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First published online March 28, 2008
Journal of Experimental Biology 211, 1289-1304 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.012336

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A functional analysis of myotomal muscle-fibre reorientation in developing zebrafish Danio rerio

Johan L. van Leeuwen^*, Talitha van der Meulen, Henk Schipper and Sander Kranenbarg

Experimental Zoology Group, Wageningen Institute of Animal Sciences, Wageningen University, Marijkeweg 40, 6709 PG Wageningen, The Netherlands

View larger version (37K): [in this window] [in a new window]   Fig. 1. (A) Drawing of myomere architecture in king salmon (after Greene and Greene, 1913). (B) Schematic drawing of curved muscle trajectories in the anterior trunk muscles of a teleost [redrawn from Alexander (Alexander, 1969)]. (C) Helical muscle fibre-trajectories (red) run over cylindrical surfaces (green). The medial plane is straight in this reference configuration (shown in blue). The central muscle-fibre trajectory (1) runs over a cylinder with a radius of zero and therefore forms a straight line in the centre of the other two cylinders. Muscle trajectory 3 has a pitch angle of 32°8' as proposed in Alexander's model (Alexander, 1969). The geodesic trajectory follows the shortest path between the end points at the medial plane. The cylinder is not fully drawn because it is cut off by the medial plane. In the reference configuration, the strain in the trajectories is assumed to be zero. Trajectory 2 makes a full turn over a cylinder with a smaller radius that almost touches the medial plane [not considered by Alexander (Alexander, 1969)]. Halfway along its length, the trajectory is parallel to the medial plane. This would lead to very small strains at this location. (D) Similar to C, but with a curved medial plane. The largest cylinder of C is shown deformed into a torus with an identical volume (Alexander's case iii). The radius of curvature of the medial plane is 10 times the maximum distance between the most lateral location on the cylinder and the medial plane in the reference position (C). The computed average strain in the peripheral trajectory is –0.039 as computed from (s–s0)/s0, against –0.02 in the central trajectory, where s0 is the trajectory length in the reference configuration and s is the length in the contracted situation.

View larger version (10K): [in this window] [in a new window]   Fig. 2. (A) Straight horizontal segment along the body of a fish. The thick central line indicates the longitudinal axis that keeps the same length during bending. The numbered red lines represent muscle-fibre segments that are assumed to lie in the horizontal section. (B) Simple beam deformation of the body segment of A, with a lateral expansion at the concave side and a lateral compression at the convex side. Material points are assumed to have a constant dorso-ventral position and an incompressibility constraint is applied. At the concave side, the muscle-fibres segments near the skin (4 and 8) are able to shorten much more than those near the central axis (3 and 7); at the convex side, the segments directly underneath the skin (1 and 5) lengthen much more than the more medial segments (2 and 6). (C) Bending with an added shear deformation is caused by oblique muscle fibres near the medial plane (see main text). The shear deformation is maximal at the central axis and zero at the skin. This deformation enables muscle fibres near the central axis to contract at the concave side if they have a suitable orientation such as fibre segment 3, in spite of the constant length of the central axis. The strain in segments 3 and 4 is approximately the same. Segments 1 and 2 lengthen by similar percentages. Fibre segments 6 and 7 do not change more in length than in B, because they are orientated parallel to the axis. The orientations of segments 6 and 7 would result in very low strain and work output and are not present in real fish. The shear deformation causes a longitudinal shift of muscle tissue and skin relative to the central axis. The shear angle near the medial plane is denoted by .

View larger version (15K): [in this window] [in a new window]   Fig. 3. (A) Coordinate systems of the Z-stack (xz, yz, zz) in blue and of the fish (x, y, z) in red. (B) Schematic representation of a portion of a muscle fibre (thick red line) that extends between two parallel optical sections (shown pink) in the (x, y, z) system at a mutual distance z. The azimuth and elevation angles and β and the projection of the fibre on the horizontal plane are indicated, as well as the differences between begin and end points in the x- and y-directions x and y. See main text for further explanation.

View larger version (33K): [in this window] [in a new window]   Fig. 4. (A) Transverse slab S through the muscle tissue (red), for one side of the body. Medial plane (dark blue) is straight (reference configuration). Global coordinate system (x, y, z) is indicated. (B) Similar to A, but with curved medial plane. (C) Muscle portion from the slab of A0, infinitesimal fibre element in red; area A0 is shown in green. (D) Similar to C, but with curved medial plane and added shear deformation. (E) Projection of the muscle portion of C onto a horizontal plane with y constant. Projection of muscle fibre segment is shown by red line. (F) Similar to E, but with added shear deformation and curved medial plane. Local Cartesian system (x', y', z') has its origin (0,0,0) in the centre of curvature. The x'-axis and the z'-axis are shown as broken lines. For further explanations, see main text.

View larger version (38K): [in this window] [in a new window]   Fig. 5. (A) Drawing of the regions in the muscle slab that illustrate the sign convention for the `added' shear function (Eqn 25 and Eqn 26) and muscle-fibre directions (black arrows, projected onto a transverse plane) that represent roughly the pattern at 51 d.p.f. The boundaries between regions are depicted by horizontal thin green lines. The function is zero at these boundaries. The left side is considered to represent the convex side of the body, and the right side the concave side. Regions with a positive value of are indicated by red + signs, and negative values by blue – signs. The medial myoseptal multilayers of connective tissue in the epaxial and hypaxial muscles (MESP and MHSP) are indicated by thick green horizontal lines. The attachment angles of the muscle fibres that insert at the dorsal and the ventral side of these layers are very different. (B) Contour plot of according to the sign convention of A.

