Cosolvent Effects on Polymer Hydration Drive Hydrophobic Collapse

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Cosolvent Effects on Polymer Hydration Drive Hydrophobic Collapse Divya Nayar, and Nico F. A. van der Vegt J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b10780 • Publication Date (Web): 14 Feb 2018 Downloaded from http://pubs.acs.org on February 20, 2018

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The Journal of Physical Chemistry

Cosolvent Effects on Polymer Hydration Drive Hydrophobic Collapse Divya Nayar and Nico F.A. van der Vegt∗

Eduard-Zintl-Institut f¨ ur Anorganische und Physikalische Chemie, Center of Smart Interfaces, Technische Universität Darmstadt, Alarich-Weiss-Strasse 10, 64287, Darmstadt, Germany ∗ e-mail: [email protected] Abstract Water-mediated hydrophobic interactions play an important role in self-assembly processes, aqueous polymer solubility, and protein folding, to name a few. Cosolvents affect these interactions however the implications for hydrophobic polymer collapse and protein folding equilibria are not well-understood. This study examines cosolvent effects on the hydrophobic collapse equilibrium of a generic 32-mer hydrophobic polymer in urea, trimethylamine-N-oxide (TMAO), and acetone aqueous solutions using molecular dynamics simulations. Our results unveil a remarkable cosolvent concentration-dependent behavior. Urea, TMAO, and acetone all shift the equilibrium towards collapsed structures below 2 M cosolvent concentration and, in turn, to unfolded structures at higher cosolvent concentrations, irrespective of the differences in cosolvent chemistry and the nature of cosolvent-water interactions. We find that weakly attractive polymer-water van der Waals interactions oppose polymer collapse in pure water, corroborating related observations reviewed by Ben-Amotz (Ann. Rev. Phys. Chem. 2016, 67, 617-638). The cosolvents studied in the present work adsorb at the polymer/water interface and expel water molecules into the bulk, thereby effectively removing the dehydration energy penalty that opposes polymer collapse in pure water. At low cosolvent concentrations, this leads to cosolvent-induced stabilization of collapsed polymer structures. Only at sufficiently high cosolvent concentrations, polymer-cosolvent interactions favor polymer unfolding.

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Introduction Hydrophobic interactions are determined by a subtle balance of direct and water-mediated interactions between non-polar solutes, 1–6 with magnitudes that depend sensitively on solute size and the nature of the attractive intermolecular interactions. 7 It is broadly accepted that hydrophobic interactions push oily molecules together and play a fundamental role in, e.g., self-assembly processes, protein folding, protein-protein and proteinligand interactions in water. Counterintuitively, recent studies have indicated that weakly attractive van der Waals interactions of large non-polar solutes such as neopentane, 8 adamantane, 8 fullerene 9 and small alcohol molecules like tert-butanol 10 with water are sufficiently strong to drive these solutes apart in aqueous solution. 1,7,10,11 Repulsive watermediated hydrophobic interactions, which are also suggested by positive osmotic second virial coefficients, 6,12–14 may thus overcompensate the direct, attractive, van der Waals interactions between the non-polar solutes, preventing their association. The interplay between direct (attractive) and water-mediated (attractive or repulsive) interactions in hydrophobic polymer collapse-unfolding equilibria is not well understood. The mechanism underlying hydrophobic polymer collapse in water has been described in terms of a dewetting or drying transition that involves evaporation of water in the vicinity of the polymer. 15 The large but rare solvent density fluctuations involved in dewetting are however sensitive to solute-solvent attractions. 16–19 Therefore, polymer collapse in real systems is governed by a delicate balance of attractive polymer-solvent interactions that oppose polymer collapse and entropic solvent density fluctuations that favor it. The Widom potential distribution theorem 20 provides a theoretical framework for quantifying these contributions in the free energy change of polymer collapse (vide infra). In this work, we explore the effects of cosolvents on hydrophobic polymer collapseunfolding equilibria. Cosolvents, or osmolytes, are small, neutral organic molecules that modulate the hydrophobic effect and, in turn, impact aqueous polymer solubility and protein folding equilibria. 21–23 On the one hand, the unfolding of a protein or a collapsed


