Force-velocity Curve Research Articles

Measurement of nonlinear normal modes using multi-harmonic stepped force appropriation and free decay

Nonlinear Normal Modes (NNMs) offer tremendous insight into the dynamic behavior of a nonlinear system, extending many concepts that are familiar in linear modal analysis. Hence there is interest in developing methods to experimentally and numerically determine a system׳s NNMs for model updating or simply to characterize its dynamic response. Previous experimental work has shown that a mono-harmonic excitation can be used to isolate a system׳s dynamic response in the neighborhood of a NNM along the main backbones of a system. This work shows that a multi-harmonic excitation is needed to isolate a NNM when well separated linear modes of a structure couple to produce an internal resonance. It is shown that one can tune the multiple harmonics of the input excitation using a plot of the input force versus the response velocity until the area enclosed by the force–velocity curve is minimized. Once an appropriated NNM is measured, one can increase the force level and retune the frequency to obtain a NNM at a higher amplitude or remove the excitation and measure the structure׳s decay down a NNM backbone. This work explores both methods using simulations and measurements of a nominally-flat clamped–clamped beam excited at a single point with a magnetic force. Numerical simulations are used to validate the method in a well defined environment and to provide comparison with the experimentally measured NNMs. The experimental results seem to produce a good estimate of two NNMs along their backbone and part of an internal resonance branch. Full-field measurements are then used to further explore the couplings between the underlying linear modes along the identified NNMs.

Solute drag on perfect and extended dislocations

The drag force exerted on a moving dislocation by a field of mobile solutes is studied in the steady state. The drag force is numerically calculated as a function of the dislocation velocity for both perfect and extended dislocations. The sensitivity of the non-dimensionalized force–velocity curve to the various controlling parameters is assessed, and an approximate analytical force–velocity expression is given. A non-dimensional parameter S characterizing the strength of the solute-dislocation interaction, the background solute fraction , and the dislocation character angle , are found to have the strongest influence on the force–velocity curve. Within the model considered here, a perfect screw dislocation experiences no solute drag, but an extended screw dislocation experiences a non-zero drag force that is about 10 to 30% of the drag on an extended edge dislocation. The solutes can change the spacing between the Shockley partials in both stationary and moving extended dislocations, even when the stacking fault energy remains unaltered. Under certain conditions, the solutes destabilize an extended dislocation by either collapsing it into a perfect dislocation or causing the partials to separate unboundedly. It is proposed that the latter instability may lead to the formation of large faulted areas and deformation twins in low stacking fault energy materials containing solutes, consistent with experimental observations of copper and stainless steel containing hydrogen.

Effect of Branching on Force-Velocity Curves and Length Fluctuations of Actin Networks

Studying dynamics of cytoskeletal filaments is crucial for understanding cell motility, wound healing and cancer invasion. In this work we study the dynamics of actin filaments against a fluctuating “wall” (membrane) under an external force. The study accounts for growth of actin network through polymerisation, depolymerisation and branching, and investigate the resulting force-velocity relationship. In our study we highlight the importance of different actin network architectures, their fluctuations, and show that the stall force for a branched actin network depends upon two factors - i) the average number of the filaments working against the resistive force on the wall and, ii) the length fluctuation of the leading edge of the filament network. We also observe that the relative measurement of wall distances and branching distances along with number of filaments touching the wall dictate the change in force-velocity relationship (e.g., convex vs. concave). Further, our results suggest that, upon introducing branching, non-monotonicity in diffusion constant is observed as a function of externally applied force. Here, the diffusion constant is a measurable quantity which helps us quantify the growth in length fluctuation of the tip.

