Power Output From a Flight Muscle of the Bumblebee Bombus Terrestris: II. Characterization of the Parameters Affecting Power Output

The usual way to determine the sustainable power available from muscle is the work loop approach. A work loop is the figure made by plotting muscle force (F) against muscle length (L) over a full cycle. The area defined by the curve from the longest length (L_max) to the shortest length (L_min) during shortening (=∫FdL, evaluated from L_max to L_min) is the work produced by the muscle during shortening; the area defined by the curve during lengthening (=∫FdL, evaluated from L_min to L_max) is the work required to re-extend the muscle. The area defined by the loop over a cycle is the difference between the work done by the muscle and that done on the muscle, or the net work per cycle. The work loops for muscle are obviously analogous to the pressure–volume loops used by cardiac physiologists in determining cardiac work.

The work loop approach for determining power output from skeletal muscle was introduced by Boettiger (1957a,b) and by Machin and Pringle (1959, 1960) in their studies of asynchronous insect flight muscles. In these and later studies with asynchronous muscles, the muscle was activated by tetanic stimulation (or, with glycerinated fibres, by immersing them in a solution containing Ca²⁺ and ATP), and the muscle was allowed to, or was forced to, shorten and lengthen cyclically. Two basic approaches were used to create work loops with asynchronous muscle: free oscillation and driven oscillation. In the free oscillation approach, the muscle is attached to a real or simulated inertial load and, as the name implies, allowed to oscillate at an amplitude and frequency determined by the muscle and the mechanical characteristics of the load. The relevant parameters which determine work output and the shape of the work loop are the elastic force, the damping and the mass of the load. If F is the load, the forces that need to be considered with free oscillations are related to F, dF/dt and d²F/dt², where t is time. In driven oscillation, the muscle is subjected to periodic strain at an amplitude and trajectory determined by the experimenter. Because it is slightly easier to implement, and because it allows greater control of the relevant parameters, driven oscillation has become the only approach used in recent studies. With asynchronous muscle, the relevant parameters with driven oscillation are the amplitude, frequency and trajectory of the applied strain, the mean muscle length and the general conditions of the experiment such as the temperature, the composition of the bathing solution and the like. Extension of the work loop approach to synchronous muscles has used forced oscillations and phasic stimulation of a chosen pattern and duration at a selected time during the length cycle (Josephson, 1985, 1993).

A major difficulty in applying the work loop approach is the large number of independent variables which must be considered. The task is a bit simpler with asynchronous than with synchronous muscles, for with asynchronous muscle tonic activation is adequate and problems of stimulus phase and pattern become irrelevant. But this still leaves muscle length, strain amplitude, strain trajectory, cycle frequency and the potential interactions between these factors as determinants of power output. The effects of these parameters and the interactions between them on the power output of bumblebee flight muscle were the topics of the following investigation.

The procedures used in preparing a muscle, stimulating it, controlling its temperature and recording force from it were generally as described in the preceding paper (Josephson and Ellington, 1997), except that the bumblebee was fixed on its back to a thin plastic disk, 16 mm in diameter, rather than to a microscope slide. This plastic disk was later attached using double-sided sticky tape to a platform on the shaft of an ergometer. The ergometer was a servo-controlled positioning device constructed from a Ling 102A shaker (Malamud and Josephson, 1991). An offset control in the electronics driving the ergometer was used to set overall muscle length. Bursts of sinusoidal position signals to the ergometer were generated by a purpose-built device. Each burst of signals began and ended at the mid-point of the sinusoidal cycle in order to avoid rapid length changes at the on and off of the burst (see Fig. 2). The maximum sinusoidal frequency available from the shaker with the bumblebee in place was more than 250 Hz at an amplitude of 1 mm peak-to-peak. After the bumblebee had been mounted on the ergometer, the force transducer, which was mounted in a manipulator, was positioned over the bumblebee and the hook from the transducer was inserted through the coupling ring attached to the ventral cuticle overlying the muscle insertion (see Josephson and Ellington, 1997). The hook and the coupling ring were then fixed together with melted dental wax to make a rigid linkage between the muscle and the transducer.

The first step in an experiment was to determine an appropriate stimulus intensity. The muscle was set at a length approximating that in an intact bumblebee as judged by the position of the cuticle overlying the muscle insertion relative to the surrounding cuticle. The muscle was then stimulated tetanically (0.5 ms shocks at 50 Hz for 500 ms) in a series of trials with a 2 min interval between trials. The stimulus intensity, which was initially at a subthreshold level, was increased in steps of 50 % from trial to trial until the maximum force in successive trials no longer increased with further increases in stimulus intensity. The stimulus intensity used in the subsequent investigation was 50 % above the minimum needed to evoke the maximum force.

