Thermodynamic Description of Hofmeister Effects on the LCST of

Article pubs.acs.org/JPCB

Thermodynamic Description of Hofmeister Effects on the LCST of Thermosensitive Polymers Jan Heyda† and Joachim Dzubiella*,†,‡ †

Soft Matter and Functional Materials, Helmholtz-Zentrum Berlin, Hahn-Meitner Platz 1, 14109 Berlin, Germany Department of Physics, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany

‡

S Supporting Information *

ABSTRACT: Cosolvent effects on protein or polymer collapse transitions are typically discussed in terms of a two-state free energy change that is strictly linear in cosolute concentration. Here we investigate in detail the nonlinear thermodynamic changes of the collapse transition occurring at the lower critical solution temperature (LCST) of the role-model polymer poly(N-isopropylacrylamide) [PNIPAM] induced by Hofmeister salts. First, we establish an equation, based on the second-order expansion of the two-state free energy in concentration and temperature space, which excellently fits the experimental LCST curves and enables us to directly extract the corresponding thermodynamic parameters. Linear free energy changes, grounded on generic excluded-volume mechanisms, are indeed found for strongly hydrated kosmotropes. In contrast, for weakly hydrated chaotropes, we find significant nonlinear changes related to higher order thermodynamic derivatives of the preferential interaction parameter between salts and polymer. The observed non-monotonic behavior of the LCST can then be understood from a not yet recognized sign change of the preferential interaction parameter with salt concentration. Finally, we find that solute partitioning models can possibly predict the linear free energy changes for the kosmotropes, but fail for chaotropes. Our findings cast strong doubt on their general applicability to protein unfolding transitions induced by chaotropes.

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concentration c on the change ΔT(c) of the LCST of PNIPAM, i.e., the temperature at the collapse transition. It was experimentally found that a series of sodium salts indeed had drastic effects on the cloud-point of PNIPAM.17,18 (Due to the sharpness of the transition the cloud-point is a good practical measure for the location of the LCST.19,20) Most of the salt effects follow the famous Hofmeister series,13,21−24 which describes the potency of the salts to precipitate proteins. However, the detailed behavior with salt type and concentration was utterly complex: the LCST was decreased or increased depending on salt type. ΔT(c) followed a simple linear behavior only for strongly hydrated anions (often called kosmotropic ions23), nonmonotonic otherwise for the weakly hydrated ’chaotropic’ anions. Interestingly, elastin-like peptides feature very similar LCST behavior in salt,25,26 despite their higher chemical heterogeneity when compared to PNIPAM. Despite significant progress in understanding salt effects on peptide-like moieties in recent years,26−29 a detailed thermodynamic understanding of these effects is still lacking. For the convenience of the reader, we plot the experimental cloud-point data of PNIPAM for a selected set of salts,18,30 which are studied in this manuscript, in Figure 1. The lines in this figure correspond to the fit function

INTRODUCTION Thermosensitive polymers have played an integral role in the last decades in the development and design of ’smart’ materials for use as drug carriers, nanoreactors, antifouling materials, and soft biomimetical model systems for cells and tissue.1−12 At their lower critical solution temperature (LCST), those polymers exhibit a sharp volume transition from extended hydrophilic states at lower temperatures to collapsed hydrophobic states at elevated temperatures. The value of the LCST itself responds sensitively to changes in the solution environment. Hence, close to the LCST, the physicochemical properties of the material are massively tunable by external stimuli. The latter typically involve a change in pH or salt concentration or the addition of cosolutes. A detailed understanding of the polymer’s response to those stimuli is of fundamental interest and essential for optimizing the future design of smart materials. A prominent place in the development of smart material applications has been occupied in the last years by the PNIPAM polymer [poly(N-isopropylacrylamide)].10−15 It possesses a LCST near room temperature and human body temperature of T0 = 32 °C, which is interesting for biotechnological applications. It has attracted much attention also due to its chemical resemblance with biological peptides.16 Thus, in additional to its potential in material design it often serves as a well-defined role model in order to understand biochemical processes such as protein denaturation and unfolding. In that respect, much interest was drawn in the past decade to the manipulating action of specif ic salts at © 2014 American Chemical Society

Received: April 29, 2014 Revised: August 22, 2014 Published: August 22, 2014 10979

dx.doi.org/10.1021/jp5041635 | J. Phys. Chem. B 2014, 118, 10979−10988

The Journal of Physical Chemistry B

Article

It is highly desirable obviously not only to categorize the salts by their effects using m-values, but also to predict, or at least interpret, the latter by using more microscopic theoretical concepts. Such a theoretical description had started already in 1960s when Tanford’s transfer model (TM) for protein denaturation/folding was introduced40 and later extended by Bolen, Rose, and co-workers.41 Here, the free energy of transfer from water to a water−cosolvent solution is first experimentally estimated for every amino acid type i. Then, in combination with the knowledge of the difference of exposed amino acid solvent-accessible surface areas (SASAs), ΔAi, between native and denatured ensembles, the free energy change of denaturation ΔΔGu(c) = c ∑i=AA GTFEiΔAi is postulated to be the sum of individual amino acids (AA) Gibbs transfer free energies per molar concentration of cosolute (GTFEi) weighted by their respected SASA differences. State-of-art single molecule FRET experiments42 and recent computer simulations43−47 have validated some of the key assumptions of the TMs, such as additivity43−46 and the proportionality of the transfer free energy to the SASA.46 Alternatively, a similar but more flexible concept than the original TM was introduced recently by Record and coworkers, where the protein is not decomposed into single amino acids but into more basic building blocks, essentially just polar (amide) and nonpolar (aliphatic and aromatic) compounds.48,49 The transfer free energy per building block is estimated by evaluating the spatial partitioning of ions in the compound solvation shell versus bulk; therefore, these models are often called Solute Partitioning Models (SPMs).50 Although the number of surface types is significantly reduced, and thus the model less specifically resolved, the stabilization and denaturation of proteins by Hofmeister salts49 and urea51 were well predicted. With the more basic surface types, the method is now applicable to a large variety of biomolecules and polymers, and therefore lends itself naturally also to the description of (and further validation by) the relatively simple PNIPAM homopolymer. However, a linear behavior of free energy changes may be applicable only for a subclass of salts in a limited concentration range. Classical solution theory36,52,53 for a three-component mixture of solvent (1), solute at infinite-dilution (2), and cosolute (3) predicts that a free energy change derivative with respect to cosolute molar concentration c3 can be expressed more generally by

