Experimental-numerical method for calculating bending moments in swimming fish shows that fish larvae control undulatory swimming with simple actuation

Abstract

Most fish swim with body undulations that result from fluid-structure interactions between the fish's internal tissues and the surrounding water. Gaining insight into these complex fluid-structure interactions is essential to understand how fish swim. To this end, we developed a dedicated experimental-numerical inverse dynamics approach to calculate the lateral bending moment distributions for a large-amplitude undulatory swimmer that moves freely in three-dimensional space. We combined automated motion tracking from multiple synchronised high-speed video sequences, computation of fluid dynamic stresses on the swimmer's body from computational fluid dynamics, and bending moment calculations using these stresses as input for a novel beam model of the body. The bending moment, which represent the system's net actuation, varies over time and along the fish's central axis due to muscle actions, passive tissues, inertia, and fluid dynamics. Our three-dimensional analysis of 113 swimming events of zebrafish larvae ranging in age from 3 to 12 days after fertilisation shows that these bending moment patterns are not only relatively simple but also strikingly similar throughout early development and from fast starts to periodic swimming. This suggests that fish larvae may produce and adjust swimming movements relatively simply, yet effectively, while restructuring their neuromuscular control system throughout their rapid development.

Fig 1. Procedure for calculating bending moments.

(A) Larval zebrafish motion is reconstructed from synchronised three-camera high-speed video. Video frames (background) from the three-camera–setup overlaid with projections (blue) of the reconstructed model fish. The legend at the top indicates which camera produced the video frame. (B) Reconstructed three-dimensional motion from the video, projected onto the x-z plane; the highlighted time instant is shown in (A) and (C–E). (C) Transparent isosurfaces of vorticity for the same motion as (B) calculated with CFD. (D) Total fluid dynamic stress distribution on the skin, the magnitude of the sum of the pressure, and shear stress contributions calculated from the flow field from CFD. (E) Fluid dynamic force distribution transformed to the two-dimensional coordinate system attached to the deformation plane. This distribution is used as input to reconstruct internal moments and forces. (F) Reconstructed bending moment distributions (colour) along the fish (abscissa) and over time (ordinate). The grey lines separate the half phases in which the bending moment was divided. The green line links a single half phase to a data point in (G). (G) The mean speed (abscissa), mean acceleration (ordinate), and body length (colours) for individual half beats in the data set (N = 285). The green data point corresponds to the highlighted half beat in (F). Underlying data for panel G can be found in S1 Data. CFD, computational fluid dynamics.

Fig 2. Comparison of the present beam model with a small-amplitude model [17,19,20].

(A–C) Bending moment distribution (colours) along the length of the fish (s, abscissa) and over the tail-beat duration (ordinate) for the ‘large-amplitude’ reference solution (A), reconstructed with the small-amplitude model (B) and reconstructed with the present method (C). (D–F) Traces of the bending moment at a single time instant along the length of the fish (left panels; corresponding to the horizontal dashed line in [A–C]) and in the middle of the centreline over the tail-beat duration (right panels; corresponding to the vertical dashed line in [A–C]) for large-amplitude (D), medium-amplitude (E), and small-amplitude (F) motion.

Fig 3. Effect of motion amplitude on reconstruction accuracy.

(A–B) Average (A) and maximum (B) bending moment reconstruction error with respect to the maximum for the small-amplitude model (red) and the present method (blue) for increasing motion amplitudes. (C–E) Motion as represented by the small-amplitude model (red) and the present method (blue) for increasing motion amplitude. Note that because the small-amplitude model only uses a lateral displacement, the centreline changes length throughout the motion. (F) Example body shape of a 3-days-postfertilisation zebrafish larva performing a C-start. Underlying data for panels A and B can be found in S1 Data.

Fig 4. Near-periodic sequence of a 3-days-postfertilisation zebrafish larva.

The larva swam at 31 ℓ s⁻¹ with a tail-beat frequency of 69 Hz. (A) Centreline motion throughout the sequence. The colours indicate half phases. The coordinates were transformed to a best-fit plane through all points along the centreline throughout the motion. (B) Motion during a single full tail beat (half beat 1 and 2) of the swimming sequence; the grey centrelines (from light to dark grey) correspond to the time points shown with horizontal lines in (C, D) and (F–‍H). The diamond (head) and dots on the centrelines correspond to points on the abscissa of (C–H). (C, D, F–H) Heat maps of distributions (colours) along the fish (abscissa) and over the phase over the tail beat (ordinate); all quantities are averaged over separate half beats; ‘negative’ half beats are mirrored for the curvature and bending moment. (C) Body curvature normalised by body length. (D) Bending moment. (E) Transverse muscle area distribution along the fish. (F) Fluid power per unit body length (power exerted by the fish to move the fluid). (G) Kinetic power (rate of change in kinetic energy) per unit body length; note the difference in scale with panel F. (H) Resultant power per unit length; the sum of the fluid and kinetic power. Underlying data for panel E can be found in S1 Data.

Fig 5. Swimming effort and vigour for all analysed half tail beats.

(A) The mean resultant power over a half beat as a function of swimming effort (N = 398). (B) Swimming vigour as a function of effort, and the optimal linear fit (N = 285). (C) Mean speed over a half beat as a function of swimming effort, with colour indicating the mean acceleration (N = 285). (D) Mean acceleration over a half beat as a function of swimming effort, with colour indicating the speed (N = 285). Underlying data for panels A–D can be found in S1 Data.

Fig 6. Bending moment patterns are similar across swimming vigour and development.

(A) Normalised bending moment pattern along the fish (abscissa) and over normalised time (ordinate) averaged over all half beats (N = 398). The dashed lines indicate slices through the pattern in time (green) and location (purple) of the centre of volume of the distribution, shown respectively at the top and right of the heat map, along with their standard deviation. The grey lines over the heat map show the zero and maximum contour line for the middle 60% along the body. (B) The SD of the normalised bending moment along the fish (abscissa) and over normalised time (ordinate) (N = 398). (C, D) The location along the body of the centre of volume of the bending moment as a function of body length ([C] N = 398) and swimming vigour ([D] N = 285). (E, F) Normalised time of the centre of volume of the bending moment as a function of body length ([E] N = 398) and swimming vigour ([F] N = 285). (G) Half-beat duration as a function of length (i.e., developmental stage), with colour indicating swimming vigour (N = 285). (H) Peak bending moment as a function of length, with colour indicating swimming vigour (N = 285). Underlying data for panels C–H can be found in S1 Data. n.s., not significant; SD, standard deviation.

Fig 7. Swimming control parameters.

(A, B, C) Individual half beats in the duration–peak bending moment landscape. The coloured background with white contour lines shows the swimming effort (N = 285). (A) Dot colour indicates acceleration. (B) Dot colour indicates speed. (C) Dot colour indicates swimming vigour. (D, E) Relative occurrence of half beats in bins (line markers) of peak bending moment (D) and half-beat duration (E) divided in 3 bins of swimming effort (colours), each containing the same number of half beats. Underlying data for panels A–E can be found in S1 Data.