How fish larvae swim: from motion to mechanics

Most of the world's 34,000 known fish species are undulatory swimmers. Their body undulations are produced by fluid-structure interaction between water and the body of the fish, powered by its muscle system. Despite these complex physics, just-hatched fish larvae can already produce effective swimming motion. How they do this is not yet fully understood. With this thesis, we aim to contribute to answering this question by examining the biomechanics of swimming of early-development larval zebrafish. With novel experimental and computational techniques, we reconstructed the dynamics of the larvae from high-speed video. These analyses highlight the challenges that larval fish face during swimming, and how the larvae have evolved to solve these challenges. In chapter 2 we reviewed the mechanics of swimming of larval fish. We examined the functional demands on the locomotory system of fish larvae: immediately after hatching, fish need to escape predators, search and hunt for food, and migrate and disperse. These demands need to be fulfilled by the larvae while undergoing large changes in their bodies, both internal and external. Furthermore, the swimming speed and size of many larvae causes them to be in the intermediate flow regime, where the nature of the flow changes considerably with changes in size or speed. In this chapter, we integrated previous literature to gain insight into how these functional demands on the locomotory system are met with the advantages and limitations of their developing bodies and the changing hydrodynamic regime. In chapter 3, we analysed near-periodic swimming of zebrafish larvae with two-dimensional inverse dynamics from motion that was manually tracked from high-speed video images. We used these data to show how the intermediate flow regime affects the swimming dynamics of fish larvae. We used the Reynolds number, which indicates the relative importance of viscous forces to inertial forces, to characterise the flow regime that the larvae swim in. Furthermore, we applied the Strouhal number, a measure of the ratio of the approximate lateral tail speed to the forward swimming speed, to express changes in swimming kinematics. We found that the Strouhal number depends inversely on the Reynolds number. Fish swimming at low Reynolds numbers tend to use relatively high Strouhal numbers, indicating that their tail-beat amplitude and frequency are high. Even the larvae swimming at the highest Reynolds numbers still use relatively high Strouhal numbers (around 0.72) compared to adult fish (typically 0.2–0.3). Swimming at intermediate Reynolds numbers is associated with high drag, requiring the larvae to use high tail-beat amplitudes and frequencies (and therefore Strouhal number) to produce sufficient thrust. This mode of swimming requires relatively high-amplitude yaw torques, resulting in large angular amplitudes and an expected high energetic cost of transport: the small size of the larvae is a burden to their swimming. Most of the previous research on fish swimming, including our chapter 3, has been done two-dimensionally. However, fish can perform complex, three-dimensional motions to escape predators, search or hunt for food, or manoeuvre through the environment. To expand our analyses to the third dimension, we developed a method to reconstruct the 3D motion of fish from multi-camera high-speed video, described in chapter 4. With an optimisation algorithm we find the 3D position, orientation, and body curvature that best fits the high-speed video frames. We demonstrated that the method allows us to reconstruct the swimming kinematics with high accuracy, while requiring minimal manual work. In addition, we developed a novel method to calculate resultant hydrodynamic forces and torques from the reconstructed motion. The described method is a valuable tool for analysing the biomechanics of swimming, providing data for future analyses of fish swimming. In chapter 5, we apply this automated tracking method to analyse fast starts of zebrafish larvae five days after fertilisation. To be able to escape predators, the main functional demands on a fast start are producing sufficient speed within a narrow time frame and being able to generate a wide range of escape directions. To investigate how these demands are met, we used a five-camera high-speed video of fast-starting zebrafish larvae with unprecedented spatiotemporal resolution. From these videos, we reconstructed the 3D motion of the larvae and the resultant hydrodynamic forces and torques. Due to their undulatory swimming style, the larvae first need to bend into a C-shape before being able to produce a propulsive tail beat. For this reason, the first stage of the start is often considered ‘preparatory’. Based on the reconstructed forces and torques, we show that the first stage of the start, in addition to its preparatory role, also serves to provide most of the reorientation of the start. After this stage, the larvae unfold their bodies, moving their tails at high speeds and thus producing large propulsive forces. The turn angle produced during a start mostly depends on the amount of body curvature in the first stage, while the escape speed mainly depends on the duration of the start. This suggests that larvae are able to independently adjust the direction and speed of their escape. Fish larvae are able to produce these escape responses and the subsequent swimming bout immediately after hatching, despite their bodies and brains still undergoing development. To understand how this is possible, we use an advanced inverse-dynamics approach, with computational fluid dynamics and a large-amplitude beam model, to reconstruct internal mechanics from the motion of the fish in chapter 6. We compute the internal bending moments from more than 100 3D-recordings of swimming over a range of developmental stages. We show that larvae use similar bending moment patterns across development, speeds and accelerations. By varying the amplitude and duration of this pattern, the larvae can adjust their swimming speed and/or acceleration. This similarity suggests that their muscle activation patterns are also similar, which would help to explain how just-hatched larvae with limited neural capacity can produce effective swimming motion across a range of speeds and accelerations. In this thesis, we demonstrated that larval fish swim in a challenging hydrodynamic regime. Despite the relatively high drag, they can produce effective swimming motions to help them survive to adulthood. We developed novel methods to quantify this motion in 3D, and from it reconstructed the external and internal mechanics. With these inverse-dynamics approaches, we show that fish larvae can likely adjust their swimming in a relatively simple way, for both fast starts and continuous swimming. Thus, complex physics do not obstruct developing larvae from swimming effectively.</p

Data from: How body torque and Strouhal number change with swimming speed and developmental stage in larval zebrafish

