In vivo force-velocity relation of human muscle: a modelling from sinusoidal oscillation behaviour

Isokinetic tests performed on human muscle in vivo during plantar flexion contractions lead to torque–angular velocity relationships usually fitted by Hill's equation expressed in angular terms. However, such tests can lead to discrepant results since they require maximal voluntary contractions performed in dynamic conditions. In the present study, another way to approach mechanical behaviour of a musculo-articular structure was used, i.e. sinusoidal oscillations during sub-maximal contractions. This led to the expression of (i) Bode diagrams allowing the determination of a damping coefficient (B_bode); and (ii) a viscous parameter (B_sin) using an adaptation of Hill's equation to sinusoidal oscillations. Then torque–angular velocity relationships were predicted from a model based on the interrelation between B_bode and B_sin and on the determination of optimal conditions of contraction. This offers the possibility of characterizing muscle dynamic properties by avoiding the use of isokinetic maximal contractions.     

Dissipative processes in human passive skeletal muscle

The measurement of damping of low-amplitude limb oscillations permits to evaluate the energy losses in passive human skeletal muscle during small length changes. The attenuation curve for the limb oscillations is quite different from the classical attenuation curve in the presence of viscous damping. Energy losses per oscillation cycle are practically frequency independent. Thus the damping properties of passive muscles at joint angular velocities up to 100% are due mostly to the velocity-independent resistance of "dry friction" type. The value of this "friction" is about 0.07 N per sm2 of muscle cross-section. The passive muscle also has marked thixotropy, as its resistance to small amplitude low velocity stretches strongly depends on time between stretches.     

Frequency-dependent distortion of meridional intensity changes during sinusoidal length oscillations of activated skeletal muscle

Bundles of intact, tetanized skeletal muscle fibers from Rana temporaria were subjected to sinusoidal length oscillations in the frequency domain 100 Hz to 3 kHz while measuring force and sarcomere length. Simultaneously, intensity of the third-order x-ray reflection of the axial myosin unit cell (I(M3)) was measured using synchrotron radiation. At oscillation frequencies <1 kHz, I(M3) was distorted during the shortening phase of the sinusoid (i.e., where bundle length was less than rest length). Otherwise, during the stretch phase of oscillations at all frequencies, during the shortening phase of oscillations above 1 kHz, and for bundles in the rigor state, I(M3) was approximately sinusoidal in form. Mean I(M3) during oscillations was reduced by 20% compared to the isometric value, suggesting a possible change in S1 disposition during oscillations. However, the amplitude of length change required to produce distortion (estimated from the phase angle at which distortion was first evident) corresponded to that of a step release sufficient to reach the maximum I(M3), indicating a mean S1 disposition during oscillations close to that during an isometric tetanus. The mechanical properties of the bundle during oscillations were also consistent with an unaltered S1 disposition during oscillations.     

Effects of volatile anesthetics on elastic stiffness in isometrically contracting ferret ventricular myocardium.

The effects of halothane, isoflurane, and sevoflurane on elastic stiffness, which reflects the degree of cross-bridge attachment, were studied in intact cardiac muscle. Electrically stimulated (0.25 Hz, 25 degrees C), isometrically twitching right ventricular ferret papillary muscles (n = 15) at optimal length (L(max)) were subjected to sinusoidal length oscillations (40 Hz, 0.25- 0.50% of L(max) peak to peak). The amplitude and phase relationship with the resulting force oscillations was decomposed into elastic and viscous components of total stiffness in real time. Increasing extracellular Ca(2+) concentration in the presence of anesthetics to produce peak force equal to control increased elastic stiffness during relaxation, which suggests a direct effect of halothane and sevoflurane on cross bridges.     

Self-oscillations of biological microbodies

Theoretical analysis of natural oscillation spectra of different kind of cells is presented. The study of received dispersive equation shows that the character of the cell natural movement depends on its size and the viscosity of internal and external liquids. If the depth of viscous wave penetration is small in comparison with the cell radius, the natural movements are weak damping oscillations. If the depth viscous wave penetration is comparable to the cell size, we have a relaxation process of cell form restoration.     