In most experiments, the muscle was stimulated tetanically and, after force had reached a plateau, subjected to 10–20 cycles of sinusoidal length change. Values for muscle force and change in length were collected with an analog-to-digital converter (sampling rate 10 kHz per channel), displayed on a computer, and stored when it was so desired. Work output per cycle was calculated as the length change during a sampling interval times the average force for the interval, summed over all the sampling intervals of a cycle, i.e. ∑(L_i+1−L_i)(F_i+F_i+1)/2, where L_i and F_i are the measured values of length and force at a given sampling time and L_i+1 and F_i+1 are the values at the next sampling time. Work per cycle, usually determined as the average work per cycle for cycles 3–7 of the imposed strain, was calculated on-line and displayed along with muscle force and length. When considering work output during imposed, sinusoidal strain, the terms ‘muscle length’ or ‘average muscle length’ will refer to the base length upon which the sinusoidal length changes were superimposed. Most of the experiments to be described were aimed at characterizing the effects of varying a selected parameter on work output. Typically, this was done in a series of trials in which the value of the relevant parameter was increased progressively from trial to trial, followed by a mirror series of trials during which the parameter was progressively decreased. Generally, trials without muscle stimulation were interposed between trials with tetanic stimulation. The interval between trials with stimulation was 2 min except where otherwise indicated. Work loops obtained during trials with stimulation will be termed ‘active loops’, those from trials without stimulation ‘passive loops’. The muscle temperature was maintained at 30 °C. This is lower than the normal muscle temperature during flight, which is approximately 40 °C (Joos et al. 1991; Heinrich, 1993), but preparations deteriorated much less rapidly at 30 °C than at 40 °C, and using the lower temperature made it possible to examine the effects of a greater range of parameter values with each preparation than would have been possible had the normal flight temperature been used.

When the experiments were complete, the muscle was set at the length found to be optimum for work output (L_opt) and fixed in situ with 70 % ethanol before it was detached from the apparatus. Muscle length, mass and area were determined from the fixed muscle as described by Josephson and Ellington (1997). Muscle stress was calculated as the ratio of force to muscle cross-sectional area, as the latter was determined for the muscle at the optimum length for work output. In these experiments, muscles were subjected to both static and rapid, oscillatory length changes. It would have been possible, but certainly inconvenient, to recalculate the cross-sectional area for each length sampled during a trial and to use the actual area at each sampling instant to calculate the true stress. At no time during an experiment did the muscle length differ by more than 10 % from the optimum length, so using muscle area at the optimum length rather than the instantaneous area did not introduce a very large error into calculations of stress.

The muscle length was set well below the estimated optimum length for work output (L_opt), and work was measured in a series of trials in which the muscle was stimulated tetanically and subjected to sinusoidal length change at 64 Hz and 0.15 mm peak-to-peak amplitude (approximately 3 % muscle length). This frequency and strain were selected for being close to the optimum values for work output found in preliminary experiments. In addition, a strain of 3 % is near the maximal strain experienced by the muscle during tethered flight (Josephson and Ellington, 1997). The onset of sinusoidal oscillation was delayed with respect to the onset of stimulation such that oscillation occurred at the peak of the tetanic contraction. The muscle length was increased from trial to trial until it was clearly beyond the optimum for work output and then decreased in a mirror series. Typically, the length steps were 0.1 mm (approximately 2 % of muscle length) at long and short lengths and 0.05 mm near the optimum length. Length changes were made immediately after each trial with tetanic stimulation, so in each stimulation trial the muscle had been at the new length for approximately 2 min when it was stimulated again. The force from an unstimulated muscle will be referred to as passive force, even though evidence suggests that some of the force of a stretched, unstimulated muscle is due to non-passive activity of muscle cross bridges (Granzier and Wang, 1993a,b; see below). The force immediately before stimulation was taken as the passive force for constructing length–tension curves. The tetanic force was the maximum increase in force above the passive level before the onset of the oscillatory strain.

The work output at L_opt and 64 Hz oscillatory strain was measured in a series of trials during which the sinusoidal length change, set initially at a low value, was increased from trial to trial in steps of either 0.02 mm or 0.025 mm and then decreased in a mirror series.

Work output at L_opt and with a sinusoidal length change of 0.15 mm was determined in a series of trials in which the cycle frequency was progressively increased from trial to trial in steps which were, in different preparations, either 15 % or 25 %; followed by a mirror series in which frequency was progressively decreased.

Work output at L_opt was determined in a series of trials in which the sinusoidal length change was increased in steps of 0.025 mm (approximately 0.5 % of muscle length) from trial to trial until it was clearly beyond the optimum value. The increasing strain series was followed by a mirror series with decreasing strain. At each strain level, work output was measured using oscillatory frequencies of 33, 50 and 75 Hz in successive trials. The goal of these experiments was to determine the optimal strain at different frequencies. Because there were two independent variables (strain and frequency), these experiments were rather lengthy. The size of the strain steps chosen was a compromise between small steps, which would increase the resolution in determining the strain optimum, and large steps, which would reduce the number of trials and the associated decline in muscle performance needed to establish the optimum.

