Comparative experimental and computational study of synthetic and natural bottlebrush polyelectrolyte solutions

Abstract

We systematically investigate model synthetic and natural bottlebrush polyelectrolyte solutions through an array of experimental techniques (osmometry and neutron and dynamic light scattering) along with molecular dynamics simulations to characterize and contrast their structures over a wide range of spatial and time scales. In particular, we perform measurements on solutions of aggrecan and the synthetic bottlebrush polymer, poly(sodium acrylate), and simulations of solutions of highly coarse-grained charged bottlebrush molecules having different degrees of side-branch density and inclusion of an explicit solvent and ion hydration effects. While both systems exhibit a general tendency toward supramolecular organization in solution, bottlebrush poly(sodium acrylate) solutions exhibit a distinctive “polyelectrolyte peak” in their structure factor, but no such peak is observed in aggrecan solutions. This qualitative difference in scattering properties, and thus polyelectrolyte solution organization, is attributed to a concerted effect of the bottlebrush polymer topology and the solvation of the polymer backbone and counterions. The coupling of the polyelectrolyte topological structure with the counterion distribution about the charged polymer molecules along with direct polymer segmental hydration makes their solution organization and properties “tunable,” a phenomenon that has significant ramifications for biological function and disease as well as for numerous materials applications.

INTRODUCTION

Aggrecan is a high molecular mass biological polyelectrolyte (10⁶ < M < 3 × 10⁶ Da). This molecular construct possesses a bottlebrush structure, consisting of an extended protein core, of approximate length 400 nm, to which many relatively stiff chondroitin sulfate and keratan sulfate (sulfated polysaccharide) side chains (≈30 nm in length) are attached.^1–3 In the presence of hyaluronic acid (HA) and link protein, aggrecan forms large aggrecan–HA aggregates in which as many as 100 aggrecan molecules are condensed on a linear HA chain. Aggrecan–HA assemblies, enmeshed in a collagen matrix, govern the resilience of cartilage and its ability to withstand high compressive loads.⁴ In cartilage, aggrecan is exposed to an environment in which both mono- and divalent ions are present. Electrostatic repulsion between highly charged bottlebrush aggrecan molecules is a determining factor of cartilage biomechanical properties.^5–7 For example, degradative changes in the size and composition of aggrecan become more progressive with age. This process can be potentially deleterious to articular cartilage function as it reduces the charges on the aggrecan molecules.^8–10 Knowledge of the effect of ions, notably calcium ions, on the structure and dynamics of aggrecan is essential to understand its function in cartilage/skeletal metabolism and bone development. To this end, it is necessary to determine how aggrecan differs from “ordinary” polyelectrolytes. As the effects of ions on polyelectrolytes are not only short-range but, by virtue of the connected structure of macromolecules, also long-range in nature, information must be obtained from extended length and time scales. In many emerging applications, such as in tissue engineering and regenerative medicine, this knowledge is required for tailoring the properties of the replacement tissue in the development of tissue-engineered cartilage implants.

Previous studies^11,12 have shown that aggrecan possesses a set of exceptional material properties that distinguish this macromolecule from other highly charged polyelectrolytes and make aggrecan well suited for its multiple biological roles in cartilage and other connective tissues. For example, cartilage load-bearing properties are attributed to the compressive properties of large aggrecan–HA complexes. However, earlier works have not focused on the physicochemical origin of this singular behavior of aggrecan solutions. In particular, it is not clear whether the osmotic properties of aggrecan solutions originate from its bottlebrush architecture or from the high charge density and intrinsic rigidity of the polysaccharide bristles. It is also unclear whether the remarkable ionic insensitivity reported for aggrecan solutions^11,12 is a common feature of charged bottlebrush polymers or it is mainly the consequence of the stiffness of the polysaccharide backbone. To address these unsolved issues, we synthesized model poly(sodium acrylate) bottlebrush polyelectrolytes (BP-PAANa) and then compared their solution properties with those of the aggrecan. Clearly, there are significant differences in the chemical nature of BP-PAANa and aggrecan that will be discussed below addressing differences in the observed organization of these polymers in solution.

Molecular dynamics simulations of a coarse-grained bead-spring model provide some insight into the general trends of polyelectrolyte organization in solution.^13–18 Simulations of highly charged polyelectrolyte solutions with low salt and explicit solvent^12,19–21 have indicated a general propensity of polyelectrolytes to undergo supramolecular assembly in solution, making these solutions essentially different from neutral solutions of linear polymers. The concepts of dilute, semi-dilute, and concentrated solutions no longer have a clear meaning in such associating liquids. We should rather expect temperature and added salt dependent “critical concentrations” for association, as in micellization, and other supramolecular self-assembly processes.

Different “universality classes” of self-assembly arise in solutions^22–24 depending on the symmetry characteristics of the associative intermolecular interactions: linear chain polymer formation, randomly branched polymer formation, or the formation of compact structures having a more regular internal structure and definite shape.^22,25–36 Shifting the balance of these interactions should also lead to changes in assembly geometry,^23,24 sometimes triggering significant changes in solution properties since extended assemblies^37,38 in the form of linear and randomly branched polymers tend to influence solution properties at the same volume fraction more than compact structures.^35,36 In practice, it is often hard to predict a priori which class of assembly process should arise because of the subtle interplay between multiple competing interactions, and for this reason, it is important to identify singular scattering signatures and solution properties that can be used to determine the supramolecular organization in any particular system. We address this basic characterization problem in our simulations below.

In connection with this general phenomenon, we have recently discovered that the relative extent to which the polymer backbone of a simulated polyelectrolyte solvates, compared to the counterions in solution, can greatly influence polyelectrolyte solution organization, and we argue below that this competitive solvation process is essential for understanding the organization of aggrecan and other naturally occurring bottlebrush polyelectrolytes in solution.¹⁹ We indeed find a qualitatively different solution structure organization between our synthetic and natural bottlebrush polyelectrolyte solutions.

This paper is organized as follows: First, we compare the osmotic behavior of BP-PAANa and aggrecan solutions in near physiological ionic environments. Then, we focus on the small angle neutron scattering (SANS) results obtained at different polymer and salt concentrations. We then identify ion-induced changes in the organization of polyelectrolyte bottlebrush solutions as a function of the length scale. The dynamic response of both systems is determined by dynamic light scattering (DLS) in the presence of different ions as a function of polymer concentration.

