The fast muscle fibres in the anterior trunk of teleost fish are primarily responsible for large amplitude undulatory swimming motions. Previous theoretical studies suggested that the near-helical arrangement of these fibres results in a (fairly) uniform distribution of fibre strain and work output during swimming. However, the underlying simplifications of these studies precluded unequivocal support for this hypothesis. We studied the fast muscle-fibre reorientation and the concomitant myotomal strain variance in a body segment near the anus during larval and juvenile development in the zebrafish. From 2 to 4 days post fertilization (d.p.f.), the measured angles between the muscle fibres and the longitudinal axis of the zebrafish were small. Yet, onset of a near-helical muscle-fibre arrangement was recognized. Juveniles of 51 d.p.f. have larger mean fibre angles and already possess the near-helical pattern of adult teleosts. We present a model that computes the distribution of the strain along the muscle fibres from measured muscle-fibre orientations, body curvature and prescribed tissue deformations. We selected the most extreme body curvatures, which only occur during fast starts and turning manoeuvres. Using the model, we identified the (non-linear) tissue deformations that yield the least variance in the muscle-fibre strain. We show that simple beam theory cannot reliably predict the strain distribution: it results in very small strains and negligible work output of the most medial fibres. In our model, we avoided these functional limitations by adding a shear deformation to the simple beam deformation. At 2 d.p.f., the predicted variance in the muscle-fibre strain for the shear deformation optimized for strain uniformity is fairly small, due to the small variation in the fibre distances to the medial plane that is caused by the relatively large spinal cord and notochord. The predicted minimal strain variance increases sharply from 2 d.p.f. to 3 d.p.f., remains relatively large at 4 d.p.f., but decreases again considerably at 15 and 39 d.p.f. The 51 d.p.f. stage exhibits the smallest variance in the fibre strains (for the identified optimal deformation), in spite of the widely varying muscle-fibre distances to the medial plane. The non-linear nature of the body deformations with the least strain variance implies an interesting optimization constraint: the juvenile muscle-fibre arrangement results in small predicted spatial strain variations at large-amplitude body curvatures, at the (modest) expense of a large coefficient of variation for small curvatures. We conclude that larval fish rapidly change their muscle-fibre orientations (probably in response to mechanical signals). Within the theoretically examined plausible range of deformations, the closest correspondence to a uniform strain field was found for the juvenile stage.

Architecture of the axial muscles in fish

The trunk muscles in cephalochordates, hagfish, lampreys, fish(Chondrichtyes and Osteichtyes) and salamanders (Urodela) are arranged in a longitudinal series of myomeres with a complex shape(Fig. 1A). The myomeres are separated by myosepta, i.e. connective tissue sheets with complex collagen-fibre networks, tendons and intermuscular bones(Gemballa et al., 2003a; Gemballa et al., 2003b; Gemballa et al., 2006; Gemballa and Röder, 2004; Gemballa and Vogel, 2002; Vogel and Gemballa, 2000). The incomplete insight into the functional significance of the myomere architecture is a long-standing problem in functional morphology(Brainerd and Azizi, 2005; Van Leeuwen, 1999; Wainwright, 1983).

Comparative overviews of the myotomal muscle-fibre arrangement in Actinopterygii and Selachii have been published(Wainwright, 1983; Gemballa and Vogel, 2002), and the morphology of the myomeres in scombroid fishes has been described(Westneat et al., 1993). In bony fish (including our present focus, the zebrafish), each myomere is folded into an anterior cone, a dorsal posterior cone and a ventral posterior cone. Thus, the longitudinal series of myomeres forms an array of nested `cone-like'structures (e.g. Greene and Greene,1913; Nursall,1956; Van der Stelt,1968; Alexander,1969). The myosepta attach to other parts of the connective tissue skeleton, viz. the notochord (in early developmental stages) or vertebrae (with vertebral centres and neural- and haemal spines), the medial septum (enforced by the vertebral spines), and the main horizontal septum(with tendons and epicentral bones). The portion of the myoseptum between the anterior cone and the dorsal posterior cone is called the epaxial sloping part(ESP), and the part between the anterior cone and the ventral posterior cone is the hypaxial sloping part (HSP) (see Gemballa and Vogel, 2002). Medial portions of the ESP and HSP are about parallel to the horizontal septum and the longitudinal series of myosepta form connective tissue multilayers in this region (the medial multilayers of the epaxial and hypaxial sloping parts of the myosepta (MESP and MHSP, see Fig. 8C). Wainwright described these layers as epaxial en hypaxial horizontal septa (Wainwright,1983), but they have also been called secondary horizontal septa(Westneat et al., 1993).

In most fish, slow twitch muscle fibres are located laterally as a longitudinally oriented band directly underneath the skin (e.g. Boddeke et al., 1959). The deeper fibres that are arranged in more complex three-dimensional arrays are of a faster type. Intermediate fibre types are often present between these two groups. Muscle fibres extend from one myoseptum to the next, and between myosepta and axial connective tissue structures(Alexander, 1969; Gemballa and Vogel, 2002). Alexander (Alexander, 1969)defined a muscle-fibre trajectory as the curve that could be plotted through the myomeres by following a fibre to its end on a myoseptum, continuing along a fibre starting directly opposite it on the other side of the myoseptum, and so on. In their pioneering studies, Van der Stelt and Alexander discussed the functional relevance of the complex myomere architecture(Van der Stelt, 1968; Alexander, 1969). Alexander described a nested helical muscle-fibre arrangement for the deep fibre system(Alexander, 1969)(Fig. 1B). However, muscle fibres near the medial plane deviate from a helical pattern, as shown for the juvenile zebrafish (Mos and Van der Stelt,1982) (see Fig. 9C). At this location, all fibres make considerable angles with the medial plane. The deviation from the helical pattern has also been described for various fish species (Van der Stelt, 1968; Alexander,1969; Van Leeuwen,1999; Gemballa and Vogel,2002). In a purely helical pattern, many muscle fibres would be(almost) parallel to the medial plane. We demonstrate that this deviation from a helical arrangement is a requirement for an (almost) uniform muscle-fibre strain during swimming, in which `muscle-fibre strain' indicates the strain component in the longitudinal direction of the muscle fibres, unless otherwise stated. In this paper, we explore the development and functional significance of this pseudo-helical pattern.

Form and function of the fast axial muscle fibres

The white muscle-fibre mass generates the high power output required to drive fast starts and turning manoeuvres. These strenuous motions are important to escape from predators and to capture preys. Due to its key role in the survival of the species, we expect that natural selection has resulted in a highly effective escape response system in many species. The high power output that is required for such a response can only be generated by the fast muscle fibres if they are activated almost synchronously (e.g. Jayne and Lauder, 1995) and are able to contract at a speed that provides (close to) maximum power output,irrespective of muscle-fibre location. We have ample evidence that the timing of the activation in the strain cycle of the muscle fibres varies along the trunk in cyclic swimming and fast starts (e.g. Van Leeuwen et al., 1990; Jayne and Lauder, 1994; Jayne and Lauder, 1995; Johnston et al., 1995). Thus,any uniform work hypothesis should be restricted to a series of narrow slabs or perhaps individual myomeres along the trunk. Indeed, equal work also implies that the myomeres are fully activated. This is expected to occur only during extreme performances such as fast starts with a minimal delay of the activation onset down the trunk. We therefore focus on the anal region of the trunk muscles and amplitudes of body curvature that are representative for turning manoeuvres and fast starts.

