Hyperbolic Force-velocity Relation Research Articles

Can Quick Release Experiments Reveal the Muscle Structure? A Bionic Approach

The goal of this study was to understand the macroscopic mechanical structure and function of biological muscle with respect to its dynamic role in the contraction. A recently published muscle model, deriving the hyperbolic force-velocity relation from first-order mechanical principles, predicts different force-velocity operating points for different load situations. With a new approach, this model could be simplified and thus, transferred into a numerical simulation and a hardware experiment. Two types of quick release experiments were performed in simulation and with the hardware setup, which represent two extreme cases of the contraction dynamics: against a constant force (isotonic) and against an inertial mass. Both experiments revealed hyperbolic or hyperbolic-like force-velocity relations. Interestingly, the analytical model not only predicts these extreme cases, but also additionally all contraction states in between. It was possible to validate these predictions with the numerical model and the hardware experiment. These results prove that the origin of the hyperbolic force-velocity relation can be mechanically explained on a macroscopic level by the dynamical interaction of three mechanical elements. The implications for the interpretation of biological muscle experiments and the realization of muscle-like bionic actuators are discussed.

Proof of Concept: Model Based Bionic Muscle with Hyperbolic Force-Velocity Relation

Recently, the hyperbolic Hill-type force-velocity relation was derived from basic physical components. It was shown that a contractile element CE consisting of a mechanical energy source active element AE, a parallel damper element PDE, and a serial element SE exhibits operating points with hyperbolic force-velocity dependency. In this paper, a technical proof of this concept was presented. AE and PDE were implemented as electric motors, SE as a mechanical spring. The force-velocity relation of this artificial CE was determined in quick release experiments. The CE exhibited hyperbolic force-velocity dependency. This proof of concept can be seen as a well-founded starting point for the development of Hill-type artificial muscles.

Proof of Concept: Model Based Bionic Muscle with Hyperbolic Force-Velocity Relation

Recently, the hyperbolic Hill-type force-velocity relation was derived from basic physical components. It was shown that a contractile element CE consisting of a mechanical energy source (active element AE), a parallel damper element (PDE), and a serial element (SE) exhibits operating points with hyperbolic force-velocity dependency. In this paper, a technical proof of this concept was presented. AE and PDE were implemented as electric motors, SE as a mechanical spring. The force-velocity relation of this artificial CE was determined in quick release experiments. The CE exhibited hyperbolic force-velocity dependency. This proof of concept can be seen as a well-founded starting point for the development of Hill-type artificial muscles.

An in vitro motility assay system retaining the steady-state force-velocity characteristics of muscle fibers under positive and negative loads

To investigate in vitro myofilament sliding in conditions close to those in muscle fibers, we developed a new assay system in which a polystyrene bead (diameter, 4 μm; specific gravity, 1.3), with a single to a few thick filaments of a molluscan smooth muscle attached, was made to slide along actin filament arrays (actin cables) in the internodal cell of an alga, mounted on the rotor of a centrifuge microscope. The bead moving along actin cables in the presence of ATP was subjected to centrifugal forces either opposite to the bead movement (positive loads) or in the same direction as the bead movement (negative loads). With positive loads increasing from zero to the maximum isometric force P 0, the velocity of bead movement decreased gradually to zero, exhibiting the hyperbolic force-velocity relation except for loads above 0.8 P 0. With further increase of positive loads above P 0, the bead was forced to move in the direction of centrifugal force, being eventually detached from actin cables at a load of ∼1.4 P 0. These features are very similar to the force-velocity relation in intact single muscle fibers. With negative loads increasing from zero to a certain value, on the other hand, the bead moved faster than the maximum unloaded velocity (1.5–3.5 μm/s, 20–25°C), until it was eventually detached from actin cables. These results indicate that our assay system is extremely promising for future combined biochemical and physiological studies on the mechanism of muscle contraction.

Dependence of the work done by ATP-induced actin-myosin sliding on the initial baseline force: its implications for kinetic properties of myosin heads in muscle contraction.