Figure 1. Experimental cloud-point data (full lines) of the PNIPAM polymer in aqueous solutions of selected salts versus molar salt concentration c measured in the Cremer group.18,30 The lines correspond to fits by eq 1. In the inset the curves for the chaotropic salts are magnified to better illustrate the strong nonlinearity in their cloud point behavior. Their initial slopes are depicted by dashed lines.

ΔT (c) = ac +

Bmax KBc 1 + KBc

(1)

which was empirically derived previously.17,18 It has three fitting parameters, a, Bmax, and KB, and was shown to provide excellent fits. The reasoning behind this particular form was partially based on the observation that cosolute effects on protein folding are often linear in the cosolute concentration,31 leading to the first term. A linear behavior alone, however, was not appropriate for the strongly nonlinear effects of the chaotropic anions; see the inset in Figure 1. Since those ions are known to have a larger potency in binding to amide or hydrophobic groups of peptides,24,32,33 the second term in eq 1 was chosen to have a Langmuir form. In the latter, KB was interpreted as a binding constant, related to the standard binding free energy between anions and a PNIPAM monomer.17 A deeper physical reasoning behind the coefficients or their thermodynamic interpretation could not be provided so far. In the realm of protein folding, cosolute effects are typically investigated by starting from a two-state description F ⇋ U, where the associated standard Gibbs free energy of denaturation (or unfolding) ΔGu = Gu − Gf describes the difference in free energies between the unfolded (u) and folded (f) states.34−36 It was empirically observed that cosolutes often change ΔGu by an amount ΔΔGu(c) = ΔGu(c) − ΔGu(0) in a simple linear fashion in cosolute concentration c over a wide range of concentrations, via31 ΔΔGu(c) = −mc

∂ΔGu(c3) k T ΔΓ23(c3)a33(c3) =− B ∂c3 c3

(3)

where ΔΓ23(c3) = Γ23,u(c3) − Γ23,f(c3) is the difference of the preferential interaction parameter Γ23(c3) in the two states of the solute. The definition of Γ23(c3) is particularly simple in molal units and open systems, where it reads Γ23(m3) = (∂m3/ ∂m2)T,μ1μ3. On a more microscopic level, the preferential interaction parameter is defined by the number of cosolute (N3) and water (N1) molecules in the solute vicinity (so-called local domain) compared to the expected (molal bulk) bulk composition Γ23(m3) = N3 − N1(mbulk 3 /m1 ). Details to this description can be found in the Supporting Information. Further, β−1 = kBT is the thermal energy, and a33(c) = ((∂ ln a3)/(∂ ln c3))T,P the activity derivative.36 For most monovalent salts, however, a33(c3) ≃ 1 to a good approximation. Since the cosolute (salt) concentration c3 is the only concentration used in this text, we coin it simply c in the

(2)

where the constant coefficient m is known as the m-value.34−36 This linear approach has been very popular for categorizing cosolute effects and provides a useful starting point for theoretical treatments of unfolding by cosolutes. While for osmolytes positive and negative values of m are assigned to denaturing and stabilizing cosolutes,24 for salts these values traditionally correspond to ’salting-in’ and ’salting-out’ behavior,32,33 respectively. Since the PNIPAM collapse transition can also be viewed as a two-state process,37−39 and given the overall resemblance of PNIPAM to peptide-like moieties, this m-value approach should naturally also serve to categorize and describe, at least partially, the cloud-point data of PNIPAM in Figure 1. 10980



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beyond free energy changes only linear in c. The derivation is based on a thermodynamic expansion of the free energy of a two-state model. The latter is motivated by the observation that the polymer volume transition at the LCST can be understood as a transition from a dense collapsed state of the polymer to an expanded, coil-like state in a bimodal free energy landscape as a function of a suitable reaction coordinate.37−39 The two states are denoted in following as the collapsed globule state (g) and the extended coil state (c). We note here that although PNIPAM behavior resembles peptides in many respects, its collapse transition (C ⇋ G) with increasing temperature is opposite to unfolding transition of proteins (F ⇋ U). However, in our thermodynamic description we follow the convention for protein unfolding (F ⇋ U ≡ G ⇋ C). Right at the LCST of the polymer in neat water (no salt), in the following called T0, the free energy difference (per monomer) between coil and collapsed states vanishes, ΔG(T0) = Gc(T0) − Gg(T0) = 0. The swelling transition between the two states is accompanied by a transition entropy ΔS0 ≡ ΔS(T0) = −∂ΔG(T0)/∂T. If we identify G with the Gibbs free energy, then the usual thermodynamic relation ΔG = ΔH − TΔS provides ΔS0 = ΔH(T0)/T0 at the LCST, where ΔH(T0) is the corresponding transition enthalpy. Changing the temperature T by a small amount ΔT or adding salt with concentration c will change the free energy balance between the two states in the (c,T) parameter space. For small changes we can apply a two-parameter Taylorexpansion of the free energy, ΔG(c,T), with respect to the variables c and T, taking the neat water state with c = 0 at T0 as the reference state. We will perform an expansion up to second order and subsequently discuss the individual terms. The Taylor expansion of the free energy difference (per monomer) reads