View larger version (54K): [in this window] [in a new window]   Fig. 6. (A) Contour plot of (the ratio of the s.d. in the strain and the mean strain) as a function of max and p for a developmental stage of 2 d.p.f. The computation is made for the concave side of the body with =5, and deformation type I (i.e. with lateral body thickening). The highest value of of is obtained for high values of max and p. (B) Idem for a developmental stage of 51 d.p.f., with the optimum for strain uniformity at p=1.4 and max=26°.

View larger version (82K): [in this window] [in a new window]   Fig. 7. Propidium iodide stained sagittal sections through trunk muscles of larval zebrafish, aged 16 (A,B), 18 (C,D), 24 (E,F) and 72 h.p.f. (G,H). For each stage, two sections are shown, one close to the medial plane and one at a more lateral position. Hs, horizontal septum; m, myoseptum; nc, notochord; nt, neural tube. Horizontal scale bars, 20 µm.

View larger version (92K): [in this window] [in a new window]   Fig. 8. (A) Cross-section through the trunk muscles of a larva of 2 d.p.f. (B) Idem, but for the 15 d.p.f. stage (see also Movie 1 in supplementary material). Note the relatively large size of the spinal cord and notochord in the 2 d.p.f. stage compared with 15 d.p.f. In the 15 d.p.f. stage many muscle fibres attach to the medial septum that is thought to undergo only very small changes in length during swimming. The measured muscle-fibre orientations and predicted strains with the lowest coefficient of variation corresponding to A are shown in Fig. 9A,D. For B, the corresponding panels are Fig. 9B,E. (C) Cross-section through a portion of the trunk of a juvenile zebrafish of 45 days [stained according to Crossmon (Crossmon, 1937)]. The epaxial (E) and hypaxial (H) multilayers of connective tissue (MESP and MHSP) are indicated. These connective tissue sheets are oriented almost parallel to the horizontal septum and are formed by a close junction of neighbouring myosepta. They have to transmit the forces of the muscle fibres that attach with very different orientations at the dorsal and ventral side of the multilayers (as shown in Fig. 9C), and thus cannot balance their forces.

View larger version (31K): [in this window] [in a new window]   Fig. 9. (A) Measured muscle-fibre orientations at 2 d.p.f. (corresponding to Fig. 8A). Note the scale differences between panels. Every muscle fibre is represented by the projection of a unit vector on a transversal plane. The length of the unit vector is shown at the bottom left-hand side. (B,C) Idem for 15 and 51 d.p.f. B corresponds to Fig. 8B. (D–F) Computed strain distribution in the muscle fibres at the convex side (left) and the concave side of the body (right), for the same developmental stages as those of (A–C), and a normalized curvature =5. Values of med and p (Eqn 25) that gave the least variation in strain (i.e. smallest vales of ||) were used. The position of each muscle fibre is indicated by a small circle, with a fill color representing the computed strain according to the labeled color bar. The muscle fibres from the left and the right-hand side were pooled for this purpose. Thus, the mirror image of all measured fibres from the left-hand side were added to the ensemble of fibres at the right-hand side, and vice versa. The asterisks in C denote regions with abrupt changes in the muscle fibre directions.

View larger version (15K): [in this window] [in a new window]   Fig. 10. Computed coefficient of variation (, blue curves for concave side of the fish, black curves for the convex side) as a function of developmental time. The local radius of curvature was set to five times the local maximum half-width of the body (R=5xmax0). (A) Results for deformation type I and (B) type II. Solid ratio curves show opt for measured fibre orientations and optimized shear values in the computational model. Dotted curves show for measured orientations but without added shear deformation. Broken curves show the results for hypothetical longitudinally oriented fibres and without added shear deformation. The variation in the computed mean strain with shear optimization (, solid red curve for concave side, dotted curve for convex side) is remarkably small during development. The results are very similar for the two deformation types.

View larger version (11K): [in this window] [in a new window]   Fig. 11. Analysis of the sensitivity of (the coefficient of variation, defined as the ratio of s.d. over mean strain) for the normalized body curvature as a function of developmental time. For each combination of developmental time and , the lowest absolute value of was computed using Eqn 25. The lower the absolute values of , the better the strain uniformity in the examined muscle-fibre population. Values of are indicated at the right side of each curve. (A) Results for deformation type I and (B) type II. Black curves show the computation for the convex side of the fish, blue curves are for the concave side. Values of are hardly influenced by body curvature at the youngest stages (up to 15 days). The best uniformity is obtained for the strongest curvatures for the two oldest stages. The results are again very similar for the two deformation types.

View larger version (137K): [in this window] [in a new window]   Fig. 12. Polarized light image of two adjacent and parallel collagen fibre layers of the epaxial multilayers of a carp (dissected form the anal region along the trunk). The muscle fibres were carefully removed by microdissection to reveal the collagen fibre bundles. The main directions of the collagen fibres are indicated by lines with arrowheads. The top layer, visible at the right hand side, is largely removed to reveal the second layer. The orientations of the collagen fibre bundles in the two layers are very different. The functional relevance of the jump in fibre orientation between the layers for the force transmission is discussed in the main text. In vivo, these layers are predicted to slide parallel to each other during muscle contraction of the muscular system (to allow for the jump in added shear, see main text). Some other fibres with deviating orientations are visible that do not form part of these layers. Scale bars, yellow: 51.24 µm; red, 46.66 µm.

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