Myosin Binding to Actin in the 3D Sarcomere Lattice

The effect of tethering molecules in 3D space on bimolecular binding kinetics is rarely addressed and occasionally incorporated in models of cell motility. The simplest system to quantitatively determine this effect is in the 3D sarcomere lattice of the striated muscle where the tethered myosin can only bind to a relatively few available actin sites, positioned within a limited range of the thermal movement of myosin head. A multiscale Monte Carlo software platform MUSICO takes into account the exact number of unbound myosin molecules interacting only with unoccupied actin sites in their proximity and provides quantitative predictions of muscle response at the macroscopic scale. As such MUSICO is well suited to determine the effect of the distortion of interacting molecules on biochemical reactions, i.e. modulation of state transition rates in the actomyosin cycle by the imposed geometrical constraints of tethered myosin molecules. The simulations provide the distribution of bound sites, preservation of species, and most importantly the departure in nanoscopic behavior of tethered myosin molecules from unconstrained myosin interactions with actin. The MUSICO simulations with the simplest actomyosin cycles show large differences in predicted force velocity curves and during early force recovery phase after step in change in length comparing to all mass-action models. The origin of the differences in predicted muscle response is rooted in the differences the fluxes of myosin binding and corresponding instantaneous crossbridge distributions. Consequently a new approach showed that fitting of the experiments require significantly different binding rates than used in mass-action models and, therefore, to correctly describe the biochemical reactions of tethered molecules and the interaction energetics. The knowledge gained by this analysis can be used for development of new mass-action and other models that can implicitly incorporate this phenomenon via correction factors.

Contribution of elastic tissues to the mechanics and energetics of muscle function during movement.

Muscle force production occurs within an environment of tissues that exhibit spring-like behavior, and this elasticity is a critical determinant of muscle performance during locomotion. Muscle force and power output both depend on the speed of contraction, as described by the isotonic force-velocity curve. By influencing the speed of contractile elements, elastic structures can have a profound effect on muscle force, power and work. In very rapid movements, elastic mechanisms can amplify muscle power by storing the work of muscle contraction slowly and releasing it rapidly. When energy must be dissipated rapidly, such as in landing from a jump, energy stored rapidly in elastic elements can be released more slowly to stretch muscle contractile elements, reducing the power input to muscle and possibly protecting it from damage. Elastic mechanisms identified so far rely primarily on in-series tendons, but many structures within muscles exhibit spring-like properties. Actomyosin cross-bridges, actin and myosin filaments, titin, and the connective tissue scaffolding of the extracellular matrix all have the potential to store and recover elastic energy during muscle contraction. The potential contribution of these elements can be assessed from their stiffness and estimates of the strain they undergo during muscle function. Such calculations provide boundaries for the possible roles these springs might play in locomotion, and may help to direct future studies of the uses of elastic elements in muscle.

Collective effects in force generation by multiple cytoskeletal filaments pushing an obstacle

We report here recent findings that multiple cytoskeletal filaments (assumed rigid) pushing an obstacle typically generate more force than just the sum of the forces due to individual ones. This interesting phenomenon, due to the hydrolysis process being out of equilibrium, escaped attention in previous experimental and theoretical literature. We first demonstrate this numerically within a constant force ensemble, for a well known model of cytoskeletal filament dynamics with random mechanism of hydrolysis. Two methods of detecting the departure from additivity of the collective stall force, namely from the force-velocity curve in the growing phase, and from the average collapse time versus force curve in the bounded phase, is discussed. Since experiments have already been done for a similar system of multiple microtubules in a harmonic optical trap, we study the problem theoretically under harmonic force. We show that within the varying harmonic force ensemble too, the mean collective stall force of N filaments is greater than N times the mean stall force due to a single filament; the actual extent of departure is a function of the monomer concentration.

Microtubule C-Terminal Tails Can Change Characteristics of Motor Force Production.

Control of intracellular transport is poorly understood, and functional ramifications of tubulin isoform differences between cell types are mostly unexplored. Motors' force production and detachment kinetics are critical for their group function, but how microtubule (MT) details affect these properties--if at all--is unknown. We investigated these questions using both a vesicular transport human kinesin, kinesin-1, and also a mitotic kinesin likely optimized for group function, kinesin-5, moving along either bovine brain or MCF7(breast cancer) MTs. We found that kinesin-1 functioned similarly on the two sets of MTs--in particular, its mean force production was approximately the same, though due to its previously reported decreased processivity, the mean duration of kinesin-1 force production was slightly decreased on MCF7 MTs. In contrast, kinesin-5's function changed dramatically on MCF7 MTs: its average detachment force was reduced and its force-velocity curve was different. In spite of the reduced detachment force, the force-velocity alteration surprisingly improved high-load group function for kinesin-5 on the cancer-cell MTs, potentially contributing to functions such as spindle-mediated chromosome separation. Significant differences were previously reported for C-terminal tubulin tails in MCF7 versus bovine brain tubulin. Consistent with this difference being functionally important, elimination of the tails made transport along the two sets of MTs similar.

