The muscle segments of fish have a folded shape, termed a chevron, which is thought to be optimal for the undulating body movements of swimming. However, the mechanism shaping the chevron during embryogenesis is not understood. Here, we used time-lapse microscopy of developing zebrafish embryos spanning the entire somitogenesis period to quantify the dynamics of chevron shape development. By comparing such time courses with the start of movements in wildtype zebrafish and analysing immobile mutants, we show that the previously implicated body movements do not play a role in chevron formation. Further, the monotonic increase of chevron angle along the anteroposterior axis revealed by our data constrains or rules out possible contributions by previously proposed mechanisms. In particular, we found that muscle pioneers are not required for chevron formation. We put forward a tension-and-resistance mechanism involving interactions between intra-segmental tension and segment boundaries. To evaluate this mechanism, we derived and analysed a mechanical model of a chain of contractile and resisting elements. The predictions of this model were verified by comparison with experimental data. Altogether, our results support the notion that a simple physical mechanism suffices to self-organize the observed spatiotemporal pattern in chevron formation.

All chordates have metameric muscle segments that are critical for locomotion. These muscles are termed myotomes, and their characteristic shape in aquatic species (Fig. 1) is referred to as a chevron (Hoar et al., 1969; Kimmel et al., 1995). Typically, one observes three folds in sagittal sections of adult fish like salmon and zebrafish (Agassiz and Vogt, 1845; Greene and Greene, 1913; Maurer, 1913; Videler, 1993), but there can be up to six (Harder, 1964). The oldest known fossils with chevrons are Pikaia (Morris and Caron, 2012) and Haikouichthys (Shu et al., 1996). They are present in amphioxus (Bone and Moore, 2008; Hatschek, 1882; Hatschek, 1893; Nursall, 1956), the most basal extant chordate (Delsuc et al., 2006), the agnathan lamprey (Langelaan, 1904; Nursall, 1956; Wickström, 1897), the tadpole of the anuran Xenopus (Harrison, 1898; von Seckendorff Hoff and Wassersug, 1986) and in salamander larvae (Bordzilovskaya et al., 1989; Francis, 1934; Gegenbaur, 1859; Mchedlishvili et al., 2007).

The myotome's locomotory function has been conserved for at least 500 million years (Bone and Moore, 2008), suggesting that the chevron shape has been optimized for this function (Hoar et al., 1969; Videler, 1993). Chevron shape and function are currently used to infer evolutionary relationships among chordates (Lacalli, 2012; Mallatt and Holland, 2013; Morris and Caron, 2012). Although the mechanics of chevron-shaped myotomes during swimming has been studied for over a century (Alexander, 1969; Chevrel, 1913; Nursall, 1956; Rauther, 1940; van der Stelt, 1968; van Leeuwen, 1999; van Leeuwen et al., 2008; Videler, 1993; Willemse, 1966; Wunder, 1936), the development of the chevron remains poorly understood.

How does the chevron pattern emerge during embryogenesis? We addressed this question in the zebrafish Danio rerio (Hamilton 1822). Its embryo facilitates investigation of morphological changes, notably somitogenesis (Durbin et al., 2000; Henry et al., 2005; Holley et al., 2000; Schröter et al., 2008; Schröter et al., 2012; van Eeden et al., 1996), myotome differentiation (Cortés et al., 2003; Currie and Ingham, 1996; Daggett et al., 2007; Devoto et al., 1996; Henry and Amacher, 2004; van Raamsdonk et al., 1974a; van Raamsdonk et al., 1978; Yin and Solnica-Krezel, 2007) and chevron formation (van Raamsdonk et al., 1974a; van Raamsdonk et al., 1974b; van Raamsdonk et al., 1977; van Raamsdonk et al., 1979). As early development progresses, somites sequentially bud off the unstructured paraxial mesoderm (Fig. 1A–C; supplementary material Movies 1, 2) and commit most of their cells to myotome formation, thereby respecting and further enforcing tissue segmentation (Kimmel et al., 1995; Morin-Kensicki et al., 2002). We will use the terms segment and segment boundary for both somites and myotomes and their boundaries, respectively.

About 35 segments form within ~15 h (Schröter et al., 2008). Initially, segments appear as cuboidal blocks (Fig. 1A). The anteriormost segments retain their cuboidal shape; more posterior segments deform into V-shaped chevrons with the apex pointing anteriorly (van Eeden et al., 1996; van Raamsdonk et al., 1974a). However, reports differ as to the timing of chevron formation (starting when seven or 12 segments have formed) (van Eeden et al., 1996; van Raamsdonk et al., 1974a). van Raamsdonk et al. (van Raamsdonk et al., 1974a) reported that segments five to 12 deform simultaneously. Tail segments (18–35) start to deform immediately after each somite has formed (van Raamsdonk et al., 1974a). The timing and the extent of shape changes of segments 13–17 have not been reported. Three weeks after hatching, W-shaped myotomes emerge (van Raamsdonk et al., 1974a). Together, these findings indicate temporal and spatial heterogeneities but their quantification is incomplete. Pioneering quantitative work measured the chevron angles of the anal segment for different time points (van Raamsdonk et al., 1979), but earlier and more frequent measurements are required to constrain mechanistic models. In this study, our first goal was to provide a definitive description of chevron formation using quantification of time-lapse microscopy of the developing zebrafish embryo.