MATERIALS AND METHODS

Bottlebrush polyelectrolytes

BP-PAANa was synthesized as previously reported.^39–41 The polymer used in this study has M_n = 678 kDa, D = 1.13 with the degree of polymerization (DP) of 400 and 12 for the backbone and side chain, respectively (Scheme 1).

SCHEME 1.

(a) Poly (sodium acrylate) bottlebrush polyelectrolyte and (b) aggrecan bottlebrush.

Aggrecan was purchased from Sigma-Aldrich (Saint Louis, MO, A 1960–1 mg, Aggrecan from bovine articular cartilage, Lot# 075 M 4012V).

Polymer solutions

BP-PAANa was dissolved in water or in D₂O (for SANS measurements). The polymer concentration ranged between 4% and 16% m/m. (Polymer concentrations are given in terms of the percentage of relative mass of the polymer to the entire solution.) Concentrations of NaCl and CaCl₂ were varied between 0 and 400 mM and 0 and 100 mM, respectively. Aggrecan solutions were prepared in 100 mM NaCl. The concentration of aggrecan was varied from 0.2% to 5% m/m. For all samples, the pH was adjusted to 7.¹¹

Small angle neutron scattering

SANS measurements were made on the NG3 and the 10 m SANS instruments at NIST, Gaithersburg MD. The solutions were prepared in D₂O, and the temperature during the experiments was maintained at 25.0 ± 0.1 °C. Measurements were carried out at two sample-detector distances, 2.5 and 13.1 m (on the NG3 instrument) and 2.0 and 10 m (on the 10 m SANS instrument), with incident wavelength 8 Å. After radial averaging, corrections for incoherent background, detector response, and cell window scattering were applied. Intensity normalization was made with NIST standard samples.⁴²

Osmotic pressure measurements

The osmotic pressures of aggrecan and BP-PAANa solutions were determined as a function of polymer concentration by bringing the solutions to equilibrium with polyvinyl alcohol (PVA) gels of known osmotic swelling pressure.^43,44 At equilibrium, the osmotic pressure of the polymer solution is equal to the swelling pressure of the gel. The high molecular mass of the dissolved polymer prevented its penetration into the gel. The size of the PVA filaments was determined by optical microscopy after equilibration in the polymer solution (≈1 day). Osmotic pressure measurements were made at 25.0 ± 0.1 °C in aqueous salt solutions.

Dynamic light scattering

DLS measurements were made with a precision detector—Expert Laser Light Scattering DLS Workstation (Bellingham, MA), equipped with a HeNe laser working at 698 nm.⁴⁵ Measurements using this instrument were performed in the angular range 45°–120° with accumulation times of 500 s. Absolute intensities were obtained by normalizing with respect to toluene. To avoid shear degradation of the high molecular mass aggrecan molecule, the solutions were not filtered.

Simulation methods

We employed two different models, one for charged bottlebrushes and one for neutral polymers. For the former, we utilize a bead-spring model of Lennard-Jones (LJ) segments bound by stiff harmonic bonds suspended in explicit LJ solvent particles, some of which are charged to represent counterions. The model has been utilized in previous studies to probe the association of polyelectrolyte chains in solution, and here we briefly outline the model and the relevant methods.^20,21,46 For computational expediency, all macro-ion segments, dissolved ions, and solvent particles are assigned the same mass m, size σ, and strength of interaction ε, and all dissolved ions are monovalent. This is clearly an idealization of real bottlebrush polyelectrolytes, but we introduce this highly simplified model in an attempt to identify general trends with molecular parameters and solution conditions.

Following standard computational practice for this type of coarse-grained polymer modeling, we set ε and σ as the units of energy and length; the cutoff distance for LJ interaction potential is r_c = 2.5 σ. These parameters for the different particle types are set equal to unity, except for the energy interaction parameter between the solvent particles and the counterions ε_c,s, which reflects the relative affinity of the counterions for the solvent. This provides the basis of a minimal model of ion hydration.

In previous papers,^23,24 we established a methodology to estimate the effective interaction of the ions with the solvent, ε_c,s/ε = 5 for Ca⁺². All model polyelectrolyte bottlebrush polymers are taken to have a linear chain backbone of length N_b = 40 segments. The interactions between the polymer segments and the solvent molecules were set by the ratio ε_p,s/ε = 5 in an attempt to model the strongly hydrophilic character of the polysaccharide side chain “bristles” of aggrecan.^2–4 In our coarse-grained model of a bottlebrush polymer, we assume f side chains of length M = 12 segments bonded along the backbone and distributed uniformly along the bottlebrush polymer chain backbone. Two different grafting densities are studied here, f/N_b = 1 and 1/4. The total number of segments per bottlebrush polymer is M_w = f M + N_b. Each segment of the side chain of bottlebrush polyelectrolytes carries a − e charge, where e is the elementary charge and the total polyelectrolyte charge is Z_p = −M_w e. We also include divalent ions to represent calcium ions (Ca²⁺). The bonds between polymer segments are connected via a stiff harmonic spring, V_H(r) = k (r − l₀)², where l₀ = σ is the equilibrium length of the spring and k = 1000 ε/σ² is the spring constant.

Notably, the present models of linear and bottlebrush polyelectrolyte chains do not include the solvation of uncharged chain segments, an effect that might be important if the chain backbone contains hydrophobic sidegroups, in addition to charged segments. This is just one way the simulations have been simplified to enable simulation of these complex polymer solutions. Although our coarse-grained model appears quite adequate for describing typical synthetic polyelectrolytes,³⁹ in which sidegroups are not present and in which the backbone is often relatively hydrophobic, our simulations below seem to indicate that additional solvation interactions of the chain backbone and sidegroups might be crucially important for understanding the solution organization and properties of aggrecan. We next describe the size of the computed system, concentrations of the various species, temperature range, the computational times considered, etc.

