Structure and Thermodynamics of ProteinâPolymer Solutions

J. Phys. Chem. B 2005, 109, 15107-15117

15107

Structure and Thermodynamics of Protein-Polymer Solutions: Effects of Spatially Distributed Hydrophobic Surface Residues Malin Jo1 nsson*,† and Per Linse‡ Biochemistry and Physical Chemistry 1, Lund UniVersity, Box 124, SE-221 00 Lund, Sweden ReceiVed: October 26, 2004; In Final Form: March 13, 2005

Protein-polymer association in solution driven by a short-range attraction has been investigated using a simple coarse-grain model solved by Monte Carlo simulations. The effect of the spatial distribution of the hydrophobic surface residues of the protein on the adsorption of weakly hydrophobic polymers at variable polymer concentration, polymer length, and polymer stiffness has been considered. Structural data on the adsorbed polymer layer and thermodynamic properties, such as the free energy, energy, and entropy, related to the protein-polymer interaction were calculated. It was found that a more heterogeneous distribution of the surface residues promotes adsorption and that this also applies for different polymer concentrations, polymer chain lengths, and polymer flexibilities. Furthermore, the polymer adsorption onto proteins with more homogeneous surface distributions displayed larger sensitivity to polymer properties such as chain length and flexibility. Finally, a simple relation between the adsorption probability and the change in the free energy was found and rationalized by a simple two-state adsorption model.

Introduction Protein chemistry and interactions between proteins and other macromolecules, such as polymers, are widely studied areas. The manifold functions of proteins can to a large extent be attributed to their surface properties and thus to the types of amino acids exposed at the surface. Typically, several types of interactions of different ranges and magnitudes are at play. In this context, a fundamental question is how multiple nonspecific interactions come together and become highly specific, resulting in recognition and self-organization in, for example, biological systems.1,2 Evaluation of protein surface properties is generally undertaken by dividing the amino acids into subgroups depending on, for example, their charge, ability to form hydrogen bonds, and hydrophobicity. The charge-charge interaction has been examined extensively, and it is essential for protein-polyelectrolyte association.3-7 Nevertheless, other studies show the importance of hydrogen-bonding ability and hydrophobic interactions in protein-polymer association, even when chargecharge interactions are present.6,8,9 In many systems, particularly with uncharged macromolecules, the hydrophobic effect is the dominant factor for protein-polymer association. The surface hydrophobicity of proteins is especially important in association with slightly hydrophobic polymers such as poly(ethylene oxide propylene oxide) (EOPO) or hydrophobically modified polymers such as hydrophobically modified polyacrylate (HMPA).10-14 Berggren et al. have determined the net surface hydrophobicity by considering the amino acid exposure as well as the degree of hydrophobicity.13,14 In aqueous two-phase systems of EOPO and dextran, the hydrophobic EOPO phase was preferred to the more hydrophilic dextran phase as the net surface hydrophobicity increased. * Author to whom correspondence should be addressed. E-mail: [email protected]. † Biochemistry. ‡ Physical Chemistry 1.

Although the hydrophobic effect in protein-polymer association is well-established as a phenomenon, effects of the spatial distribution of the hydrophobic residues are less investigated. Nevertheless, the effect of the location of hydrophobic patches on hydrophobic interaction chromatography was recently examined by Mahn et al.15 Moreover, the influence of the spatial arrangement of hydrophobic surface residues on the interaction with a slightly hydrophobic polymer has been examined by simulation methods.16 In our previous study,16 we found that the distribution of the hydrophobic surface residues played a decisive role for the adsorption of a single polymer onto the surface of a protein. In particular, it was observed that adsorption is promoted by a heterogeneous distribution of the hydrophobic residues. These results were obtained from a coarse-grain model, in which the complex hydrophobic interaction between nonpolar species molecules in water17-26 was simplified and represented by a short-range 1/r6 potential. The simplicity of the model suggests that our findings are generic. Here, we extend our previous study to include the adsorption of polymers onto a single protein with different distributions of the hydrophobic surface residues at variable polymer concentrations and for polymers with different lengths and stiffnesses. In addition, besides adsorption probabilities, the change in free energy of inserting a protein into the polymer solution was evaluated, and a correlation between structural and free energy data was established. Model A simple coarse-grain model is used to examine structural and thermodynamic properties of aqueous solutions containing a globular protein and slightly hydrophobic polymers. The protein is represented by a large hard sphere with hydrophobic surface residues, and the polymers by chains of hard spheres (segments) connected by harmonic bonds. Water is modeled implicitly and enters the model only through a short-range attractive potential between the protein surface residues and the

10.1021/jp0451288 CCC: $30.25 © 2005 American Chemical Society Published on Web 07/19/2005

15108 J. Phys. Chem. B, Vol. 109, No. 31, 2005

Jo¨nsson and Linse

Figure 1. Illustration of the hydrophobic surface potential of one representative protein from protein classes MD2, MD4, and MD6. For the color code, dark blue denotes zero potential, green intermediate attractive (negative) potential, and dark red strong attractive potential.