Small undulatory swimmers such as larval zebrafish experience both inertial and viscous forces, the relative importance of which is indicated by the Reynolds number (Re). Re is proportional to swimming speed (vswim) and body length; faster swimming reduces the relative effect of viscous forces. Compared with adults, larval fish experience relatively high (mainly viscous) drag during cyclic swimming. To enhance thrust to an equally high level, they must employ a high product of tail-beat frequency and (peak-to-peak) amplitude fAtail, resulting in a relatively high fAtail/v̅swim ratio (Strouhal number, St), and implying relatively high lateral momentum shedding and low propulsive efficiency. Using kinematic and inverse-dynamics analyses, we studied cyclic swimming of larval zebrafish aged 2–5 days post-fertilization (dpf). Larvae at 4–5 dpf reach higher f (95 Hz) and Atail (2.4 mm) than at 2 dpf (80 Hz, 1.8 mm), increasing swimming speed and Re, indicating increasing muscle powers. As Re increases (60 → 1400), St (2.5 → 0.72) decreases nonlinearly towards values of large swimmers (0.2–0.6), indicating increased propulsive efficiency with vswim and age. Swimming at high St is associated with high-amplitude body torques and rotations. Low propulsive efficiencies and large yawing amplitudes are unavoidable physical constraints for small undulatory swimmers

Biomechanics of swimming in developing larval fish

Most larvae of bony fish are able to swim almost immediately after hatching. Their locomotory system supports several vital functions: fish larvae make fast manoeuvres to escape from predators, aim accurately during suction feeding and maymigrate towards suitable future habitats. Owing to their small size and low swimming speed, larval fish operate in the intermediate hydrodynamic regime, which connects the viscous and inertial flow regimes. They experience relatively strong viscous effects at low swimming speeds, and relatively strong inertial effects at their highest speeds. As the larvae grow and increase swimming speed, a shift occurs towards the inertial flow regime. To compensate for sizerelated limitations on swimming speed, fish larvae exploit high tail beat frequencies at their highest speeds, made possible by their low body inertia and fast neuromuscular system. The shifts in flow regime and body inertia lead to changing functional demands on the locomotory system during larval growth. To reach the reproductive adult stage, the developing larvae need to adjust to and perform the functions necessary for survival. Just after hatching, many fish larvae rely on yolk and need to develop their feeding systems before the yolk is exhausted. Furthermore, the larvae need to develop and continuously adjust their sensory, neural and muscular systems to catch prey and avoid predation. This Review discusses the hydrodynamics of swimming in the intermediate flow regime, the changing functional demands on the locomotory system of the growing and developing larval fish, and the solutions that have evolved to accommodate these demands.</p

Bending moments in swimming fish

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.,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.

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

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

Automated Reconstruction of Three-Dimensional Fish Motion, Forces, and Torques.

Fish can move freely through the water column and make complex three-dimensional motions to explore their environment, escape or feed. Nevertheless, the majority of swimming studies is currently limited to two-dimensional analyses. Accurate experimental quantification of changes in body shape, position and orientation (swimming kinematics) in three dimensions is therefore essential to advance biomechanical research of fish swimming. Here, we present a validated method that automatically tracks a swimming fish in three dimensions from multi-camera high-speed video. We use an optimisation procedure to fit a parameterised, morphology-based fish model to each set of video images. This results in a time sequence of position, orientation and body curvature. We post-process this data to derive additional kinematic parameters (e.g. velocities, accelerations) and propose an inverse-dynamics method to compute the resultant forces and torques during swimming. The presented method for quantifying 3D fish motion paves the way for future analyses of swimming biomechanics

Data_J_R_Soc_Interface_rsfi20150479

This zip file contains the raw data that have been used in Van Leeuwen JL, Voesenek CJ, Muller UK, J. R. Soc. Interface 12: 20150479 (http://dx.doi.org/10.1098/fsif.2015.0479). Three text files are provided on the data formatting (data_formats.txt), an overview of the swimming sequences (overview.txt), and the measured tail beat amplitudes for the swimming sequences (tail_amplitudes.txt). Raw video data (directory: Video), digitised longitudinal axis data (directory: Axes), and three-dimensional body models (directory: Body models) are arranged in three main directories

Reorientation and propulsion in fast-starting zebrafish larvae: an inverse dynamics analysis

Most fish species use fast starts to escape from predators. Zebrafish larvae perform effective fast starts immediately after hatching. They use a C-start, where the body curls into a C-shape, and then unfolds to accelerate. These escape responses need to fulfil a number of functional demands, under the constraints of the fluid environment and the larva's body shape. Primarily, the larvae need to generate sufficient escape speed in a wide range of possible directions, in a short-enough time. In this study, we examined how the larvae meet these demands. We filmed fast starts of zebrafish larvae with a unique five-camera setup with high spatiotemporal resolution. From these videos, we reconstructed the 3D swimming motion with an automated method and from these data calculated resultant hydrodynamic forces and, for the first time, 3D torques. We show that zebrafish larvae reorient mostly in the first stage of the start by producing a strong yaw torque, often without using the pectoral fins. This reorientation is expressed as the body angle, a measure that represents the rotation of the complete body, rather than the commonly used head angle. The fish accelerates its centre of mass mostly in stage 2 by generating a considerable force peak while the fish ‘unfolds’. The escape direction of the fish correlates strongly with the amount of body curvature in stage 1, while the escape speed correlates strongly with the duration of the start. This may allow the fish to independently control the direction and speed of the escape

Fish tracker source code and example data

This repository contains the source code of the fish tracker, and its post-processor, described in the associated article. It also includes the code of a reference data generator, to create synthetic data used to test the fish tracker. Finally, it contains a set of example data of a 3 days post fertilisation zebrafish larva