Langmuir 2007, 23, 3937-3946
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On the Mechanism of Uptake of Globular Proteins by Polyelectrolyte Brushes: A Two-Gradient Self-Consistent Field Analysis F. A. M. Leermakers,*,† M. Ballauff,‡ and O. V. Borisov‡,§ Laboratory of Physical Chemistry and Colloid Science, Wageningen UniVersity, Dreijenplein 6, 6703 HB Wageningen, The Netherlands, Physical Chemistry I, UniVersity of Bayreuth, 95440 Bayreuth, Germany, and Institute of Macromolecular Compounds of the Russian Academy of Sciences, 199004 St. Petersburg, Russia ReceiVed NoVember 9, 2006. In Final Form: January 12, 2007 We present model calculations for the interaction of a protein-like inhomogeneously charged nanoscale object with a layer of densely grafted polyelectrolytes (“polyelectrolyte brush”). The motivation of this work is the recent experimental observation that proteins that carry an overall negative charge are absorbed into negatively charged polyelectrolyte brushes. Two-gradient self-consistent field (2G-SCF) calculations have been performed to unravel the physical mechanism of the uptake of protein thus effected. Our results prove that an overall neutral, protein-like object can electrostatically be attracted and therefore spontaneously driven into a polyelectrolyte brush when the object has two faces (patches, domains), one with a permanent positive charge and the other with a permanent negative charge. Using a 2G-SCF analysis, we evaluate the free energy of insertion, such that the electric dipole of the inclusion is oriented parallel to the brush surface. An electroneutral protein-like object is attracted into the brush because the polyelectrolyte brush interacts asymmetrically with the charged patches of opposite sign. At high ionic strength and low charge density on the patches, the attraction cannot compete with the repulsive excluded-volume interaction. However, for low ionic strengths and sufficiently high charge density on the patches, a gain on the order of kBT per charge becomes possible. Hence, the asymmetry of interaction for patches of different charges may result in a total attractive force between the protein and the brush. All results obtained herein are in excellent agreement with recent experimental data.
Introduction The interaction and accumulation of proteins at solid/liquid interfaces is a phenomenon of primary importance in biomedical and biotechnological applications.1,2 Typically, nonspecific protein adsorption at solid surfaces of, e.g., implants and diagnostic assays must be avoided. On the other hand, specific association and complexation of proteins with nanocolloids present a viable route to design nanobioreactors, high-performance diagnostic assays, and many other applications.2 Up to now, long non-ionic water-soluble polymeric chains, particularly poly(ethylene glycol) attached to surfaces, have been used as a means to prevent unspecific protein adsorption.3 The protein-resistant properties of the polymer-modified surfaces are due to shielding of the surface by end-grafted chains that produces a steric repulsion experienced by proteins. Extensive theoretical and numerical analysis of this behavior was presented in refs 4 and 5. A different picture appears if the grafted chains are ionic, i.e., when they are polyelectrolytes. In particular, our recent observations indicate that even proteins which in the bulk have the same charge as the chains in the brush still absorb significantly into the brush.6,7 In this regime, the brush remains charged and †
Wageningen University. University of Bayreuth. § Institute of Macromolecular Compounds of the Russian Academy of Sciences. ‡
(1) Baskin, A.; Norde, W. Physical Chemistry of Biological Interfaces; Marcel Dekker: New York, 2000. (2) Cooper J. M.; Cass, A. E. G. Biosensors; University Press: Oxford, 2004. (3) Ratner, B. D.; Hoffman, A. S.; Schoen, F. J. Biomaterial Science. An Introduction to Materials in Medicine; Academic Press: London. (4) Halperin, A. Polymer Brushes that Resist Adsorption of Model Proteins: Design Parameters. Langmuir 1999, 15, 2525-2533. (5) Fang, F.; Satulovsky, J.; Szleifer, I. Kinetics of Protein Adsorption and Desorption on Surfaces with Grafted Polymers. Biophys. J. 2005, 89, 15161533.
therefore remains swollen because of the osmotic pressure of the counterions.8-11 This is a surprising observation, because intuitively one would expect some electrostatic and steric repulsion between the brush layer and the protein. However, observations done on spherical polyelectrolyte brushes,6 as well as on planar brushes,7 demonstrated that proteins are strongly attracted to the brush layer if the ionic strength is low. If the ionic strength is high, however, virtually no adsorption takes place, and the proteins are repelled from the brush layer. Proteins absorbed at low ionic strength can be liberated again upon raising the salt concentration in the system.12 An analysis by smallangle X-ray scattering (SAXS) has demonstrated that proteins such as, e.g., bovine serum albumine (BSA) enter deeply into the brush layer and are strongly correlated to the polyelectrolyte chains.13 Moreover, absorbed enzymes retain most of their biological activity.14 All findings available so far suggest that (6) Wittemann, A.; Haupt, B.; Ballauff, M. Adsorption of Proteins on Spherical Polyelectrolyte Brushes in Aqueous Solution. Phys. Chem. Chem. Phys. 2003, 5, 1671-1677. (7) Czeslik, C.; Jansen, R.; Ballauff, M.; Wittemann, A.; Royer, C. A.; Gratton, E.; Hazlett, T. Mechanism of Protein Binding to Spherical Polyelectrolyte Brushes Studied in Situ Using Two-Photon Excitation Fluorescence Fluctuation Spectroscopy. Phys. ReV. E 2004, 69, 021401. (8) Borisov, O. V.; Birshtein, T. M.; Zhulina, E. B. Collapse of Grafted Polyelectrolyte Layer. J. Phys. II 1991, 1, 521-526. (9) Borisov, O. V.; Zhulina, E. B.; Birshtein, T. M. Diagram of States of a Grafted Polyelectrolyte Layer. Macromolecules 1994, 27, 4795-4803. (10) Pincus, P. Colloid Stabilization with Grafted Polyelectrolytes. Macromolecules 1991, 24, 2912-2919. (11) Zhulina, E. B.; Borisov, O. V.; Birshtein, T. M. Structure of Grafted Polyelectrolyte Layer. J. Phys. II 1992, 2, 63-74. (12) Anikin, K.; Ro¨cker, C.; Wittemann, A.; Wiedenmann, J.; Ballauff, M.; Nienhaus, G. U. Fluorescent Protein Binding to Individual Polyelectrolyte Nanospheres. J. Phys. Chem. 2005, 109, 5418-5420. (13) Rosenfeldt, S.; Wittemann, A.; Ballauff, M.; Breininger, E.; Bolze, J.; Dingenouts, N. Interaction of proteins with spherical polyelectrolyte brushes in solution as studied by small-angle X-ray scattering. Phys. ReV. E 2004, 70, 061403. (14) Haupt, B.; Neumann, Th.; Wittemann, A.; Ballauff, M. Activity of Enzymes Immobilized in Colloidal Spherical Polyelectrolyte Brushes. Biomacromolecules 2005, 6, 948-955.