The processes driving chevron formation during embryogenesis and larval stages were first investigated in the 1970s by van Raamsdonk et al. (van Raamsdonk et al., 1974a; van Raamsdonk et al., 1974b;

List of symbols and abbreviations

     
  • a

    distance between apex of chevron and the initially straight border (Fig. 2B)

  •  
  • E

    energy in the mechanical model

  •  
  • Fx

    forces exerted by the springs in the segments of the mechanical model

  •  
  • h

    dorsoventral height of a segment

  •  
  • hpf

    hours post-fertilization

  •  
  • i

    segment index; counted from anterior to posterior

  •  
  • K

    dimensionless model parameter indicating the stiffness of the boundary relative to the stiffness of the intra-segment springs

  •  
  • Kx

    spring constant of springs within a segment of the mechanical model

  •  
  • Kα

    resistance of the segment boundaries of the mechanical model against bending

  •  
  • L

    anteroposterior length of a segment

  •  
  • L0

    rest length of springs within a segment of the mechanical model

  •  
  • n

    number of segments in the mechanical model

  •  
  • N

    number of individuals

  •  
  • nc

    notochord

  •  
  • t

    time; the average time point of the formation of segment 1 is t=0 if not stated otherwise

  •  
  • tstart, tend

    fit parameters describing the time points at which chevron formation started and ended

  •  
  • xi

    anteroposterior extension of the spring in segment i of the mechanical model

  •  
  • α

    chevron angle between the dorsal segment boundary and the dorsoventral axis, calculated from measurement of γmin, γmax

  •  
  • αend

    fit parameter describing the final chevron angle

  •  
  • αi(t)

    chevron angle of segment i of the mechanical model at time t

  •  
  • αmin, αmax

    minimum and maximum angle α, calculated from γmax and γmin, respectively

  •  
  • γ

    measured angle between the dorsal extremity of the posterior segment boundary and the dorsal roof of the notochord

  •  
  • γmin, γmax

    minimum and maximum angle of γ (Fig. 2A)

  •  
  • Δα

    uncertainty of measurement of α

  •  
  • Δαend, Δtstart, Δtend

    errors of the fit parameters

van Raamsdonk et al., 1977; van Raamsdonk et al., 1979). Below, we introduce four competing hypotheses seeking to explain chevron formation, namely body movements, templating of the chevron from the posterior border of the head, relative tissue movements, or differential growth or tonic muscle tension associated with muscle pioneer cells.

First, given the accepted importance of the chevron shape for locomotion, the undulating lateral movements that herald the onset of locomotion were suggested to play a role in the morphogenesis of the chevron (te Kronnié, 2000; van Raamsdonk et al., 1974a; van Raamsdonk et al., 1977; van Raamsdonk et al., 1979). When testing this hypothesis by immobilizing young fish with developed chevrons, a slight straightening of the chevron angles was observed. The fish recovered the normal pattern when it could move again. Independently, and arguing against a strong role for swimming movements in chevron development, the genetically immobile homozygous nicb107 mutant zebrafish forms chevrons (Schröter et al., 2008; van der Meulen et al., 2005). However, this formation has never been quantified. Thus, a contribution from movement to the dynamics of chevron formation, or the final angle of the chevron, may yet be present.

Second, independent of motion, a templating hypothesis for chevron formation posits that the posterior border of the head already possesses the chevron shape, and this pattern can be passed to the next segment. Muscle cells may simply grow to a predefined size, so that when the muscle cells connect to the next segment's anterior boundary, this segment would also form a chevron (van Raamsdonk et al., 1979). Unless interrupted, this pattern would be passed sequentially into the tail. Inconsistent with templating as the sole determinant, chevrons develop immediately posterior to segments with experimentally disrupted shapes (Riedel-Kruse et al., 2007; Windner et al., 2012). However, a contribution to the final shape by templating is not ruled out.

The third hypothesis features relative movement of structures in the embryo (Turner, 2007). The myosepts were proposed to attach to the notochord to be pulled into chevron shape. This hypothesis can be tested by time-lapse microscopy.

Finally, tonic muscle contraction and differential growth within the myotome have also been considered as potential mechanisms (van Raamsdonk et al., 1979). The earliest developing muscle fibres, the muscle pioneers, elongate in concert with the formation of chevrons (Kimmel et al., 1995). Mutants that fail to differentiate muscle pioneers do not have normal V-shaped chevrons when examined at larval stages, instead displaying a U-shaped chevron (Halpern et al., 1993; te Kronnié, 2000; van Eeden et al., 1996). It is thus now widely believed that the muscle pioneers and/or the myoseptum are responsible for the chevron shape (Dolez et al., 2011; Koumans and Akster, 1995; Meyers et al., 2013), but this has not been directly tested, and a mechanism whereby these cells would cause the chevron shape has not been proposed.

In summary, these four hypotheses represent a range of mechanisms that are not mutually exclusive, but for which there is currently no conclusive experimental support. Our second goal was to test these hypotheses by comparing their predictions with our quantitative data.

Independent of which, if any, of the proposed mechanisms will be confirmed, it is clear that morphogenesis of the chevron is ultimately executed by mechanical forces. These might act (1) autonomously in each segment in a segment-specific and genetically encoded manner. Alternatively, by analogy to chevron patterns in abiotic systems such as rock formations in geology (Bayly, 1974) and polymers and liquid crystals in material science (Read et al., 1999; Singer, 1993), the chevron might emerge (2) collectively as a result of force fields across the tissue with initially homogeneous properties and without the need to invoke genetically encoded and segment-specific properties. Our third goal was to test whether the latter scenario is possible by formulating a mechanical model, motivated by correlations revealed by our quantitative data analysis.

