Skip to main content
. 2024 Feb 26;12:RP89968. doi: 10.7554/eLife.89968

Figure 5. Finely tuned energy storage in the supracoracoideus tendon simplifies the function of both major flight muscles.

(A) Our power model was derived by combining the external and internal force balance shown with the associated kinematic velocities (not shown); see equations in Figure 7—figure supplement 1. Approximately all internal flight forces and associated power are produced by the two primary flight muscles, the pectoralis (pink; F→pect) and the supracoracoideus (orange; F→supra), which move the mass of the body (green; mB: modeled as a point mass) and wings (green; mW: modeled as 20 point masses displayed in Figure 1C) to generate the wings' aerodynamic forces (blue; F→aero) and thus power to sustain flight. (B–D) The pectoralis (pink) generates positive power during the downstroke and acts like a brake to absorb negative power during the late-upstroke (power is normalized by pectoralis mass). The supracoracoideus (orange) generates and/or releases positive power during the upstroke. Any energy released by the supracoracoideus tendon during the upstroke must first be stored during the downstroke, which is plotted as negative power. (B) Without energy storage in the supracoracoideus tendon, there is an unfavorable mid-downstroke dip in the required pectoralis output power. (C) If energy storage is poorly timed during the downstroke, the required pectoralis output power unfavorably spikes. (D) If the timing and quantity of energy storage are finely tuned, the pectoralis work loop is more favorably shaped mid-downstroke as it more closely approximates the theoretically ideal rectangular work loop (yellow rectangle in [E]) (E) The finely tuned elastic storage shown in (D) corresponds with the black work loop and results in the highest pectoralis shape factor (0.73) for the range of elastic storage parameters we simulated in (F), with stars corresponding to the loops in (E). Numbers between parenthesis in [E]: shape factor; dark gray shading: electrical pectoralis activation; light gray shading: downstroke phase. The shape factors of both the no-storage (brown: [B]) and finely tuned storage (black: [D]) cases exceed prior experimental shape factor measurements, of doves flying similarly slow (Tobalske et al., 2003). (G) Work loop shape factor as a function of elastic energy storage timing and fraction (color codes energy storage fraction): see Figure 5—figure supplements 1–3 for a detailed overview.

Figure 5.

Figure 5—figure supplement 1. Using our energy storage model (Figure 5A), we analyze how differing amounts of energy storage in the supracoracoideus tendon would affect how the total time-resolved required muscle power (Figure 4B) is split between the pectoralis and supracoracoideus.

Figure 5—figure supplement 1.

To isolate the effect of energy storage amount, which is an important question in the literature (Tobalske and Biewener, 2008), we set the timing of the energy storage constant for now. Specifically, we vary the prescribed fraction of power that the supracoracoideus tendon stores for release during the upstroke and model the pectoralis to produce and transfer this extra power into the supracoracoideus tendon during the mid-to-late downstroke. We find that a power increase of 24.9% ± 17.5% in the pectoralis muscle would enable it to fully power the upstroke by tensioning the supracoracoideus tendon during the downstroke. The color schemes for the two positive and two negative power modes are: blue for generated power (the muscle needs to fully produce this positive power), bright green for released power (the muscle has elastic energy stored up that it can release as positive power), dark green for stored power (the muscle elastically stores negative power), and red for dissipated power (the muscle absorbs this negative power, acting as a brake). N = 4 doves; n = 5 flights each; gray region indicates second downstroke after takeoff; all power scaled by pectoralis mass; vertical dashed lines indicate pectoralis strain rate equals zero. (A–D) The mid-downstroke dip in power generated by the pectoralis (pect) is flattened as extra power is generated and transferred to tension the supracoracoideus. (E–H) When the pectoralis tensions the supracoracoideus tendon during the downstroke, the supracoracoideus (supra) does not need to generate as much power during the upstroke. (I–L) Time-resolved pectoralis (pink) and supracoracoideus (orange) total power, each scaled by the pectoralis mass, are summarized from rows 1 and 2, respectively. Total power equals the summation of the power from the four power modes. By transferring pectoralis power into the supracoracoideus tendon and storing it as potential energy, the pectoralis power buildup does not have to dip midstroke. (M–P) The shape factor (ratio of the observed area to the area of a rectangle with the same range of stress and strain; yellow rectangle shows the work loop with a shape factor of one with the same range of strain) of the pectoralis work loop improves locally as the energy storage fraction increases and flattens the mid-downstroke dip in pectoralis power. The modeled work loop of the pectoralis (black) is compared to the work loop for 67% supracoracoideus energy storage (column 3, light gray; corresponds to the dove avatar in Q and R). The portions of the work loop corresponding to electrical activation of the pectoralis are shaded in dark gray, while the downstroke is shaded light gray. We also compare our derived pectoralis work loop to the work loop measured for doves in similar conditions using a strain gauge mounted on the deltopectoral crest (DPC) of the humerus to estimate pectoralis stress (light blue) (Tobalske et al., 2003). The present pectoralis work loop shape factor is 0.70 according to our baseline recordings (no elastic storage added), whereas the earlier value based on DPC strain gauge recordings was 0.62. The strain gauge-based recordings also underestimate the required stress level to sustain the external aerodynamic and inertial power in vivo (especially considering our analysis excludes minor additional internal musculoskeletal power transfer losses). (Q) As the fraction of elastically stored supracoracoideus power increases, the stroke-averaged absorbed pectoralis power remains constant, while the stroke-averaged generated power increases by up to 24.9% ± 17.5%. The maximum generated pectoralis power (black) increases even more (31.9%), based on the peak required to tension the supracoracoideus tendon during the late downstroke. Notably the pectoralis work loop shape factor remains relatively constant throughout because its value lower than 1 is primarily caused by the simultaneous pectoralis strain rate and power buildup around the start of the downstroke (5C). (R) As the fraction of elastically stored supracoracoideus power increases, the stroke-averaged released power increases linearly, allowing the stroke-averaged generated power and maximum generated power (black) to decrease linearly. The gray stars in (Q) and (R) correspond to the four storage cases in the associated columns (A–P).
Figure 5—figure supplement 2. Using our energy storage model (Figure 5A), we find that spreading out the energy storage in the supracoracoideus tendon during the downstroke substantially improves the shape factor of the pectoralis muscle work loop by better flattening the mid-downstroke dip in required muscle power (Figure 4B), resulting in a peak shape factor of 0.73.

