Letter pubs.acs.org/JPCL
Origin of Enthalpic Depletion Forces Liel Sapir and Daniel Harries* Institute of Chemistry and The Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel ABSTRACT: Solutes excluded from macromolecules or colloids are known to drive depletion attractions. The established Asakura−Oosawa model, as well as subsequent theories aimed at explaining the effects of macromolecular crowding, attribute depletion forces to diminished hard-core excluded volume upon compaction, and hence predict depletion forces dominated by entropy. However, recent experiments measuring the effect of preferentially excluded solutes on protein folding and macromolecular association find these forces can also be enthalpic. We use simulations of macromolecular association in explicit binary cosolute−solvent mixtures, with solvent and cosolute intermolecular interactions that go beyond hard-cores, to show that not all cosolutes conform to the established entropically dominated model. We further demonstrate how the enthalpically dominated depletion forces that we find can be well described within an Asakura−Oosawa like model provided that the hard-core macromolecule−cosolute potential of mean force is augmented by a “soft” step-like repulsion. SECTION: Biophysical Chemistry and Biomolecules
A
temperature dependence.21−26 Subsequently, several models have considered additional “soft” (or “chemical”) cosolutemacromolecule attractions (i.e., finite, nondiverging interactions on the order of thermal energy).17,22,27,28 In these studies the entropic depletion attraction is then mitigated by an energy penalty due the added attractions. As in AOM, the repulsive part of the PMFcm in these studies was generally assumed to be hard-core. AOM and later models have faithfully described and predicted cosolutes’ effect on processes that involve a reduction in excluded volume, such as colloidal or macromolecular association, and protein folding. However, recent experiments also reveal depletion forces dominated by favorable enthalpy. Numerous documented examples include peptide and protein folding as well as protein−protein associations in the presence of excluded cosolutes.18,29−31 As AOM predicts, for many excluded cosolutes, the change in folding free energy due to cosolute, ΔGfold, is linear in cosolute concentration. Moreover, the associated enthalpic and entropic components for low to moderate concentrations indicate that some cosolutes, such as the molecularly large polymers dextran and PEG, exert predominantly entropic depletion.18,24 Yet for many other preferentially excluded cosolutes (such as molecularly small polyol osmolytes), the main favorable thermodynamic contribution is enthalpic.18,29−31 Counterintuitively, entropy can even disfavor depletion association or compaction. Hence, AOM and its extensions alone cannot describe the entire gamut of excluded cosolutes’ action. We show here how this apparent conundrum can be resolved by including additional interactions beyond hard-core, and by
n important effective attraction between colloids or macromolecules in solution results from the depletion of added solute (or “cosolute”) that is preferentially excluded from the macromolecular interface. This “depletion interaction” impacts the stability of macromolecules, and directs their association and assembly.1−6 Asakura and Oosawa first described this effective force and attributed it to the entropic gain associated with the reduced cosolute-excluding volume upon macromolecular association.7,8 Their model and related strategies, such as theories of macromolecular crowding, have since successfully described stabilization of folded proteins and their oligomerization by excluded cosolutes or “crowders”.9−11 In Asakura and Oosawa’s model (AOM), depletion forces arise between nondirectly interacting (macromolecular) surfaces induced by cosolutes that interact with each surface through hard-core repulsions. Thus, rather than explicitly accounting for the solvent, its integrated effect was tacitly assumed to yield a completely rigid cosolute-macromolecule potential of mean force, PMFcm. The gain in free energy associated with joining macromolecular surfaces, ΔG, was then in proportion to the reduced cosolute-excluding volume as macromolecules approach, ΔVex, and to solution osmotic pressure, Π, so that ΔG = ΠΔVex. Importantly, depletion attraction in this athermal solution is necessarily purely entropic. Addressing low cosolute concentrations and purely hard-core repulsions, AOM could be considered a limiting case of Kirkwood−Buff theory of solutions12−15 (that quantifies depletion in terms of preferential interaction coefficients) or of scaled particle theory.9,16−19 Beyond the hard-core repulsions considered by earlier models, both in implicit2,7−9 and explicit19,20 binary solvating mixtures, recent experiments with synthetic and protein crowders suggest that steric crowding alone cannot account for cosolute action and its © 2014 American Chemical Society
Received: February 9, 2014 Accepted: March 10, 2014 Published: March 10, 2014 1061
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independently, whereas σij followed the mixing rule σij = 1/2(σii + σjj). Interactions were truncated and smoothed with a cutoff of 2.5σcc + 0.5σmm. Henceforth, we shall use the reduced variables ε̂ij = εij/kT and σ̂ij = σij/σss, where k is Boltzmann’s constant. To probe equilibrium properties, we used Monte Carlo simulations with the Metropolis algorithm in the isobaric− isothermal ensemble (T = εss/k, P = 0.125 εss/σ3ss), employing periodic boundary conditions.32,33 Cell dimensions satisfied Lx ≥ 23σss and Ly, Lz ≥ 6σcc + 1.2σmm. Changes in the PMFmm between rods, . , with separation d were derived using umbrella-sampling34 (Figure 1B). The free energy change upon rod−rod association, Δ.(x) ≡ .(x , d = σmm), and the free energy difference with respect to solutions with solvent only, ΔG = Δ.(x) − Δ.(0), were determined at different cosolute fractions x = Nc/(Nc + Ns) (Figure 1B). We consider here well solvated cosolutes characterized by strong favorable interactions with solvent, macromolecules, and with each other: ε̂cm = 0.2, ε̂cs = 1.26, ε̂cc = 1.6, ε̂ss = 1.0, and ε̂sm = 1.0. Values for all interaction strengths were set in the vicinity of εss = 2.5 kJ mol−1, ensuring that all solutions were in the liquid phase at the simulated conditions, and were in the single phase region as verified in the simulations. We set σ̂mm = 2.67, so that if, for example, σss = 3.0 Å (typical molecular solvents) then σmm = 8 Å, as characteristic of biopolymers. Furthermore, a cosolute with σ̂cc = 1.6, has a partial molar volume that is about 3.6 times that of the solvent, close e.g., to the ratio of glycerol and water; similarly, a cosolute with σ̂cc = 2.3 relates to the solvent’s molar volume as trehalose does to water’s. We find that, for a wide range of cosolute sizes, σ̂cc, and at low to moderate values of x, ΔG varies linearly with concentration (Figure 1C). This linearity matches typical experiments (see e.g., ref 18), and is predicted by AOM, because in the low x limit Π ∼ x. Moreover, at low concentrations ∂ΔG/∂x varies linearly with ΔVex as estimated from the overlap of two cylinders (shown in Figure 1D, lower inset). Thus, ΔG ∼ xΔVex, as shown for many values of σ̂cc and x (Figure 1D). Moreover, the apparent change in free volume, ΔVex,app = V̅ s(∂ΔG/∂x)x→0 (where V̅ s is the solvent molar volume), varies linearly with cosolute partial molar volume, V̅ c (Figure 1D, upper inset). These scaling relations are often used as compelling evidence that depletion attraction is entropic and driven by excluded volume. We show in the following that this conclusion is too general. Insight into the underlying thermodynamic mechanism comes from dissecting free energies into the respective enthalpic and entropic contributions, ΔG = ΔH − TΔS. These components are conveniently described in entropy− enthalpy plots, where they delineate four sectors (Figure 2A, inset). Stabilizing cosolutes, ΔG < 0, appear above the diagonal of full entropy−enthalpy compensation, ΔH = TΔS (cyan). The second diagonal, ΔH = −TΔS, separates stabilization dominated by favorable enthalpy (sector I) and entropy (sector II). For low to moderate values of x, the action of the simulated cosolutes is accompanied by a favorable enthalpic gain, while entropy is decreased and disfavors the association, corresponding to sector I (Figure 2A). These results agree with experiments of osmolyte cosolutes18,29−31 but they are at variance with the AOM prediction. Similarly to the experimental trend, as cosolute molecular size increases, the