remainder of the manuscript, unless special distinction between species is needed. As seen in its microscopic definition, the preferential interaction parameter Γ23(c) describes the excess adsorption of the cosolutes over water to the solute surface and can be directly related to the local water and cosolute structure in the vicinity of the latter.36 Thus, ΔΓ23 quantifies the cosolute (salt) and solvent redistribution which occurs close to the solute (polymer) upon the collapse (swelling) transition. Consequently, eq 3 tells us that a salt that displays a larger excess adsorption of salt to the swollen form, e.g., by ion binding to specific monomer groups, will tend to shift the equilibrium in favor of that form. Alternatively, a salt that displays preferential exclusion from the swollen form (or excess adsorption to the collapsed form) will tend to shift the equilibrium in favor of the collapsed form. Such a view was also supported by recent allatom molecular dynamics (MD) computer simulations, where NaI induced swelling by preferential adsorption to the monomer, in contrast to the collapse behavior in other, from the monomer depleted salts.28 Linear behavior of the free energy changes in eq 3, as in the m-value framework, is given only if m = kBT ΔΓ′23

(4)

where ΔΓ23 ′ is a c-independent constant and the difference of the preferential interaction parameter has a linear form ΔΓ23(c) = ΔΓ′23 c. We will see, however, that a linear free energy behavior alone is not sufficient to describe the complex data in Figure 1 over the whole concentration range. Higher-order corrections to linear theoretical approaches or to the preferential interaction parameter for a more comprehensive description, to the best of our knowledge, have not been determined and discussed in detail before. The achievements of this work are manifold. We first derive a generally applicable, nonlinear expression for the cosoluteinduced change ΔT of the LCST by a thermodynamic expansion of the two-state free energy. The derived expression resembles empirical eq 1 and fits the data in Figure 1 equally well, but has physically meaningful thermodynamic fitting parameters including higher order free energy and entropy changes. The latter are quantified for the first time for various salts by fitting the experimental data for the PNIPAM polymer. The analysis demonstrates that strictly linear concepts for LCST (or free energy) predictions only apply for the kosmotropes but in general fail for chaotropes. The consequences on preferential binding mechanisms for the specific salts in various concentration regimes are discussed and elucidated. Finally, we attempt a prediction of linear free energy changes for PNIPAM by the recently introduced SPM. Due to the chemical simplicity of PNIPAM the validity of the SPM or related models can be tested here in the case of salts for a welldefined model system. If SPMs are not accurate here, they will likely be even more unreliable in the case of the more heterogeneous proteins. Our study complements the traditional descriptions of Hofmeister effects13,22−24 with new insights from a thermodynamic point of view and provides further connections to microscopic insights on ion adsorption from allatom computer simulations.26−28

ΔG(c , T0 + ΔT ) ≃ ΔG(0, T0) ⎤ ⎡⎛ ⎛ ∂ΔG ⎞ ∂ΔG ⎟⎞ ⎟ + ⎢⎜ ΔT + ⎜ c⎥ ⎝ ∂c ⎠0, T ⎥⎦ ⎢⎣⎝ ∂T ⎠0, T0 0 ⎡ ⎤ ⎛ ∂ 2ΔG ⎞ ⎛ ∂ 2ΔG ⎞ 1 ⎢⎛ ∂ 2ΔG ⎞ 2 2⎥ + ⎜ ⎟ ΔT + 2⎜ ⎟ ΔTc + ⎜ ⎟ c 2 ⎢⎣⎝ ∂T 2 ⎠0, T ⎝ ∂c ∂T ⎠0, T ⎝ ∂c 2 ⎠0.T ⎥⎦ 0

+ ...

0

0

(5)

We now employ the following identities. From the definition of the reference state it follows ΔG(0,T0) = 0. The expression (∂ΔG/∂T)T0,0 = −ΔS(0,T0) ≡ −ΔS0 defines the transition entropy, and, similarly (∂2ΔG/∂T2)T0,0 = −ΔCp(T0,0)/T0 is the change in heat capacity upon the transition. This second-order derivative in T will be neglected in the following, as for our model it is sufficient to work only in the small neighborhood of the reference state (|ΔT| ≲ 20 K ≪ T0). For polymers with larger ΔT, the full second order expansion and corresponding LCST fitting formula are presented and discussed in the Supporting Information. In addition we identify the derivative (∂ΔG/∂c)T0,0 ≡ −m(0,T0) ≡ −m, with the coefficient m which will be discussed in detail later. We also introduce the coefficients (∂2ΔG/∂c2)T0,0 = −m′, and (∂2ΔG/∂c∂T)T0,0 ≡ −(∂m/∂T)T0,0 ≡ −(∂ΔS0/∂c)T0,0 ≡ −ΔS′0, where we realize that the mixed derivative of the state function commutes. In other words, the T-derivative of coef f icient m is equivalent to the derivative of the system’s transition entropy ΔS0 with salt concentration c.