The possibility that there is a relationship between operating frequency and the optimal muscle length for work output was examined by alternately determining L_opt at 32 Hz and at 64 Hz. Sinusoidal length change was 0.15 mm. The muscle length was set on the first trial at a length judged to be slightly shorter than L_opt and increased in steps of 0.02 mm in successive trials until the net work per cycle began to decline. In these experiments, the length changes were not presented in a mirror series, the intertrial interval was 1 min, and there were no interposed trials without stimulation. Determinations of optimum length at the two frequencies were made repeatedly from a preparation until the work output began to decline significantly.

The property of asynchronous muscle that allows oscillatory contraction with positive power output is a time lag between the change in length and the associated change in force. This time lag is seen as delayed shortening deactivation and delayed stretch activation following step changes in length (Fig. 1) and as a phase shift between length change and force during sinusoidal strain, with the resulting production of counterclockwise (i.e. positive work output) work loops (Fig. 2). The delayed force changes in response to step changes in length were not investigated other than to show that they occur and that they are similar to those described from glycerinated muscle fibres from bumblebees by Gilmour and Ellington (1993a). The relationships between force and length during sinusoidal oscillation are the topic of this and the following paper (Josephson, 1997). Several important features about the length–force relationship during sinusoidal oscillation are apparent in Fig. 2. First, as has been found in other asynchronous muscles, and as was shown using a different technique in Josephson and Ellington (1997), unstimulated muscle is stiff and small changes in length result in large changes in force. Second, unstimulated muscle is quite resilient. The passive work loop (Fig. 2Bii) has a small area, indicating that most of the work required to stretch the muscle is returned to the apparatus during shortening, and there is little net work expended over a full cycle. Third, the net work output from the stimulated muscle (the area of the work loop in Fig. 2Cii) is rather small compared with the work required to extend the muscle (the area beneath the lower limb of the work loop, from minimum length to maximum length) and the work done during shortening (the area beneath the upper limb of the loop). The muscle does a substantial amount of positive work during shortening, but most of this is balanced by negative work during lengthening, leaving a relatively small residuum available to power the wings over a full cycle.

Comparable sets of data covering a similar range of length change with similar numbers of length steps within the range were obtained from 10 preparations. One of these preparations, selected as being reasonably representative of the group, is the source of most of the figures used to illustrate the general findings (see Figs 3–5, 7, 8A).

The relationships between muscle length, passive force and the increase in force above the passive level during tetanic stimulation were generally similar to those found using the slightly different approach described in the preceding paper (Josephson and Ellington, 1997). Passive force was close to zero at short muscle lengths, but rose rapidly and nearly linearly with stretch at longer lengths (Fig. 3). The slope of the linearly rising portion was equivalent to a muscle stiffness of 971±98 kN m⁻² (per 100 % length change, ±S.E.M, N=10), which is a little higher, but not statistically significantly so, than the passive stiffness found for bumblebee muscle (730 kN m⁻²) by Josephson and Ellington (1997). At short muscle lengths, the passive force became negative. This is a consequence of the configuration of the preparation, in which the muscle is essentially intact and surrounded by its normal neighbours. To achieve a short muscle length requires stuffing the muscle more deeply into the thorax; the resistance to this compression produces a force which is negative in the reference system used. Active force, the increase in force above the passive level during tetanic stimulation, declined steeply on either side of an optimum length. The optimum length for active force production coincided well with the length at which passive force began to rise rapidly with stretch. The mean active force in these experiments was 35.0±3.7 kN m⁻² (±S.E.M), which is greater than the active force reported in the preceding paper (18.0 kN m⁻²), possibly a consequence of the more rapid time course in the present set of experiments, with less time for muscle deterioration, than in the study of Josephson and Ellington (1997) in which control trials were inserted between each experimental trial.

Stretching and stimulating muscles led to a deterioration in their performance. In eight of the ten preparations, tetanic force was lower during the trials of the decreasing length series than during the equivalent trials of the preceding increasing length series (Fig. 3). The maximum active force during the decreasing length series was lower by an average of 11.9±21.6 % (±S.D., N=10) than that reached during the increasing length series. Because of the damaging effects of stretch, in most preparations the descending portion of the force–length curve was not explored much beyond the optimal length. In five of the ten preparations, the ascending curve was characterized over a range wide enough to define a half-width for the rise, defined as the distance along the length axis beginning at 50 % and ending at 100 % of the maximum force. The muscle length at which the active force was 50 % of its maximum value was estimated by linear interpolation between the two lengths bracketing the 50 % level. The average half-width for the rising curve was 5.9±1.3 % (±S.D.) of the optimum length for force production, which is essentially the same half-width as that found in Josephson and Ellington (1997) (5.2 %).