directly addressing the role of the solvent. We extend AOM to consider the association of two macromolecules in an explicit Lennard-Jones binary mixture. This simple model already includes the essential ingredients necessary to account for the full range of experimentally observed thermodynamic characteristics of depletion forces, which have so far remained elusive. We find that many cosolutes with an enthalpically dominated mechanism possess an added, finite, repulsive “shoulder” in the PMFcm. Mediated by the solvent, we show that these effective “soft” repulsive cosolute-macromolecule interactions suffice to induce attractive depletion forces that are enthalpically dominated, thereby underscoring the importance of explicit solvent consideration. With this, we suggest that extensions of AOM that consider an added soft repulsion part to the PMFcm more accurately describe the thermodynamic mechanism of depletion forces. This conclusion is general and is independent of the specifics of the underlying interaction potentials that lead to the resulting potentials of mean force. We model an association (“dimerization”) process of two parallel rod “macromolecules” (m) as they approach from a large to smaller separation d. The rods are immersed in binary solutions of Ns solvent (s) and Nc cosolute (c) particles, at temperature T and pressure P (Figure 1A). Representing stiff
Figure 1. Depletion forces as witnessed in simulations. (A) Side and top projections of a simulation snapshot. Solvent and cosolutes are in cyan and purple, respectively; some particles were removed for clarity. (B) Rod−rod potential of mean force, ., versus separation d, where . ≡ 0 for d ≫ σmm. Black and red lines are for pure solvent, and for x = 0.08 of cosolute with σ̂cc = 1.3, respectively. (C) Variation of ΔG with concentration x, for several cosolute sizes σ̂cc. (D) Scaling of ΔG with xΔVex for data in panel C. The dashed line is a guide to the eye. Upper inset: ΔVex,app versus V̅ c. Lower inset: Schematic of ΔVex.
polymers, the rods interact as hard cylinders with diameter σmm, while all other interactions are of the Lennard-Jones type, ⎡⎛ σij ⎞12 ⎛ σij ⎞6 ⎤ Uij(r ) = 4εij⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝r ⎠⎦ ⎣⎝ r ⎠
(1)
Here, εij (the potential well depth) and σij (the distance where Uij(σij) = 0) are defined for any two species i and j, and r is the interparticle distance. Values of εij, εii, and σii were set 1062
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parallel to the full compensation diagonal (Figure 2A). Overall, therefore, we find an entropic loss and energetic gain to the attractive depletion interaction. Alongside cosolute size, variations in ε̂cm and ε̂cs alter the depletion interaction, so that enthalpically dominated depletion forces can even become entropically dominated. The effect of varying ε̂cm is straightforward (Figure 3A,B). Decreasing the
Figure 2. Effect of cosolute size (σ̂cc) on rod−rod association. (A) Entropy−enthalpy plot of the cosolute mediated changes. Inset: entropy−enthalpy scheme. (B) Entropy−enthalpy plot of data in panel A including only macromolecule−solution terms. (C) Macromolecule−solution energy versus solution composition. (D) Macromolecule-solution entropy versus solution composition.
induced stabilization generally shifts toward the entropically dominated sector.18,29,30 Although seemingly contradictory, the AOM prediction and our simulations can be reconciled by further breakdown of the free energy components. Specifically, it has been shown that ΔH can be dissected into two terms: the macromoleculesolution term, ΔEm, that includes sm, cm, and mm interactions, and the remainder, ΔHsol ≡ Δ(Esol + PV) = ΔH − ΔEm, where ΔEsol sums all solution−solution (cc, cs, and ss) interactions. Importantly, it was shown that, though typically large, ΔHsol does not directly contribute to ΔG, because it exactly cancels out with a solution−solution related entropy term in the free energy, ΔHsol = TΔSsol.15,35−37 Hence, the free energy change upon association can be written as
Figure 3. Effect of interaction parameters on the different thermodynamic components. (A) Entropy−enthalpy plot for several values of ε̂cm. (B) Entropy−enthalpy plot of data in (A) for macromolecule-solution terms. (C) Entropy−enthalpy plot for several ε̂cs. (D) ΔG versus x for several ε̂cs. In all sets, σ̂cc = 1.3.