polymer segments, representing a hydrophobic attraction. We have previously employed this model,16 and below only a brief description of it will be given. Protein. The construction of the protein model was guided by several crystal structures of small- to medium-sized proteins from the Protein Data Bank (PDB).16 In our model, proteins are represented as hard spheres with radius Rp ) 16 Å and Nhs ) 30 hydrophobic surface residues. The hydrophobic residues are placed 13 Å from the center of the sphere, and hence 3 Å below the surface, merely to produce hydrophobic patches rather than hydrophobic points on the protein surface. The residues were distributed randomly on the protein surface but subjected to the restriction that any two residues should be separated by, at least, the distance Rmin. Three classes of proteins were considered, the classes being characterized by Rmin ) 2, 4, and 6 Å, and referred to as MD2 (minimum distance 2 Å), MD4, and MD6, respectively. A larger Rmin gives rise to a more uniform residue distribution. Thus, a sequence of proteins with more heterogeneous (MD2) to more homogeneous (MD6) surface properties was established. Ten different distributions of each class were generated using different sequences of random numbers. Hence, in total 3 × 10 ) 30 different proteins have been considered. These proteins with their specific residue distributions constitute a subset of those proteins previously examined.16 The hydrophobic surface potential of one representative protein from each protein class is shown in Figure 1, which also clearly illustrates the decreased heterogeneity as Rmin is increased. These representative proteins have structural and thermodynamic properties characteristic for their classes, and the selection of representative proteins was the same as in our earlier study.16 Polymers. The polymers are represented as chains of nseg ) 10, 16, 20, 25, or 40 hard spheres (segments) connected with harmonic bonds, where each segment has a radius of Rseg ) 2 Å. The number of polymers in the solution varied from Nc ) 1, 5, 10, 20, to 40. Furthermore, polymers with different persistence lengths are considered, and they were established by assessing different spring constants of angular harmonic bond potentials. Potential Energy. All interactions are assumed to be pairwise additive. The total potential energy, Utot, is divided into three terms according to

Utot ) Unonbond + Ubond + Uangle

(1)

The nonbond potential energy, Unonbond, is given by

Unonbond ) Uhs + Uhydrophob )

uhs ∑i ∑ ij (rij) - ∑ ∑ j>i i j

rij6

(2)

with

uhs ij (rij) )

{

0 rij g Ri + Rj ∞ rij < Ri + Rj

(3)

where Ri denotes the radius of particle i. Hence, the hydrophobic potential is described by an attractive 1/r6 potential. In the hardsphere term, the sum extends over protein center-polymer segment pairs as well as pairs of polymer segments, whereas in the hydrophobic potential term the sum only extends over protein residue-polymer segment pairs. Note that the interaction among polymer segments does not depend on , which simplifies the evaluation of the results at different polymer segmentprotein interactions. In addition to Unonbond, the description of the polymer includes harmonic potentials for bonds and angles. The bond potential energy, Ubond, is given as Nc Nseg-1

Ubond )

∑ ∑ k)1 i)1

kbond k - r0)2 (ri,i+1 2

(4)

k where ri,i+1 denotes the distance between segment i and i + 1 of polymer k with the equilibrium constant r0 ) 5.0 Å and the force constant kbond ) 2.4 kJ/(mol Å2). The root-mean-square (rms) segment-segment separation for connected hard-sphere segments becomes 〈Rseg,seg2〉1/2 ≈ 5.5 Å. The angular potential energy, Uangle, is represented by

Nc Nseg-1

Uangle )

∑∑ k)1 i)2

kangle (Rki - R0)2 2

(5)

k k where Rki is the angle formed by the vectors ri+1 - rki and ri-1 k - ri made by three consecutive segments of polymer k with the equilibrium angle R0 ) 180° and force constant kangle. To model a variety of different chain flexibilities, six different angular force constants, kangle ) 0, 0.0025, 0.0075, 0.015, 0.03, and 0.075 J/mol deg, were used, corresponding to the persistence

Thermodynamics of Protein-Polymer Solutions

J. Phys. Chem. B, Vol. 109, No. 31, 2005 15109

∆A(′) ) ∆Ahs + ∆Aint(′)

TABLE 1: General Data of the Model protein radius number of hydrophobic sites number of polymer chains number of polymer segments segment radius persistence length box length temperature

Rp ) 16 Å Nhs ) 30 Nc ) 1, 5, 10, 16, 25, 20, or 40 nseg ) 10, 16, 25, or 40 Rseg ) 2 Å lp ) 7, 22, 60, 114, 224, or 555 Å L ) 250 Å T ) 298 K

lengths lp ) 7, 22, 60, 114, 224, and 555 Å, respectively. General data of the model are compiled in Table 1. Simulation Details Structural and thermodynamic properties of the model system were obtained by performing Metropolis Monte Carlo simulations27 in the canonical ensemble at the temperature T ) 298 K. The protein and the polymers were enclosed in a cubic box with periodic boundary conditions. A box length of L ) 250 Å was chosen, which slightly exceeds the contour length of the longest polymer (∼220 Å). In the Metropolis sampling, the protein was subjected to translational and rotational trial moves, whereas the polymers were subjected to trial moves involving (i) translation of a single segment, (ii) pivot rotation of one end of the chain, (iii) a slithering move, and (iv) translation of an entire chain with the relative probabilities 0.80, 0.10, 0.05, and 0.05, respectively. The simulations were performed by employing 1 × 106 trial moves per particle for systems with up to 400 polymer segments, 2 × 105 for systems with 800 polymer segments, and 1.5 × 105 for systems with 1600 polymer segments. The simulations were performed using the program MOLSIM, a Monte Carlo/ molecular dynamics/Brownian dynamics simulation package.28 Properties Investigated Adsorption. A geometrical definition has been applied to determine whether a polymer is adsorbed to the protein surface or not. A polymer is regarded to be adsorbed if the distance between at least one of its segments and the protein |rseg - rp| does not exceed the contact distance, Rc, according to

|rseg - rp| e Rc ≡ Rseg + Rp + 5 Å

(6)

