Faster embryonic segmentation through elevated Delta-Notch signalling

An important step in understanding biological rhythms is the control of period. A multicellular, rhythmic patterning system termed the segmentation clock is thought to govern the sequential production of the vertebrate embryo's body segments, the somites. Several genetic loss-of-function conditions, including the Delta-Notch intercellular signalling mutants, result in slower segmentation. Here, we generate DeltaD transgenic zebrafish lines with a range of copy numbers and correspondingly increased signalling levels, and observe faster segmentation. The highest-expressing line shows an altered oscillating gene expression wave pattern and shortened segmentation period, producing embryos with more, shorter body segments. Our results reveal surprising differences in how Notch signalling strength is quantitatively interpreted in different organ systems, and suggest a role for intercellular communication in regulating the output period of the segmentation clock by altering its spatial pattern.


Supplementary Figure 5. Stability, frequency and wavelength of time-periodic wave patterns
Properties of time-periodic wave patterns as a function of the coupling delay τ and the coupling strength ε. (A) Regions where the wave pattern is stable (green), unstable (red) and where more than one solution for the wave pattern exists (dark gray). (B) Collective frequency Ω (blue), determined by Eq. (S14) in Supplementary Note 1, in the regions with a single stable solution. Other regions as in A. (C) Number of waves K (brown), Eq. (S17), in the regions with a single stable solution. Other regions as in A. (D) The wavelength of the pattern at the anterior end (cyan), Eq. (S16). Other regions as in A. Dots indicate parameter values for which the corresponding wave pattern is shown in Supplementary  Fig. 4. The blue dot marks the wildtype parameters from 2 (ε = 0.07 min -1 , τ = 20.75 min) and the green square marks a parameter set that yields a shorter anterior wavelength as observed in Damascus (ε = 0.09 min -1 , τ = 17 min). Green triangle ε = 0.07 min -1 , τ = 26 min; red diamond ε = 0.04 min -1 , τ = 20.75 min. The other parameters are ω 0 = 0.2205 min -1 , μ = 0.249 cd min -1 , x = 39 cd, and σ = 27 cd, where 'cd' denotes cell diameters 26  PCR efficiency is the fold change of product concentration from one PCR cycle to next PCR cycle. All primers were synthesized by biomers.net.

Supplementary Note 1:
Changes in the wavelength can alter the segmentation rate Here we illustrate how a change in the wavelength can increase the segmentation rate by enhancing the Doppler effect that occurs due to PSM shortening over time. 3 The wave pattern is described by a one-dimensional phase field .x; t /, which indicates the local state of oscillation at time t and position x as measured from the posterior tip along the body axis, see Fig. S4A. 1 The position of the anterior end is given by x D N x.t/. As the PSM shortens over time, N x continously decreases over time. The phase at the anterior end is given by A .t / D . N x.t /; t / and the segmentation rate R, which is proportional to the anterior frequency, can be expressed as 1, 3 where the frequency P at the posterior tip x D 0, the dynamic wavelength contribution W and the Doppler contribution D are given by Here, u D d N x=dt is the shortening speed of the PSM and is the wavelength of the pattern at the anterior end, given by ( We now assess how the segmentation rate R is affected if the wavelength of the wave pattern is altered. If describes a reference wave pattern (e.g., wildtype), then a correspondingly altered wave pattern (e.g., Damascus) can be expressed as where Â describes the differences in the pattern that may depend on time. To illustrate the effects of a changed wave pattern in a simple example, we here consider a time-independent change of the wavelength, Â D Â.x/. In this case, since @ Q =@t D @ =@t , both the posterior frequency contribution P as well as the dynamic wavelength contribution W are unaffected by such a change in the wave pattern. However, the corresponding wavelength at the anterior end is changed as Given that the shortening speed u of the PSM is the same, Eq. (2) implies that the Doppler contribution Q D D u= Q for the changed wave pattern can be expressed in terms of the Doppler contribution D of the reference wave pattern as Using Eqs. (3) and (5), the ratio of wavelengths can explicitly be given as Moreover, the segmentation rate Q R resulting from the change in the pattern is given by Here, D and R are the Doppler contribution and the anterior frequency (which coincides with the segmentation rate) as defined in Eqs. (1) and (2), and T D 500 min is the duration over which the average phase map has been determined. Using the functional form for .x; t / and the parameters provided in Ref. 3, we obtain q ' 0:22.

Supplementary Note 3:
Effects of coupling strength and coupling delays on the wave pattern In the previous sections, we have shown that an altered wave pattern can lead to a different segmentation rate but we have not addressed how such an alteration of the wave pattern can arise in the first place. Here we illustrate how changes in the strength and the delay of intercellular communication can give rise to changes in the wave pattern. For simplicity, we consider a scenario in which the PSM has a constant length throughout segmentation. Even though this is not the case in vivo, where the PSM continuously changes its length during segmentation, this scenario enables illustration of the effects of coupling on the overall shape of the wave pattern in a simple setting.