10.1021/la0632777 CCC: $37.00 © 2007 American Chemical Society Published on Web 02/22/2007
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the “polyelectrolyte-mediated protein adsorption” (PMPA) has an electrostatic origin. The immobilization of proteins thus effected does not disturb their conformational state, as the proteins remain biologically active.15 While there is clear experimental evidence for the PMPA, a consistent theoretical explanation of the protein uptake by a similarly charged polyelectrolyte brush is still lacking. A model proposed in ref 16 assumes the inversion of the protein charge when it is inserted into the brush. Although this theory provides a valid explanation for the immediate vicinity of the isoelectric point (IEP) of the protein, the model fails to explain the experimental observation of protein absorption at pH values exceeding the IEP by more than 1 unit.15 Czeslik and co-workers17 used the Poisson-Boltzmann approach to analyze the effective interaction between a protein, which is mimicked by a uniformly charged surface and a similarly charged polyelectrolyte brush. The result of this analysis suggests that a homogeneously charged finite size object can be electrostatically driven into a similarly charged polyelectrolyte brush due to “counterion evaporation”. However, our exact numerical solution of the Poisson-Boltzmann problem disproves this result and indicates purely repulsive interaction between a homogeneously charged protein-like object and a similarly charged PE brush. An alternative, qualitative argument has been proposed6,15 in which the counterion release plays the central role. In this line of argument, an inhomogeneous distribution of charges on the protein globule surface plays a central role. It is well-known that proteins having an overall negative charge may still have surface patches with a positive charge. At the isoelectric point (IEP) of a protein, the numbers of positive and negative charges on the protein surface just cancel. Globular proteins are, to a good approximation, conformationally quenched objects and may be seen as roughly spherical entities with a surface charge distribution that depends on the pH in the system. When the charges on the protein surface are strongly nonuniformly distributed, i.e., when they form welldefined and separated patches with a locally high charge density, a significant localization of counterions will occur whereby these ions will screen and locally compensate the charge of the patches. When such protein is inserted into the brush, the negatively charged chains adsorb and locally compensate the charge of the positive patches. As a result, the counterions of both the protein patch as well as those from the PE chain are released. This process is hence entropically favorable and will occur spontaneously. Here, we present a quantitative theory of the interaction of inhomogeneously charged protein-like objects with a polyelectrolyte (PE) brush. Our approach accounts for the free energy contributions due to electrostatic interactions as well as those of the excluded volume type when a protein-like object is inserted into the PE brush. More specifically, the analysis is based on the nonlinearized Poisson-Boltzmann approximation for treating the electrostatics. The self-consistent field approach is used to analyze the polymer chain conformations in the perturbed brush. Since analytical solutions of the PB equation for this problem cannot be obtained, we use a numerical two-gradient self(15) Wittemann, A.; Ballauff, M. Interaction of Proteins with Linear Polyelectrolytes and Spherical Polyelectrolyte Brushes in Aqueous Solution. Phys. Chem. Chem. Phys. 2006, 8, 5269-5275. (16) Biesheuvel, P. M.; Wittemann, A. A Modified Box Model Including Charge Regulation for Protein Adsorption in a Spherical Polyelectrolyte Brush. J. Phys. Chem. 2005, 109, 4209-4214. (17) Czeslik, C.; Jackler, G.; Steitz, R. R.; von Gruenberg, H.-H. Protein Binding to Like-Charge Polyelectrolyte Brushes by Counterion Evaporation. J. Phys. Chem. B 2004, 108, 13395-13402.
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consistent field (2G-SCF) model to quantify the relevant contributions up to high numerical accuracy. It has been shown theoretically,8-11 and it was confirmed by experiments,18-20 that a PE brush comprising long and crowded PE chains accumulates a major fraction of its counterions inside the brush. Indeed, the brush is even locally almost electroneutral: the counterion distribution follows the polymer brush charge distribution closely.21,22 Only a minor fraction of the counterions escape from the brush and participate in the formation of a diffuse ion layer on the top of the brush (Gouy-Chapman layer). The ratio between the local ion concentration and the bulk concentration is given by the Boltzmann weight which features the local excess electrostatic potential in the brush (Donnan equilibrium). The osmotic pressure of the counterions kept in the brush by the Donnan potential provides a force such that typically the brush is strongly swollen. Because of this swelling, there is enough free volume between the chains to accommodate protein molecules. To ensure the spontaneous incorporation of proteins into the brush, the attraction should be strong enough to overcome (i) translational entropy losses related to protein immobilization and (ii) excluded-volume interactions (steric repulsion) between the relatively bulky protein globule and the polymer chains in the brush. We shall demonstrate that it is possible to have a significant attraction between an in-total electroneutral object and the brush in an experimentally relevant range of the parameter space. The paper is organized as follows: First, we will outline the 2G-SCF approach and go into the (molecular) model that is used. In the results section, we will focus exclusively on the case in which the overall charge of the inserted object is zero and the dipole is parallel to the brush surface. In the discussion, we will evaluate its importance for experimental situations by putting the result in a slightly broader perspective. Finally, we give a brief conclusion.
