with the above constraints, the design parameter space is rather large, including: (i) the magnitude of the charge, q; (ii) the number of charge pairs; (iii) the location(s) of the pair(s) and their relative orientation (e.g., NNPP vs NPNP); and (iv) the strength of van der Waals interactions of the monomers. In addition to performing clustering of 3-D structures, we monitor conformational preferences by calculating contact maps (see top left panels of Figure 1a and c). The collapsed globular states do not possess specific internal structure as indicated by their essentially featureless contact map (Figure 1a). We note that contact map in Figure 1a is not a schematic but actual data from MD simulations plotted on the same scale as that of Figures 3 and 5. In contrast, for the hairpin-like structures, the diagonal of the contact map and its vicinal region would be densely populated (Figure 1c). Such hairpin structures would show a minimum in the PMF profile near Rg ≈ 0.8 nm, as shown in the bottom panel of Figure 1c. We note that both panels in Figure 1c are schematic representations. Instead of placing a charge pair (or two), as we do here, one may attempt to stabilize hairpin-like structures in water by imposing a covalent bond between monomers i and (25 - i) or by artificially increasing the strength of noncovalent LJ interactions between them. Interestingly, both strategies fail to produce hairpin-like structures (results not shown). In these cases, the homopolymer primarily collapses into a subset of compact and roughly globular structures that accommodate those constraints. Our strategy of placing charge pairs represents a consistent and noncovalent/water-mediated interactions based approach to designing heteropolymers. Also, we note that the presence of an elongated bond in the CG-24 polymer is helpful but not critical. Additional preliminary simulations showed that polymers without this bond still form hairpin structures, except that they are somewhat less stable. The elongated bond can be thought of as similar to having two glycine residues in turn regions of peptides, which make that region locally more flexible. This is different from a proline residue, which introduces a kink, and can have longer range effects on the structure. The elongated bond does not have such longer ranged effects. B. Heteropolymers with One Charge Pair. Before performing heteropolymer simulations, to determine the charge, q, we performed 40 ns long simulations of aqueous solutions of four pairs of oppositely charged free monomers in 2168 water molecules. The association of ions in water is known to be charge-density dependent.24-26 For a small value of charge, q () 0.2e), the strength of association as characterized by the ion-ion PMF is ∼1.25kT, which gradually increases with increasing q (Figure 2). For q ) 0.8, 1.0, or 1.2e, the barrier between contact and solvent-separated minimum becomes smaller, and the association is quite favorable. We used q ) 1.2e, which leads to a charge-density on the monomer roughly equal to that on a chloride ion. The choice of the magnitude of q, although important, does not alter the results significantly over a reasonable range of values (>0.8e). Eleven different heteropolymers of type NiP25-i containing one i -(25 - i) charge pair can be made, corresponding to i ) 1, 2, · · · , 11. Of these, we have studied eight, corresponding to i ) 1, 3, 5, 6, 7, 9, 10, and 11. Figure 3 shows contact maps for these heteropolymers obtained from ∼80 ns long equilibrium MD simulations. We show later that results obtained using three different sampling methodssa long MD simulation, by umbrella

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Figure 1. (a) Top: the contact map of a CG24 homopolymer in water obtained from MD simulations. Bottom: potential of mean force, W(Rg), for conformational sampling of the CG-2515 and CG-24 polymers in water. We set W(Rg ) 1.4) ) 0 for reference. Snapshots of extended and globular compact CG-24 homopolymer are shown as well; for clarity, water molecules are not shown. (b) Schematic of the heteropolymer design problem. Can placing one (or two) charge pair(s) change conformational preferences in solution significantly? (c) Top: the contact map for ideal hairpin structures shown schematically. We explore the design of heteropolymers that may display such a contact map. Bottom: PMF, W(Rg), for heteropolymers forming a hairpin in solution is shown schematically by the dashed line. (d) Nomenclature: ideal hairpin structures and nomenclature used to describe sequences of one-pair and two-pair heteropolymers are shown. Color: all neutral monomers (cyan), positively charged (red), and negatively charged (blue).

