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

Abstract

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.

Multiscale Modeling of Skeletal Muscle Active Contraction in Relation to Mechanochemical Coupling of Molecular Motors

“…Our results further quantified that the contraction velocity decreased with a larger force and slowed to zero, as the force is too large for the myosin to drive. Our observations are consistent with the V-F curve fitted from experiments, referred to as V-F of Hill in Figure 5 [2,12]. …”

“…The potentials determine the values of jump rates Fi n and Bi n in Equation (12). The jump rates can be calculated with Equations (3) and (4).…”

“…This model has been extended to different families of molecular motors, such as kinesin, dynein, and myosin. Chin et al [12] divided the biochemical cycle of myosin motor into seven states, and proposed a simulation model that allows altering the rates of transitions, mimicking the role of chemicals, including ATP, ADP, etc., between any of the states in a stochastic cycle. Wang [13] developed a mathematical model for investigating the mechanochemical coupling mechanism of molecular motors.…”

Simulation system of spinal cord motor nuclei and associated nerves and muscles, in a Web-based architecture

“…"Sag" property in fast-type motor units (Burke et al 1973) 4. Non-isometric force generation for simulations involving body segment movement (Chin et al 2006;Siebert et al 2008) 5. Initiation and extinction of single fiber action potential (at the neuromuscular junction and at tendons, respectively) and its volume conduction to the surface electrodes (Stegeman et al 2004) As most of the improvements listed above would require computer-intensive biophysically-based models, their inclusion may be feasible if one simulates a network with a relatively small number of elements or if a more powerful computer is available.…”

Studies on biomechanics of skeletal muscle based on the working mechanism of myosin motors: An overview

“…From the perspective of chemical kinetics, Lan et al [58] presented a mechanical model for steady muscle contraction based on the conformational changes of a single motor, and studied the force-velocity relation of contraction, taking the actin filament as the object. Chin et al [59] described the cycling process of molecular motors with a 7-state model and further modified the force-velocity relationship. Guo et al [60] provided a mechanochemical coupling model for the working cycle of myosin, and investigated the spontaneous vibration of muscle by describing the collective behavior of myosin motors with a set of chemical dynamical equations.…”