cosolute−macromolecule attraction exerts stronger preferential cosolute exclusion from the macromolecules’ surface, thus favoring their association. The macromolecule-related terms show positive entropy, as AOM predicts. The enthalpic contribution, however, changes from negative (at smaller values of ε̂cm) to positive (for larger values, where the cosolute can even act as a destabilizer/denaturant; Figure 3B). By contrast, changes in ε̂cs only weakly affect the total entropy− enthalpy balance (Figure 3C). Yet as ε̂cs increases, solvent− cosolute miscibility grows, effectively increasing the cosolute’s exclusion from the macromolecules and consequently |∂ΔG/∂x| grows (Figure 3D). Thus far we have explicitly considered solvation in binary (solvent/cosolute) mixtures. And yet, as we demonstrate below, enthalpic depletion forces can still be accounted for even when the solvent degrees of freedom are effectively integrated out, in the AOM spirit. Consider the following simple effective potential acting between a (flat) macromolecular interface and cosolute particle:
ΔG = ΔHsol + ΔEm − T (ΔSsol + ΔSm) = ΔEm − T ΔSm
(2)
Because the solution−solution components only implicitly affect ΔG through their effect on the free energy functional minimum, it is informative to examine the entropic and enthalpic contributions of the macromolecule−solution interactions that impact it explicitly. Figure 2B shows the entropy−enthalpy plot of the macromolecule-related terms for the data in Figure 2A, where ΔEm is the sum over all macromolecule interactions with solvent and cosolute, and entropy is TΔSm = ΔEm − ΔG. Notably, the corresponding values shift to the entropically dominated sector II. For low to moderate values of x, we find smaller values of |ΔEm| (Figure 2C) compared with larger corresponding stabilizing entropy, |TΔSm|. Furthermore, this entropy gain grows with cosolute size (Figure 2D). Interestingly, the macromolecule-solution terms match the entropically dominated AOM prediction, based on implicit solvent. The large solution−solution interaction terms, which fully compensate in ΔG, only shift the total ΔH and TΔS
⎧∞ r ≤ σ1 ⎪ Ucm(r ) = ⎨ γ σ1 < r ≤ σ1 + σ2 ⎪ σ2 < r ⎩0
(3)
where r is the sphere-interface distance (Figure 4A, inset). Here, beyond hard wall repulsion, an added square-shoulder “soft” repulsion leads to two exclusion “layers” around the macromolecule. Because the solvent particles are implicit, hereafter Ucm(r) should be understood as a temperature 1063
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force between macromolecules. Strikingly, we find a very similar PMFcm in simulations of cosolutes that reside in sector I (Figure 4A), associating this feature with the overall enthalpic driven stabilization. Moreover, as the depletion mechanism shifts to the entropic driven sector II with increasing concentration, the PMF also changes, and at short cosolute− macromolecule separation becomes more reminiscent of the AOM hard-core potential. In the simulation, this effective soft repulsion is a result of the combined interactions between solvent, cosolute, and macromolecule. With no prior assumptions, this PMF is an emergent property of the binary solution, where intermolecular interactions are explicit. In conclusion, describing depletion forces requires consideration of interactions between the macromolecule and solution, as well as changes in solution structure upon macromolecular solvation. In AOM, solvent is implicit and depletion forces are completely entropic because cosolutes are fully rigid. By considering explicit mixtures of solvent and cosolute, we find that certain cosolutes can follow the AOM prediction, only augmented by additional solvent-induced terms that impact ΔG only implicitly by contributing to cosolute exclusion. These cosolutes are characterized by an effective “soft” (noninfinite) repulsion between macromolecule and cosolute, in addition to the AOM hard-core repulsion. A simple analytic model demonstrates that even effectively onecomponent solutions, whose particles (cosolutes) possess a “soft” repulsive shoulder to the PMFs, can show overall energetically dominated depletion attractions. This result depends only on the functional shape of the cosolute− macromolecule potential of mean force, and does not depend on the details of the underlying potentials that lead to it. Our study further suggests that molecularly small excluded cosolutes tend to act enthalpically due to short-range solute−cosolute interactions that lead to “soft” (sometimes termed “chemical”) repulsions between cosolute and macromolecules. In contrast, large cosolutes tend to be dominated by more significant hardcore contributions, leading to entropic depletion forces. Finally, our simulations suggest that even when cosolute-excluding volume around macromolecules correlates well with molecular size (Figure 1D) the magnitude of these two volumes can be much different. Further resolving thermodynamic mechanisms for molecular cosolutes requires additional experimental and theoretical verifications. Specifically, it would be interesting to study the mechanisms leading to depletion forces in realistic aqueous solution, where all molecular level details are accounted for,40−43 and work along these lines is ongoing.