Figure 2. Potential of mean force between equal and oppositely charged monomers in water over a range of monomer charge states. The PMFs were calculated using W(r) ) -kT log[g(r)], where g(r) is the radial distribution function between positively and negatively charged monomers in water.

sampling, as well as by REMDsare consistent with each other. Unlike that for the homopolymer, contact maps for one-chargepair heteropolymers are not featureless. The diagonal region is populated to different extents depending on the location of the charge pair, indicating that charging a single pair of monomers can change the conformational preferences dramatically. The contact map for N1P24 polymer, where the charge pair is present near the free ends, is only slightly different from that of the hydrophobic CG-24 homopolymer. As the charge pair is moved away from the free ends, the diagonal region is sampled with higher probability. Hairpin-like structures seems to be best observed when the charge pair is placed toward the middle of the polymer (e.g., N7P18 or N9P16 polymers). As the charge pair is moved further toward the turn (e.g., N11P14), the contact is easier to form and is almost always observed. Yet, the rest of the polymer can form a smaller globule, as indicated by the blue off-diagonal regions for N10P15 and N11P14 polymers.

Another interesting feature that appears in the contact maps of one-charge-pair heteropolymers is the two vertical and horizontal white (i.e., low probability) stripes. They appear exactly at the locations of charged monomers and indicate the water-mediated repulsion between the charged monomers and the remaining (uncharged) hydrophobic monomers consistent with previous studies of free monomers.27 This water-mediated repulsion destabilizes globular structures in which the polymer folds back on itself and brings charged and uncharged monomers in close contact, thus partially dehydrating the charged monomers. If the charge, q, was much lower in magnitude, the watermediated repulsion or the dehydration penalty would be smaller, and compact globular structures would be favored. Figure 4 shows how the average number of water molecules in the hydration shell, and the average radius of gyration of the polymer changes with the location of the charge pair. These measures track each other very well. Interestingly, both Rg and Nhyd vary nonmonotonically with the location of the charge pair. If the charge pair is close to either of the 12-13 turn or the end of the polymer, globular states are favored. When the charge pair is placed in the middle of the sequence, hairpin-like structures are favored (which also minimize water-mediated repulsions between charged and uncharged monomers). These data reinforce the results shown in Figure 3. C. Heteropolymers with Two Charge Pairs. The probability of forming hairpin structures is expected to be considerably enhanced if two charge pairs are used instead of one. Within our design constraints, there are 11C2 ) 55 NNPP sequences, as well as 55 additional NPNP sequences. For NNPP polymers, we explored 10 distinct cases using extensive MD simulations. We discuss four of these below along with a heuristic approach, which may be useful for predictive purposes. To design a two-charge-pair polymer with a high propensity to form hairpin-like structures in water, one might use informa-

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Figure 3. Contact maps for NiP25-i CG-24 heteropolymers containing one charge pair. Colors indicate -kT log(Pkl), where Pkl is the probability of observing a contact between monomers k and l. The cutoff distance for defining a contact was 0.5 nm.

Figure 4. Sequence dependence of (Rg) and number of hydration shell water molecules, (Nhyd), for one-charge-pair heteropolymers. The x-axis shows, i, the location of the negatively charged monomer within NiP25-i sequences. A cutoff distance of 0.70 nm was used to define the hydration shell.

tion from simulations of one-charge-pair heteropolymers as follows:

F(l, k)

|

NiNjP25-jP25-i

≈ F(l, k)

|

+ F(l, k) NiP25-i

|

(1) NjP25-j

where F(l, k) ) -kT ln[P(l, k)] is the free energy of formation of l-k contact and P(l, k) is the corresponding probability of contact formation. The above approximation assumes that the relative free energies of contact formation are additive. This assumption is highly simplistic and ignores manybody effects that arise when the second charge pair is added. It also predicts that the contact maps for NNPP and the flipped version NPNP will be identical. Below we show that, for NNPP polymers, eq 1 provides surprisingly good estimation of the contact map but fails to capture interesting structural features that arise for the flipped NPNP versions. Figure 5a shows the predictions of eq 1 for four cases. The heuristic predicts that for N1N11P14P24 polymer, in which the ends (1-24) and the near-hinge (11-14) monomers are charged,