In this study, we quantified time-lapse light microscopy movies of chevron formation for mobile, immobile and muscle pioneer-ablated zebrafish embryos. Analysis of these data disproves three of the four previously hypothesized mechanisms and restricts any potential role for muscle pioneers to chevron maintenance. We then propose a tension-and-resistance mechanism for which chevrons form collectively and formalize it in a mechanical model that shows qualitative agreement with our data. We suggest further tests and extensions of our model and briefly discuss our results in the light of evolution.

Measurement of chevron formation in zebrafish embryos

We quantified chevron formation using time-lapse imaging of embryos from in-crosses of free-spawning heterozygous nicb107 parents (Herrgen et al., 2009; Schröter et al., 2008). Wild-type and heterozygous nicb107 mutant zebrafish show undulatory swimming movements (supplementary material Movie 1), whereas the homozygous nicb107 mutant remains still throughout the observation period (supplementary material Movie 2). We will refer to homozygous nicb107 mutant embryos as ‘immobile’, and both wild-type and heterozygous nicb107 embryos as ‘mobile’. We analysed time-lapse microscopy movies of three mobile and five immobile zebrafish from the beginning of segmentation until the end of chevron formation. The time point of the formation of segment 1 was chosen as t=0. Below, we first focus on mobile embryos.

We oriented the embryos laterally, affording a view of the segmenting paraxial mesoderm in which the forming straight somite boundaries could be seen reliably (Fig. 1A). As the somites matured into myotomes and changed their shape from a cuboidal block to a chevron, we could observe the deformation of segments (Fig. 1B,C). Through this 16 h time window, the overall shape of the embryonic axis deformed as the segmenting mesoderm converged from a flattened mediolaterally extended cross-section (Fig. 1D) to a dorsoventrally extended cross-section (Fig. 1E). In each time-lapse frame, the medially located notochord was visible as a reference line that intersected with the apex of the forming chevron (Fig. 1F).

To quantify the emergence of the chevron shape, we measured the angle γ=(γmax–γmin)/2 between the dorsal extremity of the posterior segment boundary and the dorsal roof of the notochord in each time-lapse movie frame (Fig. 2A; supplementary material Table S1 and Fig. S1). Next, we transformed the measurements of γmin and γmax to α=90 deg–(γmaxmin)/2, the angle between the dorsal segment boundary and the dorsoventral axis (Fig. 2B). The error is given by Δα=(γmax–γmin)/2. We show these data for segment 1, 7 and 15 from a representative embryo in Fig. 2C–E. Because of the high contrast of the yolk edge, the anterior-most segments were difficult to follow, leading to gaps in the time series (Fig. 2C). In contrast, more posterior segments yielded near-continuous time series of chevron angle values until the start of occasional movements at 7.3±0.2 h also introduced intermittent gaps (grey regions in Fig. 2C–E; supplementary material Movie 1).

The anterior-most segment (segment 1) did not form a chevron (Fig. 2C). Strikingly, the chevron angle of segments 6 and 15 changed smoothly over a time interval of about 2 h before reaching the final chevron shape, which remained constant until the end of the recording (Fig. 2D,E).

Based on the data for each single segment, we identified non-chevron-forming segments as those for which most of the measured angles α were close to 0 deg. For them, we calculated the final angle αend as the mean of all measured angles (Fig. 2C, orange). To capture the dynamic behaviour of each chevron-forming segment in a compact and informative manner, we described the change in the chevron angle over time with three parameters: αend, the final chevron angle, where αend=0 deg means no chevron formation and a typical posterior chevron has αend≈40 deg; and the time points at which chevron formation started (tstart) and ended (tend). These parameter values were determined for each segment by fitting a piecewise linear function to the time series (Fig. 2D,E). We restricted our analysis to trunk segments 1 to 15, as the values for αend did not change appreciably in segments 16–30 when compared with segments 8–15.

Quantitative description of chevron formation

For all quantified time series of mobile and immobile zebrafish, we first assessed which segments along the axis showed robust chevron formation. For segments 1 and 2, we did not observe embryos with significantly chevron-forming boundaries, regardless of mobility (Table 1). For both mobile and immobile embryos, segment boundaries deformed significantly at segments 4–15 (Table 1). We conclude that the two anterior segments do not form chevrons and that the transition from block-like to chevron-shaped myotomes occurs at segment number 3.

For segments 4–15, 96% of the datasets were of sufficiently high quality that the result of the curve fitting was accepted and these values are plotted in Fig. 3. For mobile embryos (green symbols), the deformation of boundaries of segments 4–7 starts nearly simultaneously, whereas the boundary deformation of segments 8–15 starts sequentially and with a small offset after the formation of the segment boundary (Fig. 3A, green symbols). Boundary deformation of all quantified segments in mobile embryos ceased 2±1 h after the start of chevron formation (Fig. 3A,B, green symbols). The data for the final chevron angle group the segments in a similar way as the chevron formation start time: whereas the final chevron angle monotonically increases for segments 1–7, the more posterior segments homogeneously acquire 40±3 deg (Fig. 3C, green symbols).

Complementing previous studies of chevron shape in later developmental stages with established chevrons (van der Meulen et al., 2005; van Raamsdonk et al., 1974a; van Raamsdonk et al., 1977; van Raamsdonk et al., 1979), the results here represent the first quantitative measurements of the dynamics of chevron formation in any species. Thus, the first goal of this study has been reached and we can now compare our data with the predictions of hypothetical mechanisms for chevron formation.