Two-Gradient Self-Consistent Field Theory The SCF model for polymer brushes is well-known. In this model, we account for all possible chain conformations of the grafted molecules. These trajectories are generated using the Edwards diffusion equation23
∂G 1 ) ∇2 - u G ∂N 6
(
)
(1)
Here, the Green’s function G, which obeys this diffusion equation, is related to the volume fraction profile of polymer chains with number of monomer units N. The self-consistent potential u is a function of these volume fractions. Both G(r) and u(r) depend on the spatial coordinates such that gradients in these quantities are accounted for. In the present calculations, all the quantities depend on two spatial coordinates, i.e., r ) (x, z), and the meanfield averaging is implemented in the y-direction (see Figure 3 for a schematic diagram). The y-direction is chosen parallel to the grafting surface. The surface is located at the z ) 0 coordinate, (18) Ahrens, H.; Fo¨ster, S.; Helm, C. A. Charged Polymer Brushes: Counterion Incorporation and Scaling Relations. Phys. ReV. Lett. 1998, 8, 4172-4175. (19) Guo, X.; Ballauff, M. The Spatial Dimensions of Colloidal Brushes as Determined by Dynamic Light Scattering. Langmuir 2000, 16, 8719-9114. (20) Guo, X.; Ballauff, M. Spherical Polyelectrolyte Brushes: Comparison between Annealed and Quenched Brushes. Phys. ReV. E 2001, 64, 051406. (21) Zhulina, E. B.; Borisov, O. V. Structure and Interactions of Weakly Charged Polyelectrolyte Brushes: Self-Consistent Field Theory. J. Chem. Phys. 1997, 107, 5952-5967. (22) Zhulina, E. B.; Klein, Wolterink, J.; Borisov, O. V. Screening Effects in Polyelectrolyte Brush: Self-Consistent Field Theory. Macromolecules 2000, 33, 4945-4953. (23) Edwards, S. F. The statistical mechanics of polymers with excluded volume. Proc. Phys. Soc. 1965, 95, 613.
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and the positive values of z are available for the grafted polymer molecules (with ranking numbers s ) 1, ‚‚‚, N), the monomeric ions, and the solvent (water). The initial condition for the diffusion equation (i.e., the grafting condition) is that segment with ranking number s ) 1 located at z ) 1, and there exists a lateral mobility of the chains in the x-y plane. Of course, only gradients in grafting densities can develop in the x-direction. Equation 1 is solved on a discrete system of coordinates (i.e., on a lattice). More specifically, we use the discretization scheme of Scheutjens and Fleer.24-26 In this approach, the differential eq 1 reduces to a propagator formalism, and the volume fraction profile of the grafted chains is computed by a composition law
φ(r) )
σLx exp[u(r)/kBT] G(N|1; 1)
N
G(r; s|1; 1)G(r; s|N) ∑ s)1
(2)
where G(rs; s|rt; t) gives the combined statistical weight of all possible and allowed conformations that start at segment t at coordinate rt and end at segment s at coordinate rs. When in a Green’s function one spatial coordinate is not specified, this means that we have summed over this coordinate, for example, G(rs; s|t) ) ∑rt G(rs; s|rt; t). In eq 2, the average grafting density (number of grafted chains per lattice site) is given by σ, and the simulation volume extends in the x-direction over Lx sites. The numbering of the sites in the x-direction runs for an odd number of sites from x ) 0, ‚‚‚, xc, ‚‚‚, Lx, and thus, the position xc ) (Lx + 1)/2 is the central spot. Here, we consider polymer chains with a fixed charge per segment in a 1:1 electrolyte solution. In the case where there are no gradients in dielectric permittivity (we will set the relative dielectric constant r ) 80 throughout the system) and when short-range interactions are purely repulsive (all Flory-Huggins interaction parameters are set to the athermal value χ ) 0), there are just two contributions to the segment potential
u(r) ) u′(r) + Reψ(r)
(3)
In this equation, u′(r) is a Lagrange field coupled to an incompressibility constraint
∑A φA(r) ) 1
(4)
where the summation runs over all segment types in the system, including the monomer units, the solvent, and the monovalent ions. The valences R ) 1 for the cation and R ) -1 for the anion, and we will use R ) -0.1 for the monomer units (on average, 1 out of 10 monomer units is charged along the chain). In eq 3, e is the elementary charge and ψ(r) is the local electrostatic potential. For the monomeric solvent, u(r) ) u′(r), because its charge is zero. The concentrations of the monomeric species follow from the Boltzmann equation φ(r) ) φb exp[-u(r)/kBT]. In this equation, the bulk volume fraction of the ions φs ≡ φb+ ) φb- is an important input variable. The bulk volume fraction of the monomeric solvent follows from the bulk incompressibility φbW ) 1 - 2φs. The default value for temperature is T ) 323 K. (24) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman and Hall: London, 1993. (25) Scheutjens, J. M. H. M.; Fleer, G. J. Statistical Theory of the Adsorption of Interacting Chain Molecules. I. Partition Function, Segment Density Distribution and Adsorption Isotherms. J. Phys. Chem. 1979, 83, 1619-1635. (26) Evers, O. A.; Scheutjens, J. M. H. M.; Fleer, G. J. Statistical Thermodynamics of Block Copolymer Adsorption. 1. Formulation of the Model and Results for the Adsorbed Layer Structure. Macromolecules 1990, 23, 52215232.
The electrostatic potential can be evaluated from the Poisson equation
∇2ψ ) -q
(5)
In eq 5, the local charge density follows from volume fraction profiles of all charged components
q(r) ) e[0.1φ(r) + φ+(r) + φ-(r)] + qp(r)
(6)
where the charge of the protein-like object is included by the last term. Equation 5 is also solved on the discrete coordinate system. The electroneutrality of the complete system is guaranteed by the reflecting boundary conditions implemented at all system boundaries except at the uncharged surface. The contribution of the protein-like object to the local charge density follows the volume fraction distribution, i.e., qp(r) ) eRp(r)φp(r), where
{{
x ) xc - 1, xc, xc + 1 φp(r) ) z ) H - (1/2), H + (1/2) 0 otherwise 1
(7)
The protein-like object has a dipolar character, the positive charges Rpp ≡ Rp(xc - 1) > 0 reside on x ) xc - 1 coordinates, and the negative charges Rpn ≡ Rp(xc + 1) > 0 on the x ) xc + 1 coordinates (cf. Figure 3). In all such cases, we choose for the symmetry condition that Rp ≡ Rpp ) -Rpn, which means that the overall charge of the object is exactly zero. Hence, by construction of the model, the dipole moment of the inclusion is oriented parallel to the surface (the orientation of the dipole normal to the brush surface is also easily implemented, but we leave this out of our present analysis for simplicity). It is clear that the center of mass of the object is at (x, z) ) (xc, H) where H is exactly in between two lattice layers, whereas the distance of the object to the surface (counted in number of lattice sites) is given by H. We will also show results for which there are two inclusions in the brush. The first one is positioned at (x1, H) with x1 < Lc and has two faces with a positive charge, i.e., R1 ) Rp(x1 - 1) ) Rp(x1 + 1) (cf. Figure 3). The second inclusion is at (x2, H), where x2 ) 2Lc - x1 and has two phases with a negative charge R2 ) -Rp(x2 - 1) ) -Rp(x2 + 1). The lateral distance between the two inclusions (in number of lattice sites) is given by D ) x2 - x1. We note that the objects that are inserted into the brush extend infinitely in the y-direction. All results typically are normalized per unit length in the y-direction, and the model strictly has only two gradient directions. As specified in eq 7, the protein-like object occupies two sites in the z-direction, and thus, the total charge on the patch (per unit length in the y-direction) equals 2Rp. The set of equations is closed and is routinely solved numerically up to high precision. For the self-consistent solution, it is possible to compute the partition function, and from this partition function, all thermodynamic quantities follow straightforwardly. Of interest is the partial open free energy Fpo defined by
Fpo([φ], [u]) ) F -
∑i niµi
(8)
in which the summation runs over all mobile components (all molecules excluded the surface with the grafted chains and the rigid inclusion). The square brackets indicate that this partial
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Figure 1. (a) Volume fraction profiles normal to the surface of polymer brushes. (b) Corresponding electrostatic potential (in volts) profile (solid lines; left ordinate) and total (dimensionless) charge density profile (dotted lines; right ordinate), for various values of the volume fraction of 1:1 electrolyte, as indicated. Other parameters: N ) 100, σ ) 1/N, R ) -0.1. The vertical arrows indicate curves that correspond to the same ionic strength. Similarly to the two-gradient calculations discussed below, we have implemented a reflecting boundary condition at z ) 60. As a result, the electric field strength E(60.5) ) 0.