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THEORETICAL MODELS Thermodynamic Expansion of the Two-State Free Energy. We will first establish an equation which formally describes the change ΔT(c) of the LCST by the addition of salt 10981



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Table 1. Coefficients {μ̃i23} Describe Transfer Free Energy (or Chemical Potential) Derivatives per SASAi for Nonpolar Aliphatic (i = np) and Polar Amide (i = p) Groups and Are Partially Adopted from Previous Work54a

For a given salt concentration c, a new LCST is reached if the balance between the populations of the collapsed and extended states are equal, ΔG(c,T0 + ΔT) = 0. We then solve eq 5 for ΔT(c) to obtain the desired equation

1/M·Å2

1

mc + 2 m′c 2 ΔΔG(c) ≃− ΔT (c) ≃ ΔS(c) ΔS0 + ΔS0′c

(6)

Equation 6 can now be used to describe the experimentally measured change of the LCST with salt concentration c of a specific salt. The parameters, determined at T0 = LCST, are the transition entropy ΔS0 of the PNIPAM polymer in neat water and, in leading order, the ion-specific coefficient m. At small salt concentrations the latter can be essentially equated to the general m-value introduced in the Introduction, for which we also followed the sign convention as used in the protein community. Second order corrections, which likely become important whenever the salt distribution is strongly correlated with itself and the polymer, are described by derivatives of the coefficient m with respect to c and T. Values of ΔS0 for pure PNIPAM have been measured or estimated in the literature with a median of about ΔS0 = −17 J/mol/K; see the SI. The remaining thermodynamic coefficients are determined in this work by fitting eq 6 to the available LCST (cloud-point) data for various salt,17,18,30 cf. Figure 1. Interestingly, the fitting formula eq 1, purely based on empirical rational, has the same functional form as eq 6. Its connection to our thermodynamic expansion will be discussed in the Results section. Solute Partitioning Model. We now briefly sketch the SPM for m-value predictions introduced by Courtenay, Pegram, and Record.48,54 In the SPM, the m-value can be estimated by a set of transfer free energy (or chemical potential) derivatives {μi23} per compound i, together with the change in solventaccessible surface area (SASA), {ΔAi}, of the molecular surface of compounds of type i upon the unfolding transition. The i derivatives {μ23 } are estimated by measuring the spatial partitioning of ions in the compound vicinity versus bulk. If ions are preferentially excluded from the surface, the coefficients {μi23} are positive; if ions are enriched (binding), the coefficients {μi23} are negative. In the framework of the SPM, PNIPAM contains only two types of compounds: nonpolar aliphatic (hydrocarbon) groups, which will be indicated as i = np in the following, and the polar amide group, i = p. With that the m-value is given by34,35,55

3 μ̃np 23·10

μ̃ p23·103

Na2SO4 NaF LiCl NaCl KCl NH4Cl NaBr NaNO3 NaI NaClO4 NaSCN

9.4 3.3 1.7 2.4 2.1 1.1 1.7 1.6 1.2 1.3 0.5

−10.0 −3.7 −3.1 −4.1 −3.8 −2.4 −3.4 −5.7 −4.9 −6.7 −4.1

a

The coefficients for additional salts not previously published were derived from single-ion partitioning coefficients according to the same recipe. See the SI for details.

SASA Calculations for PNIPAM by Molecular Simulations. In order to estimate the SASA per monomer of the polar (p) and nonpolar (np) part in the globule (g) and coil (c) p np states of PNIPAM, Apg , Anp g , Ac , Ac , respectively, we performed explicit-water MD computer simulations. We followed recent successful MD simulations of PNIPAM swelling and collapse28 and employed the GROMACS 4.5.4 software package together with the OPLS-AA force-field parametrization.56 The ensemble of collapsed states of an 80mer of PNIPAM was sampled in explicit-water simulation using the SPC/E model57 at a temperature T = 330 K, well above the LCST. Starting from an initially swollen configuration the 80mer quickly (∼50 ns) collapsed into a roughly spherical globule, with a radius of gyration of Rg = 1.4 nm. We gathered statistics in a time window of about 40 ns to sample over 80 different collapsed states. The collapsed state of polymers longer than our 80mer certainly possesses a higher number of entirely buried monomers, while only a minority of monomers form the surface of the collapsed state. Therefore, the averaged SASA per monomer of the collapsed state for a long polymer is likely to be lower than that of an 80mer, where most of monomers are still exposed to solvent on the globule surface. To avoid statistical uncertainty effects, the extended state was modeled in a fully stretched case, which was simulated long enough such that internal degrees of freedom (such as angles and torsions) could relax, but not so long that kinetically trapped locally collapsed states would occur. The SASA calculation was then performed on a single stretched conformation. To be consistent with the SPM,49,54 we employed the SurfaceRacer software58 and Richard’s set of atomic radii59 to estimate the SASA. The fully stretched state is obviously an upper limit of the amount of exposed surface in the swollen state. The results of the SASA analysis are provided in Figure 2. We have calculated distributions and mean values of polar/ nonpolar surface per NIPAM monomer for collapsed and extended ensembles of PNIPAM (conformations are in the insets). With that we have obtained for the collapsed globule 2 2 2 p np state Atotal = Apg + Anp g g = 80 Å , Ag = 8 Å , and Ag = 72 Å , and total 2 p 2 for the swollen coil state Ac = 123 Å , Ac 16 Å , and Anp c = 107 Å2. This results in SASA differences between the globule