Body movements

To test whether the normal undulating body movements resulting from rhythmic muscle contraction have an influence on early chevron formation, as has previously been proposed (van Raamsdonk et al., 1974a; van Raamsdonk et al., 1977; van Raamsdonk et al., 1979), we used our data to quantitatively relate the dynamics of chevron formation and the start of movements. As expected (van Raamsdonk et al., 1974a), we found that movements start after the anterior segments 4–17 have already started to deform (Fig. 3A; supplementary material Movie 1). Hence, movements are not required for chevron formation to start. Segments 4–13 had already finished chevron formation before movements started in wild-type embryos.

To what extent movements contribute to chevron formation dynamics and to setting the final chevron angle can also be determined by comparison with non-moving embryos. The nicb107 mutant is ideally suited for this purpose as it has a lesion in the nicotinic cholinergic receptor that prevents neurotransmitter from motor neurons from activating muscle contraction, consequently paralysing the embryo (Sepich et al., 1994; Sepich et al., 1998; Westerfield et al., 1990). The data for immobile nicb107 homozygous embryos confirm that chevrons form (Fig. 3, orange symbols), as expected (Schröter et al., 2008; van der Meulen et al., 2005). As then expected for segments 4–13, which ceased chevron formation before the start of movements in mobile embryos, mobile and immobile embryos showed very similar timing of chevron formation, tstart and tend.

Importantly, no significant difference was found between mobile and immobile embryos in the values of final chevron angle αend, based on 8 h of recording after movements of wild-type embryos had started. Strikingly, no significant differences were found for segments 14 and 15. We conclude that the movements of early rhythmic muscle contraction that precede swimming have no effect on the formation of chevrons in the zebrafish embryo.

Simple chevron templating

The data in Table 1 and Fig. 3C can also be used to test the previously proposed templating mechanism for chevron formation (van Raamsdonk et al., 1979). If the chevron shape is simply copied from a chevron-shaped surface at the back of the head, then we would expect the same chevron angle for all segments, but this was not observed at any stage of chevron formation. Thus, our new data do not support the idea that templating alone is responsible for the development of the chevron shape in the anterior trunk.

Inhibition of muscle pioneer differentiation

The data presented above rule out the body-movement, simple-templating and notochord-movement hypotheses (see Discussion). Muscle pioneer cells are thought to be responsible for chevron formation (Dolez et al., 2011; Meyers et al., 2013). Muscle pioneers can be identified by engrailed2 expression (Fig. 4A). Embryos treated with 0.2% DMSO carrier form normal chevrons, indistinguishable from wild-type embryos grown in E3 medium only, as illustrated by cb1045 expression at segment boundaries (Fig. 4B). We blocked muscle pioneer differentiation by inhibiting Shh signalling with cyclopamine (Barresi et al., 2001; Wolff et al., 2003) (Fig. 4C). As expected, segment boundaries showed a U-shape phenotype at 36 hpf (hours post-fertilization) (Fig. 4D), resembling U-mutants. We performed time-lapse imaging of these muscle pioneer-ablated embryos as described above (Fig. 4E–H; supplementary material Movie 3). Segment formation progressed along the axis at the same rate as under normal conditions (data not shown). Although the chevron angle with cyclopamine treatment had a larger measurement error, surprisingly, we found that many chevrons formed with a normal angle at around 400–500 min (Fig. 4F,H; supplementary material Fig. S2). Subsequently, the chevron shape was not maintained (Fig. 4G,H; supplementary material Fig. S2) and relaxed into a U-shape (Fig. 4D). For the majority of the analysed segments, the wild-type start- and end-times of bending were in good agreement with the transient bending of muscle pioneer-ablated embryos (orange curve in Fig. 4H).

Insufficiencies of three proposed mechanisms to explain chevron formation

Our quantitative comparison of mobile and immobile embryos rules out a role for swimming motions of the embryo in both the timing and final shape of the embryonic chevrons, thereby corroborating and extending earlier evidence. This finding does not impact the acknowledged importance of the myotome's chevron shape for swimming mechanics.

We quantified final chevron angles as function of segment index and found a monotonic increase for segments 4–7 that also start formation simultaneously, whereas segments 1 and 2 do not form chevrons (Fig. 3). An earlier report described this transition between segments 4 and 5 (van Raamsdonk et al., 1974a), instead of segment 3, which may reflect the lower spatial and temporal resolution of earlier methods. However, van Raamsdonk et al. used a slightly lower temperature, which might shift the transition. Our data rule out the simple templating hypothesis according to which each angle is just a copy of its earlier formed anterior neighbour. This observation agrees with the results of Riedel-Kruse et al. (Riedel-Kruse et al., 2007) in which correct segmentation was transiently perturbed and then rescued. Although segments 7–17 did not form correctly and thereby would be unable to pass on a chevron pattern, the more posterior segments were shaped as chevrons again. Recently, Windner et al. (Windner et al., 2012) rescued defective somitogenesis in the fss mutant by transient expression of the Fss/Tbx6 protein. They observed one or two contiguous chevrons, where the tissue immediately anterior was not chevron shaped, again inconsistent with a simple templating mechanism for chevron formation. Nevertheless, the posterior boundary of any segment is the anterior boundary for the next segment and can thereby influence its shape.

The anchoring hypothesis of Turner (Turner, 2007) requires an anterior-ward movement of the notochord at the time of chevron formation. In contrast, our time-lapse movies showed a strong posterior-ward movement of the notochord relative to the segment boundaries (supplementary material Movie 2), but this was observable only after the vacuolating cells of the notochord became visible, which occurs after completion of chevron formation. Confocal time-lapse movies revealing the posterior-ward sliding of the notochord have been reported (Barrios et al., 2003), but these were recorded slightly before chevron formation of the corresponding segments. Thus, there is no continuous anterior-ward movement of the notochord that would be consistent with the anchoring hypothesis. It remains formally possible that a wavefront of transient reversal of the direction of notochord sliding propagates along the axis in concert with chevron formation. However unlikely this proposal seems, time-lapse microscopy targeted to the notochord at this time point would be required to resolve the question.