The results do not qualitatively depend on the exact value of Rc. The probability for exactly nads (nads ) 0, 1, ...) polymers adsorbed to the protein is denoted by Pnads. The probability for at least one polymer of the Nc polymers to be adsorbed to the ∞ protein, Pads, is related to {Pnads} according to Pads ) ∑n)1 Pnads. Radial Distribution Functions. The structure of the solution near the protein has been characterized by radial distribution functions (rdfs). In particular, the protein-polymer segment rdfs have been employed. This function expresses the local density of polymer segments as a function of the radial distance from the protein center. Conventionally, an rdf is zero for short distances at which hard-sphere overlap occurs and is normalized to unity at large separations where spatial correlations are lost. Thermodynamics. To get further insight into the polymer adsorption, free energy, energy, and entropy changes associated with the insertion of a protein into a polymer solution were calculated. These changes arise from the direct protein-polymer interactions and from structural rearrangements in the polymer solution. The insertion free energy at the hydrophobic interaction strength ) ′, ∆A(′), can conveniently be divided into two contributions according to

(7)

The hard-sphere term, ∆Ahs, denotes the free energy change of inserting a hard sphere of radius Rp into the polymer solution, i.e., to create a cavity in the polymer solution, which fits the sphere. This contribution is the same for all proteins because they all have the same hard-sphere radius but depends on the properties of the polymer solution. The interaction term, ∆Aint(′) ≡ A( ) ′) - A( ) 0), represents the free energy change of turning on the short-range attraction from ) 0 to ) ′. The hard-sphere contribution was calculated by using the conventional Widom insertion method.29 In our case with only hard-sphere interactions between the inserted protein and the remaining system, we obtain

∆Ahs ) -kT ln Pacc

(8)

where Pacc is the fraction of insertions with no hard-sphere overlap. The interaction contribution was obtained by using thermodynamic integration. In this procedure, the initial system is brought continuously into the final one by the coupling parameter λ. The λ-dependence is here accomplished by substituting with λ in the second term of eq 2. The free energy difference between the systems described by λ ) 0 and λ ) 1 is then given by

∆Aint ≡ A(λ ) 1) - A(λ ) 0) ≡

∫01 ∂λ∂ A(λ) dλ ) ∫01 〈∂λ∂ Utot(λ)〉λ dλ

(9)

where 〈‚‚‚〉 denotes a canonical ensemble average and Utot is given by eq 1. The integral was determined numerically by using the trapezoidal rule with up to 17 points representing the integrand. The insertion free energy, ∆A(′), is easily separated into an energetic, ∆U(′), and an entropic, ∆S(′), contribution. This is achieved by realizing that the energy change of inserting a hard sphere is zero, and thus ∆U(′) ) Utot( ) ′) - Utot( ) 0), where Utot is directly obtained from the simulations. We are now able to consider the free energy difference between two different protein-polymer systems. Let ∆∆Ass′ denote the free energy difference between system s and s′ according to

∆∆Ass′ ) ∆As′ - ∆As

(10)

When systems s and s′ differ with respect to residue distributions or hydrophobic interaction strength, ∆∆Ass′ becomes the difference in the chemical potential of these two proteins in a polymer solution. In the case systems s and s′ differ with respect to polymer concentration, chain length, and/or chain stiffness, ∆∆Ass′ is the difference in the chemical potential of introducing a protein into two different polymer solutions. If these two polymer solutions form two different phases, then we have established a direct link to the partitioning of the protein between these two polymer phases. Finally, regarding calculated enthalpic and entropic quantities, it should be noted that in our model only the degrees of freedom associated with the polymers contribute to ∆S. Entropic contributions arising from the degrees of freedom of water are embedded in the temperature dependence of , and these entropic contributions are in the present model included in ∆U. This is in complete analogy with, for example, the temperature depen-


Jo¨nsson and Linse

Figure 2. Adsorption probability Pads() as a function of the interaction strength at the indicated number of polymers Nc for the representative protein of class MD2: nseg ) 40; lp ) 7.

dence of the dielectric permittivity for Coulomb interaction in dielectric media. Results and Discussion The adsorption of polymers to the protein surface will first be examined for a single distribution of the hydrophobic surface residues at different strengths of the hydrophobic interaction and different polymer concentrations. Thereafter, the role of the surface residue distribution on the adsorption will be considered; again different interaction strengths and polymer concentrations will be used. Then we continue to examine the influence of the polymer length and the polymer flexibility on the adsorption at different interaction strengths. Finally, a comparison across the different conditions will be given. Single Protein of Class MD2. Structural and thermodynamic properties of systems containing the representative protein of class MD2 have been determined at different and Nc. The interaction parameter was varied between 0 and a value where strong adsorption appears, and the variation of the polymer concentration was achieved by considering systems with Nc ) 1, 5, 10, 20, and 40 polymers at fixed volume. Structural Data. Figure 2 shows the adsorption probability Pads as a function of the hydrophobic interaction strength for systems with different numbers of polymers. At low , Pads is close to 0, showing that protein-polymer association is rare, whereas at high , Pads ) 1, implying the existence of a proteinpolymer complex. At intermediate , Pads is s-shaped with a marked transition at which a small change in induces an adsorption. The abrupt onset of the adsorption is a general feature of polymer adsorption to surfaces when attractive shortrange interactions are involved.30 For polymers near the protein surface, below the adsorption transition the entropic penalty, due to restricted polymer conformations, dominates over the enthalpic gain arising from the attractive protein-polymer interaction. With an increase in polymer concentration, (i) below the adsorption transition Pads becomes significant larger than zero, (ii) the transition is shifted to lower values of , and (iii) the transition becomes less sharp. All three observations are consequences of the increased osmotic pressure associated with the higher polymer concentration, which in turn makes it more favorable for the polymers to be located close to the protein surface.

