A Molecular Model of Proteoglycan-Associated Electrostatic Forces in Cartilage Mechanics

Abstract

Measured values of the swelling pressure of charged proteoglycans (PG) in solution (Williams RPW, and Comper WD; Biophysical Chemistry 36:223, 1990) and the ionic strength dependence of the equilibrium modulus of PG-rich articular cartilage (Eisenberg SR, and Grodzinsky AJ; J Orthop Res 3: 148, 1985) are compared to the predictions of two models. Each model is a representation of electrostatic forces arising from charge present on spatially fixed macromolecules and spatially mobile micro-ions. The first is a macroscopic continuum model based on Donnan equilibrium that includes no molecular-level structure and assumes that the electrical potential is spatially invariant within the polyelectrolyte medium (i.e. zero electric field). The second model is based on a microstructural, molecular-level solution of the Poisson-Boltzmann (PB) equation within a unit cell containing a charged glycosaminoglycan (GAG) molecule and its surrounding atmosphere of mobile ions. This latter approach accounts for the space-varying electrical potential and electrical field between the GAG constituents of the PG. In computations involving no adjustable parameters, the PB-cell model agrees with the measured pressure of PG solutions to within experimental error (10%), whereas the ideal Donnan model overestimates the pressure by up to 3-fold. In computations involving one adjustable parameter for each model, the PB-cell model predicts the ionic strength dependence of the equilibrium modulus of articular cartilage. Near physiological ionic strength, the Donnan model overpredicts the modulus data by 2-fold, but the two models coincide for low ionic strengths (C0 < 0.025M) where the spatially invariant Donnan potential is a closer approximation to the PB potential distribution. The PB-cell model result indicates that electrostatic forces between adjacent GAGs predominate in determining the swelling pressure of PG in the concentration range found in articular cartilage (20–80 mg/ml). The PB-cell model is also consistent with data (Eisenberg and Grodzinsky, 1985, Lai WM, Hou JS, and Mow VC; J Biomech Eng 113: 245, 1991) showing that these electrostatic forces account for ˜ 1/2 (290kPa) the equilibrium modulus of cartilage at physiological ionic strength while absolute swelling pressures may be as low as ˜ 25 – 100kPa. This important property of electrostatic repulsion between GAGs that are highly charged but spaced a few Debye lengths apart allows cartilage to resist compression (high modulus) without generating excessive intratissue swelling pressures.