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polymer chain requires a solvent density fluctuation that creates empty space needed for the chain to expand into. Cosolvents affect those solvent density fluctuations and, therefore, may shift collapse-unfolding equilibria without direct cosolvent-polymer interactions involved. On the other hand, cosolvent molecules may partition into the polymer/water interface and steer the macromolecular conformational equilibrium through other mechanisms. 24–27 A possible scenario is that preferential binding of cosolvents to the polymer reduces the role of attractive polymer-water van der Waals interactions in the water-mediated hydrophobic interactions discussed above, with so far unknown but significant consequences for hydrophobic polymer collapse. It is, therefore, intriguing to ask how water-mediated interactions and polymer hydration modulate hydrophobic polymer collapse-unfolding equilibria in aqueous systems with cosolvents that partition into the polymer/water interface. Herein, we investigate the effects of the cosolvents urea, trimethylamine-N-oxide (TMAO), and acetone on the two-state collapse-unfolding equilibrium of generic 32-mer hydrophobic polymer. 23 These cosolvents are amphiphilic and interact favorably with water as well as with the polymer and partition into the polymer/water interface. Urea-water mixtures are nearly ideal, with a homogeneous distribution of urea and water molecules at the microscopic level. TMAO can form di- or tri-hydrated complexes 28,29 and it can interact with the polymer through its methyl groups via van der Waals interactions. Acetone-water mixtures are non-ideal and are locally inhomogeneous due to acetonewater interactions being weaker than water-water interactions. 30 We show by means of umbrella sampling free-energy calculations that even though these cosolvents favorably interact with the polymer in, both, collapsed and unfolded states at all cosolvent concentrations, polymer collapse is observed at low cosolvent concentrations while polymer unfolding is observed at high cosolvent concentrations. To rationalize this observation, we demonstrate that polymer collapse in pure water is penalized by attractive polymer-water van der Waals interactions. Addition of small amounts of cosolvent however reduces this dehydration energy penalty due to cosolvent binding to the polymer/water interface. Ef3 ACS Paragon Plus Environment


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fectively, this leads to shifting the collapse-unfolding equilibrium in favor of the collapsed state. At high cosolvent concentrations, favorable polymer-cosolvent van der Waals interactions are instead found to drive polymer unfolding.

Theory The polymer studied in this work is assumed to have a two-state conformational equilibrium, E C, between unfolded or extended (E) and collapsed (C) states. These states can be differentiated based on a polymer size metric such as the radius of gyration, Rg (defined below). Figure 1 shows that each of these two states corresponds to a well-defined free energy minimum in the potential of mean force (PMF), w(Rg ), of the polymer as a function of Rg , with a free energy barrier at Rg# . The PMFs in Figure 1 have been computed with umbrella sampling simulations in the N pT ensemble (vide infra). The C-state comprises of structures with Rg < Rg# that consist of compact as well as the hairpin conformations, as shown in Figure 1. The E-state corresponds to the stretched conformations with Rg > Rg# . The Gibbs free energy change of the transition E→C, referred to as ∆GE→C , can be computed from the unbiased w(Rg ) as,

e

−∆GE→C /RT

R Rg# −w(R )/RT g e dRg = R0∞ −w(R )/RT g e dRg Rg#

(1)

with R denoting the gas constant and T the temperature. The quantity ∆GE→C contains contributions of the changes in the intramolecular E→C E→C potential energy of the chain (∆Eintra ), chain conformational entropy (∆Sconf ), and

polymer solvation (∆µ∗E→C ) (details in Appendix),

E→C E→C − T ∆Sconf + ∆µ∗E→C ∆GE→C = ∆Eintra

(2)

where ∆µ∗E→C = µ∗C − µ∗E is the difference in excess chemical potentials of the chain in the collapsed and unfolded states. The excess chemical potential, µ∗ , corresponds to the 4 ACS Paragon Plus Environment

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Figure 1: Potential of mean force profiles, w(Rg ), as a function of the radius of gyration (Rg ) of a 32-mer polymer chain in aqueous solutions with different urea concentrations at 300 K and 1 atm. Polymer structures corresponding to the minima in the PMF profile are indicated. The collapsed state consists of compact structures (Rg ≈ 0.45 nm) and hairpin structures (Rg ≈ 0.6 nm). The extended state has conformations with Rg > 0.7 nm.

reversible work associated with slowly switching on the attractive and repulsive polymersolvent interactions. It can be expressed in terms of the Widom potential distribution theorem, 20 here written in inverse form, ∗ /RT

eµ

= heφ/RT i

(3)

where φ denotes an energy that corresponds to the interactions between the polymer and the surrounding solvent molecules and h. . .i denotes an ensemble average over all configurations of the solution (that contain the polymeric solute). Equation 3 can be rearranged in the following form, 22

µ∗ = hφi + RT lnheδφ/RT i = Euv − T Suv


(4)


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with Euv ≡ hφi denoting the average polymer-solvent energy, and Suv = −R lnheδφ/RT i with δφ ≡ φ − hφi denoting the polymer-solvent (fluctuation) entropy, which is determined by solvent-density fluctuations and fluctuations of attractive polymer-solvent interactions. A further discussion of the polymer-solvent entropy can be found in reference. 22 The polymer induced changes in solvent-solvent energy and entropy exactly compensate each other and do not directly contribute to the solvation free energy. 31–34 From Eqns. 2 and 4 it follows that

E→C E→C E→C E→C ∆GE→C = ∆Eintra + ∆Euv − T [∆Sconf + ∆Suv ] E→C E→C = ∆Epv − T ∆Spv

(5)

In Eq. 4, we used the notation Euv and Suv for the solute-solvent energy and entropy. The quantities with subscript pv in Eq. 5 include the changes in polymer internal contributions in addition to the contributions deriving from polymer-solvent interactions:

E→C E→C E→C ∆Epv ≡ ∆Eintra + ∆Euv E→C E→C E→C ∆Spv ≡ ∆Sconf + ∆Suv

(6)

We note that Eq. 5 differs from Eqns. 7 and 9 in ref. 22 which account only for the solvation (uv) contribution.