Figure 4. Analytic model predicts enthalpically dominated depletion forces. (A) PMFcm for cosolute with σ̂cc = 2.0 and varying x (between 0.007 (black) and 0.115 (cyan)). Inset: Macromolecule−cosolute intermolecular potential (eq 3). (B) Delineation of thermodynamic regimes for the model PMFcm in eq 3, in terms of σ1/σ2 and γ. Numbers correspond to sectors in Figure 2A, inset.
independent PMFcm, the cosolute−macromolecule potential of mean force (PMF) in the medium of other solution molecules. While details of the particular form of the PMF (eq 3) do not qualitatively alter the subsequent conclusions, our choice is well motivated by the emerging PMF in simulations, as we discuss in the following. Moreover, with a single parameter, this PMF can be used to account for “soft” attraction (square well for γ < 0) or repulsion (repulsive shoulder for γ > 0). As usual, the net depletion of cosolute leads to free energy gains when interfaces are buried, as in macromolecular association. This “association” free energy is the difference between the free energy of solvating “monomeric” macromolecules and solvating them as a “dimer”. For ρ→0, the solvation free energy, ΔGtr, is ΔGtr = ρ
∫λ
dλ
∫r
dr
∂Ucm(r , λ) −βUcm(r , λ) e ∂λ
(4) −1 36−39
where ρ is cosolute density and β = (kT) . The integration parameter λ varies from 0, where Ucm ≡ 0, to a value for which Ucm takes its full value in eq 3. The gains in free volume for each exclusion layer upon association, per unit of interfacial area, become ΔV1 = −σ1 and ΔV2 = −σ2. The corresponding association free energy is tr tr β ΔG = β ΔGdim − 2β ΔGmon
= 2ρ[−σ1 − σ2(1 − e−βγ )]
(5)
Requiring ΔG < 0 and ΔE < −TΔS = −β(∂ΔG/∂β) (defining sector I, Figure 2A) translates to σ1 < e−βγ (2βγ + 1) − 1 σ2 (6)
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AUTHOR INFORMATION
Corresponding Author
Thus, below a critical value of σ1/σ2 ≈ 0.2, there exist values of βγ such that the depletion interaction is energetically dominated (Figure 4B). Other values of σ1/σ2 and γ correspond to entropically stabilizing cosolutes (the AOM prediction, sector II) or destabilizing cosolutes (as expected for cosolutes bearing “soft” attractions,21,27 sector III). This simple model cannot explain regions where ΔS < 0, which likely requires consideration of explicit changes in solution structure. While the first term in eq 5 represents the AOM contribution to depletion, the added “soft wall” is solely responsible for the temperature-dependent part of βΔG. Moreover, because for γ>0 the latter term represents macromolecule-cosolute repulsion, it contributes to a favorable (attractive) depletion
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank N. van der Vegt for helpful discussions. We acknowledge support from the Israel Science Foundation (ISF Grant No. 1538/13). L.S. is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. The Fritz Haber Research Center is supported by the Minerva Foundation, Munich, Germany. 1064
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