The system is composed of a total of N = 252 000 particles in a periodic cube of side L and volume V. The system includes N_p polyelectrolyte chains and N₊ = N_p |Z_p/z₊| counterions, with z₊ = 2 representing the ion valence; the number of neutral particles is N₀ = N − N₊ − N_p M_w, and we define the charge fraction as φ = (N₊ + N_p |Z_p/e|)/N. Each system has an overall neutral total charge. All charged particles interact via the Coulomb potential with a cutoff distance of r_c,c = 10 σ, and the particle–particle particle–mesh method is used for distances r > r_c,c. The Bjerrum length was set equal to l_B = (e²/ϵ_s k_B T) = 2.4 σ, where T is the temperature, k_B is Boltzmann’s constant, and ϵ_s is the dielectric constant of the medium. Our simulations were equilibrated at constant pressure and constant temperature conditions, i.e., reduced temperature k_B T/ε = 0.75 and reduced pressure 〈P〉 ≈ 0.02, and production runs were performed at constant temperature and constant volume, maintained at equilibrium by using a Nosé–Hoover thermostat. Typical simulations equilibrate for 40 000 τ, and data are accumulated over a 20 000 τ interval, where τ = σ (m/ε)^1/2 is the time unit; the time step used was ∆t/τ = 0.005. Since we encountered different supramolecular organizations in the model synthetic polyelectrolyte and aggrecan solution measurements, we also simulated model branched polymers corresponding to randomly branched polymers, which form rather extended or “open” polymeric clusters and relatively “compact” or “closed” gel particles having a network-like internal structure. These simulations were performed to aid in our interpretation of the neutron scattering data and in our assignment of the structural organization of polyelectrolyte solutions, and we describe the details of these computations in the Appendix.

RESULTS AND DISCUSSION

Osmotic pressure measurements

Figure 1(a) shows the variation of the osmotic pressure Π as a function of polymer concentration c for BP-PAANa solutions measured in the absence of added salt and at different CaCl₂ concentrations. The Π vs c plots were analyzed in terms of the scaling expression,

$$\Pi =A{c}^n,$$ (1)

where A and n are constants. For neutral polymer solutions in the semi-dilute concentration regime, n = 9/4 (good solvent condition) or n = 3 (theta condition).⁴⁷

FIG. 1.

Osmotic pressure Π as a function of polymer concentration for solutions of BP-PAANa (a) and aggrecan (b). The dashed lines through the data points are guides to the eye. Uncertainties are estimated by one standard deviation of the linear regression fit parameters.

Figure 1(a) shows that at low polymer concentration, Π increases linearly with the polymer concentration, which is also typical of dilute non-associating polymer solutions. However, the observations in Fig. 1(a) show an abrupt change in the slope at higher concentration that suggests the onset of chain association. In particular, the BP-PAANa solutions exhibit a power-law scaling where the exponent n progressively decreases with increasing CaCl₂ concentration from n = 3.3 (CaCl₂ free solution) to n = 1.93 (100 mM CaCl₂). The characteristic concentration describing the transition between the two concentration regimes takes place at around 1.5% m/m. A change in the slope also occurs in neutral polymer solutions beyond the “dilute regime” in which the chains are relatively isolated, but the transition is less abrupt.⁴⁷ As noted earlier, we avoid the traditional concentration classification scheme of uncharged polymer solutions, i.e., dilute/semi-dilute/concentrated for our charged bottlebrush polymer solutions, since there is clear evidence that these are associating fluids. We provide further evidence of chain association below.

Figure 1(b) shows Π vs c for aggrecan solutions containing different amounts of CaCl₂. At low concentration, Π increases linearly with the aggrecan concentration, but we see a transition to another regime in the vicinity of 1% m/m polymer concentration, clearly suggestive of some sort of supramolecular assembly transition, such as micelle formation.^11,12 Correspondingly, we interpret the reduction in the effective concentration scaling exponent n in the intermediate concentration range to reflect a reduction in the increase in the osmotic pressure due to self-assembly of the aggrecan bottlebrushes into dynamic branched polymers having an open branched polymer structure [see Fig. 4(b) and associated discussion]. The self-assembly can be attributed to the anisotropy of the aggrecan molecular interactions and, in particular, the difference in water affinity between the N-terminal domain of the protein core of the aggrecan bottlebrush and the highly charged hydrophilic polysaccharide side chains. This trend was found to be reproducible even in the presence of CaCl₂. Figure 1(b) indicates that self-assembly begins at around 0.8% m/m. At higher concentration, above 1.5% m/m, the osmotic pressure rises more steeply (slope ≈2) owing to the densification of the overlapping aggrecan assemblies. The osmotic pressure gradually decreases with increasing CaCl₂ concentration, but both the concentration thresholds and the power-law dependence of Π on c remain unaffected by the Ca²⁺ ions.

FIG. 4.

(a) Schematic of the molecular architecture of bottlebrush polymer. (b) Typical molecular configurations of polyelectrolyte bottlebrush polymers at grafting densities f/N_b = 1 (top) and f/N_b = 1/4 (bottom). (c) Form factor (dotted lines), P(q), and static structure factor (solid line), S(q), of the charged bottlebrushes in solution at the same polymer concentration at different grafting densities. The inset shows the scaling of the primary peak position q* in S(q) as a function of polymer concentration, q ∼ c_pol^μ, where μ is a scaling exponent; the dashed lines are guides for the eye.

In summary, the osmotic behaviors of BP-PAANa and aggrecan solutions exhibit marked differences in the polymer and salt concentration dependences. At low polymer concentrations, the osmotic pressure increases linearly with the concentration in both systems, while at higher concentrations, the power-law exponent n approaches a value observed before for ordinary linear polyelectrolytes.⁴⁸ In aggrecan solutions, we see an evident transition in scaling in the intermediate concentration range, suggesting again some type of self-assembly transition of the bottlebrush polyelectrolytes. The addition of CaCl₂ reduces the osmotic pressure in both systems, but the nature of reduction is different in the synthetic and natural bottlebrush solutions. In BP-PAANa solutions, the power-law exponent n decreases with increasing CaCl₂ concentration, while in aggrecan solutions, the general shape of the Π vs c curves and the concentration scaling exponent remains nearly unaffected by the Ca²⁺ ions. We believe that the observed differences between the osmotic pressure scaling behaviors of aggrecan and BP-PAANa solutions originate from both the high charge density and the intrinsic rigidity of the charged polysaccharide bristles. In addition, aggrecan is a hydrophilic polymer, while the backbone of BP-PAANa is hydrophobic. We elaborate on these differences in our discussion below.