open free energy is a functional of all volume fraction and all potential profiles. Below, we are interested in the free energy of interaction Fint(H) of the protein-like object with the polyelectrolyte brush as a function of the distance H of the object from the grafting surface. To obtain this interaction free energy in our calculations, we fix the overall grafting density and the bulk concentrations of all mobile components and vary the distance of the inclusion to the surface. From this, it is obvious that the Fpo is the characteristic function, and thus
Fint(H) ) Fpo(H) - Fpo(∞)
(9)
The value of the reference value can be obtained by evaluating the Fpo of the unperturbed brush plus the isolated inclusion in a bulk solution. In practice, however, we will use the largest H value used in the calculations to define the reference value for the free energy of interactions. Correspondingly, we use the free energy as a function of the lateral distance D between two inclusions in the system. In such a calculation, we also fix the height H of the two inclusions. In our calculations, we set the number of monomer units in the grafted molecules to N ) 100. The default value of the grafting density (number of grafted chains per lattice site on the surface) is σ ) 1/N, that is the value near the mushroom to brush transition for a neutral brush (overlap condition), and it is varied up to σN ) 5. The second variable is the charge density of the inclusion. Protein molecules may have a high local surface charge density. A value of Rp ) 1 means that there are two charges per unit length (in the y-direction) on each patch of the protein-like object. The third variable is the ionic strength. This quantity is varied over a large range φs ) 10-4, ‚‚‚, 10-1. Because the attraction appears strongest for the low ionic strength cases, most of the results have been obtained for the case that φs ) 10-4. As the segment length (lattice site) is set to l ) 0.5 nm, a multiplication factor of approximately 10 is needed to convert the volume fraction of salt to a molar concentration. We have fixed the box size in the x-direction to Lx ) 41 sites so that the center of the inserted object sits at xc ) 21; the box size in the z-direction is set to Lz ) 60 throughout.
Results The Isolated Polyelectrolyte Brush. Before we show results of the 2G-SCF calculations for the polyelectrolyte brush with protein-like inclusion, we will first briefly discuss the properties of the bare, unperturbed polyanionic brush. These results are obtained within a 1G-SCF analysis. In Figure 1, typical results are shown for the structural and electrostatic properties of the polyanionic brush for various ionic
strength conditions. As follows from theory8-11,21,22 and is illustrated by Figure 1a, the brush structure and chain conformations are fairly unaffected by the addition of salt as long as the salt concentration in the bulk of the solution remains lower than the concentration of counterions trapped inside the salt-free brush (osmotic regime). Taking into account the average monomer volume fraction in the salt-free brush and the fraction of charged monomers R ) -0.1, we can estimate the threshold salt concentration as φs < 10-2. This is in good agreement with results presented in Figure 1. At salt concentrations exceeding this characteristic value, the brush height progressively decreases (Figure 1a), and simultaneously, the average volume fraction of polymer segments in the brush increases as a function of salt concentration (salt dominance regime). At any salt concentration, the polymer density in the brush is a decreasing function of the distance from the surface except at the narrow region proximal to the surface, where polymer density drops down to zero. This depletion effect is attributed to the fact that polymer chains experience steric restrictions and there is thus some conformational entropy loss near the impenetrable surface. Apart from this depletion effect, the polymer density profile closely follows the Gaussian or parabolic shape at low or at high ionic strength conditions, respectively, in accordance with predictions of analytical self-consistent field theory.21,22 The electrostatic potential profile (Figure 1b, left ordinate) is roughly parabolic, irrespectively of the ionic strength,21,22 and is negative throughout the system. At low ionic strength, the electrostatic potential can reach 100 mV, but this value quickly drops for higher ionic strength situations. The charge density profiles of Figure 1b (right ordinate) prove that q(z) in the brush is extremely low. To a good approximation, the charges on the grafted chains are locally compensated by counterions, except for a narrow region just near the grafting surface, where steric restrictions play an important role. This shows that the local electroneutrality approximation, which is often used in theoretical models for PE brushes,8-11 is justified. Remarkably, at the lowest ionic strength case there is a significant diffuse layer outside the brush extending into the bulk solution. The extension of this diffuse layer at low ionic strength is controlled by the noncompensated charge per unit area of the brush and scales proportionally to the brush height.21 At higher salt concentration, the thickness of the double layer spreading beyond the edge of the brush is controlled by the Debye length imposed by the salt concentration in the solution. Hence, the local charge compensation becomes less perfect the lower the ionic strength is. In Figure 2, we present the brush characteristics at fixed and low ionic strength φs ) 10-4 for a range of relevant grafting densities give by σ ) 1/N, ‚‚‚, 5/N. As can be seen from Figure
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Figure 2. (a) Volume fraction profiles of the polyanion brush. (b) The corresponding electrostatic potential (in V) profiles (solid lines; left ordinate) and dimensionless charge density profiles (dotted lines; right ordinate) for various grafting densities, as indicated. The vertical arrows show which curves belonging to the same grafting density (only at low and high grafting densities are these arrows plotted). The ionic strength is φs ) 10-4. Other parameters are as in Figure 1.