The properties of the ATP-dependent actin-myosin sliding responsible for muscle contraction was studied using an in vitro force-movement assay system, in which a myosin-coated glass microneedle was made to slide on actin filament arrays (actin cables) in the giant algal cell with iontophoretic application of ATP. With a constant amount of ATP application, the amount of work done by the actin-myosin sliding increased with increasing baseline force from zero to 0.4-0.6 Po, and then decreased with further increasing baseline force, thus giving a bell-shaped work versus baseline force relation. The result that the maximum actin-myosin sliding velocity did not change appreciably with increasing baseline force up to 0.4-0.6 Po implies, together with the limited number of myosin heads involved, that (1) the rate of power output of actin-myosin sliding is determined primarily by the amount of external load rather than the velocity of actin-myosin sliding, and (2) the bell shaped work versus baseline force relation (and also the hyperbolic force-velocity relation) results from the kinetic properties of individual myosin head rather than the change in the number of myosin heads involved.

The instantaneous torque-angular velocity relation in plantar flexion during jumping.

Torques, angular velocities, and power of the ankle joint during plantar flexion were measured in jumping experiments in order to achieve insight into shape and magnitude of the instantaneous torque-angular velocity relation in a complex movement. Twelve trained subjects performed maximal vertical jumps from a semi-squatting position with 100 degrees of flexion in the knee joint. Ground reaction force measurements and film analyses were used to calculate instantaneous torques, angular velocities, and power outputs during plantar flexion. The shape of the instantaneous torque-angular velocity was different from the well-known hyperbolic force-velocity relation for isolated muscles. Maximal power output (2499 +/- 751 [SD] W) occurred at 60% of the mean maximal torque (301 +/- 62 N X m) and 80% of the mean maximal angular velocity (970 degrees/s). The maximal power output was six times larger than the power output reported in the literature for maximal isokinetic (monoarticular) plantar flexions. Influences like storage of energy in the series elastic component of Hill's muscle model and the role of polyarticular muscles in transporting energy from knee to ankle are discussed. It is concluded that many more selective studies will be necessary before it is possible to relate intrinsic muscle properties to the performance of muscles in poly-articular complex movements.

A model for the relation between respiratory neural and mechanical outputs. I. Theory.

A model for the conversion of respiratory neural output to mechanical output, and vice versa, is described. Neural output was expressed in terms of isometric pressure generated at passive functional residual capacity. The mechanical response time constant of respiratory muscle was assumed to be 0.06 s. The effect of volume and configuration on pressure output was modeled after the data of Grassino et al. (J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 44: 829-839, 1978). Equations were developed to examine the effect of different configurational pathways during inspiration. We utilized Hill's hyperbolic force-velocity relation to model the effect of flow on pressure output. The pressure asymptote of the hyperbola was considered to be similar to that in other skeletal muscles (0.25 isometric pressure). The flow asymptote was derived from data obtained during maximal voluntary inspiration. A major feature of the model is the dependence of volume, configuration, and flow-related pressure losses on level of inspiratory activity. The practical effect of potential errors and the overall accuracy of the model are presented in the two succeeding communications.

Analytical solution to isometric mechanogram of Hill's model of striated muscle.

The two-element muscle model considered consists of a contractile element defined by a hyperbolic force-velocity relation connected in series with an “exponential spring”. Differential equations for the isometrically developed force during a tetanic contraction and the corresponding contractile element shortening velocity are derived and their stability is investigated. Analytical solutions to both equations are obtained. Two numerical examples are given, the second chosen to illustrate pressure-induced hypertrophy of a cardiac muscle.

Analytical solution to isometric mechanogram of Hill's model of striated muscle

The two-element muscle model considered consists of a contractile element defined by a hyperbolic force-velocity relation connected in series with an “exponential spring”. Differential equations for the isometrically developed force during a tetanic contraction and the corresponding contractile element shortening velocity are derived and their stability is investigated. Analytical solutions to both equations are obtained. Two numerical examples are given, the second chosen to illustrate pressure-induced hypertrophy of a cardiac muscle.