Small Angle Neutron Scattering (SANS) measurements

An insight into the organization of dissolved bottlebrush polyelectrolytes can be obtained from SANS. This technique probes the structure of the solutions over a wide range of length scales (5 × 10⁻³ Å⁻¹ < q < 0.5 Å⁻¹). Figures 2(a) and 2(b) show the SANS responses of BP-PAANa and aggrecan solutions.

FIG. 2.

SANS profiles of BP-PAANa (a) and aggrecan (b) solutions at different polymer concentrations. The inset in the lower left corner shows the variation of the peak position q* with the polymer concentration in BP-PAANa solutions. The inset in the upper right corner illustrates the effect of CaCl₂ on the SANS response of the 4% m/m BP-PAANa solutions. The numbers shown indicate reference values of the scaling of the intensity I(q) vs q. The red symbols in (b) show the SANS signal of the 0.25% m/m aggrecan solution in 100 mM CaCl₂ solution. The drawings above the SANS curves show schematically the structure of aggrecan solution. Uncertainties in the insets of Fig. 2(a) are estimated by one standard deviation of the linear regression fit parameters.

SANS measurements on the BP-PAANa solutions reveal the existence of different scaling regimes. At low q, corresponding to relatively large length sales, we see a sharp upturn as often observed in polyelectrolyte solutions.^49,50 This low q feature, characteristic of many synthetic and natural polyelectrolyte solutions, has conventionally been interpreted in terms of formation of large-scale polyelectrolyte clusters,⁴⁴ and recent simulations with explicit solvent have confirmed this interpretation (see discussion below). The absence of a plateau in the low q regime implies that the size of the scattering objects exceeds the resolution of the SANS measurements. In the intermediate q range (0.03 Å⁻¹ < q < 0.1 Å⁻¹), the salt-free BP-PAANa solution displays a distinct peak, which is also a characteristic feature of many polyelectrolyte solutions. This “polyelectrolyte peak” is diminished in 50 mM calcium chloride solution [see the inset in the upper right corner of Fig. 2(a)]. The peak is usually attributed to a correlation length defined by the electrostatic and excluded volume induced repulsion between adjacent polymer chains within the polyelectrolyte clusters evidenced by the low q scattering.^49–51 In the absence of added salt, the electrostatic repulsion is maximized, and the addition of salt gradually screens these interactions, making the peak progressively broadened and smaller in height. The peak position q* defines an average characteristic distance between neighboring scattering centers that can be formally estimated by invoking Bragg’s equation, d = 2π/q*. We see that d decreases with increasing polymer concentration from 128 Å for a concentration of 4% m/m to 76 Å for a concentration of 16% m/m. Scaling arguments that neglect solution heterogeneities predict that the position of the polyelectrolyte peak varies as q* ∼ c^p, where p = 1/3 in dilute solutions and p = 1/2 in semi-dilute solutions.^49,50,52 In our bottlebrush polyelectrolyte solutions, we see a concentration exponent of about p = 0.43, which is the characteristic of a polymer having a somewhat compact structure (p values near 1/2 are often reported, but values as low as 1/3 are frequently found for compact polymers, e.g., dendrimers, and protein solutions).^49,53 At high values of q, the scattering intensity varies as q⁻¹, indicating that the local structure of the polymer appears rod-like. This analysis shows that the scattering behavior of synthetic bottlebrush polyelectrolyte is similar to that of linear polyelectrolytes observed in previous experimental studies. In a recent paper, we have shown that the presence of side chains makes the polyelectrolyte peak sharper than its linear counterparts,³⁹ but this is a minor change in the overall scattering characteristics of this class of polymer solutions. We next show the scattering characteristics of aggrecan solutions and thus the structural organization of these solutions, which is completely different from that of synthetic bottlebrush polyelectrolyte solutions.

The SANS response of the aggrecan solution in the low q range is superficially reminiscent of the scattering of ordinary uncharged polymers in solution where the scattering intensity decays smoothly with a power law near −2, corresponding to ideal non-interacting random coils in solution. The exponent becomes somewhat smaller for swollen neutral polymers whose fractal dimension d_f ≈ 5/3 is lower than 2. However, the approximately power-law scaling of S(q) in Fig. 2 with a power law of around 2 does not imply that the conformation of aggrecan molecules is anything like a “random coil” polymer. A closer look at the data reveals an apparent power law near −2.7, which is the characteristic of branched polymers with screened binary excluded volume interactions,^54,55 i.e., percolation clusters that have a fractal dimension near 2.5.^56,57 At high q, where we probe the small-scale structure of the polymers, we see a wave vector scaling exponent close to −1, reflecting the “rod-like” structure of the molecules, an effect that also arises in linear polymer chains with some bending stiffness.

Since the interpretation of this type of scattering pattern is important for our discussion below, we note that randomly branched polymers may be generated from a simple model of spherical particles with “sticky” groups on their surface modeling local regions of molecular binding affinity with other molecules of the same kind^54,55 or with quadrupolar interactions, which are multi-functional long-range interactions.²² Regardless of the functionality of the molecular association, apart from particles having two associative regions (“spots”) or dipolar interactions that lead to the formation of linear chain assemblies, particles tend to associate to form dynamic randomly branched polymer clusters having an “open” structure whose fractal dimension is typically near 2.5 when the clusters are not very large and the effect of excluded volume interactions has not built up so that the asymptotic value of 2 is reached in a good solvent.^54,55 The swollen branched polymer R_g exponent is exactly 1/2 in the limit of infinite-sized polymers, which also illustrates the fact that the swelling exponents are generally dependent on polymer topology. A wavevector scaling exponent for S(q) in the range between 2 and 2.5 is rather typical of open branched polymeric clusters of this type.