C57BL/6 life span study: age-related declines in muscle power production and contractile velocity.

Quantification of key outcome measures in animal models of aging is an important step preceding intervention testing. One such measurement, skeletal muscle power generation (force * velocity), is critical for dynamic movement. Prior research focused on maximum power (Pmax), which occurs around 30–40 % of maximum load. However, movement occurs over the entire load range. Thus, the primary purpose of this study was to determine the effect of age on power generation during concentric contractions in the extensor digitorum longus (EDL) and soleus muscles over the load range from 10 to 90 % of peak isometric tetanic force (P0). Adult, old, and elderly male C57BL/6 mice were examined for contractile function (6–7 months old, 100 % survival; ~24 months, 75 %; and ~28 months, <50 %, respectively). Mice at other ages (5–32 months) were also tested for regression modeling. We hypothesized and found that power decreased with age not only at Pmax but also over the load range. Importantly, we found greater age-associated deficits in both power and velocity when the muscles were contracting concentrically against heavy loads (>50 % P0). The shape of the force-velocity curve also changed with age (a/P0 increased). In addition, there were prolonged contraction times to maximum force and shifts in the distribution of the myosin light and heavy chain isoforms in the EDL. The results demonstrate that age-associated difficulty in movement during challenging tasks is likely due, in addition to overall reduced force output, to an accelerated deterioration of power production and contractile velocity under heavily loaded conditions.Electronic supplementary materialThe online version of this article (doi:10.1007/s11357-015-9773-1) contains supplementary material, which is available to authorized users.

A Comparative Study of the Damping Force and Energy Absorbtion Capacity of Regenerative and Conventional-Viscous Shock Absorber of Vehicle Suspension

This paper presents a comparative study of the damping force and energy absorbtion capacity of a typical conventional-viscous and a regenerative shock absorber for vehicle suspension. Regenerative shock absorber (RSA) is a shock absorber which can regenerate the dissipated vibration energy from vehicle suspension into electricity. In this research, a prototype of regenerative shock absorber was developed, its damping force and energy absorbtion capacity were tested, and the results were analized and compared with those of a typical conventional-viscous shock absorber. The regenerative and viscous shock absorber were compressed and extended in various excitation frequency using damping force testing equipment to obtain force-velocity and the force-displacement curves. The force-velocity and force-displacement curves indicate the damping force and energy absorbtion capacity of the shock absorber. The results show that the damping force of the typical-viscous shock absorber closed to linear at all exciation frequencies. For regenerative shock absorber, nonlinearity and large hysteresis area of the damping force occur at all excitation frequencies. Further, the energy absorbtion capacity of the typical-viscous shock absorber shows an elliptical area with the compression part bigger than the extension one, while those of the regenerative shock absorber shows an asymmetric square area, which indicates a smaller energy absorbtion capacity. These phenomena indicate the significant effect of implementing dry friction damper and elctrical damper to the characteristics of regenerative shock absorber.

Soil Damping Effects on Integrity Test of Piles

The effects of surrounding soil on pile integrity test results are still not precisely quantified despite its influence on test output. It is usually difficult to assess the integrity of piles socketed in hard soil strata due to soil damping effects. A method of calculating soil damping effects on stress wave force and velocity values during pile integrity test is presented. Theoretical model based on numerical solution of wave equation was formulated. This model incorporates soil resistance effects on wave propagation in piles. Results generated from the proposed model was compared with PIT-S software, which simulates low strain waves propagation in piles, assuming pile cast in very dense granular soil. The generated Force-Velocity curves were found to be similar comparing both methods; however, some variations were observed at pile toe due to the different procedures used for soil reaction estimation. Furthermore, the model produced results were compared with real in-situ pile integrity test and PIT-S results. The model had showed better prediction of pile toe received signal compared to PIT-S software.