The longitudinal axis bends under the influence of muscle forces, but shortens only very little, whereas more laterally considerable strain variations parallel to the axis do take place(Fig. 2). Muscle fibres adjacent to the medial plane should therefore not be arranged parallel to the medial plane if they have to shorten by more than a few percent. In addition,a deformation identical to that of a simple bending beam with uniform material properties would considerably reduce strain fluctuations near the medial plane, even if the muscle fibres make substantial angles with the medial plane(Fig. 2B). We demonstrate that sufficient muscle-fibre strains occur near the medial plane by adding a shear deformation to the simple beam deformation(Fig. 2C). The local shear deformations of a simple beam deformation are insufficient for a uniform fibre strain. We propose that the `added shear deformation' is caused by forces produced by contracting medial muscle fibres, which make an angle with the medial plane. For instance, muscle fibre segment 3(Fig. 2B,C) is attached to the medial plane and pulls when activated with its lateral end in a cranio-medial direction. A medial motion is restricted, given that muscle is nearly incompressible and dorso-ventral displacements near the medial plane are very small. The rostral force component contributes to a longitudinal shift of the muscle tissue relative to the medial plane (i.e. the `added shear'; Fig. 2C). The importance of shear deformations has already been recognized(Van der Stelt, 1968). The strain amplitude of more lateral fibres can be reduced if they are oriented at an angle with the horizontal plane, because dorso-ventral strains are expected to be close to zero or slightly positive. A medial enhancement of muscle-fibre strain and a lateral strain reduction can both contribute to a more uniform strain distribution in the trunk muscles. Both effects are implemented with a pseudo-helical arrangement of the muscle fibres.

Maximum power output is further enhanced if all muscle fibres are of a fast type, adapted to the extreme fast start. Obviously, most fish have to generate a wide variety of swimming motions, leading to a trade-off between a design for an extreme fast-start behaviour and one for swimming versatility with a spatially varying ratio between muscle-fibre strain and lateral body curvature(described also as a variation in the gearing ratio). Therefore, we do not expect to find purely uniform distributions of muscle-fibre types and strains in the trunk muscles, even if we restrict ourselves to the white muscle mass.

Theoretical prediction of function from architecture

To investigate whether the helical arrangement in teleosts could result in uniform muscle-fibre strains, Alexander calculated average strains of muscle-fibre trajectories with different radii that span several myomeres(Alexander, 1969). He prescribed the lateral curvature of the medial plane and the deformation of the muscle tissue. In the reference configuration, muscle-fibre trajectories were assumed to represent geodesics on initially straight cylinders of circular cross-section, except where the (virtual) cylinders are cut off by the medial plane (Fig. 1C). At the instant of maximal lateral bending, each cylinder was assumed to be deformed into a torus of circular cross-section, again with geodetic muscle-fibre trajectories on its surface(Fig. 1D). The dimensions of the torus were derived by assuming a constant volume of the enclosed muscle mass. For both configurations, the lengths of the muscle-fibre trajectories were calculated. Zero strain was assumed for the reference configuration. Strains were calculated for muscle-fibre trajectories from nested tori of different radii. For a typical example, Alexander found a strain of–0.02 for the initially straight central trajectory (with zero radius of the associated cylinder or torus, see trajectory 1 in Fig. 1C,D) and –0.03 for a peripheral trajectory (Alexander's case iii, with a muscle trajectory that starts and ends at the medial plane; see trajectory 3 in Fig. 1C,D). From these calculations, he concluded that the helical arrangement enables a fairly uniform strain distribution in the white muscle mass(Alexander, 1969).

We have identified several problems in Alexander's approach(Alexander, 1969). First, in the absence of a fast computer, Alexander had to simplify his computations. Using the same geometrical starting points as Alexander, we found a strain of–0.039 (instead of –0.03) for the peripheral trajectory with a more accurate numerical approximation, almost twice the value of the central trajectory. A better agreement between the mean strains of the two trajectories could be obtained either by choosing a higher pitch angle of the muscle fibres in the peripheral trajectory (as proposed by Alexander) or by moving the central trajectory to a more lateral position. Second, Alexander computed the average strain along a trajectory. This approach, however, does not guarantee that the strain along an individual muscle-fibre trajectory is constant, and may still allow fairly large spatial strain variations, even if the mean strain for different trajectories would be similar. Finally,Alexander avoided the calculation of an intermediate trajectory in which the associated cylinder would almost `touch' the medial plane, leading to fibre orientations that are parallel to the medial plane (see trajectory 2 in Fig. 1C)(Alexander, 1969). For such trajectories, Alexander's assumption of circular cross-sections of the cylinders and tori are not substantiated by quantitative descriptions of the muscle-fibre orientations close to the medial plane(Alexander, 1969). In this region, the muscle fibres were never found to be parallel to the medial plane in quantitative 3D-measurements [Poecilia reticulata(Van der Stelt, 1968); Danio rerio (Mos and Van der Stelt, 1982); Scomber scombrus(Van Leeuwen, 1999); Chimaera monstrosa, Polypterus delhezi, Salmo trutta, Channa obscura(Gemballa and Vogel, 2002)]. Muscle fibres that attach at the ventral side of the epaxial connective tissue multilayers (MESP) between the medial plane and the central muscle trajectories (corresponding to the ventro-medial portion of trajectory 2 in Fig. 1C, which does not agree with anatomical observations) are oriented caudo-laterally (in addition to the ventral component of the orientation vector). At the dorsal side of the MESP,the muscle fibres are oriented cranio-laterally (in addition to the dorsal component). Thus, the fibre directions projected in a transverse section are shown to converge at these locations (cf. Fig. 9C). Caudal to the abdominal cavity, a similar muscle fibre arrangement (more or less a mirror image of the epaxial configuration) is seen at the ventral multilayers (MHSP). We prefer to call the observed muscle-fibre arrangement pseudo-helical instead of helical. The interesting attachment angles of the muscle fibres at the multilayers have been ignored in previous quantitative strain predictions (cf. Alexander, 1969; Mos and Van der Stelt, 1982). The nested cylinders of Alexander's model imply muscle fibres that are parallel to the medial plane at a close distance from the axis (for instance,trajectory 2 shown in Fig. 1C). As explained above (see also Fig. 2), this would yield very small strain fluctuations because the length and height dimensions of the medial plane remain almost constant during bending of the body.

In conclusion, Alexander's assumptions(Alexander, 1969) would lead to very low strains and work output at particular locations close to the medial plane and a significant non-uniformity of the fibre strains in the deep muscle-fibre mass. In the present study we demonstrate that the problem of low strains near the medial plane can be avoided with the described pseudo-helical arrangement in combination with an appropriate added shear deformation.

Theoretical prediction of architecture from functional demands

The architectural consequences of the demand of uniform strain and work output have been investigated (Van der Stelt, 1968). In his quantitative modelling approach, Van der Stelt restricted himself to a two-dimensional analysis of the strain field in the horizontal plane and infinitesimal bending motions. He assumed that the myosepta could bend, but not stretch, restricting deformations that might otherwise occur during bending. He derived a differential equation that fulfils the demand of zero strain in the myosepta and uniform strain of the ensemble of muscle fibres. Van der Stelt found two distinct solutions that showed remarkable similarity with the architecture of particular horizontal sections through the trunk muscles of the lamprey (Lampetra) and the smelt (Osmerus). In the selected sections, the muscle fibres make relatively small angles with the horizontal plane. Van der Stelt showed that valid solutions can only be obtained with longitudinal displacements of muscle tissue relative to the medial plane (similar, but much smaller, to that shown in Fig. 2C), corresponding to added shear deformations.

In reality, the muscle complex is essentially three-dimensional (3D); large amplitude deformations occur and myosepta are stretchable. Thus, we still have to decide whether the nature of the deformation and the arrangement of the connective tissue and muscle fibres are compatible with a uniform strain and work hypothesis. In the present study, we present a model that is capable of computing 3D strain fields for large amplitude deformations. We demonstrate that an architectural solution that yields fairly uniform strains at a large bending amplitude may not be compatible with a good uniformity at low lateral curvatures.

