A distribution-moment model of energetics in skeletal muscle

Abstract

In this paper we develop a theory for calculating the chemical energy liberation and heat production of a skeletal muscle subjected to an arbitrary history of stimulation, loading, and length variation. This theory is based on and complements the distribution-moment (DM) model of muscle [Zahalak and Ma, J. biomech. Engng 112, 52–62 (1990)]. The DM model is a mathematical approximation of the A. F. Huxley cross-bridge theory and represents a muscle in terms of five (normalized) state variables: A, the muscle length, c, the sarcoplasmic free calcium concentration, and Q 0, Q 1, Q 2, the first three moments of the actin-myosin bond-distribution function (which, respectively, have macroscopic interpretations as the muscle stiffness, force, and elastic energy stored in the contractile tissue). From this model are derived two equations which predict the chemical energy liberation and heat production rates in terms of the five DM state variables, and which take account of the following factors: (1) phosphocreatine hydrolysis associated with cross-bridge cycling; (2) phosphocreatine hydrolysis associated with sarcoplasmic-reticulum pumping of calcium; (3) passive calcium flux across the sarcoplasmic-reticulum membrane; (4) calcium-troponin bonding; (5) cross-bridge bonding at zero strain; (6) cross-bridge strain energy; (7) tendon strain energy; and (8) external work. Using estimated parameters appropriate for a frog sartorius at 0°C, the energy rates are calculated for several experiments reported in the literature, and reasonable agreement is found between our model and the measurements. (The selected experiments are confined to the plateau of the isometric length-tension curve, although our theory admits arbitrary length variations.) The two most important contributions to the energy rates are phosphocreatine hydrolysis associated with cross-bridge cycling and with sarcoplasmic-reticulum calcium pumping, and these two contributions are approximately equal under tetanic, isometric, steady-state conditions. The contribution of the calcium flux across the electrochemical potential gradient at the sarcoplasmic-reticulum membrane was found to be small under all conditions examined, and can be neglected. Long-term fatigue and oxidative recovery effects are not included in this theory. Also not included is the so-called “unexplained energy” presumably associated with reactions which have not yet been identified. Within these limitations our model defines clear quantitative interrelations between the activation, mechanics, and energetics in muscle, and permits rational estimates of the energy production to be calculated for arbitrary programs of muscular work.