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 interacti on between glycosaminoglycan (GAG) molecules in cartilage to the shear prop erties of the tissue. We first measured the equilibrium and dynamic shear m oduli of cylindrical disks of cartilage at varying ionic concentrations (0. 01-1.0 M NaCl). We then used a molecular model of GAG electrostatic interac tions that would occur during macroscopic shear deformation to predict the dependence of the shear properties of cartilage on the ionic strength of th e bath. The molecular model of GAG interaction was based on approximating G AG segments as charged rods, i.e., the unit cell model and then changing th e 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 s hear deformation and changes in bath ionic concentration. The nonlinear PB equation was solved numerically using finite element methods (FEM) within t he unit cell, The electrostatic free energy was calculated from the numeric ally obtained electrical potential and the associated mobile ion distributi on, and the electrical contribution to the equilibrium shear modulus was ob tained using an energy method. Using physiologically relevant values for th e GAG concentration and the nonelectrical contribution to the shear modulus (the two adjustable parameters of the model), the theory predicted the obs erved dependency of the equilibrium shear modulus on ionic concentration ra ther well. These results confirmed the validity of the proposed model of th e GAG interactions under pure shear deformation and, accordingly, the impor tant role of electrostatic interactions to the shear stiffness of cartilage extracellular matrix.