Following the stability principles outlined by Van Leeuwen and Spoor(Van Leeuwen and Spoor, 1992; Van Leeuwen and Spoor, 1993),Van Leeuwen derived myomere shapes from an equilibrium analysis of external forces (due to tensile muscle-fibre forces, intramuscular pressure gradients,and forces from the skin and medial septum) acting on a single myoseptum and the elastic forces acting within the myoseptum(Van Leeuwen, 1999). He used a given pseudo-helical arrangement of the muscle fibres as input and predicted the folded myoseptal shape from mechanical equilibrium assumptions. Here, we measure the muscle-fibre arrangement in early developmental stages and compute the possible degree of strain uniformity during bending of the body.

Form and function during development

While most studies have focused on the design of adult fish, a limited number of authors have considered the architectural changes during development. Van Raamsdonk and colleagues measured myoseptal shape and changes in muscle-fibre orientations at a limited number of sites(Van Raamsdonk et al., 1979; Van Raamsdonk et al., 1982). Interventions of normal mobility patterns prevented normal development of myomere shape and muscle-fibre orientations. Van der Stelt et al.(Van der Stelt et al., 1977)and Mos and Van der Stelt (Mos and Van der Stelt, 1982) measured muscle-fibre orientations in a selection of transverse sections in the zebrafish. At 8 weeks of development, the fibre arrangement is already very similar to the adult pattern.

In the present study, we first quantify the changes in muscle-fibre directions during larval and juvenile development of the zebrafish. We then aim to explain the observed changes in the orientation patterns in the light of the functional demand of an effective muscle-fibre contraction over the myomere space. We explore the most likely shear deformations that are compatible with this functional demand. To achieve this goal, we developed a computational model that calculates muscle-fibre strains from a given 3D deformation. We are limited to a model approach because direct measurements are extremely difficult to achieve with current technology.

Histological techniques

Zebrafish larvae and juveniles were fixed in 4% paraformaldehyde (PFA) in phosphate-buffered saline (PBS) overnight (o.n.) at 4°C, then stored in 1%PFA in PBS. They were postfixed o.n. in 10% PFA in PBS and embedded in 15%gelatin in PBS and fixed again o.n. at 4°C in 4% PFA in PBS. For stages 2,3, 4 and 15 d.p.f. (days post fertilization), transverse sections (100 μm thick) in the area just behind the anus were cut on a vibratome 1500(Vibratome, St Louis, MO, USA) and stained o.n. at 4°C with propidium iodide (1 μg ml–1) in PBS. They were incubated in 25% (1 h), 50% (1 h), 75% (o.n.), and 90% glycerol in PBS (3 h) and then embedded in 90% glycerol in PBS immediately prior to examination. Using similar procedures, we made sagittal sections of fish larvae of 16, 18, 24 and 72 h.p.f. (hours post fertilization).

For our computations, we also used data from zebrafish juveniles of 6 and 8 weeks of development reared at 26°C obtained from published papers(Van der Stelt et al., 1977; Mos and Van der Stelt, 1982),where the tissue had been fixed(Karnovsky, 1965), post-fixed in osmium tetroxide in phosphate buffer (pH 7.4), transferred to alcohol and propylene oxide and embedded in Epon, and muscle-fibre orientations from serial sections of 2 and 2.5 μm measured. A lower rearing temperature leads to a slower development. We recomputed developmental time for the 6- and 8-week juveniles (by multiplication of the time by 26/28.5) such that their developmental stage corresponds approximately with the developmental rate at 28.5°C. Our conclusions are not sensitive to inaccuracies of this correction. They will be referred to as 39 and 51 d.p.f. in the rest of the paper.

Finally, cross sections through a caudal portion of the trunk of a juvenile zebrafish of 45 d.p.f. were made and stained according to Crossmon(Crossmon, 1937).

Measurement of muscle-fibre orientations

A 15 μm thick Z-stack of 1 μm thick consecutive optical sections was created using a laser-scanning microscope (Zeiss LSM-510,Göttingen, Germany) from the 100 μm thick transverse section. The Z-stack was exported as individual TIFF files to AnalySIS software(Soft Imaging System GmbH, Münster, Germany) and calibrated. Each fibre that was present as a complete cross section in 15 consecutive sections (i.e. over a distance of 15 μm from anterior to posterior) was manually tracked in sections 5, 10 and 15. This implies that fibres close to myosepta were not digitized, due to tapering. For each cross section, the centre of area (CA) in coordinates of the Z-stack was determined by the AnalySIS software. The fibre orientations in (xz, yz, zz) coordinates of the Z-stack were computed in Matlab 6.5 (The Mathworks, Inc, Natick, MA, USA) from the line that runs through the CA in the first section and the nearest CA in the second section. As a control of the validity, two such computations were made per embryo,using sections that were slightly shifted compared to the first pair. For final analyses, optical sections that were 5 μm apart were analyzed. At this distance, individual fibres can be easily identified and tracked and a straight line between corresponding CA values is a relatively accurate description of the local fibre orientation. The error in the computed orientation was determined to be less than 5° (not including tissue deformation).

In general, the computed orientation of the fibres in the(xz, yz, zz)coordinates of the Z-stack is not a fair representation of the fibres in a fish-bound (x, y, z) coordinate system, because the sections in the (xz, yz) plane are not exactly parallel to the (x, y) plane (transversal plane) of the fish(Fig. 3A). Muscle fibres are defined to run (obliquely) from anterior to posterior, at an angleθ<π/2 with the positive z-axis. Based on the left–right symmetry of the fish, the computed fibre orientations were rotated over three perpendicular axes to obtain a visually left–right symmetrical vector field. The result is a series of vectors describing the elevation and azimuth of each individual fibre in the (x, y, z)coordinate system. The elevation β is the angle between the orientation of the fibre and a horizontal plane, with y constant, see Fig. 3B). The elevation is positive if the fibre direction has a component in the positive y-direction. The azimuth α is the angle of the projection of the fibre on a horizontal plane with the positive z-axis. An anti-clockwise rotation in the (x, z) plane is considered positive. These angles were approximated for each measured muscle fibre by:
\[\ {\alpha}{\approx}\mathrm{arctan}({\Delta}x{/}{\Delta}z),{\ }{\ }{\ }{\Delta}z{>}0,\]
(1)
and
\[\ {\beta}{\approx}\mathrm{arctan}({\Delta}y{/}\sqrt{{\Delta}x^{2}+{\Delta}z^{2}}),\]
(2)
where Δx, Δy, Δz are the differences in the x, y and z-position, respectively,between the start and end locations of the muscle-fibre segment. The distance between the examined sections is equal to Δz. Finally, the angle θ between the longitudinal direction of the muscle fibre and the z-axis was calculated as:
\[\ {\theta}{\approx}\mathrm{arccos}({\Delta}z{/}\sqrt{{\Delta}x^{2}+{\Delta}y^{2}+{\Delta}z^{2}}),{\ }{\ }{\Delta}z{>}0.\]
(3)

Calculation of muscle-fibre strains

We aimed to derive muscle-fibre strains from measured fibre orientations(see previous section) and prescribed deformations of the trunk. The model serves to explore the extent to which the pseudo-helical fibre arrangement of`teleost fish' enables an equal strain and work output in an infinitesimally thin muscle slab along the trunk. In particular, we consider large-amplitude motions that are roughly based on in vivo performance measurements for fast starts and turning manoeuvres. All calculations were made for both the epaxial and the hypaxial trunk muscles of the measured developmental stages of the zebrafish. The hypaxial muscles resemble approximately a mirror image of the epaxial muscles in the post-anal region.

Linear strain ϵ is defined as

\(({\ell}-{\ell}_{0}){/}{\ell}_{0}\)
⁠, where
\({\ell}_{0}\)
is the reference length and
\({\ell}\)
the actual length. We will use the previously defined orthogonal fish-bound (x, y,z)-frame (Fig. 3A) to describe the geometry of the reference configuration with a straight medial plane. The x-axis points in the right lateral direction, the y-axis points in the dorsal direction, and the z-axis points in the caudal direction. The symbols used in the calculations are summarized in the List of symbols and abbreviations.