U-shaped myotome mutants and the role of muscle pioneers

Zebrafish mutants have been reported in which somites form normally but the myotomes display a U-shape instead of a chevron at larval stages. These phenotypes have been associated with the failure to form a specific cell type in the myotome, the muscle pioneer at the apex of the chevron. The first genetic evidence of a role for these cells in chevron patterning came from the no tail (ntl) mutant, encoding a zebrafish Brachyury/T transcription factor, in which muscle pioneers fail to form and myotomes take on a U-shape (Halpern et al., 1993). Most of this class of mutants encode components of the Sonic Hedgehog signalling pathway, which is required for the differentiation of the adaxial cells, a columnar epithelium that is apposed to the notochord (Hatta et al., 1991; Weinberg et al., 1996), into early muscle fibre types such as the muscle pioneer (Johnston et al., 2011; Stickney et al., 2000). In kny;tri double-mutants, adaxial cells do take on a muscle fibre fate, but do not show proper spanning of the segment. Even in this case the chevron shape gets weaker (Yin and Solnica-Krezel, 2007). However, the relationship between these cells and the development of the chevron has not been explored and the surgical removal of muscle pioneer precursor cells in a few segments did not result in obviously altered segment shapes (Honjo and Eisen, 2005). Similarly, the inhibition of muscle pioneers by Dorsalin-1 in two consecutive segments did not lead to altered chevrons (Du et al., 1997).

Our own cyclopamine-treated embryos closely recapitulate the appearance of the original U-mutants; they lack muscle pioneers in early segmentation stages and they exhibit U-shaped myotomes at larval stages. However, the surprising observation of transient chevron shapes that mimic the normal untreated embryo with respect to maximum chevron angle as well as bending start- and end-time speaks against a role for muscle pioneers in chevron formation. High-level Shh signalling induces further cell types of the sclerotome (Fan and Tessier-Lavigne, 1994; Hammerschmidt et al., 1996), the dermomyotome (Feng et al., 2006), the spinal cord and other embryonic tissues. Our cyclopamine treatment likely also affected these Shh-dependent cell types and, hence, a requirement for any of them in chevron formation is also unlikely. Conversely, the observed failure to maintain the chevron shape may be due to the perturbation of any of the Shh-dependent cell types, including muscle pioneers. Our findings suggest that U-class mutants may similarly exhibit a failure of chevron maintenance, not formation. A potential role for other cell types in chevron formation remains to be investigated.

A mechanical model for the self-organization of heterogeneous chevron angles

The final angle of the chevron in subsequent segments increases monotonically in the anteroposterior direction until it reaches the maximum value of 40±3 deg at segment 7, which was also reached by all posterior segments that we analysed. Why is there such a strong segment dependence of chevron angle across the anterior segments? Is this because there is a molecular or genetic heterogeneity along the axis? Is it necessary that the pattern is pre-programmed so that each segment's angle is individually specified by genetic means, perhaps through a Hox gene code mechanism (Prince et al., 1998), or could the collective behaviour of segments with homogeneous properties self-organize the observed heterogeneous angles?

To investigate the latter possibility, we propose a mechanical model that is motivated by a series of published observations about the timing of cell differentiation and its coincidence with the timing of chevron formation, as reported here. The expression of the muscle-specific markers Eng protein and myod mRNA starts simultaneously in the anterior up to seven segments in the adaxial cells. For the more posterior segments, the two genes are expressed sequentially (Hatta et al., 1991; Weinberg et al., 1996). The muscle pioneers differentiate from the adaxial cells and later give rise to the myoseptum (Daggett et al., 2007; Devoto et al., 1996; Johnston et al., 2011). The cellular shape changes and cell migration that accompany the differentiation of other muscle types also start close to the notochord and simultaneously in the first five segments (Cortés et al., 2003; Daggett et al., 2007; Stickney et al., 2000; van Eeden et al., 1996; Yin and Solnica-Krezel, 2007) but then proceed posteriorly with constant time lag after segment formation (Henry and Amacher, 2004). In summary, these observations group the segments in the same way as our data analysis does: a group of anterior segments behaves differently from the complementary posterior segments (Fig. 5).

Although there are currently no measurements of the forces involved, it is plausible that the reported cellular rearrangements accompanying muscle differentiation generate tension from within the segments. Those cellular rearrangements start medially close to the notochord and the tension would act only medially on the segment boundaries and thereby bend them. Reinforced by the extracellular matrix, these boundaries will resist bending to some degree. The emergent behaviour induced by the tension cannot be foreseen intuitively as all segments are coupled and interact simultaneously. It is possible to imagine all but the first segment acquiring equal shapes or, alternatively, multiple solutions may coexist and a fluctuation of a single initial segment shape could set up a pattern of mirrored chevrons to either side of the initial fluctuation.