Figure 3. Adsorption probability Pnads() as a function of the interaction strength for (a) Nc ) 10 and (b) Nc ) 40 polymers at the indicated number of adsorbed polymers nads for the representative protein of class MD2: nseg ) 40; lp ) 7.

The probabilities of having different numbers of polymers adsorbed to the protein Pnads as a function of the interaction strength are displayed in Figure 3 for systems with Nc ) 10 and 40 polymers. Naturally, the probability of having no polymers adsorbed P0ads ) 1 - Pads decreases as increases. The probability functions for nads ) 1 and 2 for a system with 10 polymers and for nads ) 1, 2, and 3 for a system with 40 polymers display a maximum at some , whereas the probability functions for larger nads are continuously increasing or zero for the interval investigated. Moreover, there is a considerable overlap between the curves, showing that the number of adsorbed chains fluctuates at a given value of . In more detail, for the system with Nc ) 10 polymers, the maximal probability of finding exactly one or two polymers adsorbed to the protein appears at ) 1.4 × 105 and 1.8 × 105 kJ Å6/mol, respectively (Figure 3a). Moreover, within the studied interval of the largest number of adsorbed polymers was four. At larger polymer concentrations, Nc ) 40, the maxima of the probability distributions are shifted to smaller , and the largest number of adsorbed polymers is increased to five (Figure 3b). These numbers of adsorbed polymers at large are reasonable for a polymer with a root-mean-square radius of gyration 〈RG2〉1/2


Figure 4. Protein-polymer segment radial distribution function for (a) Nc ) 10 and (b) Nc ) 40 polymers at the indicated interaction strength (given in 105 kJ Å6/mol) for the representative protein of class MD2: nseg ) 40; lp ) 7.

≈ 21 Å, which is somewhat larger than the protein hard-sphere radius Rp ) 16 Å. Figure 4 shows the protein-polymer segment rdfs, for the systems with 10 and 40 polymers. At small values of , the rdf is below unity, whereas for larger values of the rdf becomes larger than unity at distances larger than Rp + Rseg ) 18 Å. The transition from an effectively repulsive to an effectively attractive protein-polymer interaction appears at ≈ 0.8 × 105 kJ Å6/mol, somewhat below the value of at the transition in Pads. At a closer inspection of Figure 4, we find that at ) 0 the depletion zone extends approximately ξobs ) 30 Å from the protein surface, i.e., to r ≈ 46 Å. Below the overlap concentration, theoretical arguments for planar surfaces suggest ξtheory ≈ 2〈RG2〉1/2 ≈ 40 Å, in reasonable agreement with ξobs. At Nc ) 40, the polymer concentrations Fc ) Nc/L3 are still far below the overlap concentration Fc* ≈ [(4π/3)〈RG2〉3/2]-1 (Fc/Fc* ≈ 0.1 , 1). At g 0.9 × 105 kJ Å6/mol, the rdfs exhibit two maxima. The first maximum occurs at r ) 18 Å, and the second at 22 Å, 2 and 6 Å from the protein surface, respectively. Since the radius of a polymer segment is 2 Å, the results show that the polymer segments adsorb in two (still fairly intermixed) layers as the interaction strength increases.

J. Phys. Chem. B, Vol. 109, No. 31, 2005 15111 The radial distribution functions for systems with Nc ) 10 and 40 polymers do not display any larger differences (cf. Figures 4a and 4b). The insensitivity of the rdf on Nc at ) 0 is consistent with the fact that 〈RG2〉1/2 differs only by 0.1 Å between the two concentrations, and obviously at ) 0.9 × 105 kJ Å6/mol a sufficient amount of polymer segments is adsorbed to the protein to create density oscillations. Thermodynamics. The free energy change and its enthalpic and entropic components associated with the protein insertion into the polymer solution are presented in Figure 5. The free energy change is negative (except for very small and in particular for large Nc) and decreases with an accelerating slope as the hydrophobic interaction strength is increased. Similarly, both the energy change ∆U and the entropy change ∆S decrease with increasing ; hence an energy-entropy cancellation appears. The magnitude of the energy change is larger than that of the entropy change, and the resulting free energy change becomes only 20-30% of the energy change. The magnitudes of all three thermodynamic variables increase as the polymer concentration is increased. Given that there is no adsorption at small and adsorption at large , the thermodynamic data confirm the view that the adsorption process is enthalpically driven and has to overcome an entropic penalty. The hard-sphere contribution, ∆Ahs, to the insertion free energy change is small, approximately 0.01, 0.05, 0.1, 0.2, and 0.4kT for systems with Nc ) 1, 5, 10, 20, and 40 polymers, respectively. Hence, ∆A() ≈ ∆Aint() for not too small . Variation of the Hydrophobic Surface Residue Distribution. We will now focus on the role of the spatial distribution of the hydrophobic residues by considering 30 different proteins, 10 of each class. For the proteins in class MD2, the effect of the polymer concentration will be examined using Nc ) 1, 5, and 10 polymers, whereas for proteins in the classes MD4 and MD6 only systems with Nc ) 10 polymers will be used. Again, all polymer chains are fully flexible and have nseg ) 40 segments. Structure. Figure 6 displays adsorption probabilities for protein classes MD2, MD4, and MD6 as a function of the interaction strength . The probabilities are averaged over 10 proteins of each class, and vertical bars denote the spread among the proteins within a class. The s-shaped adsorption curve appearing in Figure 2 prevails for all protein classes. The maximal spread appears at 〈Pads〉 ≈ 0.5. For the most homogeneous distribution of the hydrophobic residues in class MD6, (i) the rise in Pads appears at a larger value of , and (ii) the variation of the adsorption probabilities is smaller. Evidently, a larger hydrophobic attraction is needed to adsorb a polymer as the residue distribution becomes more homogeneous. Similar findings were made for the adsorption of a single polymer to different proteins.16 There, a strong correlation between adsorption ability and heterogeneity of the surface distribution was found. Observation ii is connected to the fact that proteins become more alike as the residue distribution is made more homogeneous. The spread of Pads among the 10 proteins of class MD2 as function of for Nc ) 1, 5, and 10 polymer chains is given in Figure 7. We notice that as the polymer concentration is increased (i) the maximal variation is shifted to lower values of and (ii) the variation of Pads of the 10 proteins reduces. Since the maximal spread in Pads appears at values of where the adsorption starts to take place, issue i is consistent with the onset of the adsorption occurring at smaller as the polymer concentration is increased (cf. the discussion related to Figure 2).