Soft-tissue vibrations in the quadriceps measured with skin mounted transducers

The purpose of this study was to develop a method to characterize the frequency and damping of vibrations in the soft tissues of the leg. Vibrations were measured from a surface-mounted accelerometer attached to the skin overlying the quadriceps muscles. The free vibrations in this soft tissue were recorded after impact whilst the muscle was performing isometric contractions at 0, 50, and 100% maximum voluntary force and with the knee held at 20, 40, and 60 degrees angles of flexion. The acceleration signals indicated that the soft tissue oscillated as under-damped vibrations. The frequency and damping coefficients for these vibrations were estimated from a model of sinusoidal oscillations with an exponential decay. This technique resolved the vibration coefficients to 2 and 7% of the mean values for frequency and damping, respectively.     

Examination of spinal column vibrations: a non-invasive approach

Accelerations of vertebrae during whole-body vibration (WBV) are used in occupational biomechanics for the prediction of internal stress. To avoid invasive techniques, a method for the calculation of bone accelerations was developed using measurements on the skin. The soft tissue between spinous processes L3 and T5 and miniature accelerometers stuck to the skin over them was modelled by a simple Kelvin element, whose parameters i.e. angular natural frequency omega n and critical damping zeta, describe an approximate transfer function between the bone (input) and the skin surface (output). The parameters were determined from free damped oscillations of the accelerometer-skin complex in the Z-axis, and depended significantly on the factors "subject" and "point of measurement". In one subject, the time courses of bone accelerations during sinusoidal WBV (4.5 and 8 Hz; 1.5 m.s-2 RMS) were calculated using separate transfer functions for each of 11 different spinal levels. Since the output signals on the skin were non-sinusoidal, the skin accelerations had to be treated with an inverse transfer function in the frequency domain. A comparison of accelerations measured on the skin and predicted for the bone mainly indicates that absolute peak values of bone accelerations are smaller and occur earlier. Both kinds of acceleration hint at differences in WBV-induced internal stress within the spine.     

Within-wingbeat damping: dynamics of continuous free-flight yaw turns in Manduca sexta

Free-flight body dynamics and wing kinematics were collected from recordings of continuous, low-speed, multi-wingbeat yaw turns in hawkmoths (Manduca sexta) using stereo videography. These data were used to examine the effects of rotational damping arising from interactions between the body rotation and flapping motion (flapping counter-torque, FCT) on continuous turning. The moths were found to accelerate during downstroke, then decelerate during upstroke by an amount consistent with FCT damping. Wing kinematics related to turning were then analysed in a simulation of hawkmoth flight; results were consistent with the observed acceleration–deceleration pattern. However, an alternative wing kinematic which produced more continuous and less damped accelerations was found in the simulation. These findings demonstrate that (i) FCT damping is detectable in the dynamics of continuously turning animals and (ii) FCT-reducing kinematics do exist but were not employed by turning moths, possibly because within-wingbeat damping simplifies control of turning by allowing control systems to target angular velocity rather than acceleration.     

Fading memory model for airway smooth muscle dynamic response

This work presents the application of a fading memory model to describe the behavior of contracted airway smooth muscle (ASM) for two biophysical cases: finite duration length steps and longitudinal sinusoidal oscillations. The model parameters were initially determined from literature data on transient step length change response and subsequently the model was applied to the two cases. Results were compared with previously published experimental data on ASM oscillations. The model confirms a trend observed in the experimental data which shows that: (i) the value of tissue length change is the most important factor to determine the degree of cross-bridge detachment and (ii) a strong correlation exists between increasing frequency and declining stiffness until a certain frequency (∼25 Hz) beyond which frequency dependence is negligible. Although the model was not intended to simulate biophysical events individually, the data could be explained by cross-bridge cycling rates. As the frequency increases, cross-bridge reattachment becomes less likely, until no further cross-bridge attachment is possible.     

Spontaneous and chemoattractant-induced oscillations of cytosolic free calcium in single adherent human neutrophils