The mathematical formalization of the model is required to understand whether the system is capable of showing the emergence of the experimentally observed heterogeneous pattern (Fig. 6). Note that this model assumes that tension is generated within the segment, but it is independent of how this tension is generated and it does not attempt an explicit description of individual cellular behaviours. Our model includes the case that the reported cellular rearrangements cause tension but it also encompasses other potential causes of tension. We reduce the model to two dimensions thanks to translational symmetry in the mediolateral direction. It consists of n=7 segments. Each segment has the same anteroposterior length L and dorsoventral height h but a variable chevron angle αi(t) of its posterior boundary. The angle of the segment's anterior boundary is given as the angle αi–1(t) of the posterior boundary of the corresponding neighbouring segment. We treat the anterior-most boundary between the head and segment 1 as stiff and straight, i.e. with a constant chevron angle of 0 deg. All segment boundaries are initialized as straight (Fig. 6A′). We then add harmonic springs (spring constant Kx, rest length L0<L) in each segment that model the intra-segmental tension (Fig. 6B′). These springs contribute part of the energy of the model (see equation in Fig. 6D, blue). The resistance of the boundaries against bending is modelled by adding an energy penalty (strength Kα; see equation Fig. 6D, orange). Of course, the extension of the springs, xi, and the chevron angles, αi, are constrained by the geometry of the system (Fig. 6C′). Assuming 2Lh (true for zebrafish segments) and small αi, we express this relationship in the formula for xi (Fig. 6D, see  Appendix).

With the derived energy and the initial conditions for αi and the head boundary condition, we can calculate the final chevron angles in the steady state of the model and simulate the dynamics. We derive the steady state of our model by minimizing the energy (detailed calculations are in the  Appendix). The relevant parameters are K=Kα/(KxL2), indicating the stiffness of the boundary relative to the stiffness of the intra-segment springs and L0/L, the rest length of the spring in the segment relative to the segment length. We find that only one unique steady state exists. Independent of the chosen parameters and the number of segments, this state possesses monotonically increasing final chevron angles. The increase in chevron angle along the anteroposterior axis in the steady state is a robust property of the model (Fig. 6E). Importantly, the heterogeneous steady-state pattern does not need to be pre-determined by a complicated initial state but is found to self-organize from a completely homogeneous initial state (Fig. 6B′).

The model predicts that all segments would remain straight in the case of no intra-segmental contraction (L0/L=1) or nearly straight (chevron angle below 1 deg) for very stiff boundaries (K>60) (see  Appendix). For stiff segment boundaries (K>>1), the monotonic angle dependency can be approximated as an exponential function of segment index (see  Appendix). Because an exponential function starts to increase slowly, the most anterior segments appear to remain straight.

To investigate the temporal dynamics of the model, we derive the over-damped equations of motion for the chevron angles from the system's energy (see  Appendix). These equations can be solved numerically. The simulations show that a simultaneous onset of chevron formation is possible (Fig. 6F). This result is compatible with the experimental data. The particular shape of the time course depends on parameter values.

Taken together, we have reached our third goal: the observed spatiotemporal pattern of chevron angles could emerge from a homogeneous tissue without the need for additional genetic or biochemical differences between the segments.

Testing the mechanical model

Our mechanical model was motivated by stereotypical cellular rearrangements and differentiation of muscle fibres commencing around the time of chevron formation (Fig. 5). Importantly, the model is independent of which cellular or sub-cellular mechanisms might be at work. Direct tests of this hypothesis should seek to alter the generation or propagation of tension through the file of segments preceding the chevron transition. One possibility would be to ablate the first somite, thereby removing the fixed end of the chain, just before the onset of chevron formation at the seven somite stage. The proposed tension-and-resistance mechanism predicts that the angles of the forming chevrons will be altered: because segments will pull from both anterior and posterior sides, reversed chevron-shapes in the anterior and a central symmetrical segment are expected. From published material, a pattern with reversed chevrons was observed in frogs (Yoshigai et al., 2009). Another test might be to explant a row of somites from the embryo prior to the segment 7 transition, and ask whether a similar perturbation of chevron angles is observed as with the ablation of somite 1. If the somites could be separated from the file, or otherwise individually explanted (Beattie and Eisen, 1997; Laale, 1982), the autonomy of the chevron shape's development could be directly addressed. If somite 3 and somite 7, as examples, developed different chevron angles in isolation, our hypothesis of a collective mechanism would be contradicted.

If evidence supporting our general mechanical model was found, it would make sense to explore the cellular and molecular bases for this activity in some detail. Cellular candidates would include the rearrangement of the adaxial cells from epithelium to mesenchyme, or the development and migration of various muscle fibre types. Molecular candidates would include various force-generating systems such as cortical actin–myosin contractility, microtubule-associated processes or ion flux-regulated cellular volume changes.

Limits and extensions of the model

Our quantitative data analysis revealed a correlation between the reported onset of muscle differentiation and chevron formation. Our mechanical model of chevron formation provides an explanation of the position-dependent final chevron angles for the anterior segments. We note that we were not able to obtain a quantitative fitting to the final angle of these segments. We would have been able to fit the data by introducing a non-linear saturation term in the model, but in the absence of a specific biological motivation for this term we have avoided the unnecessary complication. Furthermore, segments 8–32 do not deform more than segment 7 and also appear sequentially, but both of these properties are currently ignored by our simple model. To account for the constant posterior segment boundary angles, we propose a model extension with time-dependent parameters that reflect the developmental maturation of muscle differentiation.

The measurements and model were made in 2D, yet the body is 3D. A potential role for tensions in the third dimension is worth considering if such tension becomes quantitatively measurable. For future studies, it will be important to quantify changes of body cross-section, cell rearrangements, muscle fibre elongation, and muscle fibre force as function of segment index and time, and to correlate these new data sets to our space–time plot of chevron formation (Fig. 5). Any observed correlations should then facilitate the identification of the mechanistic cause of the effective force field we have modelled here. Extensions into 3D are required to address the later changes towards the concentric series of cones in the adult (Greene and Greene, 1913; Shann, 1914; Videler, 1993).