The average muscle length, the length upon which sinusoidal oscillation was imposed, had a major effect on the position, shape and area (the net work output) of work loops. Fig. 4 gives an overview of the effects of average muscle length on force and work loops during sinusoidal strain. In both stimulated and unstimulated muscles, the work required to stretch a muscle, and the work done by the muscle during shortening, increased as average muscle length increased and the background force became greater. In the length range for which active loops were mostly or all counterclockwise, indicating positive work output, the active loops were maximally open, with greatest net work, at an optimum length and became narrower on either side of this length. Force tended to lag behind strain in the strain cycles from stimulated muscles, particularly so during muscle lengthening and at muscle lengths near L_opt (Fig. 4, column B). Passive loops were relatively broad and concave upwards at short muscle lengths but became narrow and straight at long lengths. At short muscle length, force in the unstimulated muscle lagged behind the strain during lengthening but led the strain during shortening (Fig. 4, column A). With increasing muscle length, the unstimulated muscle behaved increasingly like a purely elastic element with congruence between the length and force curves.

If the bumblebee muscle behaved as a linear system, composed of simple elastic and viscous elements, the force changes in response to imposed sinusoidal strain should also be sinusoidal, possibly shifted in phase with respect to the strain. In fact, at the strain amplitude used, the resulting force was not sinusoidal in either unstimulated or stimulated muscle (seen most clearly in columns A and B of Fig. 4), although it approached being sinusoidal, with a zero phase shift, in the unstimulated muscle at longer lengths. In general, the bumblebee muscle is not a linear system, at least not in the strain range at which the muscle is normally used.

The optimum length for work output from stimulated muscle (L_opt) was similar to that for active isometric force (the ratio of optimum length for work/optimum length for force=101.2±2.1 %, ±S.D., N=10). The decline in work output between the ascending and descending length series, measured at L_opt, averaged 18.9±21 % (±S.D.). The curve relating work per cycle and muscle length (Fig. 5) was even more narrow than that of muscle force versus length (Fig. 5B). In all preparations, there was a range of short muscle lengths at which tetanic stimulation produced active force, but at which work output was negative. In seven of the ten preparations of this set, the length range examined was great enough to allow direct determination of the half-width of the work versus length curve. Lengths at which work output was 50 % of the maximum were estimated by linear interpolation between lengths at which the work output bracketed the 50 % level. The average half-width of the work versus length curve was 3.9±1.2 % (±S.D.) of L_opt.

Although the force of an unstimulated muscle increased rapidly with length in the upper part of the length range tested, and therefore the work required to stretch the muscle increased rapidly with length, the net work absorbed by an unstimulated muscle over a lengthening–shortening cycle was relatively independent of muscle length (Fig. 5A). This is because the passive muscle became increasingly elastic with length. More of the increased work required to stretch the muscle with increasing length was returned to the apparatus during the subsequent shortening, and the net work per cycle remained nearly constant. An ideal elastic element should have a linear stress–strain curve with no hysteresis. A measure of the hysteresis, or the energy lost over a cycle, is resilience, defined as the ratio of the work returned by the system during relaxation to the work originally put into the system to deform it (Vogel, 1988). An ideal elastic element would have a resilience of 100 %. The resilience of the bumblebee muscle increased with muscle length, reaching values of 90–95 % in some preparations (Fig. 6). The maximum resilience for the preparations of Fig. 6 averaged 92.0±2.2 % (±S.D.). Thus, the resilience of whole bumblebee muscle approaches that of collagen (93 %) and is only a little lower than that of resilin (97 %), which is generally regarded as the best of the protein rubbers for its ability to store energy (resilience values from Table 9.5 in Vogel, 1988).

The maximum and minimum forces reached during sinusoidal oscillation changed with average muscle length in somewhat different ways in stimulated and unstimulated muscle (Fig. 7A). In both active and passive muscle, the minimum force during oscillation was negative at very short muscle lengths. With increasing average muscle length, the minimum and maximum forces increased in both stimulated and unstimulated muscle, but the peak force in the unstimulated muscle increased more steeply with increasing average muscle length than did that of the stimulated muscle. At long muscle lengths, the maximum force of the unstimulated muscle approached that of the stimulated muscle (Fig. 7A). The ratio of the change in force to the change in length during oscillatory strain will be termed dynamic stiffness. Because of the different patterns of increasing force change with increasing length, the dynamic stiffness rose more rapidly in the unstimulated muscle than in the stimulated muscle, and in all 10 preparations the dynamic stiffness was greater in the unstimulated than in the stimulated muscle at the longer lengths examined (Fig. 7B). In effect, at long muscle lengths, stimulation of the muscle reduces its dynamic stiffness! The cross-over length at which the dynamic stiffness of active muscle was less than that of passive muscle was usually slightly longer then the optimum length for work output. In seven of the ten preparations, dynamic stiffness at L_opt was greater in stimulated than in unstimulated muscles, and the average dynamic stiffness measured at L_opt was 1.66±0.25 MN m⁻² (±S.E.M.) in stimulated muscles and 1.30±0.25 MN m⁻² (±S.E.M.) in unstimulated ones.