The openness of these clusters is important for understanding the observed form of scattering from these clusters, and in contrast, we consider a representative “gel” particle formed by the lattice network (N_x = 8, N_y = 8, and N_z = 8) of flexible star polymer elementary units having f = 4 chains and M = 15 segments in each chain; thus, the total number of segments between two branched segments is 30. The form factor P(q) or structure factor in the limit of one such “perfect network” particle along with a representative snapshot conformation of this idealized gel particle is shown in the inset of Fig. 3(a). Recently, there has been significant effort aimed at making such “perfect networks” as bulk gel materials.^58–61 We see in Fig. 3(a) that P(q) exhibits a near Gaussian variation at low q, from which the average gel molecule radius of gyration can be estimated using the Guinier model,⁵⁰ and at moderate q, there is a small peak reflecting the internal network structure of this type of compact gel particle. This scattering pattern is characteristic of compact “microgel” particles,^61,62 high generation neutral dendrimers,⁶³ and numerous other polymer structures that might reasonably be described as “fuzzy spheres.” We note that this type of form factor P(q) strikingly resembles that of “ordinary” polyelectrolytes, which have been reported to form compact polymer clusters in solution.^51,64–66

FIG. 3.

Form factors generated by molecular dynamics simulations of (a) a closed/compact gel particle and (b) an open/tree-like gel particle, both having the same molecular mass; screenshots of typical configurations of each type are also shown. The arrows point to the average radius of gyration R_g of the gel particle.

The plateau-like feature in P(q) reflects regions within the gel that are relatively void of polymer segments, and the size of these regions can be considered the mesh-size of the network. Specifically, the plateau emerges at a scale that is approximately the average distance between two neighboring branching points of the network and terminates at a scale approximately twice this distance. This scattering feature, which is not accurately captured by the popular “fuzzy sphere” model of gel particles, provides important information about the internal topological structure of gel particles. We will describe this important effect in a future paper, emphasizing the impact of network defects on the structure of gel nanoparticles.

In contrast, branched polymer clusters exhibiting multi-functional association normally take the form of “open” branched polymer clusters.^22,54,55 Such clusters are notably far less symmetric than the perfect network gel particle shown in Fig. 3(a) and have an inherently more “diffuse” fractal form. We may obtain a representative gel particle having this “open” gel geometry by simply cutting the bonds that connect the elementary units at random of the compact gel particle shown in Fig. 3(a) (see the Appendix for a detailed description of this procedure). Once there is no other bond to cut, the gel starts to disintegrate and we let the structure relax to its more extended equilibrium form. We next show an example of P(q) for an open initially perfect compact polymer network before bond decimation [Fig. 3(a)] and a polymer cluster [Fig. 3(b)]. The open branched polymer structures formed by the cutting method are quite similar to the clusters formed by dynamic multi-functional particle association.^22,54,55 In particular, note that the wavevector scaling in this system is near the value of 2.7 observed in the aggrecan system [see Fig. 2(b)].

Figure 2(b) also shows the SANS profile of the 0.25% m/m aggrecan solution measured in 100 mM CaCl₂ solution. Within experimental error, the two datasets with and without Ca²⁺ ions coincide. The lack of any appreciable change in scattering intensity over the entire q range implies that replacing sodium counterions on the polyanion with calcium ions does not induce either phase separation or densification of bottlebrush aggrecan. The insensitivity of the scattering response, and thus the solution structure over the length scales probed, has numerous functional implications for aggrecan in living systems.

The salt concentration can be highly variable, and the charge valence can also vary in the biological environment of cartilage, and the sensitivity of aggrecan solution organization to such changes is a matter of great biomedical importance. This is because cartilage mechanical properties would significantly change if aggrecan were highly salt sensitive, degrading the biological function of joint tissue. It is our current hypothesis that the salt insensitivity of aggrecan solution properties arises from the intrinsic stiffness of the molecular structural components of aggrecan, in combination with its high degree of hydration, which together limit the extent to which counterions can alter chain conformation and come in direct contact with the charges of the polymer chain. Regardless of this proposed explanation, the salt insensitivity of aggrecan is a remarkable phenomenon that deserves further experimental and computational investigation to fully understand its physical origin.

The difference between the slopes at low q reveals important differences between aggregates formed in BP-PAANa and aggrecan solutions. In the aggrecan solution, the exponent is smaller than 3, i.e., surface scattering is absent. This observation implies that aggrecan does not form compact polymer domains exhibiting a polyelectrolyte peak reflecting strong interchain correlations in these domains, but it rather organizes into hierarchically branched structures having an approximately branched structure.⁶⁷ Molecular dynamics simulations have recently provided insights into why polyelectrolytes tend to exhibit these two rather distinct patterns of solution organization depending on polymer structure and solution conditions.¹⁹ We next discuss modeling branched polyelectrolyte solutions to gain insights into the behavior of aggrecan and our synthetic bottlebrush polymer, BP-PAANa.

Exploratory molecular dynamics simulations

To gain insights into the structural differences between the synthetic charged bottlebrush polymer and aggrecan solutions, we consider differences in the bottlebrush architecture that might account for the observed differences in molecular organization. In particular, we performed molecular dynamics simulation for different grafting densities and different polymer concentrations based on our coarse-grained bottlebrush polymer model with explicit solvent and counterions to assess the importance of these potentially relevant variables.

Figure 4(c) shows that the simulated static structure factor S(q) for bottlebrush solutions having f/N_b = 1 has two peaks rather than the single “polyelectrolyte peak” normally observed in scattering and simulation studies of linear chain polyelectrolyte solutions. Two peaks have been observed before in x-ray scattering measurements on neutral bottlebrush polymer melts.⁶⁸ The first peak describes an average inter-molecular distance between the bottlebrush polymers, and we designate this peak as the “polyelectrolyte peak.” While this peak in S(q) is observed for both grafting densities, it is evidently less pronounced in the case of lower side chain grafting density [see Fig. 4(c)]. A lower number of side chains along the backbone weaken the short-ranged repulsive interactions between the polyelectrolyte chains, much as reducing the number of arms of polyelectrolyte star polymers. The concentration scaling exponent of the polyelectrolyte peak changes from ≈0.46 to 1/3 as the grafting density changes from f/N_b = 1 to 1/4, respectively [see the inset of Fig. 4(c)]. This change in the exponent indicates the importance of molecular topology on the structural properties of polyelectrolyte solutions.

The second peak in the scattering signal in the higher q range is related to the average distance between the bottlebrush side chains, a feature observed and discussed earlier in neutral bottlebrush polymer melts.⁶⁸ The secondary peak contains information about the intra-molecular structure of the bottlebrush polymers, information that is also contained in the polymer form factor; see Fig. 4(c). For a bottlebrush polyelectrolyte with lower grafting density, the second peak disappears, reflecting the fact that there are significantly reduced correlations between the side chains arising from their mutual repulsions.

