Structure and interactions of aggrecans: statistical thermodynamic approach

Abstract

Weak polyelectrolytes tethered to cylindrical surfaces are investigated using a molecular theory. These polymers form a model system to describe the properties of aggrecan molecules, which is one of the main components of cartilage. We have studied the structural and thermodynamical properties of two interacting aggrecans with a molecular density functional theory that incorporates the acid-base equilibrium as well as the molecular properties: including conformations, size, shape, and charge distribution of all molecular species. The effect of acidity and salt concentration on the behavior is explored in detail. The repulsive interactions between two cylindrical-shaped aggrecans are strongly influenced by both the salt concentration and the pH. With increasing acidity, the polyelectrolytes of the aggrecan acquire charge and with decreasing salt concentration those charges become less screened. Consequently the interactions increase in size and range with increasing acidity and decreasing salt concentration. The size and range of the forces offers a possible explanation to the aggregation behavior of aggrecans and for their ability to resist compressive forces in cartilage. Likewise, the interdigitation of two aggrecan molecules is strongly affected by the salt concentration as well as the pH. With increasing pH, the number of charges increases, causing the repulsions between the polymers to increase, leading to a lower interdigitation of the two cylindrical polymer layers of the aggrecan molecules. The low interdigitation in charged polyelectrolytes layers provides an explanation for the good lubrication properties of polyelectrolyte layers in general and cartilage in particular.

FIGURE 1

Schematic drawing representing the aggrecan molecule and the proteoglycan aggregate of aggrecans and hyaluronan. The labels denote the GAG side chains and core protein (CP: core protein, CS: chondroitin sulfate, and KS: keratan sulfate). The G1, G2, and G3 are globular domains. The G1 is involved in the binding of aggrecan to hyaluronan. Drawing adapted from Ng et al. (9).

FIGURE 2

Cartoon illustrating the theoretical model employed. D is the distance from the centers of the cylinders. The dots on the polymer chains represent dissociated groups. The radius is not to scale.

FIGURE 3

Height of polysaccharides grafted to a cylindrical surface as a function of bulk pH. (A) A bulk equilibrium constant pK_a = 3.5. (B) The polymer segments have two alternating equilibrium constants of pK_a = 2 and pK_a = 3.5. In both cases, the polymer length is n = 50 and the line density σ_l = 0.25 nm⁻¹. The radius of the cylinder is R = 0.5 nm.

FIGURE 4

Polymer density as a function of the distance from a cylindrical or a planar surface. Polymer chain length is n = 50, pH = pK_a = 3.5, and a bulk salt concentration c = 0.1 M. The radius of the cylindrical surface is R = 0.5 nm. The line density for the cylinder is σ_l = 0.25 nm⁻¹, which is equivalent to the surface density for the planar surface of σ_a = 0.08 nm⁻². The inset shows the local degree of dissociation. r corresponds to radial distance for the cylindrical surface and to the perpendicular distance for the planar surface.

FIGURE 5

Polymer height as a function of the pH for a cylindrical polyelectrolyte layer. The polymer segments have two alternating acid groups. Each acid has an equilibrium constant of pK_a = 2.5 and pK_a = 7, respectively. The polymer length is n = 50, the line density σ_l = 0.25 nm⁻¹, and the radius R = 0.5 nm.

FIGURE 6

The total and individual contributions to the free energy per unit length as a function of the distance between the aggrecans (see Eq. 2) for n = 25, σ_l = 0.1 nm⁻¹, pK_a = 3.5, c = 0.1 M, and R = 0.5 nm. (A) pH = 2.5. (B) pH = 5.6. The free energy contributions of the protons, hydroxyl ions, and the chemical equilibrium reaction have been omitted. On the size of the graph, their values are too small to be visible. The inset of the right-hand panel shows the free energy contribution arising from the polymer conformational degree of freedom.

FIGURE 7

Free energy per unit length as a function of distance between model aggrecans. (A) pH = 2.5. (B) pH = 3.5. (C) pH = 5.6. In all cases n = 25, σ_l = 0.1 nm⁻¹, R = 0.5 nm, and pK_a = 3.5.

FIGURE 8

Free energy per unit length as a function of distance between model aggrecans. (A) n = 25 and σ = 0.1 nm⁻¹. (B) n = 25 and σ_l = 0.25 nm⁻¹. (C) n = 50 and σ_l = 0.25 nm⁻¹. In all cases pK_a = 3.5, pH = 3.5, and R = 0.5 nm.

FIGURE 9

Free energy per unit length as a function of distance for (A) cylindrical and (B) planar surfaces with n = 50, pK_a = 3.5, c = 0.1 M, and a grafting density of σ_a = 0.032 nm⁻². The radius of the cylinder is R = 0.5 nm; hence the number of polymers per unit length or line density is σ_l = 0.1 nm⁻¹. For the cylinder, the left-hand scale is the free energy per unit length and the right-hand axis gives the free energy per unit area which is needed for the comparison with that of the planar surface.

FIGURE 10

Polymer volume fraction (A) and polymer fraction belonging to one cylinder (B). The distance between the cylinders D = 4 nm, n = 50, σ_l = 0.25 nm⁻², pH = 3.5, pK_a = 3.5, c = 0.1 M, and R = 0.5 nm.

FIGURE 11

Polysaccharides interdigitation as a function of the distance between the two cylinders, for n = 50, σ_l = 0.25 nm⁻¹, pK_a = 3.5, and c = 0.1 M. The solid curves labeled “max” correspond to the maximum possible overlap. The inset shows the total size of the maximal overlap.

FIGURE 12

Local degree of dissociation (A) and pH (B) for a distance between the two cylinders of D = 14 nm, for n = 50, σ_l = 0.25 nm⁻¹, pH = 3.5, pK_a = 3.5, and c = 0.01 M.