Chevron patterns in other species

Chevron patterns exist in other species, including other fish, frogs, salamanders, lampreys, Amphioxus and the pre-Cambrian fossil Pikaia. Key cellular events that inspired our tension-and-resistance model are conserved. For instance, in Amphioxus, the chevron pattern forms in concert with cell elongation and muscle differentiation is delayed for anterior segments (Hatschek, 1882; Hatschek, 1893; Kaestner, 1892). In agreement with our proposed collective mode of chevron formation, several species including the tadpole of the frog Leptodactylus albilabris (Dent, 1956), Pikaia (Morris and Caron, 2012) and hagfish (Müller, 1836) show anterior straight segments and bent posterior segments. While most authors only distinguish straight anterior versus bent posterior segments, those species might actually show a monotonic increase in chevron angle. For the tadpole of the frog Xenopus, published figures show monotonically increasing final chevron angles for at least 10 anterior segments (Gray et al., 2009; Lee et al., 2010). It will be interesting to quantify angle time courses in Xenopus and to compare these with the correspondingly parameterized model.

Evolutionary aspects

The chevron shape itself and the shape's function in locomotion have been used to infer the evolutionary relationships among chordates (Lacalli, 2012; Mallatt and Holland, 2013; Morris and Caron, 2012). However, the development of the chevron, especially the mechanism underlying it, was largely ignored. The discussion of existing hypotheses and our proposed mechanism, which was motivated by quantitative correlations in fish but might be conserved across a number of other species (see above), may add to the emerging picture of chordate evolution.

Conclusion

We have quantified the time course of early chevron formation in terms of chevron formation start time, end time and final chevron angle for segments 1–15 for mobile, immobile and cyclopamine-treated zebrafish. These data reject or heavily constrain the possible contribution of a number of hypotheses. We find that muscle pioneers are not required for chevron formation but they and/or other high-level Shh signal-dependent cell types may be involved in chevron maintenance. We proposed a tension-and-resistance mechanism based on interactions between intra-segmental cell rearrangements and segment boundaries that explains the early phase of chevron development. A simple mechanical model allowed us to verify that this mechanism could account for the observed spatiotemporal pattern. A strong aspect of our model is the minimal set of ingredients it depends on and its nevertheless robust prediction of the observable monotonic increase in final chevron angle along the anteroposterior axis. An appropriate temporal order of muscle cell differentiation (Fig. 5) with uniform properties for all segments appears sufficient for the chevron pattern and its spatial dependencies to self-organize according to physical laws. Hence, evolution of segment-specific properties such as the increasing chevron angles across the anterior segments would not be necessary.

Time-lapse movies

We analysed time-lapse movies of chevron formation during zebrafish development at 28°C, which were acquired in our previous studies (Herrgen et al., 2009; Schröter et al., 2008) using embryos from in-crosses of free-spawning heterozygous nicb107 parents (Westerfield et al., 1990). Frame rate was every 5 min. The formation of segment 1 was not always clearly detectable in the time-lapse movies. Therefore, we chose to set the time of the clearly detectable formation of segment 6 to 115 min for each individual movie, such that the average time point of the formation of segment 1 is t=0 (supplementary material Movies 1, 2).

Cyclopamine treatment

Cyclopamine-KAAD (cat. no. 239804, Calbiochem) was diluted in E3 medium to 7.5 μmol l−1 from a stock solution of 5 mmol l−1 in 100% DMSO. Manually dechorionated wild-type embryos were soaked in E3 medium containing cyclopamine-KAAD from 80% epiboly or bud stage until processed for in situ hybridization or to record somite formation in long-term time-lapse movies at 28°C. Embryos treated with 0.2% DMSO were indistinguishable from wild-type embryos grown in E3 only.

In situ hybridization

Embryos were fixed for 2 h or overnight with 4% paraformaldehyde in PBS at the 18 somite stage or at 36 hpf and stored in methanol. Antisense probes were generated against engrailed2 (gift from Michael Brand) and cb1045 (Riedel-Kruse et al., 2007) and in situ hybridization using NBT/BCIP was performed as described before (Oates and Ho, 2002) except that probes were hybridized at 55 and 60°C, respectively.

Quantitative image analysis

We analysed time-lapse light microscopy movies in which the image stacks show the development of embryos from before the one-somite stage until about the 30-somite stage. The embryos are visible in lateral view (supplementary material Movies 1, 2).

To quantify the shape of the chevron, we manually measured the angle between the posterior dorsal segment boundary and the notochord, γ, for successive time points using the angle tool in the Fiji package (Schindelin et al., 2012) (Fig. 2A,B). Assuming a simplified geometrical relationship that neglects notochord curvature, we can calculate angle α for all measured angle γ, which will be used for curve fitting and comparison with our mechanical model (Fig. 2B):
formula
(1)

We determined averages αmean and standard deviations Δα for a small number of segments by repeated manual evaluation of γ. Low contrast frames and intermittent occurrence of U-shaped contours left some uncertainty in the measurements.