Computational Details The system comprised of a generic model of a hydrophobic polymer with 32 uncharged Lennard-Jones beads as developed by Zangi et al., 23 with σb = 0.4 nm and b = 1 kJ mol−1 . The bonded and angular force-field parameters for the polymer were taken from the work of Zangi et al. 23 The polymer was solvated in 4240 water molecules in a cubic box. Aqueous cosolvent solutions (urea, TMAO, and acetone) of the polymer were prepared with varying concentrations (see Table 1). The SPC/E potential for wa6 ACS Paragon Plus Environment

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ter, 35 Kirkwood-Buff force-field of urea 36 and acetone 37 and the force-field derived by Kast et al. for TMAO 38 were used. Unlike Lennard-Jones interactions were described using the Lorentz-Berthelot mixing rule. Molecular dynamics (MD) simulations were performed using the GROMACS 4.6.7 package. 39 The system energy was minimized using the steepest descent algorithm. Equilibration runs of 5 ns were carried out in the isothermal-isobaric (N pT ) ensemble at 300 K and 1 atm, using the Nosé-Hoover thermostat and the Parrinello-Rahman barostat with τT =0.5 ps and τP = 1 ps, respectively. An integration timestep of 2 fs was used. A van der Waals cutoff of 1.0 nm was used. Longrange electrostatic interactions were calculated using the Particle Mesh Ewald (PME) method 40 with a real space cutoff of 1.0 nm and grid spacing of 0.12 nm. Potentials of mean force (w(Rg )) of the hydrophobic polymer in different cosolvent solutions were computed by performing umbrella sampling simulations as implemented in the PLUMED 2.2.0 plugin. 41 The polymer radius of gyration (Rg ) was chosen as the reaction coordinate for these simulations which was defined in terms of Cartesian coordinates of Np polymer q P Np [(xi − xc )2 + (yi − yc )2 + (zi − zc )2 ], with the subscript c deatoms as, Rg = N1p i=1 noting polymer center of mass. A harmonic restraint potential, Vb (Rg ) was applied on the polymer Rg with a force constant kb of 20000 kJ mol−1 nm−2 , with Rg0 being the desired equilibrium value. Vb (Rg ) =

kb (Rg − Rg0 )2 2

(7)

Polymer conformations with Rg ranging between 0.4 to 1.2 nm were sampled with a spacing of ∆Rg = 0.025 nm between successive windows. For each window, production runs in the N pT ensemble were performed for 20 ns in pure water and for 30 ns in aqueous cosolvent solutions. The barostat and thermostat parameters used were the same as for the equilibration runs as mentioned above. Five umbrella sampling simulation sets were performed for the system with urea from which the average w(Rg ) profile and associated standard error were computed. Because the error was found to be small, for the rest of the systems, the average w(Rg ) profiles and corresponding standard error were computed



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over five splits of trajectory generated from one umbrella sampling simulation. The unbiased potential of mean force, w(Rg ) = −RT ln P (Rg ), was obtained using weighted histogram analysis (WHAM). 42 The free energy change associated with polymer collapse was computed using Equation 1 using Rg# = 0.7 nm. To check possible finite-size effects in the PMFs at low cosolvent concentrations, the PMF was furthermore calculated for the polymer in 0.5 M urea solution using a large cubic simulation box with a 7 nm linear dimension. No significant difference was found with the PMF obtained with the smaller system size (see Table 1) and ∆GE→C computed with the two system sizes was within the error bar. The free energy data for the TMAO Kast model were taken from reference 43 for 1 M, 2 M, 3 M and 4 M concentrations. Additional PMF simulations were performed for 0.5 M and 1.5 M TMAO concentrations.

Table 1: System setup of polymer aqueous solutions. Nw and Nx represent the number of water and cosolvent molecules of type x in the system, respectively. ρ is the average density of the system in units g cm−3 obtained from N pT simulations (300 K, 1 atm.), [c] denotes the cosolvent concentration in units in mol l−1 . The average side length of the box in nm is denoted by hli.