Boettiger (1960) proposed that at least two types of asynchronous muscle could be distinguished physiologically, the distinction being based in part on the position of the freely oscillating work loops with respect to the passive and active force–length curves. According to Boettiger (1960), in Type I muscles, exemplified by bumblebee flight muscle, the work loop lies between the active and passive force–length curves, while in Type II muscles, for which flight muscle of the beetle Oryctes rhinoceros was the example, the work loop overlaps the active force–length curve with its main axis rotated clockwise with respect to the curve. It turns out, however, that the position of the work loops from bumblebee muscle with respect to the force–length curves varies with muscle length, at least for driven work loops (Fig. 8). At short muscle lengths, the work loops do lie principally between the active and passive force–length curves as proposed by Boettiger (1960). As length is increased, however, the work loops come to lie more and more centrally on the active force–length curve. The work loops for the bumblebee muscle are rotated anticlockwise with respect to the force–length curve, rather than clockwise as was found for flight muscle of Oryctes rhinoceros by Machin and Pringle (1959). Boettiger may be correct in his assertion that there are two (or more) types of asynchronous muscle, but the criteria for identifying them may differ from those he proposed.

The work per cycle rose with increasing strain up to a maximum, beyond which work declined with further increase in strain (Fig. 9). The optimum strain was 3.03±0.90 % peak-to-peak (±S.D., N=8). Strain values for 50 % maximum work were determined for the ascending and descending limbs of the curves by linear interpolation between measured points. The mean strain at the 50 % work level was 1.63±0.42 % (±S.D., N=8) on the ascending limb and 4.12±1.06 % (±S.D.) on the descending limb. The mean width of the curves at the 50 % work level was 2.46 %. Thus, an increase or decrease in strain amplitude by only approximately 1.2 % (approximately 0.06 mm) from its optimal value reduces work by half.

The work per cycle absorbed by an unstimulated muscle increased monotonically with strain (Fig. 9). The mean value for passive work at the optimum strain for active work was 72±55 % (±S.D.) of the active work and of the opposite sign.

The net work per cycle increased with increasing oscillatory frequency up to an optimum frequency, beyond which work declined with further increase in frequency (Fig. 10). As expected, power output, which is the product of work per cycle and cycle frequency, reached a maximum at a somewhat higher frequency than did work per cycle. In the seven preparations in which the effects of frequency were adequately characterized, the optimum frequency for work per cycle was 63±13 Hz (±S.D.); the optimum frequency for power output 73±12 Hz (±S.D.). The work per cycle done (absorbed) by an unstimulated muscle either increased monotonically with frequency (Fig. 10A) or remained relatively constant with increasing frequency up to a rather high frequency, 100 Hz or so, beyond which it increased rapidly with further increase in frequency.

The relationship between oscillatory frequency and optimal strain was examined in four preparations. In all four, the optimal strain was nearly equal, approximately 3 %, at 33, 50 and 75 Hz (Fig. 11). In none of these preparations was the optimum sinusoidal length change at any one frequency more than one step (0.025 mm) from that at the other two frequencies. The relationship between maximum work per cycle and cycle frequency was somewhat variable, with 75 Hz giving maximum work in two of the four preparations, 50 Hz in one, while in the remaining preparation the net work per cycle was nearly equal at all three frequencies. In all the preparations, the net work output from unstimulated muscle became increasingly negative with increasing strain and with increasing cycle frequency (Fig. 11).

The optimal muscle length for net work output per cycle was greater at 32 Hz than at 64 Hz, but the difference in optimal lengths at the two frequencies was small and, because of trial-to-trial variability, difficult to quantify. In a total of 15 cases from four preparations in which the cycle frequency was increased from 32 to 64 Hz, the optimal length at the higher frequency was shorter by one or two length steps in 11 instances and at the same length in four (one length step=0.02 mm, approximately 0.4 % of muscle length). In 17 cases in which the cycle frequency was reduced from 64 to 32 Hz, the optimal length at the lower frequency was one or two length steps longer in 12 instances and unchanged in five.

Work output was strongly dependent on the muscle length upon which sinusoidal oscillations were imposed. Increasing or decreasing the average muscle length by only 2 % from its optimum value reduced the net work per cycle by 50 % or more (Fig. 5). Changes in average muscle length had a similarly large effect on the work output from living beetle muscle (Machin and Pringle, 1960) and from glycerinated fibres from flight muscle of the bugs Lethocerus cordofanus and L. annulipes (Pringle and Tregear, 1969). Judging by Fig. 4 in Gilmour and Ellington (1993a), work output from glycerinated fibres of bumblebee muscle is less dependent on average length than is true for living bumblebee muscles.