As mentioned briefly above, we have found^19–21 that the relative strength of solvation of the charged species and the polymer backbone can greatly influence the structure of polyelectrolytes in solution, and we think that this phenomenon can explain the observed trend for aggrecan vs our synthetic polyelectrolyte solution. In particular, tuning the relative strength of solvation interactions in the simulations of linear polyelectrolyte solutions led to the emergence of two different types of effective attractive interactions between the polyelectrolytes—one type of attraction operating on length scales on the order of the entire polymer chain and the other acting on the order of the monomer size. In each case, the strength of the effective attractive interactions results in the formation of polyelectrolyte chain clusters whose presence in solution is clearly evidenced by numerous previous neutron and light scattering measurements where the sharp upturn in the low q scattering intensity related to the clustering can be followed to large scales with light scattering measurements.⁵¹ Importantly, for the present work, we found that if both the degree of ion solvation and polymer solvation are high, as likely in the case of aggrecan, the polyelectrolyte peak disappears.¹⁹ Although the resulting form of S(q) under such strong solvation conditions superficially resembles a neutral polymer solution due to the smooth power-law decay of S(q) and the absence of a noticeable polyelectrolyte peak, simulations suggest that the polymer chains are exhibiting supramolecular assembly into open branched polymeric structures. The hierarchical nature of these polymer clusters “washes out” the polyelectrolyte peak on self-assembly into compact structures.

Previous measurements on linear polyelectrolytes with different backbone chemistry gave insight into the origin of the difference in the polymer organization of aggrecan and BP-PAANa solutions. In a previous SANS study of the solution organization of chondroitin sulfate and HA solutions,¹⁹ the main components of the aggrecan–HA supramolecular complex, we found that HA solutions likewise exhibit no polyelectrolyte peak, while in contrast, the chondroitin sulfate solution exhibited a conventional polyelectrolyte peak. This difference was explained in terms of the different degrees of hydration/solvation of the chain backbone based on the current computational model.¹⁹

We illustrate the effect of the ion and polymer backbone solvation on S(q) in Fig. 5, where we also compare to the scattering of neutral flexible polymers having the same polymer mass as a natural point of reference. In Fig. 5, we see that increasing the strength of the polymer ion (ε_p,s/ε = 5) and counterion (ε_c,s/ε) solvation relative to the water–water interaction strength (see the section titled Simulation methods for the definition of these interactions) leads to a progressive decrease in the polyelectrolyte peak, which is highly prevalent when these solvation energies are weak. The lower panel in Fig. 5 shows the changes in the organization of the polyelectrolyte solution that accompanies these changes in the polyelectrolyte charge and counterion solvation. The hydration of the uncharged species is entirely neglected in our model, and the presence of hydrophobic groups on the chain backbone, as might be expected for aggrecan, is another interaction that should clearly be incorporated into a minimal model of this type of natural polyelectrolyte bottlebrush polymer.

FIG. 5.

(a) Structure factor of linear chain polyelectrolyte solution as the strength of solvent interactions with the charged species increases; see Ref. 19 for more details. (b) Screenshots of the polyelectrolyte and neutral polymer solutions; the polymer segments are in red color, the counterions are in semi-transparent blue color, and the solvent particles are rendered invisible for clarity.

Note also that while S(q) for the simulated highly solvated polyelectrolytes superficially resembles S(q) of neutral polymers, the structure of these solutions is clearly rather distinct. We next discuss the change in the nature of the interactions responsible for this change in polyelectrolyte solution organization.

Physically, the increased chain backbone solvation changes the nature of the inter-polyelectrolyte interaction¹⁹ from a long-ranged attractive interaction to a relatively short-ranged attractive associative interaction as the degree of chain solvation is increased. This change in the interaction type is responsible for the dramatic change in polyelectrolyte solution organization, seen in the lower panel of Fig. 5. For polyelectrolytes having a strong degree of chain solvation, as we may reasonably expect to be the case for aggrecan and other natural polyelectrolytes, this relatively short-range and multi-functional “sticky” interaction gives rise to the formation of hierarchically organized fractal chain aggregates¹⁹ that are more similar to an incipient gel (open randomly branched polymers) than compact polymer clusters having a well-defined average size determined by the balance of charge and van der Waals interactions.^18,22 One expected signature of the formation of such clusters is the loss of the polyelectrolyte correlation peak because of the highly fluctuating internal structure of these hierarchically organized fractal clusters. A structure of this kind is entirely consistent with our scattering observations on aggrecan solutions.

Compact clustering of polyelectrolytes has been studied previously by real space scanning electron microscopy imaging in the case of clusters of charged antibody proteins,¹⁸ and optical microscopy evidence for polyelectrolyte clustering in sulfonated polystyrene solutions at low ionic strength has also been reported.⁶⁹ Although artifacts can arise in attempts to directly image polymer nanoclusters in solution are a valid concern in these measurements, there are extensive neutron and light scattering data consistent with the existence of dynamic polymeric clusters in linear highly charged polyelectrolyte solutions under low salt conditions.⁵¹ This phenomenon should be revisited by advanced imaging methods and small angle neutron scattering measurements, such as ultra-low angle neutron scattering and single molecule or cluster imaging, along with simulations of polyelectrolytes with explicit solvent, to better characterize the nature of these compact polyelectrolyte “domains” and the physical factors that control their shape, size, and topology.

Dynamic light scattering measurements

Osmotic pressure measurements and SANS yield valuable information on the thermodynamic behavior and the static structure of bottlebrush assemblies and make it possible to identify important differences between BP-PAANa and aggrecan solutions. However, these static scattering measurement methods do not provide information on the dynamic properties of the systems. In the context of the cartilage function in the joints, the dynamic response of the constituents is particularly important because the timescale of slow joint movement is significantly different from that of rapid joint movement.^70–72 In the case of relatively slow motion of joints, the dynamics of joint movement is governed by the viscoelastic complex fluid nature of cartilage, while in the rapid motion of joints, the elastic nature of cartilage becomes prevalent.