Studies with fluorescent Ca2+ indicators in large populations of neutrophils in suspension reveal a stable base line followed by a rapid agonist-induced elevation of cytosolic free calcium, [Ca2+]i, concomitant with other parameters of cellular activation. To study the role of adhesion in cell activation, we monitored [Ca2+]i in single neutrophils adhered to albumin-coated or fibronectin-coated glass coverslips before and after stimulation with the chemotactic peptide N-formyl-L-methionyl-L-leucyl-L-phenylalanine (fMLP). Human neutrophils loaded with 2 microM fura 2/AM were allowed to adhere to coverslips for 15-20 min at 37 degrees C. [Ca2+]i was monitored with a dual excitation microfluorimeter with a time resolution of 200 ms. Statistical analysis was performed using an algorithm allowing to detect significant [Ca2+]i peaks. 54% of the cells showed spontaneous [Ca2+]i oscillations. The amplitude of these [Ca2+]i peaks averaged 77 +/- 10 nM above basal levels (mean value of 110 +/- 20 nM), and their mean duration was 28 +/- 5 s; periods of [Ca2+]i bursts could last up to 15 min. In "silent" cells exhibiting a stable [Ca2+]i base line without spontaneous oscillations, low concentrations of fMLP (10(-10)-10(-9) M) could induce sustained [Ca2+]i oscillations. By contrast, higher agonist concentrations (10(-6) M) induced a single [Ca2+]i transient followed by a stable base line. 47% of the cells showing spontaneous [Ca2+]i oscillations did not respond to fMLP. Spontaneous [Ca2+]i oscillations depended on the continuous presence of extracellular Ca2+. Therefore: (i) spontaneous oscillations of [Ca2+]i occur in neutrophils adherent to various substrata; (ii) these oscillations do not preclude and can be dissociated from the response to fMLP; (iii) neutrophil functions might be controlled by [Ca2+]i oscillations rather than by sustained alterations of [Ca2+]i.     

A physical model of ATP-induced actin-myosin movement in vitro.

The nature of the mechanism limiting the velocity of ATP-induced unidirectional movements of actin-myosin filaments in vitro is considered. In the sliding process two types of "cyclic" interactions between myosin heads and actin are involved, i.e., productive and nonproductive. In the productive interaction, myosin heads split ATP and generate a force which produces sliding between actin and myosin. In the nonproductive interaction "cycle," on the other hand, myosin heads rapidly attach to and detach from actin "reversibly," i.e., without splitting ATP or generating an active force. Such a nonproductive interaction "cycle" causes irreversible dissipation of sliding energy into heat, because the myosin cross-bridges during this interaction are passive elastic structures. This consideration has led us to postulate that such cross-bridges, in effect, exert viscous-like frictional drag on moving elements. Energetic considerations suggest that this frictional drag is much greater than the hydrodynamic viscous drag. We present a model in which the sliding velocity is limited by the balance between the force generated by myosin cross-bridges in the productive interaction and the frictional drag exerted by other myosin cross-bridges in the nonproductive interaction. The model is consistent with experimental findings of in vitro sliding, including the dependence of velocity on ATP concentration, as well as the sliding velocity of co-polymers of skeletal muscle myosin and phosphorylated and unphosphorylated smooth muscle myosins.     

Computer simulation of damped oscillations during peroxidase-catalyzed oxidation of indole-3-acetic acid

Oscillation patterns in horseradish peroxidase (HRP)-catalyzed oxidation of indole-3-acetic acid (IAA) at neutral pH were studied using computer simulation. Under certain conditions, such as the presence of a reaction promoter and continuous intake of oxygen from the gaseous phase, the simulated system exhibits damped oscillations of the concentrations of oxygen in the aqueous phase, [O(2)](aq), and of all the reaction intermediates. The critical concentration of oxygen in aqueous phase, [O(2)](cr)(aq), was used to describe the nature of the oscillations. The critical concentration is the concentration at which the system abruptly changes its properties. If [O(2)](aq) is higher than [O(2)](cr)(aq) then the reaction develops as an avalanche, otherwise, the reaction stops. The nature of oscillations is accounted for by the interaction of two processes: the consumption/accumulation of oxygen and the accumulation/consumption of reaction intermediates. Oscillations are always damped. Neither HRP or umbelliferone (Umb) deactivation nor IAA consumption can account for the damping. The nature of the damping is determined by the termination reactions of free radical intermediates and ROOH. The three major parameters of oscillations: period of oscillations, initial amplitude of oscillations and the rate of damping were studied as functions of: (i) oxygen concentration in the gaseous phase, (ii) initial oxygen concentration in aqueous phase, (iii) the concentration of IAA and (iv) the initial concentration of HRP.     