Jo¨nsson and Linse

Figure 6. Adsorption probability averaged over 10 proteins 〈Pads()〉 for each class as a function of the interaction strength for indicated protein classes. The vertical bars denote the spread among the 10 proteins according to σPads ≡ 〈P2ads - 〈Pads〉2〉1/2 where 〈‚‚‚〉 denotes an average over the 10 proteins: Nc ) 10; nseg ) 40; lp ) 7.

Figure 7. Spread of the adsorption probability σPads() as a function of the interaction strength among the 10 proteins of class MD2 at the indicated numbers of polymers Nc. σPads ≡ 〈P2ads - 〈Pads〉2〉1/2 where 〈‚‚‚〉 denotes an average over the 10 proteins: nseg ) 40; lp ) 7.

Figure 5. (a) Free energy difference A()/kT, (b) energy difference U()/kT, and (c) entropy difference -TS()/kT as a function of the interaction strength . The systems contained Nc ) 1, 5, 10, 20, and 40 polymers (a and b) from top to bottom and (c) from bottom to top and the representative protein of class MD2: nseg ) 40; lp ) 7.

Thermodynamics. The insertion free energy change for each class as a function of is shown in Figure 8. Again data for each class is averaged for 10 proteins, and the vertical bars denote the spread within a class. As for the single protein of class MD2, the free energy change increases in magnitude at

increasing interaction strength. It is clear that the free energy is greatly affected by the spatial distribution of the interaction residues. For values of corresponding to strong adsorption, the difference in free energy for different proteins, ∆∆Ass′() (eq 10), can amount to several kT. In agreement with Pads, the change in free energy indicated stronger adsorption for proteins with heterogeneous surface residue distributions. However, unlike Pads, the free energy does not reach a limiting value, and consequently the variation between different proteins increases continuously as is increased. Figure 9 shows the spread of ∆A/kT among the 10 proteins of class MD2 as function of for three polymer concentrations. In contrast to Pads, the variation in ∆A becomes larger as the polymer concentration is increased. This increase is significant as the number of polymers is increased from one to five. At higher polymer concentrations, σ∆A/kT remains essentially independent of the polymer concentration, in particular at large values of where the protein surface is saturated by polymers already at intermediate polymer concentrations.


Figure 8. Free energy difference averaged over 10 proteins 〈∆A()/ kT〉 for each class as a function of the interaction strength for the indicated protein classes. The vertical bars denote the spread among the 10 proteins according to σ∆A/kT ≡ 〈(∆A/kT)2 - 〈∆A/kT〉2〉1/2 where 〈‚‚‚〉 denotes an average over the 10 proteins: Nc ) 10; nseg ) 40; lp ) 7Å.

Figure 9. Spread of the free energy difference σ∆A/kT() as a function of the interaction strength among the 10 proteins of class MD2 at the indicated numbers of polymers Nc. σ∆A/kT ≡ 〈(∆A/kT)2 - 〈∆A/kT〉2〉1/2 where 〈‚‚‚〉 denotes an average over then 10 proteins: nseg ) 40; lp ) 7Å.

Variation of Polymer Length. The effect of the polymer length was investigated at one fixed segment concentration using the representative protein from each of the three classes MD2, MD4, and MD6 (Figure 1). Systems with Ncnseg ) 400 polymer segments with polymers chains having nseg ) 10, 16, 20, 25, and 40 segments have been considered. In the following, only results for the shortest and longest polymers will be presented. Structure. Figure 10 shows Pads as a function of for the three representative proteins with the longest (solid curves) and shortest (dashed curves) polymers selected. The Pads curves of the polymers with the intermediate chain lengths fall between the curves of nseg ) 10 and 40. Our observations are that (i) the previous conclusion that the onset of adsorption occurs at higher for more homogeneous residue distributions remains for the shorter polymer, (ii) the transition becomes smoother with very weak or no s-shaped dependence for the shorter polymer, and (iii) the influence of the polymer length on the


Figure 10. Adsorption probability Pads() as a function of the interaction strength for systems containing Ncnseg ) 400 polymer segments with polymer length nseg ) 40 (solid curves) and nseg ) 10 (dashed curves) for representative proteins belonging to the indicated protein classes: lp ) 7.