Below, we compare the dynamic response of the two bottlebrush polymers measured by DLS. This technique probes the viscoelastic properties over a broad range of time scales spanning from a few microseconds to several seconds. DLS measures the relaxation rate Γ of the concentration fluctuations. In general, interpenetrating structures exhibit collective modes and display a q² dependent exponential relaxation.^43,73

Figure 6(a) shows that the correlation function that g(t) of BP-PAANa solutions displays two characteristic relaxation rates. This behavior is typical for solutions of linear polymers.^73,74 We analyzed the experimental data by using the following equation:

$$g(t)=a\exp (-{\Gamma }_{\mathrm{fast}}t)+(1-a)\exp [-{({\Gamma }_{\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{w}}t)}^{\mathrm{\mu }}],$$ (2)

where amplitudes a and (1 − a) are the relative intensities of the fast and slow relaxation modes, respectively. Γ_fast and Γ_slow are the corresponding relaxation rates and μ is a constant. The fast mode is governed by the thermodynamic concentration fluctuations described by the first term of Eq. (2), while the slow mode is due to the presence of large clusters.^73,75 In polymer solutions in which no unique correlation length is distinguishable, the relaxation process can be represented by a stretched exponential decay with an exponent μ ≈ 2/3. The trend in Fig. 4(a) shows that the intensity of the slow component (1 − a) increases with decreasing scattering angle θ, indicating that the clusters are very large. This finding is consistent with the increase in intensity observed at small values of q in the SANS responses. The inset illustrates that Γ_fast varies as q², i.e., in BP-PAANa solution, the fast relaxation component is a diffusive process.

FIG. 6.

Angular dependence of the intensity correlation function g(t) of light scattered by solutions of BP-PAANa [4% m/m, (a)] and aggrecan [0.25% m/m, (b)]. The inset in (a) represents the variation of the fast relaxation rate Γ_fast with q. The inset in (b) represents the variation of the relaxation rate Γ with q. Uncertainties in the insets are estimated by one standard deviation of the linear regression fit parameters.

In Fig. 6(b) are shown the intensity correlation functions for aggrecan solution. The major difference between the BP-PAANa and aggrecan is that the latter displays only one characteristic relaxation mode, which resembles the slow mode in the BP-PAANa solution. The curves in Fig. 6(b) cannot be described by a simple exponential relationship, indicating that the correlation function of the aggrecan solution exhibits a wide range of relaxation times, extending from about 10⁻²–10 ms. We describe the relaxation behavior of the aggrecan solution in terms of a stretched exponential form

$$g(t)=\mathrm{\beta }\exp [{(-\Gamma t)}^{2/3}],$$ (3)

where β (≈1) is the optical coherence factor defined by the light scattering geometry.

The data show that the relaxation rate scales as Γ ∼ q^3.11. This implies that aggrecan clusters are so large that the q range of the DLS is within the cluster and only intra-cluster dynamics is observed.⁴⁷ In such systems, the characteristic length that defines the dynamics is the resolution of the observation, i.e., 1/q. This behavior is characteristic of large particle aggregate structures,^73,75 confirming the conclusions drawn from osmotic and SANS observations discussed in the first part of this paper.

A q³ scaling is the characteristic of DLS observations on the scale smaller than the size of large-scale clusters in solution, and this type of scaling is the characteristic of scattering near a critical point for phase separation in diverse fluids.^76–80 Of course, one has to be careful in this interpretation because this same type of scaling is also observed even in dilute neutral high molecular mass polymer solutions when one is looking at segmental scales within the polymer chain.⁸¹

In general, Γ represents the relaxation of the thermal concentration fluctuations, which is governed by the collective diffusion coefficient D,⁴⁷

$$D=\Gamma /q^2.$$ (4)

In Fig. 7(a) is plotted the diffusion coefficient obtained from the fast relaxation rate as a function of c for BP-PAANa solutions. Over the entire concentration range (4% m/m ≤ c ≤ 16% m/m), D obeys a power-law dependence of the form

$$D_{\mathrm{f}\mathrm{a}\mathrm{s}\mathrm{t}}=D_{\mathrm{0}}{c}^m,$$ (5)

with m = 0.54. The experimental exponent is lower than that predicted by de Gennes for uncharged polymers of high molecular mass under good solvent conditions (m = 3/4).⁴⁷ Similar low values for the exponent m have been observed in other polyelectrolyte solutions.^74,75 Figure 7(a) also shows that D_slow decreases with increasing polymer concentration as D_slow ∼ c^−0.64.

FIG. 7.

Dependence of D_fast and D_slow measured by DLS on the polymer concentration in BP-PAANa solutions (a). The slopes of the power-law fits are 0.54 (D_fast) and −0.64 (D_slow). The inset in (a) represents the variation of D_fast and D_slow as a function of CaCl₂ concentration for 4% m/m BP-PAANa solutions. (b) Variation of D with the aggrecan concentration. D as a function of CaCl₂ concentration in 0.25% m/m aggrecan solution. The data uncertainties are estimated by one standard deviation of the linear regression fit parameters.

The effect of Ca²⁺ ions on the dynamics of 4% m/m BP-PAANa solution is illustrated in the inset of Fig. 7(a). D_fast decreases and D_slow increases with increasing CaCl₂ concentration. The continuous lines show the fits to the following equation:

$$D\propto {(c_{\mathrm{c}\mathrm{a}\mathrm{c}{\mathrm{l}}_{\mathrm{2}}})}^m,$$ (6)

with m = −0.57 (D_fast) and m = 0.52 (D_slow).

Figure 7(b) shows the corresponding results for aggrecan solutions. In this system, D decreases with increasing aggrecan concentration and increases with increasing CaCl₂ content. The observed trends are similar to the behavior of the slow component in BP-PAANa solutions.

CONCLUSIONS

Osmotic pressure measurements, SANS, DLS, and molecular dynamics simulations provide complementary information on the effect of ions on the supramolecular structure and dynamics of BP-PAANa and aggrecan solutions.

SANS and DLS reveal striking differences between the organization of bottlebrush polyelectrolytes in the two systems. In salt-free BP-PAANa solutions, a distinct scattering peak (polyelectrolyte peak) is observed, which disappears at c_CaCl2 > 50 mM. This peak indicates cluster formation due to electrostatic repulsion of the charged groups on the polymer chains. The peak position varies with the BP-PAANa polymer concentration as q* ∼ c^0.43. By contrast, no polyelectrolyte peak is present in the aggrecan solutions and the SANS response is practically invariant to the presence of CaCl₂ over the entire q range explored in the present study. SANS indicates the presence of percolating branched polymer clusters as expected from scattering of weakly associated bottlebrush polyelectrolytes. At high q, the SANS intensity decreases as 1/q in both systems due to scattering from the extended side chains.