The sensitivity of work output to muscle length emphasizes a problem, identified by Darwin and Pringle (1959), in obtaining mechanical measurements from intact, asynchronous muscles of insects. Most such muscles, including the bumblebee muscle used in this study, originate and insert on broad areas of cuticle. It is very difficult to avoid some distortion of the muscle when isolating it and mounting its ends so as to make mechanical measurements from it. If an inadvertent cant is given to the cuticular attachment of the muscle, some of the muscle fibres will be shorter and others longer than their individual optimal lengths, with consequent degradation of their performance, even when the muscle as a whole is at its optimal length. Much of the variability in work output from preparation to preparation in the present study is probably a consequence of varying amounts of fibre distortion and resulting heterogeneity throughout the muscle in fibre loading and strain.

The optimum strain for work output per cycle (3 % peak-to-peak) was greater than the strain measured from bumblebee flight muscles during tethered flight (approximately 2 %, range <1–3 %, Josephson and Ellington, 1997). It is possible that muscular effort and strain are lower in tethered flight than in free flight, and that during free flight the muscle strain is approximately 3 %. However, the wing stroke amplitude during tethered flight is similar to that during free flight, suggesting that muscle strain may also be similar. Nor can the discrepancy between actual strain and optimal strain be accounted for on the basis of the lower temperature and operating frequency during the strain measurements as opposed to those that pertain during normal flight. In fact, optimal strain is, if anything, greater at the temperature and frequency of normal flight (Josephson, 1997). Nor can it be argued convincingly that bumblebees normally operate at suboptimal strain and increase the strain to optimal levels when more power is required as, for example, when crop loading increases during foraging. In fact, increased power output in bumblebees is usually achieved by increased wing stroke frequency rather than increased wing stroke amplitude (Cooper, 1993), and it is wing stroke amplitude, among the kinetic parameters of flight, that is most likely to reflect muscle strain. Thus, the available evidence indicates that bumblebees use suboptimal strain during flight, but why they do so is unclear.

Oscillation frequency has long been recognized as a primary determinant of the power output from asynchronous muscles (e.g. Machin and Pringle, 1960; Jewell and Rüegg, 1966; Steiger and Rüegg, 1969). At 30 °C, the net work per cycle from the bumblebee muscle was maximal at an oscillation frequency of 63 Hz, and power output was maximal at 73 Hz (Fig. 10B). The optimum frequency for power output of living bumblebee muscle was similar to that for glycerinated fibres from bumblebees (approximately 59 Hz at 30 °C, estimated from Fig. 7 of Gilmour and Ellington, 1993a). During normal flight, the thoracic temperature in bumblebees is approximately 40 °C and the wing stroke frequency is approximately 150 Hz (Heinrich, 1993; Joos et al. 1991; Cooper, 1993). In glycerinated fibres from bumblebees, the Q₁₀ for the optimum oscillation frequency for power output is approximately 2 (Gilmour and Ellington, 1993a). Allowing a Q₁₀ of 2 for the kinetic properties which determine the optimum operating frequency of living bumblebee muscle would raise the expected optimum frequency for power output at 40 °C to 146 Hz, which is, essentially, the normal flight frequency.

The optimum strain for work output was not dependent in a consistent way on the oscillatory frequency, at least not in the frequency range examined (33–75 Hz, Fig. 11). In general, the optimum strain for work per cycle has not been found to be strongly dependent on oscillatory frequency in asynchronous muscles. Steiger and Rüegg (1969), using glycerinated fibres from Lethocerus maximus, and Gilmour and Ellington (1993a), using glycerinated bumblebee fibres, found that the optimal strain was essentially independent of frequency. In some of the experiments with glycerinated fibres from Lethocerus cordofanus and L. annulipes described by Pringle and Tregear (1969), there was a clear, inverse relationship between optimum strain and frequency (Fig. 5 of that paper), but in other experiments there was not (their Fig. 6). Pybus and Tregear (1975) reported a consistent inverse relationship between optimum strain and frequency for glycerinated Lethocerus cordofanus fibres, but here an increase in strain by a factor of 10, from 0.2 to 2 %, resulted in only a 36 % reduction in optimum frequency (average of values in Table 1 of that paper).