Figure 11. Free energy difference ∆A()/kT as a function of the interaction strength for systems containing Ncnseg ) 400 polymer segments with polymer length nseg ) 40 (solid curves) and 10 (dashed curves) for representative proteins belonging to the indicated protein classes: lp ) 7.

adsorption curve is largest for the most homogeneous residue distribution. Issue ii is consistent with previous theoretical results that the cooperativity of the adsorption process increases with the chain length and becomes a first-order process in the limit of an infinitely long chain.30 The interpretation of observation iii is that for a heterogeneous distribution short chains still find areas on the protein with several residues close to each other (“hot spots”) to attach to, making the adsorption cooperative, whereas for the most homogeneous distribution the typical distance between the residues becomes sufficiently long to reduce the cooperativity. For a regular position of the surface residues, the distance between two adjacent attractive positions for segments becomes ∼12 Å as compared to 〈Rseg,seg2〉1/2≈ 5.5 Å. Hence, only every second segment would be attached. Thermodynamics. The corresponding averages of the free energy changes are given in Figure 11. We notice that (i) the free energy change for a given interaction strength is smallest for proteins with a more homogeneous distribution, (ii) the free

15114 J. Phys. Chem. B, Vol. 109, No. 31, 2005 energy change for a given interaction strength becomes smaller for shorter chains, and (iii) as for Pads the influence of the polymer length is largest for the most homogeneous distribution. Variation of Polymer Stiffness. Similar to the variation of the chain length, the effect of the chain stiffness was investigated using the representative proteins of each protein class. The bare persistent lengths of the chains considered were lp ) 7, 22, 60, 114, 224, and 555 Å. In the following, only results from chains with persistent lengths lp ) 7, 60, and 555 Å will be presented. These three types of chains will be referred to as being flexible, semiflexible, and stiff. Throughout this section, only one polymer concentration and one chain length will be considered, Nc ) 10 chains each having nseg ) 40 segments. Structure. The adsorption probabilities Pads as a function of for the three representative proteins with the flexible (dotted curves), semiflexible (dashed curves), and stiff (solid curves) chains are shown in Figure 12. Independent of the protein, the adsorption probability changes nonmonotonically as the chain stiffness is increased. This is noticeable by, for example, considering the interaction strength ) 0.5 at which Pads ) 0.5. As the flexible chains are made semiflexible, 0.5 decreases but increases again for the stiffest chains. Through the use of data for all flexibilities, the smallest 0.5 appeared for lp ≈ 60 Å. At > 0.6, this nonmonotonic behavior remains, but now the stiffest chain displays the least propensity for adsorption. As discussed in connection with Figure 6, the onset of the adsorption occurs for higher for proteins with more homogeneous distributions. This is also the case independent of the chain stiffness (cf. Figure 12; note the different horizontal scales). Furthermore, for the stiff chain, the cooperativity becomes less accentuated, in particular for the protein with the most homogeneous residue distribution, in a similar fashion as for the shorter chains. A comparison across the parts of Figure 12 also shows that the influence of the persistent length on the adsorption curves is largest for the most homogeneous residue distribution. Thermodynamics. Finally, the corresponding free energy changes for the variation of the chain flexibilities are shown in Figure 13. They confirm the pattern observed for the adsorption probabilities. For small , less than ∼1.2, 1.4, and 1.6 × 105 kJ Å6/mol for the classes MD2, MD4, and MD6, respectively, the free energy change is smallest for the flexible chain. At higher , similar crossing among the curves appears, and at strong interaction energies, the free energy change is smallest for the stiffest chain, its deviation from the less stiff chains being largest for protein class MD6. The semiflexible chain, which started to adsorb at the smallest , displays the largest free energy changes for any given . In other words, of all the investigated persistent lengths, the adsorption ability is largest for systems with a semiflexible polymer, independent of the protein class. The explanation originates from the fact that for a flexible polymer the conformational entropy of the chain opposes an adsorption to the protein surface. As the chain becomes semiflexible, the loss of conformational entropy associated with adsorption becomes less. For stiff chains, however, the adsorption is limited by the fact that a rodlike chain has only a few contacts with the spherical protein. Consequently, an optimum in adsorption ability should be present and appears here at the persistent length lp ≈ 60 Å. The optimal persistent length of the chain is typically on the same order as the linear dimension of the spherical particle to which the chain is adsorbed. A similar conclusion was previously seen for adsorption driven by electrostatic attraction.31

Jo¨nsson and Linse

Figure 12. Adsorption probability Pads() as a function of the interaction strength for polymers with persistent length lp ) 7 (dotted curves), 60 (dashed curves), and 555 (solid curves) for representative proteins belonging to protein class (a) MD2, (b) MD4, and (c) MD6: nseg ) 40; Nc ) 10.

The difference in adsorption between flexible and stiff chains increased as the residues were more homogeneously distributed. With a homogeneous distribution, we anticipate that the polymer has to wrap around a substantial part of the protein to make a sufficient number of contacts with hydrophobic sites. This



Figure 14. Adsorption probability Pads as a function of the free energy difference ∆A/kT (symbols) for 10 proteins from each of the classes MD2, MD4, and MD6 at different values of in the range 0.6-2.0 × 105 kJ Å6/mol, different chain lengths, and different chain flexibilities (in total 350 points): Ncnseg ) 400. The solid curve denotes the relation Pads ) 1 - exp(∆A/kT).