2a, the average volume fraction increases slightly less than linearly with the grafting density, whereas the height is a much weaker function of the grafting density. This is in accordance with the scaling analysis, which points to the absence of a power-law dependence for the salt-free polyelectrolyte brush height,8-11 whereas a more refined SCF analytical approach21 predicts a weak logarithmic increase of the brush height as a function of the grafting density. For polyelectrolyte brushes at high salt concentration, the height is known to scale as ∼σ1/3 and the average density scales with the grafting density as σ2/3, similar to those for a neutral brushes under good solvent conditions. The electrostatic potential in the brush (Figure 2b, left ordinate) increases with grafting density, and the local deviation from full charge compensation in the brush is only a very weak function of the grafting density (Figure 2b, right ordinate; dotted lines). Results presented in Figures 1 and 2 serve here as a reference for the two-gradient calculations for polyelectrolyte brush with protein-like inclusion. As the 2G-SCF calculations have limitations with respect to the system size, we have presented results of Figures 1 and 2 also for relatively small system sizes. As a result, the electrostatic potential profile above the PE brushes does not always completely decay to zero at the system boundary. Indeed, this occurs for the low ionic strength cases. We mention once again that in the 2G-SCF calculations the coordinate system is such that the mean-field averaging is implemented in the y-direction, and gradients in local properties (e.g., volume fractions, electrostatic potential, etc.) are possible only along the surface in the x-direction and perpendicular to the surface in the z-direction. It is understood that the brush remains homogeneous in the x-direction as long as there is no inclusion present. Under these conditions, the distributions of polymer density and electrostatic potential in the z-direction do not deviate from those presented in Figures 1 and 2. The Polyelectrolyte Brush with a Dipolar Inclusion. Let us first refer to Figure 3, wherein various aspects of the following calculations are schematically depicted. Our interest is to understand the free energy effects of path number one (P1): the insertion of the protein-like object from the bulk of the solution into the PE brush. In particular, we like to know how the free energy of insertion is affected by the governing parameters in our model. The alternative route(s) P2 f P3 f P4 will be discussed below, and the aim is to learn more about the physics of the process. To obtain the free energy of interaction of a protein-like inclusion with the brush, we follow P1 (see Figure 3). In this procedure, we varied the position of the inclusion above the grafting surface, H, while other parameters are fixed. The free energy of interaction as defined in eq 9 is presented in Figure 5a for a set of ionic strength conditions φs ) 10-4, ‚‚‚, 10-1. As in Figure 4, the grafting density is σ ) 1/N, the chain length N
Figure 3. A schematic illustration (in the x-z plane) of the insertion of protein-like inclusions into a negatively charged polyelectrolyte brush in the presence of a 1:1 electrolyte solution. The charges along the chain are indicated by the small open spheres. The salt ions are depicted by the small gray spheres. The protein-like object is given by a pair of larger spheres. Each subunit is given two charges (Rp ) 1). P1 is the path of interest; it involves the direct insertion of the protein from the outside into the brush. The distance H of the protein to the surface is indicated. The alternative route is given by P2 f P3 f P4. Here, P2 is the separation of the bipolar object into two homogeneously charged objects (here, we take the charges to be -2 and 2, whereas in the calculations, this charge was doubled). The lateral distance D between the units is indicated. P3 is the insertion of the positive object as well as the negative one into the brush, and finall,y P4 represents the reuniting of the two oppositely charged objects inside the brush such that the original bipolar object is recovered. The total system is electroneutral: In the bulk, the concentration of cations equals that of the anions. Around a positive patch of the protein, an equal number of anions are present as a diffuse layer; the same applies for the negatively charged patch. Each negative charge from the grafted PE chains is locally neutralized by a cation. Not shown are adsorption and depletion effects of polymer chains near the inserted protein nor its subunits.
) 100, and the degree of charging of the brush is R ) -0.1. The free energy found at the distance H ) 50 is used as the reference (offset) value of the free energy of interaction. The effective interaction potential curves presented in Figure 5 clearly indicate that the free energy of interaction between the object with a zero net charge and a polyelectrolyte brush can indeed be negative. This finding proves that the object can be attracted into the brush provided that the ionic strength in the
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Figure 4. (a) The polymer volume fraction distribution in the x-z plane of a polyanionic brush with an inclusion (linear rodlike object aligned in the y-direction) with a negatively charged side at x ) 22 and a positive side on the x ) 20 coordinate, at a distance H ) 15 from the surface. (b) The corresponding electrostatic potential profile in volts. Parameters: φs ) 10-4, N ) 100, R ) -0.1, σ ) 1/N, and Rp ) 1. There are reflecting boundary conditions at both edges of the system in the x-direction as well at the z ) 60 boundary.
bulk of the solution is sufficiently low. This attraction decreases strongly with increasing ionic strength. For φs ) 10-2, there is a very weak minimum of -0.01kBT, which is found near H ) 17. For higher ionic strength, the interaction is fully repulsive, indicating that the excluded volume effects dominate the interaction. Remarkably, attraction is switched to repulsion at a salt concentration roughly corresponding to crossover between osmotic (salt-free) and salted brush regimes, i.e., when salt concentration in the bulk of the solution reaches the value of the counterion concentrations in the osmotic salt-free brush. This is in full agreement with experimental findings.15 At sufficiently low ionic strength, we see that the free energy of interaction is negative over the whole range of H values used with the exception of the proximity of the surface. Near the surface, the excluded volume repulsion arises due to the high local polymer concentration. A key observation from the comparison of Figures 1 and 5a is that the shape of the interaction curve follows that of the electrostatic potential. This means that, especially for the case of low ionic strength, attraction extends beyond the region where the chains of the PE brush reside and operates already in the diffuse part of the double layer on top of the brush. In Figure 5b, the free energy of interaction is presented for the case in which the charge densities on the patches of the inclusion are varied, while the ionic strength is kept constant and low, i.e., φs ) 10-4. In line with expectations, we see that when the charge density on the patch is reduced, the free energy of interaction is also significantly less attractive. Recall that the total charge of each patch (positive or negative) per unit length of the inclusion is twice Rp. From Figure 5b, we may conclude that we need at least 1 charge per nm2 before the attraction can overrule the excluded volume interactions. Charge densities at the surface of proteins can be much larger than this, and thus, we conclude that our calculations provide a reliable lower estimate for attraction free energy for typical proteins.