structures will display some globular nature. When the charge pairs are located in the middle, that is, for the N6N10P15P19 polymer, almost perfect hairpin-like structures would be stabilized, as indicated by the contact map dominated by the diagonal elements. Figure 5b shows results from MD simulations for heteropolymers considered in Figure 5a. We note that equilibrium sampling using different methodssextensive MD simulations, umbrella sampling, as well as replica exchange simulationssprovides results that are consistent with each other (Figure 6). This lends support to our use of long unconstrained MD simulations to explore conformational preferences of polymers studied here. These simulations indicate that the assumption of additivity of free energies of contact formation works quite well for sequences N1N11P14P24 and N6N10P15P19 and moderately well for N3N7P18P22 and N3N9P16P22. The centers of dominant clusters obtained from cluster analysis are also shown next to their corresponding contact maps. The success of the simple additive free energy scheme suggests that similar approaches may be useful as a first-pass generator of sequences that form specific structures. Orientations of one of the charge pairs can be “flipped” without changing the location of either of the pairs, corresponding to the change in sequence from NiNjP25-jP25-i to NiPjN25-jP25-i (see Figure 1d). A comparison of contact maps from MD simulations of NNPP and NPNP versions shows that there are subtle differences between them. Specifically, although the diagonal elements are still populated in the NPNP versions, there also appear two blue bands parallel to the diagonal, indicating the formation of an additional turn in each arm of the original hairpin. These additional turns can, but do not necessarily have to, form simultaneously. The locations of the parallel bands depend on the exact sequence. Formation of such turns is not possible in NNPP versions due to electrostatic repulsion. Interestingly, predictions based on the additive approach are the same for NNPP and NPNP versions and fail to capture the subtle structural features that appear in the flipped sequences, thus highlighting the limitations of such a simplistic approach.

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Figure 5. Contact maps for two-charge-pair heteropolymers (a) predicted using the additive free energy approximation, eq 1, and (b) calculated using extensive MD simulations. Panels in (c) show contact maps for two “flipped” versions: N3P7N18P22 and N3P9N16P22. Additional blue bands parallel to the main diagonal are visible in c. Representative conformations of the polymers obtained from cluster analysis are shown for all four subpanels in b.

Figure 6. PMF, W(Rg), along the radius of gyration order parameter for heteropolymers N3N7P18P22 and N5N11P14P20 obtained using unconstrained MD simulations (blue), from umbrella sampling simulations on the Rg order parameter (red), and from REMD simulations (green).

Figure 7. Probability distribution, P(Rg), of the radius of gyration of the N3N7P18P22 heteropolymer and its purely repulsive WCA analogue. The P(Rg) curve for the CG-24 hydrophobic homopolymer is shown for reference. A representative snapshot of the WCA N3N7P18P22 heteropolymer in a compact state is also shown.

D. Effect of Monomer-Water Attractions: LJ versus WCA Monomers. Studies of hydrophobically driven collapse have many times employed polymers comprising monomers that are idealized hydrophobes, that is, they interact with purely repulsive interactions with water and with themselves.4,7 Reducing the monomer-water attractive interactions makes the polymer more hydrophobic and enhances the stability of globular states. In the study of the CG-25 polymer by Athawale et al.7 (from which our model is derived), switching off van der Waals interactions of the polymer increases the depth of the minimum in the polymer PMF, W(Rg ≈ 0.55 nm), from ∼30 kJ/mol to ∼120 kJ/mol. Clearly, attractive interactions provide another handle to tune the stability of folded polymers. To quantify the

