The self-oscillation paradox in the flight motor of Drosophila melanogaster

Abstract

Tiny flying insects, such as Drosophila melanogaster, fly by flapping their wings at frequencies faster than their brains are able to process. To do so, they rely on self-oscillation: dynamic instability, leading to emergent oscillation, arising from muscle stretch-activation. Many questions concerning this vital natural instability remain open. Does flight motor self-oscillation necessarily lead to resonance-a state optimal in efficiency and/or performance? If so, what state? And is self-oscillation even guaranteed in a motor driven by stretch-activated muscle, or are there limiting conditions? In this work, we use data-driven models of wingbeat and muscle behaviour to answer these questions. Developing and leveraging novel analysis techniques, including symbolic computation, we establish a fundamental condition for motor self-oscillation common to a wide range of motor models. Remarkably, D. melanogaster flight apparently defies this condition: a paradox of motor operation. We explore potential resolutions to this paradox, and, within its confines, establish that the D. melanogaster flight motor is probably not resonant with respect to exoskeletal elasticity: instead, the muscular elasticity plays a dominant role. Contrary to common supposition, the stiffness of stretch-activated muscle is an obstacle to, rather than an enabler of, the operation of the D. melanogaster flight motor.

Keywords: Drosophila; insect flight; self-oscillation; stretch-activated muscle.

Figure 1.

Schematic of the flight motor of D. melanogaster, based on [7,18], with 1 d.f. and planar oscillator modelling assumptions illustrated.

Figure 2.

Motor models fitted to source data for D. melanogaster—quadratic (β) and linear (d, ζ) models, illustrated for the most lightly damped (β = 0.90) and most heavily damped (β = 1.65) load requirement work loops in the source meta-dataset of Pons et al. [12]. The linear model is expressed in ζ rather than d for ease of interpretation.

Figure 3.

Ex vivo muscular data, leading to estimates of muscular negative loss tangent, r. Sinusoidal testing dataset from Viswanathan et al. [46], with associated fit and process-wise composition (Processes A, B, C): (a) storage modulus, (b) loss modulus, (c) negative loss tangent. (d) Strain rate impulse response data for stretch-activation (SA) and shortening-deactivation (SD) from Loya et al. [42], with associated time-independent impulse-response (TIIR) estimate of r overlaid on (c). This TIIR estimate is the largest among the impulse-response data for D. melanogaster surveyed in table 1.

Figure 4.

The self-oscillation paradox in the flight motor of D. melanogaster. (a) Critical levels of damping for the nonlinear model, c_crit, as a function of the exoskeletal stiffness, (ω₀/Ω_*)², and compared with estimates of c for the flight motor (figure 2). (b) Critical levels of damping for the linear model, d_crit, as a function of the exoskeletal stiffness, compared with estimates of motor d (figure 2). (c) The critical muscular scale factor, N_E,crit, required for self-oscillation under ω₀/Ω_* = 0, compared with the estimate of N_E from motor properties (equation (3.4)). Note that the ratios N_E,crit/c_crit and N_E,crit/d_crit are independent of ω₀/Ω_*. (d) The self-oscillation paradox illustrated in empirical nonlinear data: work loops for the motor [12] and muscle [38,44,56], illustrating the difference in relative damping.

Figure 5.

Relationship between self-oscillation frequency and effective resonant frequencies in linear and quadratic motor models, alongside the ex vivo experimental results of Machin and Pringle [4]. (a) Predictions of self-oscillation frequency for the linear model, alongside motor effective resonant frequencies; identified ζ for D. melanogaster; and Machin and Pringle's results. (b) Close-up of the comparison with Machin and Pringle's results, with Cases A–D indicated. (c) Fit results for the work loops recorded by Machin and Pringle, used to identify r for model prediction. (d) Predictions of self-oscillation frequency for the quadratic model, alongside motor effective resonant frequencies.