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Figure 5. The free energy of interaction of a protein-like inclusion with a PE brush in units of kBT per unit area of the brush as a function of the distance of the center of the inclusion to the surface H (a) for four ionic strength values, (b) for various values of the charge density on the patch, and (c) for different values of the grafting density as indicated. All other parameters are as in Figure 4.
In Figure 5c, the free energy as a function of H is shown for various values of the grafting density of the PE brush. From this figure, it is seen that the increase in grafting density increases the attraction somewhat. The fact that the interaction becomes slightly longer-ranged must be attributed to the increase in the brush height with increasing grafting density (cf. Figure 2a). Clearly, a further increase in grafting density would enhance excluded volume repulsions and impose steric restrictions for penetration of protein-like objects in the interior of the brush. In the latter case, the accumulation of protein-like objects is expected to occur predominantly in the periphery of the brush and in the diffuse part of the electric double layer above the edge of the brush. We do not study this effect in detail here. In order to rationalize the physical origin of the attraction between the electroneutral (in total) object and the polyelectrolyte brush, it is instructive to present the monomer unit volume fraction distribution of a polyanionic brush in the x-z plane with a proteinlike inclusion. We choose the condition of low ionic strength φs ) 10-4 and a high charge density on the patches Rp ) 1 for this example, and the position of the inclusion is set to H ) 15, somewhere halfway inside the brush (the edge of the brush is localized around z ) 37; see Figure 1). Under these conditions, a significant local perturbation of the brush occurs, as is easily seen in Figure 4a. On the positively charged patch, the monomer units strongly accumulate, whereas the chains avoid the negatively charged patch. This perturbation rapidly decreases going toward lower and higher x-coordinates, and the unperturbed profile is found at x ) 1 and x ) 41 (cf. Figure 1a). The corresponding electrostatic potential profile is presented in Figure 4b. As the brush is negatively charged, the overall electrostatic potential is of course negative. However, just near the positive patch, the potential is locally positive. On the other side of the inclusion, the electrostatic potential significantly drops
Polyelectolyte Brush Uptake of Globular Proteins
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Figure 6. (a) The free energy of interaction between the PE brush and inclusions that have the same charge (positive or negative) on their two faces. The dashed line corresponds to the case where the object has the same charge as the brush, and the continuous line corresponds to the oppositely charged situation. (Path P3 cf. Figure 1.) (b) Free energy of interaction as a function of the distance D between the objects in units of lattice sites lengths (in units of kBT per unit area of the brush) wherein two objects have been placed on H ) 40 (P2), 15 (P4), and 10 (P4), as indicated (the reference value for the free energy is chosen at D ) 20). (c) The number of cations in the system (normalized to the number of cations that are present in the system when H ) 50) as a function of the distance of the inserted objects to the surface corresponding to view graph (a). (d) The number of cations in the system (normalized to the value present at D ) 20) when the two objects are pushed together at constant height of the objects as indicated, corresponding to view graph (b). Parameters: φs ) 10-4, Rp ) 0.4, N ) 100, R ) -0.1, and σ ) 1/N.
below the local (negative) brush value. Again, far from the inclusion, i.e., at x ) 1 and x ) 41, the electrostatic potential profile has little gradient in the x-direction, and the profile in the z-direction closely follows that given in Figure 1b. The analysis of profiles presented in Figure 4a,b suggests that the origin of the overall attraction between a protein-like inclusion and the brush is in the asymmetry of interaction between positively and negatively charged poles of the inclusion with the polyelectrolyte brush. Essentially, the bipolar charge distribution of the charge on the inclusion is manifested in the interaction with the brush, because the electrostatic screening length in the brush is comparable to the characteristic size of the charge patches on the inclusion. In the following section, we shall analyze in more detail the asymmetry of interaction between the PE brush and probe charges of different sign. The most important aspect of the calculations presented in this section is the proof that there exists an attractive interaction between the PE brush and an inhomogeneously charged object with a zero net charge. The attraction can, especially at low ionic strength condition (approximately 1 mM salt), be on the order of kBT for a typical protein molecule, and this suggests that there is an electrostatic mechanism responsible for the uptake of proteins in the PE brush on the wrong side of the IEP. The Polyelectrolyte Brush with Monopole Inclusions. The remainder of the following results have the purpose of helping to understand the mechanism of the free energy gain upon insertion of a dipolar object with a zero net charge into the PE brush. Referring again to Figure 3, we modify the model and consider two inclusions that are placed at the same height H above the surface such that they are at a distance D apart. One of the objects has two faces that are both positively charged; the other one has two faces that are both negatively charged. In Figure 6a, we present the free energy of interaction of homogeneously charged (both faces have the same charge) inclusions with the PE brush at low ionic strength (φs ) 10-4). With reference to Figure 3, these results implement path P3. The solid line corresponds to the positively charged object, whereas
the dashed line corresponds to the negatively charged one.28 As expected, the free energy of interaction for an oppositely charged object (the solid line) is negative for all H values, indicating a strong attraction, whereas the free energy of interaction for a similarly charged object (dashed line) is positive, indicating a slight repulsion. The important point is that there is a clear asymmetry: The oppositely charged object is much more strongly attracted into the brush than the negatively charged object is repelled from it. Another point of interest is that the absolute value of the sum of the free energy of interaction presented in Figure 6a is significantly larger than that for a bipolar object, Figure 5a, showing the correlation effect arising due to close proximity of two oppositely charged poles in the dipolar object. To quantify this correlation, we have placed the two oppositely charged objects in the brush at equal heights and recorded the free energy as a function of lateral spacing D between the objects and plotted the result in Figure 6b. Results were obtained for three heights H ) 40 (effectively outside the brush in the diffuse part of the electric double layer above the brush edge, corresponding to P2) and H ) 15 and 10 inside the brush (representative for P4). For H ) 40, the interaction between the objects is longerranged and the magnitude of the free energy changes is larger than inside the brush. This trend can be explained by a stronger lateral screening of the electrostatic interactions inside the brush than in the diffuse part of the electrical double layer outside the brush. This high screening is due to the locally higher concentration of mobile counterions and some additional contribution to the screening is coming from the PE chains. Since the asymmetry in interaction between the PE brush and the oppositely or similarly charged objects appears to be a key issue for understanding the net attraction between the brush and (27) Lyklema, J. Fundamentals of Interface and Colloid Science. Volume I: Fundamentals; Academic Press: London, 1991. (28) We should have chosen the reference state for both curves by the same free energy value, i.e., the free energy at large distance H f ∞. At H ) 50, this value was not reached, and we have chosen the average of the free energies at H ) 50 for the two situations to obtain an estimate for the free energy of infinite separation.
