Non-axial muscle stress and stiffness

A generalization is developed of the classic two-state Huxley cross-bridge model to account for non-axial active stress and stiffness. The main ingredients of the model are: (i) a relation between the general three-dimensional deformation of an element of muscle and the deformations of the cross-bridges, that assumes macroscopic deformation is transmitted to the myofibrils, (ii) radial as well as axial cross-bridge stiffness, and (iii) variations of the attachment and detachment rates with lateral filament spacing. The theory leads to a generalized Huxley rate equation on the bond-distribution function, n(zeta, theta, t), of the form [equation: see text] where the Dij are the components of the relative velocity gradient and rho and ñ are functions of the polar angle, theta, and time that describe, respectively, the deformation of the myofilament lattice and the distribution of accessible actin sites (both of these functions can be calculated from the macroscopic deformation). Explicit expressions, in terms of n, are derived for the nine components of the active stress tensor, and the 21 non-vanishing components of the active stiffness tensor; the active stress tensor is found to be unsymmetric. The theory predicts that in general non-axial deformations will modify active axial stress and stiffness, and also give rise to non-axial (e.g., shearing) components. Under most circumstances the magnitudes of the non-axial stress and stiffness components will be small compared with the axial and, further, the effects of non-axial deformation rates will be small compared with those of the axial rate. Large transverse deformations may, however, greatly reduce the axial force and stiffness. The theory suggests a significant mechanical role for the non-contractile proteins in muscle, namely that of equilibrating the unsymmetric active stresses. Some simple applications of the theory are provided to illustrate its physical content.