To decrease the number of measurements, we approximate the averages αmean and standard deviations Δα through only measuring the two extreme angles γmin and γmax (Fig. 2A). We define:
formula
(2)
where αmax and αmin are calculated from the angles γmin and γmax, respectively (Fig. 2A). For the tested segments, α approximates the average of repeated measurements, αmean, well. To calculate the uncertainty, we assumed the probability distribution for picking the true value αt to be a Gaussian distribution around αmean. Repeated manual evaluation of the likely angle α from the same image fell within the interval (αmin, αmax) in more than 95% of cases. Therefore, following the three-sigma rule, the standard deviation of the Gaussian is given by:
formula
(3)

Curve fitting

We considered all segments for which more than 80% of the data points fulfilled αmean–Δα≤0 as non-chevron-forming segments (Fig. 2C). For the remaining segments, we fitted the data for each chevron-forming segment with a piecewise linear function (Fig. 2D,E). Hence, we will describe the chevron formation of these segments with the three parameters of the piecewise linear function: chevron formation start and end times, tstart and tend, respectively, and a final chevron angle αend that is reached after chevron formation ceased. We determined the fit parameters and their uncertainty using the bootstrap method (Press, 1996). For each segment, we calculated random samples of α for each time point using the Gaussian distribution defined by αmean and Δα. We generated 100 random data sets from the measured data for each segment and fitted each data set with the piecewise linear function. Thus, for each of the three parameters, we get N realizations from which we can calculate the mean, tstart, tend, αend, and the standard deviation, Δtstart, Δtend, Δαend.

Fitting was done with a grid search using each parameter triple generated from all possible combinations of:
formula
(4)
formula
(5)
formula
(6)
where tmin is the formation time of the current segment boundary and tmax=740 min. Then the parameter combination with the minimal sum of squared residuals was chosen.
We considered fitting results insignificant if parameter uncertainties were too high. The criteria we chose for significance are:
formula
(7)
formula
(8)
formula
(9)
Furthermore we also considered fits invalid if the end of chevron formation might not have been in the observed period:
formula
(10)

We considered only segments with significant fitting results as chevron-forming segments, whereas we ignored the results for other segments and treated them as missing data.

We are grateful to P. Brösamle and F. Schröder for help with image analysis, to O. Chara, A. Deutsch, T. Krüger, D. Richmond, J. Starruß and A. Voß-Böhme for fruitful discussions and to L. Bajard, P. Currie, S. Devoto, T. Lacalli, A. Webb and two anonymous reviewers for helpful comments on an earlier version of this manuscript.

APPENDIX

Constraints for αi and xi

For our mechanical model, the spring extensions, xi, and the segment chevron angles, αi, are geometrically constrained (Fig. 6C′). We will now deduce the relationship (Fig. 6D). In our time-lapse movies we measured the anteroposterior segment length, L, and the dorsoventral height, h, for various segments. The measurements showed:
formula
(A1)
with a deviation smaller than 30%. We introduce the ‘boundary extension’, ai (Fig. 2B). Simple trigonometry leads to:
formula
(A2)
Assuming small ai and making use of Eqn A1, we get:
formula
(A3)
For the first segment, the segment length is simply the sum of spring extension x1 and ‘boundary extension’ a1:
formula
(A4)
For each following segment, similar reasoning leads to:
formula
(A5)
Defining a0≡0, we can summarize the last two equations:
formula
(A6)
Using Eqn A3, we can write this as:
formula
(A7)

Calculating the minimal energy configuration

We will minimize the energy function (Fig. 6D):
formula
(A8)
to find the equilibrium state. The second equation in Fig. 6D formalizes our assumption that all segments have the same fixed length:
formula
(A9)
We now introduce the dimensionless quantities:
formula
(A10)
With these, we can write Eqns A8 and A9 as:
formula
(A11)
formula
(A12)
Substituting the x~i in Eqn A11 with Eqn A12 we get:
formula
(A13)
We will leave out the tildes for better readability from now on. A necessary condition for En to be minimal is the disappearance of all partial derivatives of En with respect to all αj. We therefore calculate:
formula
(A14)
This is a system of linear equations. Defining ϕ=2+K we can write the augmented matrix of the system:
formula
(A15)
We solve the system via Gaussian elimination and defining:
formula
(A16)
we get:
formula
(A17)
Eqn A16 is an order 2 linear homogeneous recurrence relation. It is solved by:
formula
(A18)
formula
(A19)
With Eqn A18 we can write Eqn A17 as a closed form expression for ai only, depending on the ratio of the spring constants K and the number of segments n:
formula
(A20)
where:
formula
(A21)
which is a constant for a chain with n segments (Fig. 6E). For all K>0, this is a monotonically increasing function of segment index. Hence, the predicted chevron angles αi always increase in the anteroposterior direction, independently of the free parameters Ka, Kx and L0.
For stiff boundaries, Ka>>Kx, hence ϕ>>2 and therefore with Eqn A19:
formula
(A22)
formula
(A23)
and thus the predicted chevron angles αi grow exponentially with segment index for stiff boundaries:
formula
(A24)
We definitely would not be able to experimentally detect chevron angles smaller than 1 deg. Hence, all segments would be detected as straight if all αi<1 deg. With:
formula
(A25)
it follows that all segments appear straight for:
formula
(A26)

Overdamped dynamics of the model

Assuming viscous damping, the overdamped equations of motion of our model system are:
formula
(A27)
where we introduced the viscous damping coefficient c. Using the earlier defined dimensionless energy (Eqn A10) and defining the dimensionless time,
formula
(A28)
the dimensionless equations of motion read:
formula
(A29)
As all segments are formed straight, the initial conditions read:
formula
(A30)
A numerical solution of the equations of motion for seven segments is shown in Fig. 6F.

Funding

This work was supported by the German Ministry for Education and Research [0315259 to F.R., L.B.]; the Max Planck Society [to C.E., C.S., A.C.O.]; the European Communities 7th Framework Programme [FP7/2007–2013, 207634 to C.S., A.C.O.]; the German Research Foundation [DFG OA21 to C.E.]; the Medical Research Council [MC_UP_1202/3 to A.C.O.]; and the Wellcome Trust [WT098025MA to A.C.O.]. Deposited in PMC for release after 6 months.

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Competing interests

The authors declare no competing financial interests.

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