Estimating the instability parameters of plasmid-bearing cells. I. Chemostat culture

What determines the stability of plasmid-bearing cells in natural and laboratory conditions? In order to answer this question in a quantitative manner, we need tools allowing the estimation of parameters governing plasmid loss in different environments. In the present work, we have developed two methods for the estimation of the instability parameters of plasmid-bearing cells growing in chemostat. These instability parameters are: (i) selection coefficient (or cost of the plasmid)alpha and (ii) the probability of plasmid loss at cell division tau(0). We have found that generally selection coefficient alpha changes during elimination of plasmid-bearing cells due to changes in substrate concentration; hence, methods which assume constancy of alpha are intrinsically imprecise. Instead, one can estimate selection coefficient at the beginning and the end of cultivation when the substrate concentration is approximately constant. Applying developed techniques to two sets of experimental data, we have found that (i) the cost of the plasmid pBR322 depended on the dilution rate in chemostat and was higher at low dilutions; (ii) high levels of plasmid gene expression led to a high cost of the plasmid pPHL-7; (iii) the probability of plasmid loss was lower at high levels of plasmid gene expression and independent of the dilution rate. We have also discussed the application of our results to understanding the basic biology of bacterial plasmids.     

A hybrid static optimisation method to estimate muscle forces during muscle co-activation

The general static optimisation (GSO) process is one of various muscle force estimation methods due to its low computational requirements. However, it can show biased muscle force estimation under muscle co-contraction. In the present study, we introduced a novel hybrid static optimisation (HSO) method to estimate reasonable muscle forces during muscle co-activation movements using more specific equality constraints, i.e. agonist and antagonist muscle moments predicted from a new correlation coefficient approach. The new method was evaluated for heel-rise movements. We found that the proposed method improved the potential of antagonist muscle force estimation in comparison to the GSO solutions. The proposed HSO method could be applied in biomechanics and rehabilitation, for example.     

The natural frequency of the foot-surface cushion during the stance phase of running

Researchers have reported on the stiffness of running in holistic terms, i.e. for the structures that are undergoing deformation as a whole rather than in terms of specific locations. This study aimed to estimate both the natural frequency and the viscous damping coefficient of the human foot-surface cushion, during the period between the heel strike and the mid-stance phase of running, using a purposely developed one degree-of-freedom inverted pendulum state space model of the leg. The model, which was validated via a comparison of measured and estimated ground reaction forces, incorporated a novel use of linearized and extended Kalman filter estimators. Investigation of the effect of variation of the natural frequency and/or the damping of the cushioning mechanism during running, using the said model, revealed the natural frequency of running on said foot-surface cushion, during the stance phase, to lie between 5 and 11 Hz. The "extended Kalman filter (EKF)" approach, that was used here for the first time to directly apply measured ground forces, may be widely applicable to the identification process of combined estimation of both unknown physiological state and mechanical characteristics of the environment in an inverse dynamic model.     

Relating intramuscular fuel use to endurance in juvenile rainbow trout

This study examined fuel depletion in white muscle of juvenile rainbow trout sprinted to fatigue to determine whether the onset of fatigue is associated with a measurable metabolic change within the muscle and whether muscle glycogen levels influence endurance. In this study, "fuels" refer to any energy-supplying compounds and include glycogen, phosphocreatine (PCr), and ATP. Fuel depletion in white muscle was estimated by the calculation of the anaerobic energy expenditure (AEE; in micromol ATP equivalents g(-1)) from the reduction of PCr and ATP and the accumulation of lactate. Progression of fuel use during sprinting was examined by sampling fish before they showed signs of fatigue and following fatigue. Most of the AEE before fatigue was due to PCr depletion. However, at the first signs of fatigue, there was a 32% drop in ATP. Similarly, when fish were slowly accelerated to a fatiguing velocity, the only significant change at fatigue was a 30% drop in ATP levels. Muscle glycogen levels were manipulated by altering ration (1% vs. 4% body weight ration per day) combined with either daily or no exercise. Higher ration alone led to significantly greater muscle glycogen but had no effect on sprint performance, whereas sprint training led to higher glycogen and an average threefold improvement in sprint performance. In contrast, periodic chasing produced a similar increase in glycogen but had no effect on sprint performance. Taken together, these observations suggest that (i) a reduction in ATP in white muscle could act as a proximate signal for fatigue during prolonged exercise in fish and (ii) availability of muscle glycogen does not limit endurance.     

Viscous Damping of Vibrations in Microtubules

Pokorný et al. have recently suggested that metabolic processes drivemicrotubules in a cell to vibrate at Megahertz frequencies, but the theorydoes not explicitly consider dissipative effects which will tend to damp outthe vibrations. To examine the effects of viscous damping on the structure,we determine viscous forces and rate of energy loss in a cylinderundergoing longitudinal oscillations in water. A nondimensional expressionis obtained for the viscous drag on the cylinder. When applied to amicrotubule, the results indicate that viscous damping is several orders ofmagnitude too large to allow resonant vibrations.     