Simplifying assumptions

We make the following simplifying assumptions.

  1. The medial plane is allowed to bend laterally with radius of curvature R, but is assumed to be strain free. Constant lateral curvature along the fish is assumed for the relatively short longitudinal region of interest.

  2. The muscle tissue is incompressible.

  3. Either the dorso-ventral strain or the medio-lateral strain is zero(considered as type I and type II deformation in the rest of this paper). Incompressibility demands that zero dorso-ventral strain during lateral bending must lead to lateral thickening at the concave side and thinning at the convex side. The consequences of these two extremes for the uniformity of muscle-fibre contraction will be considered.

Quantification of deformation and strain

The orientation of a tissue element such as a muscle fibre is defined by its azimuth α and elevation β, as defined above. Let dz be an infinitesimal distance along the longitudinal axis of the fish(Fig. 4C). During lateral bending, dz is assumed to be constant(Fig. 4B,D). We will examine how a fibre traverses a transverse tissue slice S of thickness dz in the reference configuration. The distance covered by the fibre element in the slice in the transverse direction is (anti-clockwise rotation is considered positive):
\[\ \mathrm{d}x_{0}=\mathrm{d}z\mathrm{tan}{\ }{\alpha}.\]
(4)
The projection length onto the horizontal plane is:
\[\ \mathrm{d}{\ell}_{\mathrm{pf}0}=\mathrm{d}z{/}\mathrm{cos}{\alpha}{>}0{\ }\mathrm{for}{\ }-\frac{{\pi}}{2}{<}{\alpha}{<}\frac{{\pi}}{2}.\]
(5)
The covered dorso-ventral distance is:
\[\ \mathrm{dy}_{0}=\mathrm{d}{\ell}_{\mathrm{pf}0}{\ }\mathrm{tan}{\ }{\beta}{\ }\mathrm{for}{\ }-\frac{{\pi}}{2}{<}{\beta}{<}\frac{{\pi}}{2},\]
(6)
and the length of the fibre element is:
\[\ \mathrm{d}{\ell}_{\mathrm{f}0}=\mathrm{d}{\ell}_{\mathrm{pf}0}{/}\mathrm{cos}{\ }{\beta}{>}0{\ }\mathrm{for}{\ }-\frac{{\pi}}{2}{<}{\beta}{<}\frac{{\pi}}{2}.\]
(7)
We shall use
\(\mathrm{d}{\ell}_{\mathrm{f}0}\)
to compute the strain in the fibre element in the deformed state during bending. The fibre element extends between positions x0 and x0+dx0 relative to the medial plane before bending. The green shaded area in Fig. 4E has a surface of:
\[\ A_{0}=x_{0}\mathrm{d}z.\]
(8)
If the fish axis bends locally into a circular arc with radius R,then the infinitesimal opening angle is(Fig. 4F):
\[\ \mathrm{d}{\phi}=\mathrm{d}z{/}R.\]
(9)
The curvature of the medial plane is defined as c=1/R. The radius of curvature varies along the longitudinal axis of the fish. We assume however that R varies so slowly with z that for the present purposes we can take a constant value of R for the considered thin tissue slice.

We distinguish two extreme types of deformation. First, we assume lateral thickening at the concave side of the body, but without any changes in the dorso-ventral position of material points (type I). The distances to the medial plane are reduced at the convex side. Thereafter, we will consider a deformation with constant distances of the material points to the medial plane(type II). We are not able to use the in vivo geometry changes because internal deformations are exceedingly difficult to measure in larval and juvenile fish. However, if our conclusions about the functional significance are similar for the two extremes, they are also likely to be applicable to the in vivo situation.

For the strain calculation of deformation type I, we need to compute the new locations of the end points of the infinitesimal muscle-fibre segment. Due to the assumed incompressibility of the tissue, A0 should be constant. For the deformed state, we have:
\[\ A_{0}=(R^{2}-(R-x_{1})^{2})\mathrm{d}{\phi}{/}2=(R^{2}-(R-x_{1})^{2})\mathrm{d}z{/}(2R),\]
(10)
where x1 is the distance of the medial (first) end point of the fibre element to the medial plane at the concave side of the body(x1>0). By equating Eqn 8 and Eqn 10, x1can be calculated as:
\[\ x_{1}=R-\sqrt{R^{2}-2Rx_{0}}.\]
(11)
Similarly, for the lateral (second) end point we obtain:
\[\ x_{2}=R-\sqrt{R^{2}-2R(x_{0}+\mathrm{d}x_{0})}.\]
(12)
Similar equations can be derived for the convex side of the body. For the calculation of the muscle-fibre strain, we define a local right-handed Cartesian frame of reference(x′,y′,z′), with the origin located at the centre of curvature, and y′ parallel to y. We assume that the location y′=y=0 is located halfway between the most dorsal point of the tissue slab(y′=y=ymax) and the most ventral point (y′=y=ymin). The positive x′-axis is assumed to run through the centre of the medial boundary of the considered muscle segment. The z′-axis is parallel to the tangent of the centre of this medial segment boundary. Fig. 4F shows a projection in a horizontal plane, where y′ is constant. The(x′,z′)-position of the medial end point (point 1) is given by:
\[\ (x_{1}^{{^\prime}},z_{1}^{{^\prime}})=\left((R-x_{1})\mathrm{cos}\left(\frac{\mathrm{d}{\phi}}{2}\right),(R-x_{1})\mathrm{sin}\left(\frac{\mathrm{d}{\phi}}{2}\right)\right).\]
(13)
The position of the second end point is more complicated to derive because we allow for an added shear deformation in planes with a constant value of y′. Let (x2, z2) be the position of the second end point without added shear (i.e. simple beam deformation):
\[\ (x_{2}^{{^\prime}},z_{2}^{{^\prime}})=\left((R-x_{2})\mathrm{cos}\left(\frac{\mathrm{d}{\phi}}{2}\right),-(R-x_{2})\mathrm{sin}\left(\frac{\mathrm{d}{\phi}}{2}\right)\right).\]
(14)
Let (x3′, z3′) be the shifted location due to the local added shear. To derive the strain in muscle-fibre segment we need to compute (x3′, z3′). The shifted position should lie on the circle with radius (Rx2), centred around the y′-axis. With shear angle γ (considered positive for an anti-clockwise direction in the xz-plane) and the assumption of an infinitesimal length of the muscle-fibre segment, it follows that (see also Fig. 4F):
\[\ (x_{3}^{{^\prime}},z_{3}^{{^\prime}}){\approx}\left(x_{2}^{{^\prime}}+(x_{2}-x_{1})\mathrm{tan}{\gamma}\mathrm{sin}\frac{\mathrm{d}{\phi}}{2},z_{2}^{{^\prime}}(x_{2}-x_{1})\mathrm{tan}{\gamma}\mathrm{cos}\frac{\mathrm{d}{\phi}}{2}\right),\]
(15)
\[\ {\approx}[x_{2}^{{^\prime}},z_{2}^{{^\prime}}+(x_{2}-x_{1})\mathrm{tan}{\gamma}].\]
(16)
The approximations of Eqn 15 and Eqn 16 are allowed because of the infinitesimal size of the fibre element, resulting in
\(R{\gg}|x_{2}-x_{1}|\)
, sin(dϕ/2)≈0, and cos(dϕ/2)≈1. The x′- and z′-differences of the positions of the end points of the fibre element are:
\[\ \mathrm{d}x^{{^\prime}}=x_{1}^{{^\prime}}-x_{3}^{{^\prime}},\]
(17)
\[\ \mathrm{d}z^{{^\prime}}=z_{1}^{{^\prime}}-z_{3}^{{^\prime}}.\]
(18)
The length of the fibre element can now be calculated as:
\[\ \mathrm{d}{\ell}_{\mathrm{f}}=\sqrt{\mathrm{d}x^{{^\prime}_{2}}+\mathrm{d}y_{0}^{2}+\mathrm{d}z^{{^\prime}_{2}}}.\]
(19)
Finally, we obtain the longitudinal strain of the fibre element:
\[\ {\varepsilon}_{\mathrm{f}}=(\mathrm{d}{\ell}_{\mathrm{f}}-\mathrm{d}{\ell}_{\mathrm{f}0}){/}\mathrm{d}{\ell}_{\mathrm{f}0}.\]
(20)
For a given curvature of the medial plane, a prescribed shear angle distribution γ(x, y) and measured fibre-angle distributionsα(x, y) and β(x, y), the muscle-fibre strains can now be calculated as a function of (x, y, R) over the transverse muscle slab. This allows us to evaluate the effects of muscle architecture on the strain distribution.
We shall now derive similar equations for the deformation with a constant distance from the medial plane (type II). Thus, due to the incompressibility requirement of muscle, we should now consider the displacements of tissue in the y-direction (i.e. the dorso-ventral direction). The distances of the end points to the medial plane are now given by:
\[\ x_{1}=x_{0},\]
(21)
\[\ x_{2}=x_{0}+\mathrm{d}x_{0}.\]
(22)
The locations of (x1′, z1′), (x2′, z2′), (x3′, z3′) are chosen equivalent to the type I deformation. We can obtain the location of (x1′, z1′) from Eqn 13. The values for (x3′, z3′), dx′ and dz′ are found by application of Eqn 16, 17, 18. To derive the vertical distance between the end points of dy′, we need to compute the vertical tissue displacement. The incompressibility constraint demands that:
\[\ \mathrm{d}y^{{^\prime}}=\frac{(y_{0}+\mathrm{d}y_{0})R}{R-(x_{0}+\mathrm{d}x_{0})}-\frac{y_{0}R}{R-x_{0}}{\approx}\mathrm{d}y_{0}{/}(1-(x_{0}{/}R)).\]
(23)
The length of the fibre element can now be calculated as:
\[\ \mathrm{d}{\ell}_{\mathrm{f}}=\sqrt{\mathrm{d}x^{{^\prime}_{2}}+\mathrm{d}y^{{^\prime}_{2}}+\mathrm{d}z^{{^\prime}_{2}}}.\]
(24)
Eqn 20 can again be used to derive the longitudinal fibre strain.

