A program for developing a comprehensive mathematical description of the crossbridge cycle of muscle

We describe a computer modeling system for determining the changes of force, fraction of attached crossbridges, and crossbridge flux rate through a specifiable transition in response to length changes imposed on a crossbridge model of muscle. The crossbridge cycle is divided into multiple attached and detached states. The rates of transition from one state to another are defined by rate coefficients that can either be constant or vary with the position of the crossbridge relative to the thin-filament attachment site. This scheme leads to a system of differential equations defining the rates of change for the fractions of bridges in each state. Solutions for this system of equations are obtained at specified times during and after a length change using a method for systems with widely varying time constants (C. W. Gear, 1971, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ). Crossbridges are divided into discrete populations that differ both in their axial displacement with respect to thin filament attachment sites and with respect to the twist of the actin helix. Separate solutions are made for the individual populations and are then averaged to obtain the ensemble response. Force is determined as the sum of the product of the force associated with each state multiplied by the fraction of bridges in that state. A measure of metabolic rate is determined as the net flux through one of the crossbridge transitions. When the force-extension characteristics of the individual crossbridges are linear and the filaments are noncompliant the fraction of attached bridges is equivalent to sarcomere stiffness. To illustrate the operation of the program, we also describe here some results obtained with a simplified scheme.