Optical and acoustical dynamics of microbubble contrast agents inside neutrophils

Acoustically active microbubbles are used for contrast-enhanced ultrasound assessment of organ perfusion. In regions of inflammation, contrast agents are captured and phagocytosed by activated neutrophils adherent to the venular wall. Using direct optical observation with a high-speed camera and acoustical interrogation of individual bubbles and cells, we assessed the physical and acoustical responses of both phagocytosed and free microbubbles. Optical analysis of bubble radial oscillations during insonation demonstrated that phagocytosed microbubbles experience viscous damping within the cytoplasm and yet remain acoustically active and capable of large volumetric oscillations during an acoustic pulse. Fitting a modified version of the Rayleigh-Plesset equation that describes mechanical properties of thin shells to optical radius-time data of oscillating bubbles provided estimates of the apparent viscosity of the intracellular medium. Phagocytosed microbubbles experienced a viscous damping approximately sevenfold greater than free microbubbles. Acoustical comparison between free and phagocytosed microbubbles indicated that phagocytosed microbubbles produce an echo with a higher mean frequency than free microbubbles in response to a rarefaction-first single-cycle pulse. Moreover, this frequency increase is predicted using the modified Rayleigh-Plesset equation. We conclude that contrast-enhanced ultrasound can detect distinct acoustic signals from microbubbles inside of neutrophils and may provide a unique tool to identify activated neutrophils at sites of inflammation.     

Model-based prediction of fusimotor activity and its effect on muscle spindle activity during voluntary wrist movements

Muscle spindles provide critical information about movement position and velocity. They have been shown to act as stretch receptors in passive muscle, however, during active movements their behavior is less clear. In particular, spindle responses have been shown to be out-of-phase or phase advanced with respect to their expected muscle length-sensitivity. Whether this apparent discrepancy of spindle responses between passive and active movements is due to fusimotor (γ-drive) remains unresolved, since the activity of fusimotor neurons during voluntary non-locomotor movements are largely unknown. We developed a computational model to predict fusimotor activity and to investigate whether fusimotor activity could explain the empirically observed phase advance of spindle responses. The model links a biomechanical wrist model to length- and γ-drive-dependent transfer functions of type Ia and type II muscle spindle activity. Our simulations of two wrist-movement tasks suggest that (i) experimentally observed type Ia and type II activity profiles can to a large part be explained by appropriate, i.e. strongly modulated and task-dependent, γ-drive. That (ii) the empirically observed phase advance of type Ia or of type II profiles during active movement can be similarly explained by appropriate γ-drive. In summary, the simulation predicts that a highly task-modulated activation of the γ-system is instrumental in producing a large part of the empirically observed muscle spindle activity for voluntary wrist movements.     

Relationship between asynchronous Ca2+ waves and force development in intact smooth muscle bundles of the porcine trachea

Fluctuations in intracellular calcium concentration ([Ca2+]i) constitute the main link in excitation-contraction coupling (E-C coupling) in airway smooth muscle cells (ASMC). It has recently been reported that ACh induces asynchronous recurring Ca2+ waves in intact ASMC of murine bronchioles. With the use of a novel technique allowing us to simultaneously measure subcellular [Ca2+]i and force generation in ASMC located within an intact tracheal muscle bundle, we examined a similar pattern of Ca2+ signaling in the trachea. We found that application of ACh resulted in the generation of recurring intracellular Ca2+ waves progressing along the longitudinal axis of the ribbon-shaped intact ASMC. These Ca2+ waves were not synchronized between neighboring cells, and induction of wave-like [Ca2+]i oscillations was temporally associated with development of force by the tracheal muscle bundle. By comparing the concentration dependence of force generation and the parameters characterizing the [Ca2+]i oscillations, we found that the concentration-dependent increase in ACh-induced force development by the tracheal smooth muscle bundle is achieved by differential recruitment of intact ASMC to initiate Ca2+ waves and by enhancement in the frequency of [Ca2+]i oscillations and elevation of interspike [Ca2+]i once the cells are recruited. Our findings demonstrate that asynchronous recurring Ca2+ waves underlie E-C coupling in ACh-induced contraction of the intact tracheal smooth muscle bundle. Furthermore, in contrast to what was reported in enzymatically dissociated ASMC, Ca2+ influx through the L-type voltage-gated Ca2+ channel was not an obligatory requirement for the generation of [Ca2+]i oscillations and development of force in ACh-stimulated intact ASMC.