Redefining the Latch-State in Human Airway Smooth Muscle

Rationale: The latch-state describes the ability of tonic smooth muscle to maintain force at reduced ATP consumption and crossbridge cycling rate. In animal models this is evidenced by reduced unloaded shortening velocity (Vmax), presumably indicative of crossbridge cycling rate, and reduced myosin light chain phosphorylation (pMLC). In this study we aim to assess this phenomenon in human airway smooth muscle (hASM).Methods: hASM tissues were dissected from trachea procured from the International Institute for the Advancement of Medicine. Force-velocity curves and pMLC measurements were performed at several time points in multiple prolonged methacholine 10-6M contractions. Rat ASM strips were tested with the same protocol. A mathematical model was developed to explain the findings.Results: Despite a 40% drop in pMLC between the 1st to the 20th min after the peak contraction, hASM Vmax did not show any change. Conversely, rat ASM Vmax decreased by 50%. A mathematical model of the cross-bridge cycle, combining strain-dependent ADP affinity of myosin with actin regulated binding of dephosphorylated myosin heads, was capable of reproducing our findings.Conclusion: As our results contradict the classical latch-bridge theory, we suggest an alternative model. We propose that Vmax, as calculated from standard force-velocity curves, is not indicative of the crossbridge cycling rate during latch. Instead, Vmax is reduced due to drag caused by dephosphorylated myosin heads binding to actin, with their binding force regulated by actin regulatory proteins and their activation state. The increase in dephosphorylated myosin late in contraction decreases the measured Vmax, unless its binding force is down-regulated by regulatory proteins. Our modelling further shows that force maintenance can be explained by phosphorylated myosin heads remaining attached to actin in a locked configuration, unable to release ADP because of strain dependence of the ADP affinity of myosin.

Muscle Shortening Velocity is Modulated by Alternative Myosin Converters

The myosin converter has been proposed to be critical for setting muscle shortening velocity by influencing the amplification of small conformational changes at the ATPase site into the larger lever arm swing. To test this hypothesis, we exploited the natural variation of the converter region found in Drosophila muscle types. In Drosophila, a single myosin heavy chain gene is alternatively spliced to generate myosin isoforms. The converter region is encoded by five alternative versions of exon 11 that are expressed in different muscle types. Through genetic modification we forced the expression of the alternative versions in the Drosophila jump muscle and found that two of the three versions tested to date, 11a (native to indirect flight muscles) and 11e (embryonic muscles), caused faster maximum shortening velocities of skinned jump muscle fibers relative to the native control 11c. Increases in velocity were primarily responsible for 70.9% and 79.0% higher power outputs for 11a and 11e fibers, respectively, because maximum force generation was not significantly different from control fibers. The fibers expressing 11a and 11e also exhibited a straighter force-velocity curve, as shown by increased Hill equation parameters a (95.4% and 84.5% higher, respectively) and b (72.6% and 65.5% higher, respectively) relative to control fibers, indicating that the converter modulates load dependent cross-bridge kinetics. The higher power and shortening velocity of 11a and 11e fibers enabled increased jumping ability, 76.0% and 48.8% farther, respectively. Converter homology models, using the new Drosophila myosin S1 crystal structure (pdb 4QBD), revealed minimal tertiary structure differences suggesting residue specific interactions with the relay, N-terminus, or essential light chain are critical. Overall, our data support the hypothesis that the converter helps set muscle shortening velocity under loaded and unloaded conditions and thus contributes to muscle fiber type diversity.

Branching influences force-velocity curves and length fluctuations in actin networks

We investigate collective dynamics of branched actin networks growing against a rigid movable wall constrained by a resistive force. Computing the force velocity relations, we show that the stall force of such networks depends not only on the average number of filaments touching the wall, but also on the amount of fluctuation of the leading edge of the network. These differences arise due to differences in the network architecture, namely, distance between two adjacent branching points and the initial distance of the starting filament from the wall, with their relative magnitudes influencing the nature of the force velocity curves (convex versus concave). We also show that the introduction of branching results in nonmonotonic diffusion constant, a quantity that measures the growth in length fluctuation of the leading edge of the network, as a function of externally applied force. Together our results demonstrate how the collective dynamics of a branched network differs from that of a parallel filament network.

Effect of acute exposure to moderate altitude on muscle power: hypobaric hypoxia vs. normobaric hypoxia.