⎛ ∂ΔΔG ⎞ ⎟ mSPM = −⎜ ⎝ ∂c ⎠T ,0 0

= −RT (μ23 ̃ np ΔAnp + μ23 ̃ p ΔAp) =−RT ΔAtotal (μ23 ̃ np w np + μ23 ̃ p w p),

salt

(7)

where the wi denotes the area ratios wi = ΔAi/ΔAtotal. The coefficient normalized per SASA of compound i, that is, μ̃ i23 = μi23/RTSASAi for the salts relevant here are summarized in Table 1. More details behind the derivation on the coefficients {μi23} can be found in the SI. With the set of {μi23} coefficients and SASA estimated in collapsed and extended states of the polymer, the m-value for the PNIPAM polymer can be predicted for every salt solution. In order to estimate the SASA of the collapsed and swollen states of PNIPAM, we have employed all-atom MD computer simulations of an 80mer of the polymer, as will be detailed in the next section. 10982



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Table 2. Correspondence between Fitting Parameters from Empirical eq 8 Taken from Previous Work17,18 and the Thermodynamic Coefficients in eq 6 Derived in This Worka thermodynamic model

empirical model

−m/ΔS0 −m′/ΔS0 ΔS0′ /ΔS0

a + BmaxKB 2aKB KB

a

Thermodynamic derivatives are all to be taken at the reference state c = 0 and T = T0.

available and is set to ΔS0 ≃ −17 ± 3 J/mol/K in this work; see the SI. 2. For other salts, such as sodium salts of Br−, NO3−, I−, ClO4−, and SCN−, KB ≠ 0 and Bmax ≠ 0, and the function ΔT(c) exhibits appreciable curvature. According to Table 2, the curvature can be interpreted by originating from second order corrections to the free energy change ΔΔG or transition entropy ΔS0 with concentration c. A purely linear approach thus breaks down for PNIPAM in the case of those ions for a wide range of concentrations. Only for very small (c ≲ 30 mM) or very large (c ≳ 0.5 M) concentrations the LCST behavior can be described as linear over a larger concentration window. 3. At our level of theory the two thermodynamic coefficients m′ and ΔS′0 appear, which were never discussed in detail previously. The fitting to the cloud-point data (or equivalently the comparison in Table 2) make them directly accessible from experimental data and ready for further interpretation. All resulting thermodynamic parameters are summarized in Table 3. We start our interpretation by introducing some general relations between those thermodynamic parameters and the exclusion and binding of ions to polymer compounds. For this we combine our thermodynamic expansion eq 6 with the general expression for the free energy derivative from solution theory in eq 3 in terms of thein general, T- and cdependentpreferential interaction parameter difference ΔΓ23(c,T). We neglect nonideal activity contributions, which is a good approximation since a33 ≃ 1 ± 0.2 for our salts.60 We further assume that ΔΓ23(c) = ΔΓ23 ′ c + ΔΓ23 ″ c2/2 has only linear and square dependencies in c. With that we can express all thermodynamic coefficients by derivatives of the preferential interaction parameter via

Figure 2. Probability distribution of solvent accessible surface areas (SASA) of PNIPAM for collapsed states (top figure) and extended states (bottom figure). The bars describing the polar SASA are colored red and the nonpolar SASA brown. The mean values are shown as dotted filled bars. Selected collapsed and extended conformations are shown as insets, where the polar/nonpolar parts of the polymer are colored red/brown, respectively.

and coil states of ΔAtotal = Atotal − Atotal = Atotal = 43 Å2, ΔAp = 8 c g g Å2, and ΔAnp = 35 Å2. The area ratios wi = ΔAi/ΔAtotal are thus wp = 0.19 and wnp = 0.81, which are finally employed in the SASA-based prediction eq 7 of the salt-induced changes ΔT(c) of the LCST of PNIPAM. Our SASA predictions are in accord with the values found on average for protein folding, where the fractions wnp ≃ 0.75 ± 0.05 and wp ≃ 0.25 ± 0.05 were estimated experimentally.41,51 Due to the peptide-like chemical structure of PNIPAM this agreement reassures that our estimates are very reasonable ones.

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RESULTS AND DISCUSSION Quantification and Interpretation of the Thermodynamic Coefficients. By rewriting the empirical eq 1 as ΔT (c) =

(a + Bmax KB)c + aKBc 2 1 + KBc

(8)

we quickly recognize that it has the same functional form as the thermodynamic expansion model in eq 6 in terms of powers of the concentration c. Apart from ascribing a good physical intuition to the authors of eq 1, this directly leads to two important consequences: First, since the empirical eq 1 (analogously eq 8) fits the experimental data ΔT(c) extremely well, so will our thermodynamic model expressed by eq 6. Second, by comparing both approaches we can relate the coefficients in eq 1 by the thermodynamic coefficients in eq 6 in front of the respective powers of c, which provides the numerical values for thermodynamic coefficients and the starting point for further interpretation. In Table 2 we are summarizing the relation between the coefficients from eq 8, which are a, Bmax, KB, and the coefficients of our thermodynamic model eq 6 which are ΔS0, m, m′, and ΔS′0 at the reference state T = T0 and c = 0. From such a comparison three important pieces of conclusion can already be drawn: 1. If ΔT(c) is purely linear in c as found for more than half of the salts, per construction Bmax = 0 and KB = 0 in the empirical equation, and thus the parameter a = −m/ΔS0. In other words, the slope of ΔT(c) is given by the parameter m divided by the pure PNIPAM transition entropy. The latter is experimentally