effect of monomer-water attractions, we performed simulations of a two-charge-pair heteropolymer, N3N7P18P22, with monomers interacting with water and with each other via purely repulsive Weeks-Chandler-Andersen interactions (WCA).28 The effects of turning off the attractions are dramatic. Figure 7 shows the probability distributions of the radius of gyration, Rg, for the N3N7P18P22 polymer and its WCA analogue obtained from a long MD simulation. N3N7P18P22 favors relatively open, hairpin structures as described previously. However, when dispersive interactions are switched off, hydrophobic interactions dominate, and the WCA analogue collapses into globular conformations. For a different model of heteropolymers, Ashbaugh recently observed a similar collapse into globular states when the effective hydrophobicity was increased by increasing the fraction of hydrophobic monomers.6 The P(Rg) distributions are not identical for homo- and WCA heteropolymer. There is some residual probability of forming open, higher Rg structures for the WCA heteropolymer. Also, the collapsed globular states of WCA heteropolymers seem to contain some internal structure (e.g., see Figure 7) that will be worthy of further investigation. However, it is clear that the increased driving force for collapse for the WCA heteropolymer is almost entirely able to compensate for the partial dehydration of charged monomers, favoring mostly globular structures. The above results highlight an important difference between homo- and heteropolymers with regards to monomer-water attractions. For homopolymers, turning off attractions simply stabilizes the globular conformations further, whereas for heteropolymers, it changes equilibrium conformations from solvent exposed hairpin-like to compact globular structures. Above we have only considered the effects of decreasing monomer-water (and correspondingly monomer-monomer) attractions on conformational equilibria. If the ε parameter for monomers is increased, both monomer-monomer and monomer-water interactions become favorable (due to mixing rules). Thus, their solvation by water also becomes gradually more favorable leading to the unfolding of such polymers for sufficient attraction strength. E. Effect of Salt. The structure and stability of marginally stable macromolecules can be manipulated by the addition of small molecules to solution.29,30 Here, we consider the effect of a common salt, NaCl. Salts increase the surface tension of water and thus increase the strength of hydrophobic interactions. This

Designing Heteropolymer Structure

Figure 8. Probability distribution, P(Rg), of the radius of gyration of the N3N7P18P22 heteropolymer in water and in ∼1.7 and ∼5.0 M aqueous solution of NaCl. P(Rg) for the uncharged hydrophobic homopolymer, CG-24, is shown for reference. A representative snapshot of the heteropolymer in 1.7 M salt solution is also shown.

effect has been well-studied in the literature: salts increase the free energy of hydration of hydrophobic solutes, reduce their solubility,31-34 strengthen the contact minimum in the pair PMFs between small hydrophobic solutes,27,35 and at larger length scales, significantly stabilize the globular states of hydrophobic polymers.27,36 In this sense, salt effects are expected to be simple and predictable. For the heteropolymer considered here, there can be several competing factors, making the problem nontrivial. The increased strength of hydrophobic interactions in salt solutions would favor globular states of the polymer. Salt ions could also destabilize hairpin-like structures either by specifically interacting or binding with charged monomers or simply screening the charge pair interactions. Figure 8 shows the probability distribution of the radius of gyration, P(Rg), of the heteropolymer N3N7P18P22 in water and NaCl solutions. In water, as described previously, stable hairpin structures are formed with a Rg ∼ 0.80 nm. Addition of ∼1.76 M NaCl perturbs the structure of the polymer significantly, favoring more open conformations (see Figure 8). This indicates that favorable interactions of the ions with the charged monomers dominate the strengthening of hydrophobic interactions in the presence of salt. However, hydrophobic interactions begin to play a larger role at higher concentrations of salt. At about 5 M NaCl concentration, the P(Rg) distribution shifts toward more compact structures but is nevertheless nowhere close to that observed for an uncharged polymer in water. Thus, the added salt primarily acts to better solvate the charged monomers, thereby stabilizing more open structures of the heteropolymer. IV. Conclusion We have presented results from extensive MD simulations of model heteropolymers in water. The results highlight that the structural preferences of hydrophobic homopolymers that collapse into structureless globules in water can be significantly altered by the charging relatively few monomers. Specifically, we find that simple hairpin structures can be stabilized by manipulating water-mediated Coulombic interactions. However, the stability of such structures is sensitive to the heteropolymer sequence. Two pairs of oppositely charged monomers placed away from the ends of the hairpin form the most stable folds. Charged heteropolymers, for which the attractive van der Waals interactions are turned off, collapse primarily into compact structures in spite of the presence of charged mono-