[c] 0.0 0.5 1.0 1.5 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.8 11.2

Acetone Nw Na ρ 4240 0 0.998 4000 38 0.994 4000 86 0.989 4000 124 0.987 4000 167 0.982 4000 264 0.974 4000 376 0.964 3700 480 0.955 3700 628 0.927 3500 748 0.934 3300 904 0.922 2700 1506 0.889 2100 2018 0.866

hli 5.035 5.005 5.072 5.118 5.193 5.313 5.467 5.490 5.663 5.743 5.869 6.359 6.697

Nw 4240 3800 3200 3100 3000 2900 2700 2700 2600 2400 2100 -

Urea Nu ρ 0 0.998 35 1.005 61 1.012 93 1.026 123 1.044 191 1.059 256 1.070 347 1.090 439 1.090 521 1.108 581 1.126 -

hli 5.035 4.888 4.662 4.654 4.643 4.676 4.663 4.764 4.820 4.831 4.768 -

Nw 4240 3929 3696 3000 2800 2500 3500 -

TMAO Ntmao ρ 0 0.998 31 1.001 79 1.009 127 1.015 195 1.021 257 1.036 108 1.051 -

hli 5.035 4.967 4.940 4.904 4.712 4.737 4.695 -

Separate sets of MD simulations were performed to sample flexible C and E polymer 8 ACS Paragon Plus Environment

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ensembles to compute the polymer intramolecular and polymer-solvent interaction energies. A harmonic restraining potential on Rg was used with a force constant of 2000 kJ mol−1 nm−2 , which acted and restrained the polymer conformation to the C- or E-state only when Rg exceeded or fell below the Rg# value of 0.7 nm respectively. The polymersolvent energy, polymer intramolecular energy and other properties were then averaged only over the unbiased or unrestrained polymer conformations. These simulations followed the same procedure for equilibration and production runs as described above. The production runs in the N pT ensemble were carried out for 400 ns each for C and E polymer ensembles in aqueous urea and for 200 ns each for C and E polymer ensembles in aqueous TMAO and acetone solutions.

Results Free energy of polymer collapse Figure 2 shows the free energy change of polymer collapse (∆GE→C ) as a function of cosolvent concentration. In pure water, it is unfavorable to stretch the polymer and the C-state is slightly favored over the E-state as indicated by the ∆GE→C < 0. In the water-cosolvent mixtures, a remarkable non-monotonic dependence of ∆GE→C on the cosolvent concentration is observed. Two concentration regimes can be classified based on the slope of the free energy change. Below 2 M (light purple region), a decrease in ∆GE→C with cosolvent concentration implies that the cosolvents shift the collapseunfolding equilibrium in favor of the C-state. Compared to the other cosolvents, acetone shows the largest effect at low concentrations. Above 2 M (yellow region), ∆GE→C starts increasing with further addition of cosolvent, indicating denaturant cosolvent behavior and a corresponding unfolding of the chain. This inflection in the concentration dependence of ∆GE→C is supported by an independent analysis of the cosolvent preferential binding coefficients Γ to the C- and E-states



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of the polymer. The preferential binding coefficients were calculated using 44 Γ=

Nc − nc (r) nw (r) nc (r) − Nw − nw (r)

(8)

where, nx (r) denotes the number of cosolvent or water molecules at proximal distance r from the polymer surface calculated as the minimum distance from the center of mass of the solvent molecule to the polymer surface. Nx is the total number of water or cosolvent molecules in the system. Figure 3 shows that the preferential binding coefficient is larger in the C-state than in the E-state at low concentrations, while the opposite trend is observed at higher concentrations. Because the cosolvent shifts the collapseunfolding equilibrium to the state to which it preferentially binds, we conclude that the data in Figure 3 comply with the slope change observed in Figure 2. Significantly, positive preferential binding 24 is observed at all concentrations, i.e. direct cosolvent binding to the polymer leads to collapse at low cosolvent concentrations while it leads to unfolding at the higher concentrations. To understand these effects better, we examine the polymer-solvent energetic and entropic contributions to ∆GE→C at different cosolvent concentrations, as described below.

Cosolvent-induced hydrophobic collapse The opposite cosolvent effects in the two concentration windows of Figure 2 are next anaE→C E→C E→C lyzed by considering the quantities ∆Epv and T ∆Spv defined in Equation 6. ∆Epv

is obtained from unrestrained simulations of the C- and E-states (see Computational DeE→C E→C = ∆Epv − ∆GE→C is subsequently obtained, using ∆GE→C tails), while T ∆Spv

from the umbrella sampling calculations. Figure 4(a) shows the polymer-solvent energy E→C E→C change (∆∆Epv ) presented versus the polymer-solvent entropy change (T ∆∆Spv )

at cosolvent concentrations < 2 M. 22,45 The notation ’∆∆’ is used to represent the process of polymer collapse in the systems with cosolvent, relative to polymer collapse in E→C E→C pure water. ∆∆Epv and T ∆∆Spv both decrease with increasing cosolvent concen-