When ascending to a higher altitude, changes in air density and oxygen levels affect the way in which explosive actions are executed. This study was designed to compare the effects of acute exposure to real or simulated moderate hypoxia on the dynamics of the force-velocity relationship observed in bench press exercise. Twenty-eight combat sports athletes were assigned to two groups and assessed on two separate occasions: G1 (n = 17) in conditions of normoxia (N1) and hypobaric hypoxia (HH) and G2 (n = 11) in conditions of normoxia (N2) and normobaric hypoxia (NH). Individual and complete force-velocity relationships in bench press were determined on each assessment day. For each exercise repetition, we obtained the mean and peak velocity and power shown by the athletes. Maximum power (Pmax) was recorded as the highest Pmean obtained across the complete force-velocity curve. Our findings indicate a significantly higher absolute load linked to Pmax (∼3%) and maximal strength (1RM) (∼6%) in G1 attributable to the climb to altitude (P<0.05). We also observed a stimulating effect of natural hypoxia on Pmean and Ppeak in the middle-high part of the curve (≥60 kg; P<0.01) and a 7.8% mean increase in barbell displacement velocity (P<0.001). No changes in any of the variables examined were observed in G2. According to these data, we can state that acute exposure to natural moderate altitude as opposed to simulated normobaric hypoxia leads to gains in 1RM, movement velocity and power during the execution of a force-velocity curve in bench press.

Bypass Rotary Magnetorheological Damper for Automotive Applications

A novel concept of bypass rotary Magneto-rheological (MR) damper is studied in the automotive suspension application. The proposed design is developed based on a vane-type damper structure with an MR control valve located outside of the damper. The design is intended to enhance the ease of maintenance and to reduce the thermal effect from the coil to the MR fluid. The valve is the key component of the damper that exploits the advance characteristics of MR fluid in which have sensitive rheological properties to magnetic field. The ability of the valve to modify the strength of magnetic field has given an advantage that the valve can be operated without any moving parts. The elimination of these parts in the throttling mechanism of the valve will provide benefit in terms of product lifetime and responsiveness. The main objective of this paper is to elaborate the advantages of the bypass rotary MR damper and to demonstrate the damper performance through force-velocity characteristics. The analytical model of the damper is developed and used in the determination of the force-velocity curves and the equivalent damping coefficients.

The force-velocity relationship at negative loads (assisted shortening) studied in isolated, intact muscle fibres of the frog.

The study was undertaken to explore the force-velocity relationship under conditions where the myofilament system is subjected to an external force that serves as a negative load and assists the shortening movement. The experiments were carried out on single muscle fibres isolated from the anterior tibialis muscle of Rana temporaria. The fibres, being operated under load-clamp control, were released to shorten during tetanic stimulation at sarcomere lengths where the fibres carried different degrees of passive tension. The shortening thus occurred while the sarcomeres were subjected to a force that may be characterized as a 'negative load', that is, a force assisting the shortening movement. The force-velocity relationship below zero load was found to be a smooth continuation of the force-velocity curve recorded at positive loads with the shortening velocity increasing steeply at loads below zero. A negative load amounting to merely 10% of the isometric force, thus raised the shortening velocity to a level two to three times higher than V0 , the velocity recorded at zero load. The results provide evidence that, even in the presence of a longitudinal compressive force, the speed of shortening of the muscle fibre is determined by the cycling rate of the interacting cross-bridges. The force-velocity relationship at negative loads may play a relevant part during fast movements of striated muscle as pointed out in the discussion.

Passenger Seat Vibration Control of Quarter Car System with MR Shock Absorber

Semi-active Fuzzy control of quarter car system having three degrees of freedom and assembled with magneto- rheological (MR) shock absorber is studied in present paper. First, experimental work was performed on an MR shock absorber under different excitation conditions to obtain force-displacement and force-velocity curves. Then, for the application of experimental data in semi-active quarter car system, a polynomial model was selected. Finally, Fuzzy logic controller was designed having the combination of Forward fuzzy controller and Inverse fuzzy controller for integration in secondary suspension system of concerned model. The proposed controlled quarter car model was compared with uncontrolled system using simulation work under bump type of road excitation. Results obtained by simulation work shows the effectiveness of fuzzy controlled suspension system in improving the ride comfort and safety of travelling passengers compared to uncontrolled suspension system. Keywords—MR shock absorber, three degrees of freedom, quarter car model, fuzzy controller.