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Figure 7. (a) The free energy change upon bringing two charged objects (with different sign of charge-charge per site is indicated) to a negatively charged surface. As a reference value for the free energy of interaction, the average value of the free energy of the two curves at H ) 35 is chosen. (b) The corresponding change in the number of positive ions in the system as a function of the distance H of the object to the surface. This quantity is normalized to the value at H ) 35. All parameters are similar to those in Figure 6 except that the brush is replaced by a negatively charged surface with surface charge density σ0 ) -0.5 charges per lattice site. φs ) 10-4.
the in-total electroneutral dipolar object, we aim to get a better understanding in the origin of this asymmetry. Indeed, if the insertion of a charged object into the interior (or proximity) of the PE brush would not perturb the distributions of the electrostatic potential or the distribution of the mobile ions, then the interaction free energies should be symmetrical with respect to the charge of the inclusion. This is clearly not the case, and the asymmetry in the interaction is related to some perturbation exerted onto the PE brush upon the insertion of the charged object. One of the most important consequences of the interaction is the change in the number of mobile ions trapped inside (or in the proximity) of the brush and the corresponding change in the translational entropy in the system upon the insertion. In our calculations, the chemical potentials (the volume fractions) of the ions in the bulk of the solution (in the reservoir) are fixed. However, the number of ions in the calculation box may change, depending on the position of the inclusion. This adjustment of the number of mobile ions (consistent with the fixed chemical potentials µ(r) ) (ψ(r)e + kBT ln φ((r) affects the distribution of the electrostatic potential in the system. In more detail, the number of cations may depend on whether they are needed to compensate the charge in the brush or to form a diffuse layer near the negatively charged patches on the proteinlike object. Note that, when the number of cations in the system changes, the number of anions also changes by the same amount. The reason of course is that the system as a whole has to remain electroneutral. In Figure 6c, we present the change in the number of cations in the system for the same process as depicted in Figure 6a, i.e., the transport of a positively or a negatively charged object into the anionic brush. Inspection of Figure 6c shows that a small number of cations has to be added into the box to keep their chemical potential constant upon the insertion of the negatively charged object into the brush. However, a large number of cations is released from the box when the positively charged inclusion is inserted into the brush. Hence, there is a qualitative correlation between the release or uptake of counterions and a reduction or increase of the free energy and vice versa. For a quantitative relation, one would expect curves in Figure 6a,c to follow each other. This is not the case. When a positively charged object approaches the anionic brush at a low salt concentration of φs ) 10-4, the change in the number of counterions occurs mostly before the object enters the brush, i.e., when it passes the diffuse part of the double layer at the periphery of the brush. Per unit length of the inclusion, there are in this case 4 × 0.4 ) 1.6 charges on one inclusion. When the polymer chains complex with the inclusion, the corresponding number of counterions of the brush become redundant. Although the localization of counterions (anions) near positively charged patches of the protein-like object is relatively weak, as compared to the trapping
of counterions in the brush, they provide an additional contribution to the total counterion release. This means that the maximum counterion release can be estimated (from above) by 3.2 charges per unit length in the y-direction. The result of Figure 6c shows that only approximately 1/3 of the maximum number of counterions is released. In Figure 6d, we show similar ion release data corresponding to the free energy results presented in Figure 6b. Here, the separation D between two oppositely charged objects is varied at a fixed value of H. Inspection of Figure 6d shows that only when the oppositely charged objects are separated outside the brush is there a significant change in the number of cations in the system. In this case, the shape of the free energy curve resembles the ion release curve. The total number of charges that are compensated by the other object by decreasing D is 0.8 for each protein. From Figure 6d, the maximum change in the number of cations is slightly less than 0.3, and again, approximately 1/3 of the theoretical maximum number of cations is released. As follows from results presented in Figure 6, a significant asymmetry in the interaction of a PE brush with objects with charges of different sign is observed even at distances from the surface exceeding the brush thickness. This can be attributed to the asymmetry of the interaction of charged objects with the electrical double layer extending beyond the edge of the brush. However, some additional stretching of grafted chains induced upon approach of an oppositely charged object (“bridging”) cannot be ruled out. The Poisson-Boltzmann equation has extensively been used in the context of a charged interface immersed in an electrolyte solution.27 In the linearized form, i.e., the Debye-Huckel regime, it is known that the co-ions are expelled equally strongly as the counterions are attracted. In the nonlinearized form, i.e., at high electrostatic potentials, this is no longer the case, as the coions are much more strongly attracted than the counterions are repelled. In order to get a better insight into the mechanism of interaction of charged objects with the PE brush at large distances, we therefore performed a set of calculations wherein the brush was replaced by a uniformly charged flat surface. In the brush with σN ) 1 and R ) -0.1, the corresponding surface charge density (number of negative charges per surface site) of σ0 ) -0.1. In Figure 7, we present results for a 5-fold larger surface charge density. Again, the property of interest for us is the free energy change upon bringing a homogeneously positively or negatively charged object toward such charged surface. Inspection of Figure 7a shows that the sign of the free energy change is negative (attraction) for the oppositely charged object and positive (repulsion) for the similarly charged one. Similarly to the case of the PE brush, there is a significant asymmetry in the magnitude of the interaction. The attraction is larger (for the chosen
Polyelectolyte Brush Uptake of Globular Proteins
parameters set by approximately a factor of 3) than the repulsion in a wide range of H-values. The change in the number of counterions corresponding to the results of Figure 7a is presented in Figure 7b. From this figure, we see that there are many more counterions released when the oppositely object is inserted than ions taken up when the similarly charged object is placed near the surface. Again, the number of counterions that are released (for the oppositely charged case) saturates at a value of about 1.2. This number is approximately the same as that found when the same object was inserted in the PE brush. As follows from our calculations, the release of counterions takes place due to the local electrostatic potential, and it is not so relevant whether there are polymer chains nearby.