Optimization of the deformation for strain uniformity

We will vary the deformation within a meaningful range and compute the resulting strains for all measured muscle-fibre orientations. We will assume that the shear angle γ will vary between γmed at the medial plane and zero at the half width xmax0 of the fish in the reference configuration according to:
\[\ {\gamma}(x,y)={\gamma}_{\mathrm{med}}\left(\frac{x_{\mathrm{max}0}-x_{0}}{x_{\mathrm{max}0}}\right)^{p},\]
(25)
where γmed is defined as:
\[\ {\gamma}_{\mathrm{med}}(y)={\pm}{\gamma}_{\mathrm{max}}\left(\mathrm{cos}\left(-\frac{{\pi}}{2}+{\pi}\frac{y-y_{\mathrm{low}}}{y_{\mathrm{up}}-y_{\mathrm{low}}}\right)\right)^{q},y{\in}[y_{\mathrm{low}},y_{\mathrm{up}}],\]
(26)
where γmax is the maximum value of γ. We can define how rapidly the shear angle reduces from the medial plane to the skin by assigning a particular value to p. We have chosen a power function to be able to study the effects of a non-linear variation of γ with the distance from the medial plane on the variation ϵf.

We used a fixed value of 0.1 for q in our computations. This choice guarantees a nearly constant value of γmed except close to the lower and upper limits of the interval [ylow, yup]. The value of γmax is a positive real number and constant for each computation of a strain field for one half of the considered body slice. Different values are generally chosen for the concave side (with negative muscle-fibre strains) and the convex side (with positive strains) of the body. We distinguish a series of five intervals between ymin (the minimum y-value) and ymax (the maximum y-value) that are based on anatomical features. The ± sign stands for a minus or a plus sign according to the convention of Fig. 5A. The sign of Eqn 26 is constant over a region, but swops at the transition between neighbouring regions because the muscle fibres abrubly change their direction at these transitions. Discontinuities in the considered tissue slab at the transitions are avoided because at these locations γ=0.

The central and by far the largest region [yhyp1, yep1] is located directly dorsal and ventral to the horizontal septum and represents in the juvenile stage the ventral half of the main epaxial muscle-fibre trajectories and the dorsal half of the main hypaxial muscle-fibre trajectories. It is interesting to note that in this region the muscle fibres run from the medial plane in a caudo-lateral direction. Fibre shortening is enhanced by a positive value of γ, as adopted in our sign convention (Fig. 5A). The anti-clockwise sense of the epaxial trajectories and the clockwise sense of the hypaxial trajectories induce a positive γ upon contraction at the concave side of the body, leading to a rostral shift of the lateral muscle portions of the slab (similar to the tissue deformation shown in Fig. 2C). A similar rostral motion in a connected series of myomeres supports the lateral motion of the caudal peduncle and tail fin toward the concave side of the body.

In the region delimited by [yep1, yep2], the muscle fibres run in the medio-caudal direction. In this case, a negative value of γ enhances muscle-fibre shortening near the medial plane at the concave side. Thus, muscle material is expected to shift in this region in a caudal direction relative to the medial plane. A similar situation occurs in the region with y∈[yhyp2, yhyp1]. Finally, two very small regions with positive values of γ are present near the most ventral and dorsal extension of the tissue slab([yep2, ymax] and[ymin, yhyp2]). The latter regions are only clearly present in the juvenile stages. Fig. 5B shows an example of aγ-distribution that was used for stage 51 d.p.f.

We varied both γmax and the exponent p, and computed for each combination the strain distribution of the muscle fibres,the mean strain –ϵf and the standard deviation σ. We defined the coefficient of variation as:
\[\ {\eta}={\sigma}{/}\overline{{\varepsilon}}_{\mathrm{f}}.\]
(27)
The closer η is to zero, the smaller the variance in the strain and the closer a uniform strain distribution is approximated. The optimum value ofη was identified for the (γmax, p) parameter space. Fig. 6 shows two computed examples of the variation of η with p andγ max.

Sensitivity analysis

The present analysis follows a mainly theoretical approach because direct measurements are still too difficult. To explore the extent to which our conclusions depend on the choice of parameters, we considered how the two extreme types of deformation (type I and type II, see above) and a range of curvature amplitudes of the medial plane affect η. The normalized radius of curvature =R/xmax0was varied between 5 (representing a relatively large curvature) and 20 (for a small curvature). The maximum normalized curvature (body length/R)reported (Müller and Van Leeuwen,2004) is about 8 in the anal region for the fastest starts recorded. From this value and the known values for xmax0,it can be derived that larval zebrafish can reach at least a minimal value of 10 for . We do not know whether these records represent the actual maximum performance, but it is unlikely that larval and juvenile stage could generate a value lower than 5. The higher values of that we also used are reached in less extreme performances or can be interpreted as intermediate values in a maximum performance event.

All computations were made with custom written routines in MATLAB 7.0.4(the MathWorks Inc.).