Geared up to stretch: pennate muscle behavior during active lengthening

Many locomotor activities require muscles to actively lengthen, dissipate energy and decelerate the body. These eccentric contractions can disrupt cytoskeletal structures within myofibrils and reduce force output. We examined how architectural features of pennate muscles can provide a protective mechanism against eccentric muscle damage by limiting fascicle lengthening. It has been previously shown that the angled fibers of pennate muscles change orientation when shortening. This change in fiber orientation can amplify fascicle shortening, resulting in a velocity advantage at the level of the muscle-tendon unit (MTU) that is characterized by a gear ratio (MTU velocity/fascicle velocity). A muscle's architectural gear ratio (AGR) has been shown to vary as a function of force during shortening, while AGR during lengthening remains largely unknown. We independently measured fascicle length and MTU length in vitro in the bullfrog plantaris. We characterized the muscle's force-velocity curve and AGR during both shortening and lengthening across a broad range of forces (10-190% peak isometric force). AGR was measured during the isotonic portion of each contraction, to eliminate possible contributions of series elasticity to MTU length changes. We found that gear ratio varies with force during both shortening and lengthening contractions. The highest AGR was observed during lengthening contractions, indicating that lengthening of the MTU can occur with relatively little stretch of the fascicle. As fascicle strain is considered an important determinant of muscle damage, a high gear ratio may afford pennate muscles protection against the damaging effects of active lengthening.

Load-Sharing Mechanism of Cytoplasmic Dynein

Cytoplasmic dynein is a homodimeric AAA+ motor protein responsible for minus end-directed transport along microtubules. The two heads of dynein are not strictly coordinated during processive motility, unlike those of dimeric kinesins and myosins. Therefore, tail-tethered optical trapping experiments yield insufficient information about the force production of individual heads. Here, we connected a trapped bead to a dynein dimer through a rigid DNA tether and directly measured the stepping motion of a single head under load. We observed that even though the step size of a head decreases under load, the head only stalls at half the stall force of the dimer and must rely on its partner head to walk against near-stall loads. The bulk of the force is produced when a head is bound to the microtubule, and stepping direction in the unbound state can be reversed by as little as 0.5 pN of backward load. Force-velocity curves indicate that the forward stepping rate of a dynein head is strongly inhibited by external force, while the backward rate is slow and nearly force-independent. Our results provide evidence for tension gating in dynein motility and lead us to propose a load-sharing mechanism between the two heads of a dynein dimer.

Enhancement of Force Generated by Individual Myosin Heads in Skinned Rabbit Psoas Muscle Fibers at Low Ionic Strength

Although evidence has been presented that, at low ionic strength, myosin heads in relaxed skeletal muscle fibers form linkages with actin filaments, the effect of low ionic strength on contraction characteristics of Ca2+-activated muscle fibers has not yet been studied in detail. To give information about the mechanism of muscle contraction, we have examined the effect of low ionic strength on the mechanical properties and the contraction characteristics of skinned rabbit psoas muscle fibers in both relaxed and maximally Ca2+-activated states. By progressively decreasing KCl concentration from 125 mM to 0 mM (corresponding to a decrease in ionic strength μ from 170 mM to 50 mM), relaxed fibers showed changes in mechanical response to sinusoidal length changes and ramp stretches, which are consistent with the idea of actin-myosin linkage formation at low ionic strength. In maximally Ca2+-activated fibers, on the other hand, the maximum isometric force increased about twofold by reducing KCl concentration from 125 to 0 mM. Unexpectedly, determination of the force-velocity curves indicated that, the maximum unloaded shortening velocity Vmax, remained unchanged at low ionic strength. This finding indicates that the actin-myosin linkages, which has been detected in relaxed fibers at low ionic strength, are broken quickly on Ca2+ activation, so that the linkages in relaxed fibers no longer provide any internal resistance against fiber shortening. The force-velocity curves, obtained at various levels of steady Ca2+-activated isometric force, were found to be identical if they are normalized with respect to the maximum isometric force. The MgATPase activity of muscle fibers during isometric force generation was found not to change appreciably at low ionic strength despite the two-fold increase in Ca2+-activated isometric force. These results can be explained in terms of enhancement of force generated by individual myosin heads, but not by any changes in kinetic properties of cyclic actin-myosin interaction.