ABSTRACT
Several recent accounts have estimated the maximum sustainable mechanical power available from a skeletal muscle based on measured or assumed force–velocity characteristics of the muscle and the assumption that the muscle, when active, shortens constantly at the velocity that is optimal for power output. Curtin and Woledge (1988) used this approach in estimating how much mechanical power is available from the myotomal muscles of a dogfish to power swimming. The muscles of a fish operate cyclically, shortening and doing work for half a cycle and being lengthened, possibly absorbing work, for the other half-cycle. Curtin and Woledge allowed for the periodic nature of power output by assuming that only half the muscles were active at any time, which is equivalent to assuming that the sustainable power is half the peak power. During swimming, the shape of the trunk of a fish changes sinusoidally with time (Hess and Videler, 1984), and the shortening velocity of a trunk muscle fiber must also vary approximately sinusoidally. Thus shortening velocity cannot be at the optimum velocity for power output through the entire shortening half-cycle: the assumption of shortening at a constant, optimum velocity must overestimate the power output. Weis-Fogh and Alexander (1977) and Pennycuick and Rezende (1984) have also estimated the maximum sustainable power output of muscle based on the assumption of shortening at constant velocity. The principal muscles considered in the latter two studies were flight muscles. For flight muscles, as for swimming muscles, a sinusoidal length trajectory and a sinusoidally varying shortening velocity are more realistic assumptions than a linear length change and a constant velocity of shortening. The following analysis was begun to determine the magnitude of the error that is likely to enter into estimates of power output from the assumption that muscle shortening during normal locomotion is at constant velocity rather than at a velocity which varies sinusoidally with time. It is shown that the assumption of shortening at constant velocity overestimates maximum power output by 20 % or less.
Expected muscle force during sinusoidal shortening (left) and during shortening at constant velocity (right). In both cases it is assumed that the muscle is fully active during the shortening half-cycle, and fully inactive and totally compliant during muscle lengthening. The values used in constructing these curves were: frequency 5 Hz; aF0−l, 0·2; Vmax, 5Ls−1, where L is muscle length. The total strain for the sinusoidal trajectory (14%) is the optimum for work output, as is the shortening velocity (1·45 Ls−1) in the linear model.
Expected muscle force during sinusoidal shortening (left) and during shortening at constant velocity (right). In both cases it is assumed that the muscle is fully active during the shortening half-cycle, and fully inactive and totally compliant during muscle lengthening. The values used in constructing these curves were: frequency 5 Hz; aF0−l, 0·2; Vmax, 5Ls−1, where L is muscle length. The total strain for the sinusoidal trajectory (14%) is the optimum for work output, as is the shortening velocity (1·45 Ls−1) in the linear model.
Linear shortening
Sinusoidal shortening
The total power output is the work done per cycle, Ws, times the operating frequency. It should be noted that Ws is inversely related to operating frequency, and that it is a function of the strain per cycle (total strain per cycle=2ΔS0). It has been shown that there is an optimum strain per cycle for work output (Josephson and Stokes, 1989).
The work output at a given frequency was evaluated for a set of strain values using equation 4 to determine empirically the optimum strain and the maximum work output and the maximum power output at that frequency. The work output was calculated with MathCAD (MathSoft, Inc., Cambridge, MA), a computer program which allows evaluation of integrals. Optimum strain and maximum power output were determined for a set of values of operating frequency, of a (expressed as oF0−1) and of maximum shortening velocity (Vmax=bF0a−1). Vmax was chosen as a measure of velocity characteristics rather than b because it is a more familiar parameter. The range of parameters examined encompasses the values likely to be encountered in animal muscles and locomotion: frequency 1, 2, 5,10,20,50 Hz; aF0−l 0·05,0·2,1; Vmax 1,5,20 muscle lengths per second (Ls−1).
The work per cycle with sinusoidal shortening declined with frequency, but the maximum power output was nearly independent of frequency (Fig. 2, Table 1). For linear shortening at a constant velocity the strain per cycle is necessarily inversely proportional to cycle frequency. With sinusoidal shortening the optimal strain for work output was approximately inversely proportional to frequency. As expected, the maximum power output during sinusoidal shortening was less than that which would be obtained with linear shortening at the optimum velocity, by the difference in power output between linear and sinusoidal length trajectories was not great. Through almost the entire range of parameters examined the maximum power output during sinusoidal shortening was 83–92 % of that during linear shortening. Given the usual uncertainties about values determined in force-velocity measurements, a 10–20% error due to an imperfect model is not very large. Half the peak instantaneous power from a force-velocity curve is a reasonable estimate of the maximum sustainable power during sinusoidal shortening.
ACKNOWLEDGEMENTS
This work was supported by US National Science Foundation grant no. DCB88-11347. I want to thank A. Bennett for helpful comments on the manuscript.