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10 ∆GE → C (kJ mol-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5 0 -5 urea TMAO acetone

-10 -15 0

2

4 6 8 concentration (M)

10

12

Figure 2: The free energy change upon polymer collapse (∆GE→C ) in water and different cosolvent aqueous solutions as a function of cosolvent concentration (300 K, 1 atm). The purple and yellow background colors demarcate the regimes of low and high cosolvent concentration effects. The error bars are within the point symbol size.

tration (as indicated by the arrow in Figure 4 (a)). Hence, cosolvent-induced polymer collapse observed in this concentration window is driven by a favorable polymer-solvent energy change but opposed by an unfavorable polymer-solvent entropy change. Because E→C E→C ∆∆Epv overcompensates T ∆∆Spv , we observe ∆∆GE→C < 0 (Figure 2) at low

cosolvent concentrations. E→C E→C and T ∆Spv in Figures 4 (b) and (c) shows that a large and Inspection of ∆Epv E→C >0) penalizes polymer collapse in positive change in polymer-solvent energy (∆Epv

pure water. This unfavorable energy change is however overcompensated by the polymerE→C solvent entropy (T ∆Spv >0) which favors polymer collapse. Interestingly, this implies

that changes in polymer-water van der Waals interactions, associated with desolvating and collapsing the chain, are energetically unfavorable in pure water. With the addition of cosolvent, this energy penalty decreases rapidly, but this occurs at the expense of the E→C E→C entropy, as indicated by the decreasing trend of ∆Epv and T ∆Spv with increasing E→C cosolvent concentration. Although polymer collapse is entropically favored (∆Spv >0),

this driving force decreases with increasing cosolvent concentration and, therefore, does 11 ACS Paragon Plus Environment


15

Γ

10 5 (a) urea 0.5 M 1.0 M 2.0 M 4.0 M

0 -5 0.0

40 35 30

Γ

25

0.5

1.0 r (nm)

1.5

2.0

1.0 r (nm)

1.5

2.0

(b) acetone 0.5 M 1.0 M 2.0 M 4.0 M

20 15 10 5 0 0.0

0.5

10 8 6 4 Γ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

2 (c) TMAO 0.5 M 1.0 M 2.0 M 4.0 M

0 -2 -4 0.0

0.5

1.0 r (nm)

1.5

2.0

Figure 3: Preferential binding coefficient (Γ) for the collapsed and extended states of the polymer in (a) urea, (b) acetone and (c) TMAO aqueous solutions at different concentrations. The dotted and solid lines represent the data for the extended and the collapsed states respectively.

not explain the decrease in ∆GE→C at low cosolvent concentrations (Figure 2). Hence, E→C the dominant force for cosolvent-induced collapse (< 2 M) is ∆Epv . E→C To understand this better we consider the polymer-cosolvent (∆Ep−cs ), polymer-


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→C T∆∆SE (kJ mol-1) pv

15 10 5

(a) urea TMAO acetone

0 -5 -10 -15 -20

-15

-10 -5 0 5 →C -1 ∆∆EE (kJ mol ) pv

10

15

30 →C ∆EE (kJ mol-1) pv

25 20 15 10 5 0

(b) urea TMAO acetone

-5 -10 0.0

0.5

1.0 1.5 concentration (M)

2.0

30 →C T∆SE (kJ mol-1) pv

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


25 20 15 10 (c) urea TMAO acetone

5 0 0.0

0.5

1.0 1.5 concentration (M)

2.0

Figure 4: (a) Polymer-solvent energy-entropy map in the low-concentration region E→C E→C E→C (< 2 M) showing ∆∆Epv ≡ ∆Epv (cosolvent/water) − ∆Epv (water) versus E→C E→C E→C T ∆∆Spv ≡ T ∆Epv (cosolvent/water) − T ∆Spv (water). The arrow indicates inE→C ) and (c) creasing cosolvent concentration. (b) Polymer-solvent energy change (∆Epv E→C entropy change (T ∆Spv ) upon polymer collapse in water and in different cosolvent solutions below 2 M concentration. The data points for pure water system are shown with the symbol ’*’ at 0 M.

E→C E→C E→C water (∆Ep−w ), and polymer intramolecular (∆Eintra ) energy contributions to ∆Epv ,

E→C E→C E→C E→C ∆Epv = ∆Eintra + ∆Ep−cs + ∆Ep−w


(9)


50 →C -1 ∆EE p-cs (kJ mol )

40 30 20 10 0 -10 (a) urea TMAO acetone

-20 -30 0

2


90 →C ∆EE (kJ mol-1) p-w

10

12


80 70 60 50 40 30 20 10 0 0

2


10

12

-20 -25 →C -1 ∆EE intra (kJ mol )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-30 -35 -40 -45 -50

(c) urea TMAO acetone

-55 -60 0

2


10

12

E→C E→C Figure 5: The change in (a) polymer-cosolvent, ∆Ep−cs , (b) polymer-water, ∆Ep−w , E→C and (c) polymer internal, ∆Eintra (non-bonded and angular intra-polymer interactions) interaction energies upon polymer collapse in different concentrations of cosolvent solutions. The data points for pure water are marked with the symbol ’*’ at 0 M.

