Effect of electrostatic interactions between glycosaminoglycans on the shear stiffness of cartilage: A molecular model and experiments

This study focuses on the contribution of repulsive electrostatic interaction between glycosaminoglycan (GAG) molecules in cartilage to the shear properties of the tissue. We first measured the equilibrium and dynamic shear moduli of cylindrical disks of cartilage at varying ionic concentrations (0.01-1.0 M NaCl). We then used a molecular model of GAG electrostatic interactions that would occur during macroscopic shear deformation to predict the dependence of the shear properties of cartilage on the ionic strength of the bath. The molecular model of GAG interaction was based on approximating GAG segments as charged rods, i.e., the unit cell model and then changing the shape of the unit cell in a manner consistent with the macroscopic shear deformation. The Poisson-Boltzmann equation was incorporated into the unit cell model (PB unit cell) to predict physical phenomena such as the change in electrical potential and mobile ion distribution caused by macroscopic shear deformation and changes in bath ionic concentration. The nonlinear PB equation was solved numerically using finite element methods (FEM) within the unit cell. The electrostatic free energy was calculated from the numerically obtained electrical potential and the associated mobile ion distribution, and the electrical contribution to the equilibrium shear modulus was obtained using an energy method. Using physiologically relevant values for the GAG concentration and the nonelectrical contribution to the shear modulus (the two adjustable parameters of the model), the theory predicted the observed dependency of the equilibrium shear modulus on ionic concentration rather well. These results confirmed the validity of the proposed model of the GAG interactions under pure shear deformation and, accordingly, the important role of electrostatic interactions to the shear stiffness of cartilage extracellar matrix.