1

ΔT (c) = −

ΔΓ′23 c + 2 ΔΓ″23 c 2 ΔS0 kBT0

+

(

ΔΓ ′23 T0

+

∂ΔΓ ′23 ∂T

)c

(9)

In particular, we recognize that the entropy changes, important for the exact description of the LCST behavior, are given by the T-dependence of the preferential adsorption or exclusion from the polymer surface in the respective globule or coil states. With this at hand, we can interpret the thermodynamic coefficients in terms of preferential binding or exclusion of the salt (and their derivatives with respect to T and c) to the solute. In the following, all salts which exhibit a purely linear LCST behavior we call kosmotropes, all others chaotropes. Let us first discuss the thermodynamic action of the kosmotropic ions. For those strongly hydrated entities it is generally accepted that they are preferentially excluded from the polymeric surface. In a radial distribution function between the monomer and the salt this would be reflected by an iondepleted zone (or volume) close to the monomer, see, e.g., 10983



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Table 3. Thermodynamic Coefficients m, m′, and ΔS0′ Obtained by Fitting to the Thermodynamic Expansion Model eq 6a empirical fitting formula (1)

SASA-predicted m-value

thermodynamic expansion model

salt

a [K·M−1]

Bmax [K]

KB [M−1]

m [J/(mol·M)]

m′ [J/(mol·M2)]

ΔS′0 [J/(mol·M·K)]

mSPM [J/(mol·M)]

Na2SO4 Na2CO3 NaH2PO4 NaF LiCl NaCl (I) NaCl (II) KCl NH4Cl NaBr NaNO3 NaI NaClO4 NaSCN

−36.2 −49.7 −23.1 −17.9 −7.84 −10.4 −12.4 −12 −7.87 −7.4 −6.9 −2.9 −12.6 −2.7

0 0 0 0 0 0 0 0 0 1.4 1.1 2.1 3.6 3.1

0 0 0 0 0 0 0 0 0 2.6 2.9 4.3 5.1 4.3

−615.4 −844.9 −392.7 −304.3 −133.3 −210.8 −176.8 −204 −133.8 −63.9 −63.1 104.2 97.9 180.7

0 0 0 0 0 0 0 0 0 −655 −423 −424 −2184 −394

0 0 0 0 0 0 0 0 0 −44 −49 −73 −87 −73

−403.6 N.A. N.A. −214.0 −79.8 −119.7 −119.7 −106.9 −50.4 −77.4 −21.4 −5.2 20.0 37.1

a

The values can also directly be obtained from the coefficients a, Bmax, and KB in the empirical formula (1) by Zhang et al.18 using the equalities in Table 2. The predicted m-values in the rightmost column are results from the application of the SPM54 (eq 7) as discussed in the Methods section. For NaCl we have two different sets of independent experimental data, labeled I18 and II.30.

for NaClO4 where the maximum of the LCST curve can be found already at about c ≃ 0.05 M. Eq 9 shows that those chaotropes thus have a significant c-dependence of the excess adsorption difference and ΔΓ″23 < 0, while ΔΓ′23 > 0. Thus, for the most chaotropic salts, the dif ference in preferential interaction changes sign with c; in other words, the salts are preferentially adsorbed on the coil over the globule for small concentrations, while they are preferentially excluded f rom the coil over the globule for large concentrations. Interestingly, such a turnover behavior was observed also in the theoretical modeling of the surface tension of NaI at hydrophobic model surfaces,61 where the incremental interfacial tension was negative at small c due to preferential adsorption of the iodine anion, while it rose again to reach positive values for larger c due to ion−ion correlation effects. This turnover also appears for the LCST of polyethylene glycol (PEG)62 and uncharged elastin-like peptides25,26 in chaotropic environments. Note that in the present LCST data a purely linear behavior appears again at large concentrations c ≳ 0.5 M. From our equations we find that the slope of this limiting linearity can be expressed by an apparent m-value mapp = ΔS0m′/(2ΔS′0) = −ΔS0a. As we see, this apparent m-value is related to thermodynamic derivatives of the m-parameter, but not connected to the latter itself. Let us now discuss the trends and numbers in Table 3 in more detail. We first focus on the parameter m, describing the linear behavior at small concentrations. In the protein community,34,35,55 negative and positive m-values assign stabilizers and destabilizers, respectively. Inspection of Table 3 shows that this qualitative behavior is also found for the PNIPAM polymer at small concentrations. Large negative values for the parameter m are found for the kosmotropic salts which are preferentially excluded from the polymer. For the mildly chaotropic ions Br− and NO3− the values of m are still negative but 1 order of magnitude smaller than for the kosmotropes. For the known destabilizers I−, ClO4−, and SCN− the m is even positive, pointing to a salting-in effect and a considerable binding affinity and therefore the positive slope in the cloud-point curve in Figure 1. The values of m′ and ΔS0′ for the chaotropic salts are all of similar magnitude apart for perchlorate, where m′ =