E→C These energy components are shown in Figure 5. We note that ∆Eintra in Figure 5 (c)

corresponds to the polymer internal energy change upon collapse in the solvent phase. We E→C < 0, indicating that the observe that in the low cosolvent-concentration window ∆Ep−cs E→C C-state is stabilized by polymer-cosolvent interactions. Significantly however, ∆Ep−cs


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→C ∆EE (kJ mol-1) pv

25 20 15 10 (a) urea TMAO acetone

5 0 0

2


28

10

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26 →C T∆SE (kJ mol-1) pv

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


24 22 20 18 16 14 12 0

2


10

12

E→C E→C Figure 6: The change in (a) polymer-solvent energy (∆Epv ) and (b) entropy (∆Spv ) upon polymer collapse in water and different cosolvent solutions above 2 M concentration. The data for pure water are marked with ’*’ at 0 M.

increases, i.e. becomes less favorable, with increasing cosolvent concentration. This is at odds with the fact that ∆GE→C decreases with the addition of cosolvent in the same concentration window (Figure 2). Therefore, favorable direct cosolvent-polymer interacE→C tions do not explain the minimum observed in ∆GE→C . Note that ∆Eintra < 0 favors

polymer collapse, but also the cosolvent concentration-dependence of this quantity does not explain the minimum observed in ∆GE→C . The key contribution instead comes from E→C the polymer-water energy. The data in Figure 5 show that ∆Ep−w is positive (opposing

collapse), but this energy penalty rapidly decreases with the addition of cosolvent. This 15 ACS Paragon Plus Environment


occurs because the cosolvent preferentially binds at the polymer/water interface (Figure 3) in the E-state as well as in the C-state. As a consequence, the cosolvent expels water into the bulk in both conformational states, and therefore removes the dehydration energy penalty that strongly opposes polymer collapse in pure water. We thus find that weakly attractive polymer-water interactions oppose hydrophobic polymer collapse in pure water. Small amounts of amphiphilic cosolvent added to the system reduce the role of polymer-water attractions and, therefore, cause polymer collapse.

Cosolvent-induced unfolding E→C E→C at higher cosolvent concentrations and T ∆Spv We now discuss the trends in ∆Epv

(> 2 M) as shown in Figure 6. We retain the notation E → C for computed thermoE→C dynamic quantities, even though unfolding is observed at these concentrations. ∆Epv

increases with increasing cosolvent concentration, favoring polymer unfolding. The trends E→C E→C increases up to 4 are cosolvent-specific as shown in Figure 6 (b). T ∆Spv in T ∆Spv

M for the systems with urea and TMAO, opposing unfolding. This effect is however compensated by a favorable change in polymer-solvent energy, driving polymer unfolding. In E→C decreases with concenacetone solution, and at higher concentrations of urea, T ∆Spv

tration and therefore favors unfolding, reinforcing the favorable energy change to unfold the polymer. In acetone solution, this energy-entropy reinforcement mechanism is due to relatively weak acetone-water interactions and correspondingly larger solvent-density fluctuations that facilitate unfolding of the polymer chain. Such an energy-entropy reinforcement behavior in acetone solutions has also been observed for solvation (salting-in) of small non-polar solutes. 46–48 We conclude that unfolding of the polymer above 2 M is driven by favorable polymer-solvent interactions in all cosolvent solutions. At these high cosolvent concentrations, stretching the chain does not provide an equally favorable gain in polymer-water attractive interactions as it occurs in pure water, however, the formation of energetically favorable polymer-cosolvent contacts compensates for this, as 16 ACS Paragon Plus Environment

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supported by the data shown in Figure 5. The favorable change in polymer-solvent energy upon unfolding the chain compensates for, or is reinforced by, the change in polymer-solvent entropy, depending on the type of cosolvent solution. These thermodynamic driving forces are system-specific. In one of E→C increases up to 4 M urea, driving the our previous studies, we showed that ∆Spv

collapse of a Poly-N-isopropylacrylamide (PNIPAM) chain in urea-water mixtures with E→C 26 no significant contribution from ∆Epv . In the present study, we find that in the same E→C window of urea concentrations, ∆Spv also increases with urea concentration but is E→C overcompensated by ∆Epv that drives polymer unfolding.