Figure 13. Free energy difference ∆A()/kT as a function of the interaction strength for polymers with persistent length lp ) 7 (dotted curves), 60 (dashed curves), and 555 (solid curves) for representative proteins belonging to protein class (a) MD2, (b) MD4, and (c) MD6: nseg ) 40; Nc ) 10.

implies a larger bending penalty than for an adsorption on a protein with a heterogeneous residue distribution, where the same adsorption energy can be achieved with a fewer polymer segments involved.

Correlations between ∆A and Pads. We have so far consistently found that both ∆A() and Pads() are monotonic functions of . In Figure 14, Pads is plotted versus ∆A for systems containing 1 protein and 400 polymer segments for different proteins, chain lengths, chain flexibilities, and values of . In total, 350 points are given. At a low interaction strength, Pads and ∆A are close to zero, whereas at a larger interaction strength ∆A is negative and its magnitude exceeds a few kT and Pads approaches unity. Interestingly, the data points collapse very well onto a single master curve, which, moreover, is accurately given by the relation Pads ) 1 - exp(∆A/kT) (solid curve in Figure 14). In more detail, at high polymer concentrations the Pads values for small (below the adsorption transition) slightly depend on the number of polymer chains of the system, and the best fit to the analytical relation is given by the systems with one polymer only. This weak density dependence in the nonadsorbing regime is simply due to an increase in the average segment density near the protein at increasing overall polymer concentration. In the Appendix, we consider a simple model involving a solution divided into two subvolumes, of which one is in contact with an adsorbing surface with a constant surface potential. There it is shown that the relation Pads ) 1 - exp(∆A/kT) holds trivially in the strong adsorption limit as well as in the weak adsorption limit provided that the subvolume in contact with the surface is small as compared to the remaining volume. We regard the universal behavior found in our protein-polymer systems to be firmly related to the validity of this relation under such general conditions. Comparison with Other Investigations. Very few experiments have been performed to investigate the effect of the spatial distribution of hydrophobic surface residues on protein-polymer association. However, in a recent study by Mahn et al.15 the protein retention time in hydrophobic interaction chromatography was measured for proteins with similar average surface hydrophobicity but with different distributions of the hydrophobic surface residues. The retention times were correlated to the existence of hydrophobic patches, and the relation between “uneven distribution” and “long retention time” was suggested. Unlike our model, these experiments utilize very short ligands (butyl) bound to a stationary phase as compared to our longer


Jo¨nsson and Linse

polymers dissolved in solution. Even so, our predictions are consistent with these experimental findings. Other experimental and theoretical studies investigating the effect of site distributions are usually dealing with Coulomb interactions. In these cases, conclusions were made that certain charge distributions enhance complexation between proteins and polyelectrolytes.3,7,8 These systems are, however, somewhat different from ours since the charge-charge interaction can be either attractive or repulsive. Furthermore, the range and magnitude of the Coulomb interaction is far longer and stronger than the hydrophobic one. Moreover, both computer simulations and theoretical studies have considered pattern recognition of heteropolymers to heterogeneous surfaces.32-34 In these studies, the polymer and surface were built of both attractive and repulsive interaction sites. Unlike our study, the loading of the surface interaction sites was altered instead of the interaction strength. Nonetheless, similar results of a large spread in Pads, σPads(), was found by Bratko et al.34 Conclusions Previously, it has been concluded that the adsorption of slightly hydrophobic polymers to proteins is augmented by proteins with more heterogeneous (clustered) residue distributions as opposed to more homogeneous distributions.16 In this study, we have shown that this also applies for solutions with variable polymer concentration and for polymers of different lengths and chain flexibilities. Moreover, the influence of the persistence length and polymer chain length on the adsorption is greatest for proteins with the most homogeneous residue distribution. The free energy of inserting a protein into a polymer solution was also examined. As expected, this free energy change is positive in the limit of zero hydrophobic interaction but becomes negative at such interaction strengths where adsorption starts to take place. A master behavior between the adsorption probability and the free energy of insertion was found. This behavior was well-described by the relation Pads(,nseg,lp) ) 1 - exp(∆A(,nseg,lp)/kT), which also was derived from a simple two-state adsorption model. Appendix Consider a system composed of a particle in a volume V at the temperature T. The volume V is divided into two subvolumes, V0 and V1, of which the latter is in contact with a surface. Two conditions will be considered, absence (i) and presence (f) of a negative surface potential u (u < 0) on the particle in subvolume V1. In the latter case, the particle is regarded as adsorbed if it is located in subvolume V1. Our aim is to assess the conditions for the validity of the relation

Pads ) 1 - exp(∆A/kT)

(11)

applied on this simple two-state model. In eq 11, Pads is the probability that the particle is adsorbed in the presence of the adsorbing surface potential, and ∆A is the free energy difference of the system with and without the adsorbing surface potential. The canonical partition functions of the system at the two conditions become

Qi ) V0 + V1

(12a)

Qf ) V0 + V1 exp(-u/kT)

(12b)

and the corresponding free energies become

Ai ) -kT ln[V0 + V1]

(13a)

Af ) -kT ln[V0 + V1 exp(-u/kT)]

(13b)

From eq 13, the free energy change of introducing the adsorbing potential can be expressed as

∆A ≡ Af - Ai ) - kT ln

[

]

V0 + V1 exp(-u/kT) V0 + V 1

(14)