simulation results for kosmotropes near the PNIPAM methyl group.26,28 In a highly simplified perspective this leads to an effective description of the adsorption by simple excludedvolume effects. In such a view the transfer of a polymer monomer from the collapsed interior of the globule to the solvent is accompanied by a change of the cosolventinaccessible (i.e., kosmotropic salt-inaccessible) volume ΔV(c,T). The accompanying change in the excess adsorption is thus ΔΓ23(c) ≃ −cΔV with the corresponding free energy change ΔΔG ≃ kBTcΔV. From MD simulations we find that a reasonable estimate of the excluded volume of the first solvation shell of a PNIPAM monomer is on the order of magnitude of ΔV ≃ ΔAtotall ≃ 100 Å3, where we assumed that a monomer of SASA difference ΔAtotal ≃ 100 Å2 has a layer depleted of salt of about l ≃ 1 Å. This leads to a negative coefficient m = −kBTΔV on the order of magnitude of about ∼ −100 J/mol/M in accordance with the values found in Table 3. Furthermore, for kosmotropes the derivatives m′ = 0 and ΔS′0 = −kBΔV is negligible compared to ΔS0 at 1 M of salt. Consequently, the volume change ΔV for kosmotropes is essentially c and T-independent as expected for hard-core like excluded-volume interactions. The strongly hydrated ions seemingly act effectively as simple hard spheres with specific excluded volumes and weak T-dependencies. Hence, for kosmotropes we can write ΔTkosmo(c) = −

k T ΔΓ′ c k T ΔVc mc ≃ B 0 ≃ − B 0 23 ΔS0 ΔS0 ΔS0

(10)

where we identified the constant ΔΓ′23 ≃ −ΔV and ΔΓ″23 = 0. Clearly, such a simple mechanistic picture does not resolve all the interactions (e.g., dispersion) in the system, but apparently they can be adsorbed in an effective interaction volume of reasonable magnitude for the kosmotropes. The situation is quite different for chaotropic salts judging from the nonvanishing derivatives in Table 3. The LCST curves start first in a linear fashion but exhibit curvature already for very small salt concentrations, c ≃ 50 mM. For three of the chaotropes this first slope is even positive (that is, positive parameter m) before the LCST turns over at a maximum to negative slopes ΔT′(c) < 0. This ’turnover effect’ is strongest 10984



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−2184 J/mol/M2 stands out. This large value is the reason for the highly nonlinear behavior of the cloud-point curve for perchlorate at small salt concentrations; for larger concentrations c ≳ 0.5 M the chaotropes all apparently act as stabilizers. NaClO4 exhibits the most substantial effect, solely even crossing the LCST lines of other kosmotropic salts such as LiCl. This curious behavior of the salt NaClO4 is somewhat unexpected given its far-right position in the Hofmeister series,21,22 which points to a very strong destabilizing rather than any stabilizing behavior. Additionally, the average saltingin constants for the peptide group in NaClO4 are among the highest found in the literature.24 One resolution for the perchlorate controversy may be given by recent findings in the studies of generic polymer collapse in highly attractive cosolutes63 and peptide folding in perchlorate64 that strongly attractive cosolutes indeed induce dense collapsed states, however, not by exclusion, but by collective binding and weak cross-linking effects. In such a view, the strong preference for PNIPAM to assume collapsed states in NaClO4 may have very different reasons than the collapsed states induced by the overall excluded kosmotropes. The observed sign reversal in the preferential interaction parameter dif ference ΔΓ23(c) could thus be due to a stronger binding of perchlorate salt by the globule at larger salt concentrations, not due to larger exclusion f rom the coil state. In an alternative view,17,18 one could argue that the strongly binding perchlorate saturates all possible binding sites on the polymer already at very small NaClO4 concentrations, and for larger concentrations the salt acts effectively as a simple, excluded kosmotrope with an apparent m-value. In both interpretations, however, the molecular structure of the collapsed state in perchlorate could be very different when compared to the one in simple kosmotropes, as in the former case the collapsed globule is saturated with cosolvent, in the latter fully depleted. Similar ideas about NaClO4 effects may apply to the cosolute-induced molten globule state of proteins,64−67 whose detailed structure remains obscure and controversially debated. We hope future experiments or computer simulations will be able to shed more light on these interesting issues. Prediction of the Parameter m by the SPM. Since the kosmotropes exhibit strictly linear LCST behavior, our thermodynamic parameter m can be readily equated to the more general m-value. For chaotropes, where the LCST curves are literally curved, the comparison is not straightforward. Let us also make the most reasonable assumption here that our parameter m can also be equated to the common m-value as employed in the SPM. The m-values predicted by the SPM, mSPM, in eq 7 are summarized in the rightmost column in Table 3. If we look first at the qualitative trends only, we find the series

critical view of this comparison shows that a few positions are exchanged, which is not exactly satisfying for a predictive theory. However, at least for the kosmotropes Na2SO4, NaF, NaCl, KCl, and LiCl, the order seems fine within the 20% deviation window, and, by looking again at Table 3 a quantitative agreement could also be reached by rescaling the predicted mvalues roughly by a factor of 1.5. For this, one should recall the inaccuracies that could have appeared in the SASA-based SPM. Two uncertainties are the estimates of the transition entropy, where we assumed ΔS0 = −17 J/mol/K, and the total change in SASA ΔAtotal. Both the entropy ΔS0 as well as ΔAtotal rescale all m-values by a constant factor, which could help to improve the quantitative agreement between the respective series. Similarly, the determination of the chemical potential derivatives {μi23} in the SPM is based on the empirical microscopic parameters (such as a universal hydration density b1 ), therefore contributing to some degree of uncertainty; see the discussion in SI. The comparison for the chaotropes NaBr, NaNO3, NaI, NaClO4, and NaSCN, however, shows that a simple rescaling does not yield a better prediction of m-values by the SPM. Even the interpretation of the mSPM-value as being related to the linear slope at large salt concentrations does not improve the situation. For instance, SPM-predicted slopes for the anions ClO4− and SCN− are positive, while the measured ones are all negative for large c. Based on our analysis, a possible reason for the failure of the SPM for chaotropes could be the violation of the linearity and additivity assumptions. We found strongly nonlinear behavior in ΔΔG for polymer collapse already at small concentrations of chotropes. Given the similarity of the LCST behavior between PNIPAM and elastin-like peptides the performance of the SPM to describe free energy changes of proteins in chaotropic salts solutions, we believe, is questionable and should be reevaluated.