The lack of a strong, inverse relationship between cycle frequency and the strain which maximizes work per cycle in asynchronous muscles is surprising, since an inverse relationship is to be expected on theoretical grounds for muscles operating with ordinary force–velocity properties (Josephson, 1989). The shape of the force–velocity curve for muscles is such that there is an optimum shortening velocity for the mechanical power output of a muscle (Hill, 1938). The total work done per cycle is greatest when the muscle shortens throughout the shortening half-cycle at its optimal velocity. The distance of shortening (=strain) for a muscle contracting at its optimal velocity, and thus producing maximum work per cycle, is directly proportional to cycle duration and inversely proportional to cycle frequency. The shape of the force–velocity curve of muscle is thought to be the result of repetitive cross-bridge steps which are asynchronous throughout the muscle, each step involving cross-bridge attachment and detachment (Huxley, 1957). The lack of a relationship between oscillatory frequency and optimum strain in asynchronous muscles suggests that shortening in these muscles may involve mechanisms of shortening other than the usual ones. Perhaps, as has been previously suggested (Gilmour and Ellington, 1993b), oscillatory shortening in asynchronous muscles may involve a single cross-bridge step per cycle, presumably in the bumblebee a step creating a strain equivalent to a total shortening of approximately 3 %.

Those asynchronous muscles that have been examined have been found to be relatively inextensible, even when unstimulated and at or near their normal in vivo length (Machin and Pringle, 1959; Boettiger, 1960; White, 1983; Granzier and Wang, 1993a,b). High resting stiffness appears to be a general feature of the asynchronous mechanism. The active stress generated when asynchronous muscle is stimulated is relatively low compared with that in synchronous striated muscles (discussed in Josephson and Ellington, 1997), and the force of extension due to stiffness is high; consequently, the force attributable to stiffness is a major component, and often the principal component, of the force experienced by asynchronous flight muscles undergoing length changes similar to those of normal flight (e.g. Fig. 4).

Muscle stiffness is a complex variable, in the mathematical sense of the term. The relationship between strain and stress can be expressed as the resultant of a component for which stress and strain are in phase (the elastic modulus) and a component for which stress and strain are in antiphase (the viscous modulus; Machin and Pringle, 1960). Both the elastic and the viscous moduli, and consequently the complex stiffness, are functions of frequency. In the living bumblebee flight muscle, the viscous modulus became quite small at longer muscle lengths under the conditions at which it was examined (64 Hz, approximately 3 % strain), and the muscle behaviour approached that of a purely elastic element (Figs 4, 6). This elastic behaviour may not be entirely passive. Stretching an activated, asynchronous muscle can result in increased activation as judged by increased ATPase activity (reviewed in Tregear, 1975). It is possible that stretching the unstimulated bumblebee muscle resulted in partial muscle activation and an increased force during the following shortening phase of the cycle; force which partially negated viscous losses and resulted in a force trajectory during shortening that lay close to that during lengthening. The high resilience of bumblebee muscle may be a dynamic consequence of length-dependent activation, rather than a passive consequence of the properties of molecules contributing to the stiffness.

Muscle stiffness is most often assayed with large-amplitude, very low-frequency strain, as in the construction of length–tension curves, or with small-amplitude strain at frequencies sufficiently high (typically greater than 1 kHz) relative to cross-bridge kinetics that the measured stiffness reflects the number and properties of attached cross bridges. In asynchronous muscle fibres from the bug Lethocerus (L. griseus, L. uhleri and L. indicus), the low-frequency, passive stiffness is basically unaffected by chemical removal of actin filaments. The low-frequency stiffness appears to be due almost entirely to the properties of C filaments, composed of minititin, which connect the thick filaments to the Z-line (Granzier and Wang, 1993a,b). High-frequency, low-amplitude stiffness, in contrast, is reduced by actin filament removal or by blocking actomyosin interaction. At high frequency, the stiffness is determined by both C filaments and weak cross bridges formed between thick and thin filaments. In living bumblebee flight muscle, the dynamic stiffness, assayed at an intermediate frequency (64 Hz) and at a moderate strain (approximately 3 %), is, at long lengths, lower in stimulated than in unstimulated muscles. This reduction in stiffness with activation indicates that not all the components responsible for the stiffness are passive elements parallel to and independent of the contractile apparatus. Rather, some part of the passive stiffness must reflect contributions of components within the muscle fibre whose properties change upon stimulation, presumably the cross bridges between the thick and thin filaments.

Two, not mutually exclusive, hypotheses can be offered. (1) Much of the stiffness in an unstimulated bumblebee muscle, as in passive Lethocerus muscle, is due to cross bridges formed between the thick and thin filaments. The increased Ca²⁺ level attendant on activation changes the kinetics of the cross-bridge interactions such that, at the test frequency and strain amplitude, cross bridges dissociate more rapidly and contribute less to stiffness in stimulated than in unstimulated muscles. (2) Stretch of unstimulated muscle activates it, increasing the number of cross bridges formed between the thick and thin filaments. Stretch is less effective in activating stimulated than unstimulated muscles because in stimulated muscles more of the potentially active cross bridges are already participating. The relatively greater increase in cross-bridge activation with stretch in unstimulated muscles results in a greater increase in force with stretch in the unstimulated muscles and a greater ratio of change in force to change in length, which is interpreted as increased stiffness.