Discussion When a protein molecule is at pH significantly above the IEP of the protein in the bulk, the overall charge of the protein can be the same as that of the polyacid brush, and still the protein may be electrostatically attracted into the brush. This phenomenon is observed experimentally (see the review in ref 15), and above we have collected evidence that this effect can be rationalized if we assume that the protein globule has a heterogeneously charged surface. Then, the attractive force acting on the globule, which possesses a net zero or weakly negative charge, arises due to an asymmetry in the interaction of a positively and negatively charged patch with the PE brush. Explicitly, a patch with opposite (with respect to PE chains) charge is effectively strongly attracted into the PE brush, whereas the remaining similar charges are relatively weakly repelled. In order to prove the existence of this attraction mechanism, we have investigated by means of the 2G-SCF approach a particular case in which the protein-like object is overall electroneutral and the electric dipole is directed parallel to the PE brush. The asymmetry (with respect to the sign of the charge) in the interaction of macro-ions in solution containing monovalent mobile counterions (and also salt) arises as a nonlinear effect, and thus requires a nonlinearized PB treatment. It is directly related to the localization (“condensation”) of mobile counterions in the vicinity of a heavily charged macro-ion. The “condensed” counterions strongly reduce the electrostatic field created by the macro-ions. Therefore, upon the approach of two oppositely charged macro-ions, the net change in the energy of the electrostatic field is relatively small in comparison to the increase in translational entropy of the counterions’ “counterion release” mechanism). A high concentration of counterions and monomer units of polyelectrolyte chains in the brush ensures the screening of electrostatic interactions on the length scale comparable to (or smaller than) the size of the dipolar object. This is essential for the manifestation of an asymmetry in the interaction of positively and negatively charged faces of the inclusion with the polyelectrolyte brush. Above we have focused mostly on the case in which the ionic strength is relatively low, because the attraction appeared to be the strongest in this limit. In this case, the charges of the chains forming the PE brush are not completely compensated by counterions localized inside the brush. A significant diffuse double layer is found on top of the brush, i.e., on the aqueous side next to the brush. This diffuse layer is similar to the Gouy-Chapman layer that forms next to a charged surface (cf. Figure 7). Before the protein molecule enters the brush, it has to travel through this diffuse layer. In line with the results of Figure 7, the free energy gain is already significant before the dipolar object reaches the edge of the brush. In this process, a significant number of
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counterions are released from the diffuse layer. Then, the protein enters the brush, i.e., a region where the excess electrostatic potential is high, but exhibits relatively little spatial variation. When the protein moves deeper into the brush, there are very few extra counterions released. The free energy gain proceeds because the polymer chains adsorb onto the oppositely charged patch and deplete from the similarly charged one. We conclude that (at least in the low ionic strength cases) there are significant contributions to the free energy reduction: • The counterion release mostly takes place in the GouyChapman layer on top of the brush. • The PE chain adsorption contributes significantly when the protein is inside the brush. The asymmetry of interaction of opposite and similar charges, as with any nonlinear effect, vanishes upon the increase in salt concentration. Therefore, the attraction of the dipolar object into the PE brush becomes weaker when the ionic strength is increased. Here, the concentration of counterions trapped inside the saltfree osmotic brush determines the threshold value for the salt concentration in the bulk of the solution beyond which attraction drops down significantly. In this case, the Gouy-Chapman layer on top of the brush is less developed. As a result, most of the free energy reduction occurs when the protein enters the brush. Again, counterions are released, but this occurs in concert with the adsorption of the polymer chains onto the oppositely charged patch. In the above, we only considered the electric dipole object oriented parallel to the brush surface. We showed that the overall effect could be understood by the asymmetry that exists when a positively charged object or a negatively charged object was injected into the brush. In general, the strength of the attraction must also depend on the orientation of the dipole with respect to the brush surface. The strongest attraction is expected when the positive patch has a lower z-coordinate than the negatively charged patch. In order to calculate the exact interaction potential, one has to average over different possible orientations with proper statistical weights. Our result gives a first-order (conservative) estimate for this orientation-averaged potential for interaction between the protein and the brush. It is important that, depending on the size of the protein and a number of positive charges on its surface, the overall gain in the free energy upon transfer of the protein into the interior of the brush can easily be on the order of a few kBT.
Conclusions In this paper, we have shown that the two-gradient selfconsistent field theory can prove and quantify the attraction of a globular protein molecule into a charged polymer brush. This attraction exists even when the sign of the net charge on a protein is zero or of the same sign as that of the PE chains in the brush. When the charge distribution of the protein surface is sufficiently inhomogeneous and comprises positively and negatively charged patches, an attractive contribution to the interaction free energy arises. The attraction gets stronger with decreasing ionic strength and increasing charge density on the patches and also increases slightly with grafting density of the PE brush. In agreement with experimental observations,15 the attraction therefore strongly decreases when salt concentration in the bulk of the solution reaches the value of counterion concentration in a salt-free PE brush (crossover between osmotic and salted brush regimes). All findings are traced back to the asymmetry found in the variation of the free energy upon bringing a homogeneously charged object toward a similarly charged surface (weak repulsion) or toward an oppositely charged surface (strong
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attraction). This asymmetry is universal and is found for all cases where the local electrostatic potential is sufficiently high. At low ionic strength conditions, the counterion release is significant when the oppositely charged object travels from the bulk through the Gouy-Chapman layer. As soon as the inclusion arrives at the brush periphery, the counterion release saturates. A further decrease of the free energy is found when the inclusion is placed deeper inside the brush. This is attributed to the asymmetric interaction of the PE chain with the positively charged patches (attraction) and negatively charged patches (depletion). Both free energy effects, adsorption/depletion of the PE chain, and the counterion release can be on the same order of magnitude. Hence, both effects should be included when calculating the strength of interaction of a protein molecule to a PE brush at a pH larger than the IEP of the protein. Concluding this paper, we state that our model provides a valuable explanation for polyelectrolyte-mediated protein adsorption, caused by the electrostatic interactions between the
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patches with a positive charge with the negatively charged chains of the brush layer. Acknowledgment. This work has been partially supported by EUROCORES program on Self-Organized NanoStructures (SONS) within the project 02-PE-JA016-SONS-AMPHI and by the EC Sixth Framework Program within MC RTN POLYAMPHY. The authors acknowledge also financial support from Dutch National Science Foundation (NWO) and Russian Foundation for Basic Research (RFBR) through joint project 047.017.026 “Polymers in nanomedicine: design, synthesis and study of interpolymer and polymer-virus complexes in search of novel pharmaceutical strategies”. O.V.B. acknowledges financial support of the Alexander von Humboldt Foundation. M.B. acknowledges financial support by the Deutsche Forschungsgemeinschaft, SFB 481, Bayreuth, and by the Fonds der Chemischen Industrie. LA0632777