J. Phys. Chem. B, Vol. 114, No. 42, 2010 13287 mers. This highlights importance of attractive monomer-water interactions to stabilizing more “open, but folded” structures. Surprisingly, adding simple salts such as NaCl at moderate concentrations unfolds the heteropolymer hairpins. This is in contrast to what is observed in homopolymer models, where hydrophobic interactions are strengthened in salt solutions and compact globular states are further stabilized. Our results may be useful in understanding factors stabilizing secondary structural motifs in water23,37 as well as the mechanism of formation of those secondary structures. More importantly, our study provides one- and two-charge-pair heteropolymer models for studies of structure, thermodynamics, and dynamics in inhomogeneous interfacial13,15,38-40 and confined systems. Acknowledgment. S.G. gratefully acknowledges partial financial support of NSF, Nanoscale Science and Engineering Research Center Grant, DMR-0642573 and New York State NYSTAR program to Rensselaer Nanotechnology Center. References and Notes (1) Lau, K. F.; Dill, K. A. Macromolecules 1989, 22, 3986–3997. (2) Dill, K. A.; Bromberg, S.; Yue, K. Z.; Fiebig, K. M.; Yee, D. P.; Thomas, P. D.; Chan, H. S. Protein Sci. 1995, 4, 561–602. (3) Shakhnovich, E. I.; Gutin, A. M. Proc. Natl. Acad. Sci. U.S.A. 1993, 90, 7195–7199. (4) ten Wolde, P. R.; Chandler, D. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 6539–6543. (5) Hall, C. K. AIChE J. 2008, 54, 1956–1962. (6) Ashbaugh, H. S. J. Phys. Chem. B 2009, 113, 14043–14046. (7) Athawale, M. V.; Goel, G.; Ghosh, T.; Truskett, T. M.; Garde, S. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 733–738. (8) Sun, L.; Siepmann, J. I.; Schure, M. R. J. Phys. Chem. B 2006, 110, 10519–10525. (9) Ferguson, A. L.; Debenedetti, P. G.; Panagiotopoulos, A. Z. J. Phys. Chem. B 2009, 113, 6405–6114. (10) Chopra, M.; Reddy, A. S.; Abbott, N. L.; de Pablo, J. J. J. Chem. Phys. 2008, 129, 135102. (11) Vitalis, A.; Wang, X. L.; Pappu, R. V. J. Mol. Biol. 2008, 384, 279–297. (12) Goel, G.; Athawale, M. V.; Garde, S.; Truskett, T. M. J. Phys. Chem. B 2008, 112, 13193–13196. (13) Jamadagni, S. N.; Godawat, J. S.; Dordick, R.; Garde, S. J. Phys. Chem. B 2009, 113, 4093–4101. (14) Miller, T. F.; Vanden-Eijnden, E.; Chandler, D. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 14559–14564. (15) Jamadagni, S. N.; Godawat, R.; Garde, S. Langmuir 2009, 25, 13092–13099. (16) Straatsma, T. P.; Berendsen, H. J. C. J. Chem. Phys. 1988, 89, 5876–5886. (17) Hummer, G.; Pratt, L. R.; Garcia, A. E. J. Phys. Chem. 1996, 100, 1206–1215. (18) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. J. Phys. Chem. 1987, 91, 6269–6271. (19) Essmann, U.; Perera, L.; Berkowitz, M. L.; Darden, T.; Lee, H.; Pedersen, L. G. J. Chem. Phys. 1995, 103, 8577–8593. (20) Berendsen, H. J. C.; Postma, J. P. M.; Vangunsteren, W. F.; Dinola, A.; Haak, J. R. J. Chem. Phys. 1984, 81, 3684–3690. (21) Lindahl, E.; Hess, B.; van der Spoel, D. J. Mol. Model. 2001, 7, 306–317. (22) Berendsen, H. J. C.; Vanderspoel, D.; Vandrunen, R. Comput. Phys. Commun. 1995, 91, 43–56. (23) Bystroff, C.; Garde, S. Proteins: Struct., Funct., Bioinf. 2003, 50, 552–562. (24) Collins, K. D. Biophys. J. 1997, 72, 65–76. (25) Fennell, C. J.; Bizjak, A.; Vlachy, V.; Dill, K. A. J. Phys. Chem. B 2009, 113, 6782–6791. (26) Fennell, C. J.; Bizjak, A.; Vlachy, V.; Dill, K. A.; Sarupria, S.; Rajamani, S.; Garde, S. J. Phys. Chem. B 2009, 113, 14837–14838. (27) Ghosh, T.; Kalra, A.; Garde, S. J. Phys. Chem. B 2005, 109, 642– 651. (28) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237–5247. (29) Hua, L.; Zhou, R. H.; Thirumalai, D.; Berne, B. J. Proc. Natl. Acad. Sci. U.S.A. 2008, 105, 16928–16933. (30) Rossky, P. J. Proc. Natl. Acad. Sci. U.S.A. 2008, 105, 16825–16826.

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