Figure 5—figure supplement 2.

Whereas in Figure 5—figure supplement 1, we varied the energy storage fraction, now we hold energy storage constant at 67% (column 3 of Figure 5—figure supplement 1) and vary the less considered and more nuanced timing of the energy storage in the supracoracoideus tendon during the downstroke. The format of this figure is the same as Figure 5—figure supplement 1; column 4 of this figure is identical to column 3 of Figure 5—figure supplement 1: dove avatar in (Q) and (R) indicates identical parameters from column 3 of Figure 5—figure supplement 1. (E–H) Vertical dotted lines are added to highlight the varying timing of energy storage in the supracoracoideus tendon. (A, E, I, M) If the supracoracoideus tendon is tensioned rapidly (10% the length of the stroke; 17% the length of the downstroke), this causes a costly late-downstroke spike in pectoralis power generation. (B, F, J, N) If the supracoracoideus tendon is tensioned over a longer portion of the downstroke (14% the length of the stroke; 23% of the downstroke), late-downstroke spike in pectoralis power generation is lessened. (C, G, K, O) This trend continues as the supracoracoideus tensioning is spread out more (20% the length of the stroke; 33% of the downstroke). (D, H, L, P) If the supracoracoideus tendon is tensioned over 28% of the stroke (46% of the downstroke), the spike in pectoralis power needed for energy storage overlaps the mid-downstroke dip in required muscle power (Figure 4B). Hence, the mid-downstroke pectoralis power is smoother and more closely resembles direct pectoralis power measurements for other birds (Tobalske and Biewener, 2008). (Q, R) The stroke-averaged power remains constant when all that changes is the timing of the elastic storage. However, the peak generated pectoralis power is higher when the elastic storage region is short. (Q) The pectoralis work loop shape factor is maximized when the supracoracoideus storage time is more spread out over the downstroke. The gray stars in (Q) and (R) correspond to the four storage cases in the associated columns (A–P).
Figure 5—figure supplement 3. Using our energy storage model (Figure 5A), we find that for supracoracoideus energy storage fractions above 35%, a pectoralis work loop shape factor of over 0.72 can be achieved by spreading the storage time in the supracoracoideus tendon out over 31% of the stroke.

Figure 5—figure supplement 3.

To gain insight into the tradeoffs between generated muscle power and elastic energy release, we vary two parameters in this model: the fraction of supracoracoideus power which is elastically stored and released (x-axis; columns in Figure 5—figure supplement 1 correspond to stars along the horizontal lines; 100 grid points), and the time period over which energy is stored in the supracoracoideus tendon (y-axis; columns in Figure 5—figure supplement 2 correspond to stars along the vertical lines; 51 grid points). (A) The stroke-averaged pectoralis-generated power is mostly a function of the energy storage fraction and increases gradually as energy storage is increased. (B) The stroke-averaged supracoracoideus-generated power is only a function of the energy storage fraction and decreases linearly as energy storage is increased. (C) The maximum pectoralis-generated power rapidly peaks when the energy storage fraction is high and the energy storage time period is short. (D) The maximum supracoracoideus-generated power is only a function of the energy storage fraction and decreases linearly as energy storage is increased. (E) Pectoralis work loop shape factors (ratio of the observed area to the area of a rectangle with the same range of stress and strain) near the maximum shape factor of 0.73 can be achieved at multiple different energy storage fractions. The maximum shape factor for each energy storage fraction is plotted as a thick black line.