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 th rough a specifiable transition in response to length changes imposed o n a crossbridge model of muscle. The crossbridge cycle is divided into multiple attached and detached states. The rates of transition from o ne state to another are defined by rate coefficients that can either b e constant or vary with the position of the crossbridge relative to th e thin-filament attachment site. This scheme leads to a system of diff erential equations defining the rates of change for the fractions of b ridges in each state. Solutions for this system of equations are obtai ned at specified times du ring and after a length change using a metho d for systems with widely varying time constants (C. W. Gear, 1971, Nu merical Initial Value Problems in Ordinary Differential Equations, Pre ntice-Hall, Englewood Cliffs, NJ). Crossbridges are divided into discr ete populations that differ both in their axial displacement with resp ect to thin filament attachment sites and with respect to the twist of the actin helix. Separate solutions are made for the individual popul ations and are then averaged to obtain the ensemble response. Force is determined as the sum of the product of the force associated with eac h 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 cr ossbridge 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 so me results obtained with a simplified scheme.