A Distribution-Moment Approximation for Kinetic Theories of Muscular Contraction

A rational mathematical procedure is proposed for approximating solutions to the partial differential equations of cross-bridge kinetics in theories of muscular contraction of the type first proposed by A. F. Huxley. The essence of the procedure is to approximate the exact bond-distribution functions by distributions of prescribed form, and this leads to a set of first-order ordinary differential equations on the low-order moments of the approximate distributions. Thus the procedure effectively results in a lumped-parameter model of muscle approaching the structural simplicity of the classic two-element model, but one which exhibits more realistic behavior. The approximation is worked out in detail for Huxley’s original (1957) two-state model (modified slightly to produce a more realistic stretch response). compared with exact solutions of the model, and used to predict muscle behavior under various conditions. It is anticipated that this approximation, with its attendant conceptual and computational simplifications, will make recent theoretical advances in molecular contraction mechanics more accessible for applications in macroscopic muscle dynamics, Generalizations of the procedure to the case of length-dependent behavior are discussed.