Nonaffine response of skeletal muscles on the ‘descending limb’

Abstract

Tetanized muscle myofibrils are often modeled as one-dimensional chains where springs represent half-sarcomeres (HS). The force–length relation for individual HSs (isometric tetanus) is known to have a ‘descending limb’, a segment with an apparently negative stiffness. Despite the potential mechanical instability on the descending limb, the isometric tetanus is usually interpreted as describing an affine deformation. At the same time, active stretching during tetanus around the descending limb is known to produce non-affine sarcomere patterns. In view of this paradox, the question whether the mechanical behavior of a myofibril can be interpreted as a response of a single contractile unit has been a subject of considerable controversy over the last 50 years. In this paper we question the claim that the isometric tetanus describes homogeneous configurations of the HS chain. To distinguish between the multitudes of non-affine equilibrium states available to this mechanical system, we propose to use the concept of a stored mechanical energy. While the notion of energy is natural from a mechanical point of view, physiologists have resisted it so far on the grounds that the contractile elements are active. We discuss how this objection can be overcome and show that the appropriately defined stored energy of a tetanized myofibril with N contractile units has exponentially many local minima. We then argue that the ruggedness of the ensuing energy landscape is responsible for the experimentally observed history dependence and hysteresis in the mechanical response of a tetanized muscle near the descending limb. A nonlocal extension of the chain model, accounting for surrounding tissues, shows that both the ground states and the marginally stable states are fine mixtures of short and long HSs. These mixtures are homogeneous at the macro-scale and inhomogeneous at the micro-scale and we show that the negative overall slope of the step-wise tetanus can coexist with a positive instantaneous stiffness. A salient feature of the nonlocal model is that the variation of the degree of non-uniformity with elongation follows a complete devil’s staircase.