Finally, in the presence of the adsorbing surface, the probability that the particle will be adsorbed is directly obtained from eq 12b according to

Pads )

V1 exp(- u/kT)

(15)

V0 + V1 exp(- u/kT)

At this stage, we are ready to see how well eq 11 is fulfilled for our two-state model. With the use of eq 14, the left-hand side of eq 11 becomes

1 - exp(∆A/kT) ) 1 -

[

]

V0 + V1 ) V0 + V1 exp(- u/kT) V1[exp(-u/kT) - 1] V0 + V1 exp(-u/kT)

(16)

A comparison of eq 16 with eq 15 shows that eq 11 holds when (i) exp(-u/kT) . 1 (strong adsorption limit) for all values of V1/V0 and (ii) u/kT approaches zero (nonadsorption limit) provided that V1/V0 , 1. Thus, we conclude that eq 1 holds for the simple adsorption model at all values of the adsorption potentials, including the limit of a nonadsorbing surface under the condition V1/V0 , 1. Since the adsorbing volume typically is much smaller than the volume of the solution, this condition is normally fulfilled. References and Notes (1) Zemb, T.; Blume, A. Curr. Opin. Colloid Interface Sci. 2003, 8, 1. (2) Turgeon, S. L.; Beaulieu, M.; Schmitt, C.; Sanchez, C. Curr. Opinion Colloid Interface Sci. 2003, 8, 401. (3) Tsuboi, A.; Izumi, T.; Hirata, M.; Xia, J.; Dubin, P. L.; Kokufuta, E. Langmuir 1996, 12, 6295. (4) Sato, T.; Kamachi, M.; Mizusaki, M.; Yoda, K.; Morishima, Y. Macromolecules 1998, 31, 6871. (5) Takahashi, D.; Kubota, Y.; Kokai, K.; Izumi, T.; Hirata, M.; Kokufuta, E. Langmuir 2000, 16, 3133. (6) Carlsson, F.; Linse, P.; Malmsten, M. J. Phys. Chem. B. 2001, 105, 9040. (7) de Vries, R. J. Chem. Phys. 2004, 120, 3475. (8) Hattori, T.; Hallberg, R.; Dubin, P. L. Langmuir 2000, 16, 9738. (9) Xia, J.; Dubin, P. L. Macromolecules 1993, 26, 6688. (10) Furness, E. L.; Ross, A.; Davis, T. P.; King, G. C. Biomaterials 1998, 19, 1361. (11) Tribert, C.; Porcar, I.; Bonnefont, P. A.; Audebert, R. J. Phys. Chem. B 1998, 102, 1327. (12) Porcar, I.; Cottet, H.; Gareil, P.; Tribet, C. Macromolecules 1999, 32, 3922. (13) Berggren, K.; Egmond, M. R.; Tjerneld, F. Biochim. Biophys. Acta 2000, 1481, 317. (14) Berggren, K.; Wolf, A.; Asenjo, J. A.; Andrews, B. A.; Tjerneld, F. Biochim. Biophys. Acta 2002, 1596, 253. (15) Mahn, A.; Lienqueo, M. E.; Asenjo, J. A. J. Chromatogr., A. 2004, 1043, 47. (16) Jnsson, M.; Skep, M.; Tjerneld, F.; Linse, P. J. Phys. Chem. B. 2003, 107, 5511. (17) Pangali, C.; Rao, M.; Berne, B. J. J. Chem. Phys. 1979, 71, 2975. (18) Pratt, L. R.; Chandler, D. J. Chem. Phys. 1980, 73, 3430. (19) Rossky, P. J.; Friedman, H. L. J. Phys. Chem. 1980, 84, 587. (20) Jorgensen, W. L.; Buckner, J. K.; Boudon, S.; Tirado-Rives, J. J. Chem. Phys. 1988, 89, 3742.

Thermodynamics of Protein-Polymer Solutions (21) Jorgensen, W. L.; Severance, D. L. J. Am. Chem. Soc. 1990, 112, 4768. (22) Linse, P. J. Am. Chem. Soc. 1992, 114, 4366. (23) Smith, D. E.; Haymet, A. D. J. J. Chem. Phys. 1993, 98, 6445 (24) Linse, P. J. Am. Chem. Soc. 1993, 115, 8793. (25) Hummer, G.; Garde, S.; Garca, A. E.; Pohorille, A.; Pratt, L. R. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 8951. (26) Chipot, C.; Jaffe, R.; Maigret, B.; Pearlman, D. A.; Kollman, P. A. J. Am. Chem. Soc. 1996, 118, 11217. (27) Allen, M. P.; Tildesley, D. J. Computer Simulations of Liquids; Oxford University Press: New York, 1987.

J. Phys. Chem. B, Vol. 109, No. 31, 2005 15117 (28) Linse, P. MOLSIM, version 3.21; Lund University: Lund, Sweden, 2001. (29) Widom, B. J. Chem. Phys. 1963, 39, 2808. (30) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman & Hall: London, 1993. (31) Akinchina, A.; Linse, P. Macromolecules 2002, 35, 5183. (32) Chakraborty, A. K.; Bratko, D. J. Chem. Phys. 1998, 108, 1676. (33) Bratko, D.; Chakraborty, A. K.; Shakhnovich, E. I. Chem. Phys. Lett. 1997, 280, 46. (34) Bratko, D.; Chakraborty, A. K.; Shakhnovich, E. I. Comput. Theor. Polym. Sci. 1998, 8, 113.

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