Morphological observations

For the timing of development of the pseudo-helical muscle-fibre arrangement at 28.5°C, larvae were sectioned sagittally(Fig. 7). From these sections,it is apparent that the slow fibres, which develop adjacent to the notochord,are present in bundles parallel to the notochord already at 16 h.p.f., 2 h before the first movements of the larvae are observed(Fig. 7A). The pattern of fast-fibre orientation is still quite variable(Fig. 7B). At 18 h.p.f., the time of first movements, still only a medial portion of the somite contains fibres with an orientation in a preferential direction, with all fibres oriented more or less parallel to the notochord(Fig. 7C). These muscle pioneer cells perform the initial contractions(Melancon et al., 1997). More laterally located muscle fibres do not show a preferential direction at this stage (Fig. 7D). At 24 h.p.f.(6 h after the first movements), substantial changes can be observed. Judging from the section, medial fibres are at larger angles with the myosepta(Fig. 7E) than lateral fibres(Fig. 7F). The most medial section shows that the epaxial muscle fibres are oriented caudo-ventrally, and the hypaxial fibres are oriented caudo-dorsally. In the lateral section, the epaxial muscle fibres are oriented caudo-dorsally, and the hypaxial fibres caudo-ventrally. The different muscle-fibre orientations in both para-sagittal planes agree with a pseudo-helical muscle-fibre arrangement. At 72 h.p.f., the differential arrangement over the myotome has become more prominent(Fig. 7G,H), and also agrees with a pseudo-helical arrangement.

Measured muscle-fibre directions

Two examples of optical sections of 2 d.p.f. and 15 d.p.f. that were used in the analysis are shown in Fig. 8A,B. Conspicuous differences occur between those stages. The fraction of the section that is occupied by the spinal cord and notochord is largest at 2 d.p.f. (Fig. 8A),whereas the number of muscle fibres is largest at 15 d.p.f.(Fig. 8B). Many muscle fibres are adjacent to the medial septum in the 15 d.p.f. stage.

From 3 d.p.f. onwards, the mean angle of the muscle fibres θ with the longitudinal z-axis of the fish increases with age from 7.8° to 20.7° in the measured cross-sections(Table 1). The value for the 2 d.p.f. stage is similar to that of 4 d.p.f. (about 12.9°). The largest angles are found for 51 d.p.f., with a maximum of 40.7°. All stages have fibres that are (almost) parallel to the z-direction. Vector plots of the muscle-fibre angles are shown for stages 2, 15 and 51 d.p.f. in Fig. 9A–C. At 2 d.p.f.,fairly large values of β (top-right of Fig. 9A) and α (left hypaxial region) are present adjacent to the skin.

In the earliest stages, the left–right symmetry is still poorly developed (some of the left–right difference may be caused by histological procedures). The best left–right symmetry is found in the 51 d.p.f. stage. The fibre directions deviate from a purely helical pattern. At 15 and 39 d.p.f., and especially 51 d.p.f., all fibres close to the medial plane have a significant angle with the medial septum. Close to the medial plane, stage 51 d.p.f. shows regions in the left and right epaxial musculature with muscle fibres that are either oriented medio-ventrally (the more dorsal fibres) and latero-ventrally (the more ventral fibres) in the vector plots of in Fig. 9C (regions indicated by asterisks). A histological inspection shows that these regions are divided by the epaxial multilayered septa (MESP) that serve as attachment sites for the two differently oriented muscle fibres groups in these regions (see Fig. 8C). These multi-layered septa are formed by a junction of consecutive myosepta(Fig. 8C) and probably transmit the forces produced by the attaching dorsal and ventral muscle fibres to the medial plane (see also the Discussion).

Optimization of the coefficient of variation

Fig. 6 shows contour plots for the coefficient of variation η (γmax, p) for stages 2 d.p.f. (Fig. 6A) and 51 d.p.f. (Fig. 6B), for the concave side of the body and a normalized body curvature R of 5. The smaller the value of |η|, the smaller the relative variance in the strains of the examined muscle-fibre ensemble. We variedγ max between 0° and 50°; values above 50° are extremely unlikely. Table 1shows an overview of the optimal (γmax, p)combinations for all stages. At 2 d.p.f., the minimum of |η|is obtained with the maximum examined value of γmax of 50° (Fig. 6A). Maximum values of γmax yield the lowest variance in the strain for 2 and 3 days, for both the convex and the concave side. Given the relatively small values of α in these stages, it would be very unlikely that the fish larvae could produce such a considerable added shear. Thus, the actual added shear angle is probably less than the computed optima would suggest. Hence, the value of |η| is expected to be larger than the computed optimum values. The sensitivity of η for variations inγ max is, however, relatively small for the youngest stages(2–4 d.p.f.) due to the relatively small values of α. For the 51 d.p.f. stage, a clear optimum is obtained at γmax=26° and p=1.4.

Predicted strain distributions

Fig. 9D–F shows computed strain values for stages 2, 15 and 51 d.p.f. Deformation type II(with constant distances of material points to the medial plane) was used. The results for deformation type I (constant dorso-ventral position of material points) are very similar. The results represent the value of maximum shear angle γmax and exponent p(Eqn 25), for which the smallest coefficient of variation in the strain is obtained (i.e. lowest value ofη). For these calculations, we chose a rather extreme body curvature, =5, which could occur during fast starts and turning manoeuvres. These extreme curvatures are particularly interesting because they represent the highest functional demand for the fast muscle-fibre mass. From the kinematic analyses of Müller and Van Leeuwen it is clear that larval zebrafish can generate values of at least =10(Müller and Van Leeuwen,2004). The left-hand side of each panel of Fig. 9D–F represents strain values at the convex side of the body, the right-hand side represents the concave side. For the 2 d.p.f. stage, the largest strain amplitudes occur at lateral positions, close to the skin. The smallest strains occur close to the medial plane. We conclude that the fibre orientations do not (yet) result in a very good strain uniformity within the examined deformation space. The differences between the predicted lateral and medial strain values are significantly smaller in the 15 d.p.f. stage(Fig. 9E) and are least for the 51 d.p.f. stage (Fig. 9E). A few outliers are present near the medial plane in the hypaxial region of the 51 d.p.f. stage that could be caused by measurement errors [we were not able to check this since the measurements were made many years ago by another research group (cf. Van der Stelt,1977; Mos and Van der Stelt,1982)].

Fig. 10 shows how the computed mean strain

\(\overline{{\varepsilon}}_{\mathrm{f}}\)
and coefficient of variation in the strain η vary over time using again =5 as model input. For each stage,the deformations with the least strain variance were used for this analysis. Fig. 10A,B shows the results for deformation types I and II, respectively. The curves in both panels are strikingly similar, indicating that the model predictions are insensitive to the assumed deformation type. The numbers given in this paragraph refer to type I deformation; similar values are found for type II deformations. The mean strain
\(\overline{{\varepsilon}}_{\mathrm{f}}\)
varies relatively little with developmental time on both the convex and the concave sides. In contrast, the values of η (optimized in the γmed, p parameter space) vary considerably with time. Smaller absolute values of η represent a closer approximation of a uniform strain field. At the concave side of the body, the mean muscle-fibre strain is negative,causing η to be negative, while η is positive at the convex side. Immediately after hatching, η becomes more negative on the concave side from –0.25 at age 2 d.p.f. to –0.37 and –0.36 at 3 and 4 d.p.f. (solid blue curve in Fig. 10). The strain uniformity is again improved at 15 d.p.f. and 39 d.p.f. (η=–0.27 and –0.28). The latest stage (51 d.p.f.)yields the best strain uniformity (η=–0.20). A similar trend in the variation of η is found for the convex side (with a reversed sign, solid black curve in Fig. 10).

Effects of added shear and fibre orientations on strain uniformity

For comparison with the shear-optimized values of η, we also plotted the η-curves for γmed=0, i.e. without an added shear deformation (dotted curves in Fig. 10). As expected, the absolute value of η is higher than for the optimized case for all developmental stages. The largest difference occurs for the oldest stages, which show the largest muscle-fibre angles with the longitudinal axis. An appropriate shear deformation is vital for the predicted relatively small strain variance that can be obtained with the fairly regular pseudo-helical pattern of the oldest stage. Finally, we computed η for a hypothetical arrangement with all muscle fibres arranged parallel to the longitudinal axis of the fish (broken curves in Fig. 10). In this arrangement,the added shear deformation would have no effect on the strain uniformity and was therefore not further considered in the analysis. Similar deviations from the strain-optimized curves are obtained as for the observed fibre orientations with zero added shear deformation. In the youngest stages(2–5 d.p.f.), the muscle fibre directions do not yet deviate much from an axial orientation (i.e. γ is small). These stages are therefore relatively insensitive to changes in the added shear, but have also the largest values for |η|. Thus, we predict that a relatively uniform strain field can be achieved by the right combination of initial fibre orientations and added shear deformation.