Discussion and Conclusions In this study we examined the effects of urea, TMAO, and acetone on the two-state collapse-unfolding equilibrium of a generic 32-mer hydrophobic polymer in pure water and in water-cosolvent mixtures. We report the potential of mean force of the polymer as a function of its radius of gyration, as well as the free energy change of polymer collapse, including the polymer-solvent energy and polymer-solvent entropy contributions defined by the Widom potential distribution theorem. We show that in pure water, polymer collapse is opposed by weak polymer-water van der Waals interactions which are responsible for the formation of an energetically stable hydration shell around the extended chain. This result supports ideas in earlier studies, where weakly repulsive water-mediated interactions were found to oppose association of non-polar solutes. 7,10,11 Significantly, we find that urea, TMAO, and acetone partition into the polymer/water interface and expel water into the bulk, thereby effectively reducing the dehydration energy penalty to collapse the polymer. As a result, polymer collapse occurs in these cosolvent-water mixtures with a more favorable free energy change than in pure water. We have supported this picture with a detailed analysis of polymer-solvent energy contributions to the free energy of polymer collapse. It emphasizes that cosolvent-induced polymer collapse in these



water-cosolvent mixtures is determined by cosolvent effects on the polymer hydration shell rather than by a mechanism that involves the direct cosolvent-polymer (van der Waals) interactions themselves. Polymer collapse driven by positive preferential cosolvent binding has been observed in several previous reports. 24–27,49,50 Liao et al. have recently shown that TMAO induces the collapse of a hydrophobic elastin-like polypeptide (ELP) below 2 M TMAO concentration. 27 They propose that ”TMAO acts as a unique surfactant” that interacts with the chemically heterogeneous surface of the collapsed ELP. Effects of TMAO on ELP-water interactions have however not been considered. In another recent work, the role of polymer hydration in PNIPAM cononsolvency in methanol-water mixtures has been investigated. 51 It could be shown that at low concentrations, methanol drives polymer collapse by geometrically frustrating the hydrogen bonding between PNIPAM side chain amide protons and water, in agreement with ideas proposed earlier by Graziano and Pica. 52 With a large-scale conformational sampling approach, involving 1600 independent 100 ns long molecular dynamics simulations, Dalgicdir et al. were able to show that methanol reduces the PNIPAM dehydration energy penalty, which opposes PNIPAM collapse in pure water. 51 Although such PNIPAM-water hydrogen bonding interactions are stronger than polymer-water van der Waals interactions, it highlights the role of polymer-water interactions in inducing hydrophobic polymer collapse at low cosolvent concentrations. Interestingly, in the present work we observe a nonmonotonic dependence of the free energy of polymer collapse on the concentration of acetone in water. This observation resembles the cononsolvency phenomenon (mind however that water is not a good solvent for the hydrophobic polymer) and is determined by polymer-water interactions at low acetone concentrations, and by polymer-acetone interactions at high acetone concentrations. This example underscores the important role of polymer-water interactions in theoretical models for polymer collapse and cononsolvency. 53–56 As for the acetone-water system mentioned above, the results described in this paper show that above 2 M cosolvent concentration, the hydrophobic collapse-unfolding equilibrium shifts 18 ACS Paragon Plus Environment

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towards unfolded polymer conformations in systems with urea and TMAO, too, driven by favorable polymer-cosolvent van der Waals interactions. We finally point out that energetic stabilization of the folded state of peptides by osmolyte depletion has also been observed. Gilman-Politi and Harries showed that, upon folding a model peptide, the depleted osmolyte (sorbitol) induces changes in peptide-water hydrogen bonds, and peptide-peptide internal hydrogen bonds, leading to an energetic stabilization of the folded structure. 57 The observations made for model hydrophobic polymer systems reported herein may have implications for the osmolyte stabilization mechanisms in biomolecular systems where weakly attractive van der Waals interactions of water with non-polar groups of biomolecules are important.

Appendix: Free energy of polymer collapse We consider a single, flexible polymer chain at an arbitrary, fixed position in an isotropic N -particle solvent phase at pressure p and temperature T . The configurational partition function of this system is written as Z Q=

Z dΓ

Z dV

N )+pV

drN e−[E(Γ,r

]/RT

(A.1)

where we use Γ to denote all polymer internal degrees of freedom, V to denote the volume, and rN to denote all solvent configurational variables. The configurational energy, E(Γ, rN ), is assumed to contain additive contributions from polymer internal interactions, Eintra (Γ), polymer-solvent interactions, φ(Γ, rN ), and solvent-solvent interactions, Ess (rN ), E(Γ, rN ) = Eintra (Γ) + φ(Γ, rN ) + Ess (rN )

(A.2)

We will distinguish two subspaces, C and E, of collapsed and extended chains, respectively, based on a polymer size metric (e.g. the polymer radius of gyration) and define the



Page 20 of 28

corresponding configurational partition functions QC and QE Z

Z dΓ

QC = ZC QE =

Z

drN e−φ(Γ,r

Z

drN e−φ(Γ,r

dV Z

dΓ

dV

N )/RT

e−[Eintra (Γ)+Ess (r

N )+pV

]/RT

N )/RT

e−[Eintra (Γ)+Ess (r

N )+pV

]/RT

(A.3)

E

where

R C

dΓ and

R E

dΓ are integrals over the subspaces of C- and E-chains, respectively.