A major component in the energy budget of insect flight is inertial work: the work required to accelerate the mass of the wings, and the mass of the air which is coupled to and moves with the wings, during the first half of an upstroke or downstroke (Weis-Fogh, 1961, 1973; Ellington, 1984; Casey and Ellington, 1989). The costs of flight would be minimized if the kinetic energy acquired by the wings during their acceleration were to be converted into aerodynamic power during the deceleration phase of the upstroke or downstroke, or if the kinetic energy were to be stored elastically during the deceleration so as to be available to accelerate the wing during the first half of the following return stroke. Measurements of power output and efficiency of flight muscle in comparison with the aerodynamic power requirements of flight indicate that there is substantial elastic storage of the inertial energy of the wings during their deceleration (Josephson and Stevenson, 1991; Dickinson and Lighton, 1995). Pads and tendons of the rubber-like protein resilin appear to be major inertial energy storage elements in the flight systems of locusts and dragonflies which use synchronous muscles (Weis-Fogh, 1960). The results from the bumblebee muscle, which show the flight muscle to be stiff and remarkably resilient, confirm earlier suggestions that a major site of elastic storage in asynchronous muscle is the muscle itself (Alexander and Bennet-Clark, 1977; Ellington, 1984; Alexander, 1995). Further, as was emphasized by Ellington (1984), inertial work is not simply an unavoidable consequence of moving the wings, but rather it is essential for oscillatory contraction.

The force traces of Figs 2 and 4 will provide the context for considering elastic storage and inertial forces in the flight system. During tethered flight, and presumably during normal flight as well, the wing muscles of bumblebees undergo nearly sinusoidal length changes with a peak-to-peak strain of 1–3 % (Josephson and Ellington, 1997). The muscle of Figs 2 and 4 was subjected to sinusoidal strain at approximately 3 % peak-to-peak amplitude, so the strain amplitude and trajectory are like those of flight. The muscle temperature, operating frequency and total power output during the trials which provided these figures were lower than those of a free-flying animal, but the shape of the force traces and the work loops in these figures were quite similar to those obtained from preparations in which conditions mimicked closely those of free-flying animals (see Josephson, 1997). It is reasonable to assume that the strains, forces and work loops from muscles in intact, flying animals are similar to those of the stimulated muscle in Figs 2 and 4. The questions to be addressed are as follows. What properties of the flight system lead to the regular, oscillatory changes in force and length? What sustains these forces and prevents them from dying out? How is it that the maximum force during oscillation reaches values substantially greater than the isometric force level?

As a simplified approximation, the wing of a flying insect can be regarded as pendulum which is being pulled upon by antagonistic muscles whose mechanical properties are largely those of a pair of moderately stiff springs (Fig. 12). At the equilibrium position, the force in the two springs will be identical and, if the muscles have been activated, the force will be the isometric tetanic force of the muscles at the equilibrium length. If the pendulum is displaced and released, it will oscillate at a frequency which depends on the mass of the pendulum and the stiffness of the springs. In this report, I have not shown that the wing stroke frequency is the resonant frequency of the wing and thorax, but this interpretation is strongly supported by the many demonstrations that mechanically unloading the wings by surgically shortening them in insects with asynchronous flight muscles changes the wing stroke frequency in the way to be expected for a resonant system (Sotavalta, 1947, 1952, 1953; Roeder, 1951; Greenewalt, 1960; partial wing ablation has little effect on frequency in insects which use synchronous flight muscles). During oscillation, each of the springs will be alternately stretched beyond and allowed to shorten below the equilibrium position. It is the inertial force of the oscillating pendulum which stretches a spring beyond its equilibrium position and results in force greater than the equilibrium (=isometric) level. Because of viscous forces acting on the pendulum, the oscillations of the pendulum would be damped and the pendulum would eventually come to rest if the system were purely passive. Viscous forces are related to, and in the simplest case proportional to, the velocity of movement. Because viscous forces depend on velocity rather than length, there is a phase shift between viscous forces acting on the pendulum and the displacement of the pendulum; the viscous force leads the displacement. If the oscillation is to be sustained, something must counteract the viscous force. In the flight system, it is delayed shortening deactivation and delayed stretch activation in the wing muscles that oppose viscous damping. Delayed shortening deactivation and stretch activation cause force to lag behind the length change (Fig. 4), rather than to lead the length change as in ordinary viscous systems. Shortening deactivation and stretch activation act, essentially, as negative viscosity. Ordinary viscous forces dissipate the kinetic energy of a system. Negative viscosity increases the kinetic energy, acting effectively as a motor, and provides the mechanism by which metabolic energy from ATP becomes converted into mechanical energy and movement.

This study was supported by grant IBN-9104170 from the National Science Foundation. I want to thank Dr A. Cooper for being a valuable source of information on bumblebees, Dr J. Malamud for helpful comments on the manuscript and especially Dr C. P. Ellington for his continued support and encouragement.