Sensitivity analysis

Fig. 11 shows the effect of the normalized radius of curvature on the predicted optimal value of η. The examined values for were 5, 10, 15 and 20. For the youngest stages, the curvature has very little influence on η. After 15 days, the best uniformity is obtained for the largest curvatures. A relatively poor strain uniformity for small curvatures is predicted for the latest stage,which is in contrast to the relatively good uniformity for the largest amplitudes. The large amplitude performance of the fast muscle mass is probably most important for the survival of the fish. The architecture in the juvenile fish seems to be optimized for large amplitude deformations at the expense of the strain uniformity at low curvatures that are primarily powered by the thin peripheral layer of slow muscle fibres. An optimization for both low and small amplitudes is impossible due to the non-linear properties of the required deformation. Similar predictions are made for the concave and the convex side of the body. The results for type I and type II deformations are again similar. These results demonstrate the necessity of a large amplitude analysis.

Advantages and disadvantages of current approach

We attempted to implement several advantages of previous quantitative analyses of the complex muscle-fibre arrangement in fish, and to avoid most of the pitfalls of those attempts. We included large amplitude body curvatures and a 3D approach [advantages of Alexander's model(Alexander, 1969)]. We calculated strains of infinitesimal muscle-fibre elements and introduced an added shear deformation [advantages of Van der Stelt's model(Van der Stelt, 1968)]. We avoided the strain-averaging approach of Alexander(Alexander, 1969) and the infinitesimal deformations, incompliant myosepta and 2D limitations of Van der Stelt's model (Van der Stelt,1968). The currently available computational power allowed us to include an exploration of developmental stages, to compute a coefficient of variation for the muscle-fibre strain of the white muscle mass for a relevant family of deformations, and to consider the sensitivity of our predictions to parameter variations.

We were limited by the accuracy of the measured muscle-fibre orientations. Errors arise as a result of tissue preparations and are very hard to quantify. We expect that the youngest stages are affected most, which could have a negative effect on the predicted strain uniformity, especially of the youngest stages. It is unlikely, however, that the relatively large average angles with the long axis that (according to our theory) are required for a uniform strain field would have been masked by the histological techniques in the youngest stages (2–4 d.p.f.).

We were limited to thin tissue slabs in our analysis. The computations could have been easily extended to a larger muscle region. However, the current measurement technique of the muscle-fibre angles involved a considerable amount of manual processing. Measurements of the muscle-fibre orientations in complete fresh specimens are preferable. We have attempted to apply the diffusion tensor MRI (Napadow et al., 2001) to whole specimens, but were hampered by a currently still too low signal-to-noise ratio provided by this technique (with the available equipment and resources).

A reasonable range of possible deformations was prescribed in our approach,and the deformations with the least variance in the strain were selected as predictions of the capability of the examined stages to operate with similar strains and work output over the considered muscle volume. This is the best estimate to date of the in vivo performance in the absence of a feasible way to test the predictions. A better approach would be a forward dynamics model that computes the deformation from the stress distribution in the tissue. This was beyond the present scope (see also Perspectives).

Interpretation of the predicted strain variance from architecture

As explained in the Introduction, a fairly uniform strain distribution is likely to lead to a good performance in strenuous behaviours such as fast starts. The necessity for more versatility in locomotion patterns may counteract the selective pressure for uniform strains and may reduce the size of the muscle regions over which fairly good uniformity occurs.

At 2 days of development, the muscle fibres still have relatively small angles with the longitudinal direction. The relatively small standard deviation (s.d.) in the strain at this stage (compared with 3 and 4 d.p.f.)can be explained by the relatively large neural tube and notochord (see Fig. 8A), leading to a relatively small variation in the distance of the muscle fibres to the medial plane that is assumed to keep zero strain values during bending. At 3 and 4 d.p.f., the relative sizes of the neural tube and notochord have decreased,resulting in a more variable relative distance from the medial plane. This leads to an increase in |η| that is not compensated by the right combination of muscle-fibre reorientations and shear deformation. The spinal cord and vertebral column have a relatively small size in the 51 d.p.f. stage, with many muscle fibres attaching to the medial septum. Nevertheless,for this stage we predict the smallest strain variance that could be achieved with our model.

The pseudo-helical muscular organization is characterized by relatively large azimuth angles for the fibres near the medial plane. This allows large enough strain amplitudes in this region in combination with relatively high shear angles in this region. At a relatively large distance from the medial plane (the lateral region), relatively large elevation angles β are present. High values of β reduce the strain amplitudes compared with the strain field caused by bending a simple beam with uniform material properties. The muscular arrangement leads to increased strains near the medial plane and reduced strains in the lateral region with an appropriate shear angle distribution, and therefore promotes a uniform strain distribution. A perfect nested helical arrangement of the muscle fibres would not lead to a uniform strain field because the azimuth of the muscle fibres between the central muscle trajectory [positioned at an imaginary cylinder of zero radius in Alexander's model (Alexander,1969)] and the medial plane would be too small to allow a horizontal shear deformation to have a significant effect on the muscle-fibre strain.

At the location of the epaxial and hypaxial myoseptal multilayers (MESP and MHSP), we prescribed rapid changes in the direction of the added shear (in a dorso-ventral direction, cf. Fig. 5). This was required to allow shear deformations that are in agreement with the very different muscle fibre orientations at both sides of the multilayers. The multilayer architecture presumably allows the layers to slide parallel to one another and hence accommodate the high spatial gradient in `added' shear. Due to the different attachment angles of the muscle fibres at the dorsal and ventral sides of the multilayers, the tensile muscle force at both sides cannot balance each other. We made dissections of corresponding epaxial locations of another cyprinid fish, carps (Cyprinus carpio)of 15–20 cm standard length, and found two parallel myoseptal layers(one dorsal from the other) at a very close distance with collagen fibre bundles of about 40–200 μm thick. We visualized the collagen layers by polarized light microscopy (see Fig. 12). In the dorsal layer, the collagen fibre bundles run obliquely in a medio-caudal direction to the medial plane whereas in the ventral layer the fibre bundles are oriented medio-rostrally towards the medial plane. These directions are in agreement with a transmission of the tensile forces of the muscle fibres at respectively the dorsal and the ventral side of the multilayers.

Internalization of slow muscle fibres

Slow red fibres generally exhibit larger strains at a given lateral body curvature than those of white fibres. This condition is easily guaranteed if the fibres are positioned at a large distance from the medial plane, a solution seen in many fishes. Quite interestingly, simulations with our model show that large strain amplitudes are also possible close to the medial plane if a high enough `added' shear deformation occurs in this region. This allows an `internalization' of red muscle tissues with a similar function as the slow lateral fibres, without a drastic restructuring of the tissues. This would make it a likely option in evolutionary transitions. An internalization of slow muscles is indeed present in some taxa, such as tunas, with specific mechanical properties (Syme and Shadwick,2002).

Effects of immobility on muscle-fibre arrangement

In nicb107 mutant embryos, muscle fibres are mechanically intact and able to contract, but neuronal signalling is defective and the fibres are not activated, rendering the embryos immobile(Sepich et al., 1994; Sepich et al., 1998; Westerfield et al., 1990). Pseudo-helical muscle fibre arrangements are generated despite the immobility(Van der Meulen et al., 2005). Although the initial development of curved muscle fibre trajectories does not depend on active muscle, the architectural fine tuning in later stages (2 d.p.f. and beyond) for a low strain variance in the fast muscle mass requires active swimming movements, as supported by the abnormal or retarded muscle development induced by various interventions that prevent normal locomotion(Van Raamsdonk et al., 1977; Van Raamsdonk et al.,1979).

