# Bending moments in swimming fish

## Citation

Voesenek, Cees J.; Li, Gen; Muijres, Florian T; van Leeuwen, Johan L. (2020), Bending moments in swimming fish, Dryad, Dataset, https://doi.org/10.5061/dryad.2280gb5p6

## 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 3D 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 3D 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.

## Methods

**For a detailed description, see our PLOS Biology paper (some parts copied from this paper) and its S1 Text supplement, including references. **

**Video analysis of swimming fish larvae.** We used three batches of 50 zebrafish larvae from 3–12 days post fertilisation (dpf). We recorded free-swimming larvae at 2000 fr/s with three synchronised high-speed video cameras (raw video recording available upon request from the corresponding author). We reconstructed the swimming kinematics from the high-speed videos, using previous published in-house developed automated 3D tracking software in MATLAB (R2013a, The Mathworks)(doi:10.1371/journal). For every time point of the video sequence, the best fit for the larva's 3D position, orientation and body curvature was calculated. These parameters served to calculate the position of the motion of the larva's central axis and its outer surface.

**Near-periodic swimming. **We calculated phase-averaged quantities for an individual swimming sequence to study the generated bending moments and powers. We selected (subset of a) sequence that was near-periodic. For every possible subset of a swimming sequence, we calculated the sum of absolute difference with a time shifted version of the curvature, similar to an autocorrelation. We then calculated extrema in this function – if extrema are detected, their maximum value determines the “periodicity” of the sequence. We then selected the longest possible subsequence that has a periodicity value higher than a threshold of 35 for a swimming sequence of a 3 dpf fish. We divided this sequence in half-phases based on peaks in the body angle. Body curvature, bending moment, fluid power, kinetic power, and resultant power were then phase-averaged based on these subdivisions.

**Analysis of aperiodic swimming.** We divided each swimming bout in half tail-beats, for each of which we determined a mean swimming speed and acceleration. We mirrored all half-beats towards the left side of the fish such that all extracted half-beats were in the same direction. This allowed us to systematically analyse aperiodic motion. We used lateral bending moment patterns to divide the swimming motion into half tail-beats—we defined the start of each half tail-beat as the moment at which the bending moment at 0.5ℓ (0.5 body length) crossed the zero line. Because some of these zero crossings are related to noise, we evaluated every possible permutation of zero crossings per sequence on several criteria with a custom MATLAB (R2018b, The Mathworks) program (see our PLOS Biology paper for details). Out of 113 swimming sequences, we selected 398 half-beats with this procedure, and calculated the duration, mean speed, and peak bending moment. Mean acceleration was calculated as the difference in mean speed (i.e. velocity magnitude) between the following and current half-beat and dividing by the time difference between the tail-beat mid-points. Because we could not calculate mean acceleration for the last half-beat in each sequence, 285 half-beats remained for which we computed all quantities.

**Calculation of fluid forces on the fish.** We used computational fluid dynamics (CFD) to solve the full Navier-Stokes equations numerically. This results in an accurate representation of the fluid force distributions, as all flow scales can be represented numerically. We developed an add-on to the open-source immersed boundary method implementation IBAMR to calculate fluid force distributions around swimming fish. To validate this method and assess its accuracy when calculating internal forces and moments, we used a second, experimentally validated solver (details of the methodology are described in the PLOS Biology paper and its supplement).

**Calculation of bending moments (see S1 Text of our PLOS Biology for details). **To calculate bending moments, we represented the fish by its central axis only. Effects of muscles, spine, and other tissues were combined for every transversal slice along this axis. This simplification allowed us to describe the fish as a non-linear, one-dimensional beam in two-dimensional space. We derived the equations of motion for this beam in an accelerating and rotating coordinate system attached to the fish’s head. We modelled the fish as a beam with varying cross-sections, undergoing arbitrarily large deformation. Plane cross-sections are assumed to remain plane and perpendicular to the neutral line (no shear deformation), but axial deformation is allowed. We assumed that the fish deforms in a single plane (with an arbitrary 3D position and orientation). We used a two dimensional beam model to represent the deformation of the fish in this plane. Because we attached the non-inertial reference frame to the head of the fish, the model contains both translation and rotational accelerations with respect to the inertial reference frame. We took these accelerations into account using fictitious forces. In summary, we modelled the fish as a beam undergoing large bending deformations in two dimensions. We validated the bending moment computations with reference data based on a known external force distribution and internal moment distribution. We also computed the resultant fluid dynamic power on the beam from the derived fluid-dynamic forces, as well as the kinetic energy of the beam.

**Further descriptions can be found in the readme files for each of the supplied data sets.**

## Funding

Aard- en Levenswetenschappen, Nederlandse Organisatie voor Wetenschappelijk Onderzoek, Award: NWO/ALW-824-15-001

Aard- en Levenswetenschappen, Nederlandse Organisatie voor Wetenschappelijk Onderzoek, Award: NWO/ALW-824-15-001

Aard- en Levenswetenschappen, Nederlandse Organisatie voor Wetenschappelijk Onderzoek, Award: NWO/VENI-863-14-007