Continuum theory of coupled oscillators with coupling delays
To describe the dynamics of the phase field .x; t / introduced in the previous section, we use a continuum theory of coupled phase oscillators that takes into account delays in intercellular communication. 2 The phase dynamics is given by where .x; t/ D .x; t / is the time-delayed phase variable with being the coupling delay, !.x; t / is a position-dependent frequency profile, " is the coupling strength, and is the 'lattice spacing', which corresponds to the typical cell diameter. In our simplified model, body axis elongation proceeds with a constant velocity v. We transition into a comoving reference frame, in which x D 0 marks the posterior tip of the PSM and the x-axis points in anterior direction, see Fig. S4A. The position x D N x marks the position of the anterior end of the PSM. In this reference frame, the phase field is given by '.x; t/ D .x vt; t /. The dynamics for ' is given by where we have neglected higher order corrections in the lattice spacing . Following Refs. 2 and 4, we employ a frequency profile of the form where N x is the length of the PSM and is a length scale that determines the shape of the frequency profile, see Fig. S4B. Such a frequency profile describes a gradual slowdown of oscillations across the PSM in anterior direction, which leads to the generation of waves in anterior direction. 1, 2, 4-6

Time-periodic wave patterns
For a constant PSM length, this system attains a state with a time-periodic wave pattern, 2 '.x; t/ D t C .x/ ; (13) whose recurrence period T D 2 = coincides with the period of segmentation. The time-independent phase profile describes the spatial distribution of the waves along the PSM. The collective frequency of the pattern is determined by the equation 2 This implicit equation does not possess a closed solution in . Depending on the value of , Eq. (14) can have multiple solutions corresponding to different collective states with different frequencies.
The phase profile is determined by the equation with boundary condition .x/j v <x<0 D 0. 2 From Eq. (15), it can be seen that both the coupling strength " as well as the coupling delay determine the shape of the phase profile and thus the wave pattern.
The wave pattern can be characterized by its local wavelength and the total number of waves in the PSM. Here we focus on the wavelength at the anterior end x D N x, which is experimentally evaluated in this study, see Fig. 3B in the main text and Fig. S2. The total spatial extension ƒ of the anteriormost wave in the direction of wave propagation is determined by the relation If the wavelength , defined by Eq. (3), does not vary strongly over the length ƒ, both quantities coincide as can be seen from expanding Eq. (16) to first order in ƒ.
The number K of waves that simultaneously travel across the PSM is given by

Stability of the pattern
Not all solutions for wave patterns are robust under small perturbations of the phase field. For certain configurations of the phase field, small perturbations exponentially grow and drive the system into a different dynamical state. Such unstable wave patterns can immediately be ruled out as biologically irrelevant.
A full stability analysis for general solutions of the nonlinear delay equation (11) is a formidable task. However, for a given set of parameters and a specific solution for the collective frequency, an indication of whether the corresponding pattern is stable or not can be obtained by considering the case of a flat phase profile, .x/ D 0, which is a solution for a flat frequency profile !.x/ D ! 0 . In this case, linear stability analysis of Eq. (15) reveals that the solution is stable if and only if > 0, where This condition is derived in the paragraph below. The frequency profile (12) exhibits a roughly flat region in the vicinity of the posterior tip x D 0 (the tailbud region) for which the thus derived stability criterion approximately holds.
Linear stability analysis. We perform a standard linear stability analysis of the system defined by Eq. (11) for the homogeneous state '.x; t / D t . For the phase field, we make the ansatz where is a small perturbation. Substituting this ansatz in Eq. (11) and expanding to first order in yields the governing equation for the perturbation, where has been defined in Eq. (18). To proceed, we express the perturbation in terms of its spatial Fourier transform,

Numerical results
Fig . S5A shows the stability of the time-perodic state as a function of the coupling delay and the coupling strength ", indicating unstable regions in red ( < 0), stable regions in green ( > 0), and multistable regions in dark gray (multiple solutions of Eq. (14) in ). The chosen set of parameters has been used earlier to describe segmentation in zebrafish. 2 We consider as biologically relevant only such parameter regions that enable a single stable solution (green regions in Fig. S5A), which corresponds to a robust pattern forming system. 8 Fig. S5B shows the collective frequency of the time-perodic state, determined by Eq. (14), in regions where there exists a unique solution. Both the coupling strength and the coupling delay regulate the collective frequency. Fig. S5C shows the number of waves K of the wave pattern in the PSM, Eq. (17). Fig. S5D shows the length ƒ of the anteriormost wave of the pattern at the anterior end of the PSM, Eq. (16). The anterior wavelength exhibits a non-monotonic depedence on both the coupling strength and the coupling delay. Fig. S5D illustrates the wave patterns for selected points in parameter space. Differences in the wavelength and the number of waves are clearly visible. The blue dot marks the wildtype parameters from Ref. 2 (see also Fig. S4C). The green square marks a wave pattern that follows the same trend as the wave pattern observed in Damascus with a shorter anterior wavelength and a larger total number of waves in the PSM (see also Fig. S4D).
Altogether, these results demonstrate that changes in coupling strength and coupling delays can alter the tissue-level wave pattern observed in the PSM. In particular, it can lead to changes in the anterior wavelength and the number of waves as observed in Damascus.