Perspectives

Strain measurements are very hard to make in larval and juvenile fish during free swimming. We therefore had to adopt a theoretical approach to provide the best possible interpretation of the architectural changes during development. We studied the effects of muscle-fibre arrangement, body curvature and deformation on the variance in the muscle-fibre strain. In principle, it still has to be shown that the predicted optimal deformations for the best strain uniformity are close to the actual deformations. An important further step toward a better understanding could be the construction of a forward dynamics model that computes the muscle deformation and strains from the active state of the muscle fibres, the internal architecture and material properties of muscles and connective tissues, and the physical interactions between fish and water.

In the present paper, we measured muscle-fibre orientations and computed optimal deformations with the least variance in muscle-fibre strain. The current model could also be used to predict possible muscle-fibre orientations from prescribed body curvatures and internal deformations. It is well known that distinct muscle architectures occur in different chordate clades, such as Cephalochordates, Myxinoidea, Petromyzontida, Elasmobranchii, Holocephali, and Osteichthyes (Nursall, 1956; Van der Stelt, 1968; Alexander, 1969; Gemballa and Vogel, 2002). We expect that a comparison between predicted muscle arrangements and the actual morphology found in these clades could be a useful extension of current phylogenetic and comparative analyses that focus on architectural descriptions(Gemballa and Vogel, 2002).

  1. The relative size of the notochord and the neural tube play an important role in reducing the variation of the muscle strain in the youngest larval fish (2–4 d.p.f.). According to our theoretical predictions, this allows them to swim already effectively with fibres that have on average relatively small angles with the longitudinal axis of the body.

  2. From 3 d.p.f. onwards, the predicted coefficient of variation of the muscle-fibre strain for the selected deformation with the least strain variance reduces. The present analysis suggests that this is a consequence of the continuous muscle-fibre reorientation during development, probably under the influence of mechanical factors.

  3. The present theory predicts that close to the medial plane the muscle-fibre arrangement should deviate from a helical pattern to enable similar strain amplitudes to those at more lateral locations. This prediction is supported by the observed morphology. We predict also that in juveniles, a pseudo-helical arrangement of the muscle fibres in combination with an appropriate shear deformation can (potentially) lead to a fairly uniform strain distribution in the white muscle mass during strenuous activities. It has to be experimentally tested whether this deformation actually occurs.

  4. We predict that a close to uniform strain field for large body curvatures is likely to be achieved with the pseudo-helical arrangement in juveniles at the expense of a large coefficient of variation for small amplitudes. The predicted sensitivity of the coefficient of variation for the amplitude of body curvature is smallest for the youngest stages. Challenging experiments are required to test these hypotheses.

LIST OF SYMBOLS AND ABBREVIATIONS

     
  • A0

    area used to compute the lateral displacement of muscle tissue in deformation type I

  •  
  • c

    curvature of the medial plane c=1/R

  •  
  • CA

    centre of area

  •  
  • \(\mathrm{d}{\ell}_{\mathrm{f}}\)

    length of infinitesimal muscle-fibre element

  •  
  • \(\mathrm{d}{\ell}_{\mathrm{f}0}\)

    reference length of infinitesimal muscle-fibre element

  •  
  • \(\mathrm{d}{\ell}_{\mathrm{pf}0}\)

    projected length of infinitesimal muscle-fibre element on horizontal plane

  •  
  • d.p.f.

    days post fertilization

  •  
  • dx0

    distance of infinitesimal muscle-fibre element in x-direction

  •  
  • dy0

    distance of infinitesimal muscle-fibre element in y-direction

  •  
  • dz

    infinitesimal distance along the longitudinal axis of the fish

  •  
  • opening angle as defined in Eqn 9

  •  
  • ESP

    epaxial sloping part

  •  
  • h.p.f.

    hours post fertilization

  •  
  • HSP

    hypaxial sloping part

  •  
  • \({\ell}\)

    length of a particular element

  •  
  • \({\ell}_{0}\)

    reference length

  •  
  • MESP

    epaxial multilayered myosepta

  •  
  • MHSP

    hypaxial multilayered myosepta

  •  
  • n

    number of muscle fibres measured for a particular developmental stage

  •  
  • o.n.

    overnight

  •  
  • p

    exponent in Eqn 25

  •  
  • PBS

    phosphate-buffered saline

  •  
  • PFA

    paraformaldehyde

  •  
  • q

    exponent in Eqn 26

  •  
  • normalized radius of curvature of the medial plane

  •  
  • R

    radius of curvature of the medial plane (assumed to be constant over the considered short distance along the trunk)

  •  
  • s

    muscle-trajectory length

  •  
  • s0

    muscle-trajectory length in reference configuration

  •  
  • S

    transverse tissue slice of thickness dz

  •  
  • (x′,y′,z′)

    Cartesian system of coordinates used to compute the strain in lateral bending for a fish (Fig. 4F). The origin of the system is located at the centre of curvature. The y′-direction is parallel to the positive y-axis

  •  
  • (x, y, z)

    Cartesian system of coordinates for a straight fish(Fig. 3). The x-coordinate represents the local distance to the medial plan (left side is considered positive), the y-coordinate represents the distance from the horizontal septum in the dorsal direction. The z-coordinate represents the distance along the central axis of the fish

  •  
  • (xz, yz, zz)

    Cartesian system of coordinates for the Z-stack of the confocal microscope (Fig. 3)

  •  
  • x1′, x2

    x′-locations of the end points of a fibre element without added shear

  •  
  • x1′, x3

    x′-locations of the end points of a fibre element with added shear

  •  
  • x0

    distance to the medial plane of one of the end positions of a muscle-fibre element in the reference configuration

  •  
  • x1, x2

    distances to the medial plane of the end positions of a muscle-fibre element during bending

  •  
  • xmax0

    maximum distance of muscle from medial plane in reference configuration

  •  
  • yhyp1, yhyp2, yep1, yep2

    y-positions used in the sign convention ofγ med (see Fig. 5)

  •  
  • [ylow, yup]

    interval in y-direction, used to define γ

  •  
  • ymax

    maximum y-value

  •  
  • ymin

    minimum y-value

  •  
  • z1′, z2

    the z′-locations of the end points of a fibre element without added shear

  •  
  • z1′, z3

    the z′-locations of the end points of a fibre element with added shear

  •  
  • α

    azimuth of muscle fibres (angle between projection of fibre on horizontal plane and z-axis)

  •  
  • β

    elevation of muscle fibres (angle between muscle fibre and horizontal plane)

  •  
  • γ(x, y)

    shear angle in horizontal plane

  •  
  • γmax(y)

    maximum shear angle in horizontal plane at medial plane(x=0)

  •  
  • γmed(y)

    shear angle in horizontal plane at medial plane (x=0)

  •  
  • Δ

    denotes difference

  •  
  • ϵ

    linear strain

  •  
  • ϵf

    linear strain in longitudinal direction of muscle fibre

  •  
  • η

    coefficient of variation(

    \({\eta}={\sigma}_{{\varepsilon}}{/}\overline{{\varepsilon}}_{\mathrm{f}}\)
    ⁠)

  •  
  • θ

    angle of muscle fibre with z-axis

  •  
  • σϵ

    standard deviation in longitudinal strain of examined muscle fibres

  •  
  • σθ

    standard deviation in θ of examined muscle fibres

Dr Mees Muller, Dr Ulrike Müller and two anonymous referees are thanked for useful comments on a draft of this paper. Dr Felix Vogel helped to set up the technique of the measurements of muscle-fibre orientations.

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