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Mathematical Simulation of Muscle Cross-Bridge Cycle and Force-Velocity Relationship

Leslie Chin ^* , Pengtao Yue  , James J. Feng   and Chun Y. Seow ^* 

^* Department of Pathology/Laboratory Medicine, Department of Mathematics, Department of Chemical and Biological Engineering, The James Hogg iCAPTURE Centre, University of British Columbia, Vancouver, British Columbia, Canada

Correspondence: Address reprint requests to Chun Y. Seow, Tel.: 604-806-9268; E-mail: cseow{at}mrl.ubc.ca.

Muscle contraction underlies many essential functions such as breathing, heart beating, locomotion, regulation of blood pressure, and airway resistance. Active shortening of muscle is the result of cycling of myosin cross-bridges that leads to sliding of myosin filaments relative to actin filaments. In this study, we have developed a computer program that allows us to alter the rates of transitions between any cross-bridge-states in a stochastic cycle. The cross-bridge states within the cycle are divided into six attached (between myosin cross-bridges and actin filaments) states and one detached state. The population of cross-bridges in each of the states is determined by the transition rates throughout the cycle; differential equations describing the transitions are set up as a cyclic matrix. A method for rapidly obtaining steady-state exact solutions for the cyclic matrix has been developed to reduce computation time and avoid the divergence problem associated with numerical solutions. In the seven-state model, two power strokes are assumed for each cross-bridge cycle, one before the release of inorganic phosphate, and one after. The characteristic hyperbolic force-velocity relationship observed in muscle contraction can be reproduced by the model. Deviation from the single hyperbolic behavior at low velocities can be mimicked by allowing the rate of cross-bridge-attachment to vary with velocity. The effects of [ATP], [ADP], and [P_i] are simulated by changing transition rates between specific states. The model has revealed new insights on how the force-velocity characteristics are related to the state transitions in the cross-bridge cycle.