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SUMMARY AND CONCLUDING REMARKS In summary, we have introduced a thermodynamic expansion approach for the description and interpretation of the saltspecific LCST data of the popular PNIPAM polymer. Our study offers a novel view on the thermodynamic action of salt and complements the traditional descriptions of Hofmeister effects.13 The obtained thermodynamic coefficients have a straightforward meaning and are given by the parameter m, which describes linear free energy changes close to the pure water LCST, and its higher thermodynamic derivatives. In the framework of solution theory these coefficients can all be directly related to the difference of preferential interaction parameter (excess adsorption) of salt to the solute in the globule over the coil state. By fitting our approach to the experimental cloud point data all ion-specific coefficients have been determined for the first time and interpreted. While it is found that for kosmotropes the LCST behavior can be explained by strictly linear (excluded-volume) behavior the situation is complex for the chaotropes, for which nonlinear effects can already be large for small or moderate salt concentrations in the submolar range. This nonmonotonic behavior can be assigned to substantial changes in the difference of the excess adsorption of the salt to globule over coil states with increasing salt concentration, even leading to a sign inversion of the difference in adsorption. The latter leads to a ’turnover’ or collapsed−swollen−collapsed ’reentrant’ behavior of the LCST with concentration c for the chaotropes.

Na 2SO4 < NaF < KCl ≃ NaCl < NH4Cl ≃ LiCl < NaBr ≃ NaNO3 < NaClO4 ≃ NaI < NaSCN

for the experimentally determined parameter m. We set the symbol ’ ≃ ’ whenever the relative difference is less than 20%. The SASA-predicted m-values mSPM can be ranked in the order Na 2SO4 < NaF < NaCl ≃ KCl < LiCl ≃ NaBr < NH4Cl < NaNO3 < NaI < NaClO4 < NaSCN

As we can see some of the rough trends concerning the position of kosmotropes and chaotropes agree, while a more 10985



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Such a behavior seems general for some chaotropes as it can be observed for interfacial tensions of aqueous electrolytes at hydrophobic model surfaces61 or the LCST of elastin-like peptides25,26 or PEG.62 In that respect the behavior of PNIPAM’s LCST is very interesting in methylated urea.68−70 Here nonlinear effects appear for large cosolutes concentrations, c ≃ 2M, but, interestingly, the curvature changes from negative (m′ < 0) to roughly zero (m′ ≃ 0) to positive values (m′ > 0) in order of increasing degree of methylation 0, 1, and 2, respectively. Thus, given the very different thermodynamic response, the microscopic binding and structural correlation mechanisms seem to differ considerably among the various methylated cosolutes.68,69 While the preferential binding concept provides a bridge from thermodynamics to an overall binding or exclusion picture of salt to the polymer, the exact microscopic structural reasons for this behavior remain obscured and need to be further addressed by all-atom computer simulations26−28 and experiments,13,26 which hint at very specific binding mechanisms between ions and local monomer groups. We also suspect ion− ion correlations effects to play a role which effectively increases surface tensions,61 and also collective binding effects of chaotropes highly attracted to the globular state63 cannot be ruled out. Such a curious behavior may also be relevant for the interpretation of cosolute-induced molten globule state of proteins by perchlorate.64−67 We have attempted to predict the linear behavior of the LCST at small concentrations with a transfer model, the solute partitioning model, SPM, as employed for salts and osmolytes on proteins and DNA-duplex.49,51 In our case of the relatively simple PNIPAM polymer, the SPM was found useful only for the kosmotropes. The SPM appeared not applicable for chaotropes, where m is sufficiently constant only for a very limited concentration range. It is interesting to note that the nonlinearity in the free energy changes of protein unfolding presented in recent experimental data49,51 has been interpreted solely by nonspecific electrostatic effects. Our study indicates, however, that for chaotropes also other nonlinear effects come into play. This finding is strongly supported by the LCST behavior of the temperature-responsive charge-neutral elastinlike peptides,25,26 where those reentrant effects are even more pronounced. One of such ’chaotropic’ effects, recently introduced and employed in MC study of coarse-grained γDCrystallin protein, may be the amino acid-specific titration by chaotropic ions, which is in turn electrostatically regulated.71 We finally note that the connection of our thermodynamic expansion model to the effects of (nonlinear) nonspecific electrostatic screening by salts has been recently successfully established.72 The latter may hopefully provide a comprehensive thermodynamic description of cosolute effects on the collapse transitions of charged polymers and proteins, which shall be left as an interesting task for future studies.

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Jabir Kherb and Paul Cremer for providing the unpublished LCST shown in Figure 1, Yan Lu and Matthias Ballauff for inspiring discussions, and the Deutsche Forschungsgemeinschaft (DFG) and the Alexander-von-Humboldt (AvH) Stiftung, Germany, for financial support.

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ASSOCIATED CONTENT

S Supporting Information *

Details of the theoretical model, such as full second order expansion, role of the heat capacity; determination of the transition entropy ΔS0 from experimental data and of μ23 coefficients; example of the ΔLCST analysis in terms of preferential interaction parameter. This material is available free of charge via the Internet at http://pubs.acs.org. 10986



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