The free energy of polymer collapse, ∆GE→C , is obtained from

e−∆G

E→C /RT

=

QC QE

(A.4)

∆GE→C contains contributions of changes in polymer internal energy, polymer conformational entropy, and solvation. To describe these contributions, we introduce the identity QC QC,0 QC /QC,0 = · QE QE,0 QE /QE,0

(A.5)

in which QC,0 and QE,0 are configurational partition functions corresponding to subspaces of C- and E-chains, respectively, without polymer-solvent coupling terms Z dΓe

QC,0 = =

C QpC

−Eintra (Γ)/RT

=

In Eq. (A.6), QpC ≡

R

dΓe E QpE

Z

drN e−[Ess (r

Z

drN e−[Ess (r

dV

N )+pV

]/RT

N )+pV

]/RT

· Q0

Z QE,0 =

Z

−Eintra (Γ)/RT

Z dV

· Q0

(A.6)

dΓe−Eintra (Γ)/RT and QpE ≡ C

R E

dΓe−Eintra (Γ)/RT are the configu-

rational partition functions of collapsed and extended vacuum chains, respectively, and R R N Q0 ≡ dV drN e−[Ess (r )+pV ]/RT is the configurational partition function of the pure solvent. The quantities QC /QC,0 and QE /QE,0 in Eq. (A.5) define the polymer-solvent coupling free energies (excess chemical potentials) of collapsed chains and extended chains,


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respectively. The excess chemical potential of C-chains (µ∗C ) follows from 20 ∗

QC QC,0 R R R N N dΓ dV drN e−φ(Γ,r )/RT e−[Eintra (Γ)+Ess (r )+pV ]/RT C R R R = dΓ dV drN e−[Eintra (Γ)+Ess (rN )+pV ]/RT C

= e−φ/RT C,0

e−µC /RT =

(A.7)

and corresponds to the reversible work of introducing polymer-solvent interactions. The mean value denoted h· · · iC,0 is an isothermal-isobaric average and may be interpreted analogous to the Widom particle insertion equation: Suppose that in a N -particle solvent and a vacuum C-chain, both in thermal equilibrium, the atomic motions are suddenly frozen such that all atomic positions become rigidly fixed in one of the infinitely many configurations characteristic of the equilibrium solvent phase and vacuum C-chain. Now add the frozen C-chain at an arbitrary, fixed position amidst the fixed solvent molecules and measure the total energy of interaction φ of all polymer atoms with the fixed solvent molecules and the function exp[−φ/RT ]. Then, the average of this function over the infinitely many configurations characteristic of the equilibrium solvent phase and the vacuum C-chain is symbolized h· · · iC,0 . Equivalently, for extended chains, we write ∗

e−µE /RT =

QE = e−φ/RT E,0 QE,0

(A.8)

Using Eqs. (A.5)-(A.8), we can rewrite Eq. (A.4) as

e−∆G

E→C /RT

=

QpC −∆µ∗E→C /RT ·e QpE

(A.9)

E→C in which QpC /QpE = exp −∆AE→C vacuum /RT and ∆Avacuum denotes the (Helmholtz) free energy difference between collapsed and extended vacuum chains. If we furthermore use E→C E→C the relation ∆AE→C vacuum = ∆Eintra − T ∆Sconf , we arrive at the final expression

E→C E→C − T ∆Sconf + ∆µ∗E→C ∆GE→C = ∆Eintra


(A.10)


E→C In Eq. (A.10), ∆Eintra is the change in polymer internal energy upon chain collapse, E→C ∆Sconf is the corresponding change in polymer conformational entropy, and ∆µ∗E→C =

µ∗C − µ∗E is the solvation contribution. The dependence of the latter quantity on the solvent-cosolvent composition can be studied with molecular simulations using a KirkwoodBuff approach, 26 therefore allowing to quantitatively describe cosolvent effects on aqueous polymer solubility.

Acknowledgements The authors thank Dr. Francisco Rodr´ıguez-Ropero for useful discussions and for providing free energy data for TMAO systems. Computations for this work were performed on the Lichtenberg High Performance Computer of Technische Universität Darmstadt, Germany. This research work was supported by the German Research Foundation (DFG) within the Collaborative Research Center ”Multiscale Simulation Methods for Soft Matter Systems” (SFB-TRR146).

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TOC graphic


Cosolvent Effects on Polymer Hydration Drive Hydrophobic Collapse

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