The osmotic properties of the two systems show certain similarities. In BP-PAANa solutions, the osmotic pressure exhibits power-law scaling behavior with increasing polymer concentration. At low concentration (dilute regime), the power-law exponent is n ≈ 1, while at higher polymer concentration, n > 2 and the addition of CaCl₂ reduces the value of n. Osmotic pressure observations made on aggrecan solutions reveal similar concentration dependencies at both low and high aggrecan concentrations. However, in the intermediate concentration range, n < 1 and the osmotic pressure measurements indicate association of the aggrecan molecules into hierarchically structured supramolecular assemblies. Addition of CaCl₂ reduces the osmotic pressure of the aggrecan solution, but the shape of the curves remains similar.

The autocorrelation functions of BP-PAANa solutions exhibit a fast and a slow relaxation mode. The relaxation rate of the fast mode varies as q², indicating that this mode is a diffusive process. The fast mode, associated with the collective diffusion coefficient, varies as D_fast ∼ c^0.54. Introducing CaCl₂ increases D_fast and decreases D_slow. In the aggrecan system, however, only one relaxation mode is observed. The q dependence of the relaxation rate Γ indicates that aggrecan assemblies exhibit microgel-like behavior. D decreases with increasing aggrecan concentration and increasing with increasing CaCl₂ concentration. This behavior resembles the slow mode of the BP-PAANa solution. Apparently, the increased chain backbone solvation changes the nature of the inter-polyelectrolyte interaction from the long-ranged to short-ranged associative interaction. Molecular dynamics simulations indicate that this change affects the organization of the polymer chains in solution. For polyelectrolytes having a strong degree of chain solvation, such as aggrecan, this relatively short-range and multi-functional “sticky” interaction gives rise to the formation of fractal-like open gel particles.

Aggrecan and other bottlebrush polyelectrolytes⁵³ such as mucins exhibit a significantly different solution organization from common linear polyelectrolytes. This behavior can be expected to have large ramifications for biological function and disease as well as for the design of synthetic bottlebrush polymers intended to mimic natural bottlebrush polymers. The insensitivity of the structure of aggrecan solutions to changes in salt concentrations is another striking difference of aggrecan from most synthetic polyelectrolytes that directly impacts the stability of these materials under a wide range of solutions encountered in living systems. Further computational and experimental studies are needed to address the role of competitive polymer and counterion hydration/solvation on the solution organization of polyelectrolytes having different topological structures and degrees of hydration in order to fully understand how synthetic bottlebrush polyelectrolytes differ from their natural counterparts.

APPENDIX: MODEL OF OPEN AND CLOSED GEL PARTICLES

To model the geometry of these limiting forms of “gel particles” and the corresponding scattering properties of these structures in solution, we utilize a bead-spring model of segments bound by stiff harmonic bonds suspended in an implicit athermal solvent, i.e., the gel particles are swollen in a “good” solvent. All particles are assigned the same mass m, size σ, and strength of interaction ε. The segmental interactions are described by the cut-and-shifted Lennard-Jones (LJ) potential with a cutoff distance of r_c = 2^1/6 σ, corresponding to an athermal solvent. The segments along a chain are connected with their neighbors via a stiff harmonic spring, V_H (r) = k (r − l₀)², where l₀ = 0.99 σ is the equilibrium length of the spring and k = 2500 ε/σ ² is the spring constant. It is emphasized that this model does not include the presence of hydration interactions of the polymer segments. We first describe our model and simulation conditions for “compact” gel particles having an ideal network internal structure, as often imaged for macroscopic polymer gels. This is followed by a description of randomly branched polymer structures that often formed in solution when molecules associate by random multi-functional associations. This latter type of “open gel” particle has a greater resemblance to percolation clusters than compact gel particles, and the difference between these closed and open gel particle types is clearly apparent in their scattering signatures. The elucidation of these differences is the main point of these simulations.

The elementary structural unit of a “perfect” polymer network is a regular star polymer in which f chains are attached to a core particle “node” or core particle, where each of the f chains is composed of M segments.^58–60 The total number of interaction centers per star polymer is then M_w,star = f M + 1. In particular, a “perfect” gel particle is composed of star polymers topologically connected in a cubic lattice of finite extent in which two or more of the free ends of the stars are connected with those of neighboring stars. The number of branched points in each direction (x, y, and z) defining the topology of the network is labeled N_x, N_y, and N_z. The total molecular mass of a nanogel particle is then M_w = (N_x N_y N_z) x M_w,star. The type of gel particle takes the form resembling a “sponge-like” particle, and we refer to these “particles” as being “compact gel” particles.

Of course, the networks formed by branched polymers are not always organized into perfect networks, and it is natural to define “open gel” particles arising from the random association of physical bonds along the polymer chain, such as those encountered in polymer solutions forming thermally reversible gels. We model this type of randomly branched polymer or “open gel” particle starting from the same “compact gel” particle model discussed above, but we randomly cut bonds in the network with the restriction that the gel particle retains its overall connectivity and mass. This process is continued until there remain no bonds left to cut that preserve the polymer networks integrity, i.e. For the purposes of our study, the open gel structures contain a bending potential between the segments, V_b (r) = k_b (θ − θ₀)², where θ₀ = π is the equilibrium angle and k_b = 2 ε/rad² is the spring constant. The systems were equilibrated at constant temperature k_B T/ε = 1.0, maintained by using a Nose–Hoover thermostat. Typical simulations equilibrate for 20 000 τ, and data are accumulated over a 50 000 τ interval, and the time step used was ∆t/τ = 0.005.

We find that the open gel clusters formed using this decimation method exhibit a geometry that is the characteristic of randomly associated molecules in solution.^54,55 This construction has the advantage that the mass of the open and closed type gel polymer clusters can be generated much more efficiently and allows us to compare open and closed gel particles having the same mass to assess the impact of topological disorder on the structure and scattering properties of these distinct classes of gel particles.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.