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1 doi: /nature11804 a Tailbud after cutting PSM after cutting b mean intensity ROI1 (TB) ROI2 (PSM) time (h) Supplementary Fig.1 Lfng gene activity waves can be classified as kinematic, as physical separation does not block wave progression. a, The tailbud was surgically separated from the remaining PSM. Pulses and waves of gene activity are still visible in both fragments with no obvious alteration of directionality. b, Plot of intensity in fragments reveals regular oscillations with overall maintained temporal order. 1

2 Before cutting a space c d b After cutting time Supplementary Fig.2 Kinematic waves of Lfng expression in the mpsm assay. After 15 hours of culture (a), cells of the mpsm were removed (b) in order to generate a physical gap within the mpsm, breaking its continuity along the direction of Lfng activity waves. Kymographs were generated before (c) and after (d) the gap was introduced along the red lines shown in a and b, respectively. Despite the gap, propagation of activity waves continues in the same direction. 2

3 Paraxis a Meox1 b Msgn1 d c Hes7 Axin2 e Sox2 g Uncx4.1 f Cldn6 Pitx2 h i Supplementary Fig.3 Molecular analysis of the cell culture assay with in situ hybridization after 18-24h of culture. Mesoderm differentiation markers Paraxis (a, 5/5), Meox1 (b, 3/3) and Uncx4.1 (c, 3/3) are expressed in the peripheral, segmented part. The presomitic marker Msgn1 is expressed in the centre of the culture (d, 9/10). Oscillatory genes Axin2 (e, 12/13) and Hes7 (f, 7/7) are expressed centrally in undifferentiated tissue. Axin2 also shows expression in the presumptive posterior segment compartment (arrow) in agreement with its in vivo pattern. Expression of neural (Sox2, g, 0/4) and lateral plate (Pitx2, h, 0/3) markers could not be detected. The endodermal marker Cldn6 (i, 6/7) shows variable, patchy expression patterns mostly in the centre of the ex vivo culture. Scale bar = 100µm. W W W. N A T U R E. C O M / N A T U R E 3

4 min period (min) Length of mpsm ( m) Supplementary Fig.4 Oscillation period in the central mpsm stays constant irrespective of mpsm size. Oscillation periods are plotted againt the length of the mpsm. The periods of n=17 oscillations from 4 individual samples were measured. The solid line indicates the average period (132 min) the dashed lines depict the variance of one standard deviation (±11 min). 4

5 900 a b rads rads Space (µm) α y α φ=-3.14 rads φ=-1.35 rads Δφ= 6.28 rads φ=+0.85 rads φ=+3.14 rads 1 rads 0 rads -1 rads -2 rads Space (µm) rads Time (h) Time (h) A signal intensity kymograph (intensity is colour-coded, a) was used to generate the phase kymograph (b) by calculating the instantaneous phase using a Hilbert transformation 1. We restrict the phase kymograph to the area exhibiting oscillatory activity, e.g. from the start of a wave to the point at which it halts (characterized by the peak intensity, indicated by the line in a). In addition, only intensities higher than 75% of the mean image intensity were taken into account, in order to exclude the central, not oscillating area. On the resulting phase kymograph (b) phase values (from rad to 3.14 rad) are colour-coded. By connecting the most posterior and most anterior point with a value of rad, velocities could be detergiven time point. For the selected time point (i.e. all locations are along one vertical line), the individual phase value can be retrieved for all phase dimension corresponding to a total 2π change. 5

6 Msgn1 + Mesp2 Aldh1a2 6h Dusp4 + Mesp2 d 21h g Supplementary Fig. 6 Time-course of PSM marker expression in the ex vivo cell culture assay using ISH. After 6h posterior PSM markers Dusp4 and Msgn1 (for both n=5/5) are detected throughout the cell culture. After 21h posterior PSM markers Dusp4 and Msgn1 are localized to the centre of the culture (n=13/14 for Dusp4 and n=9/10 for Msgn1). Only at these later time points, Mesp2 expression is visible in the periphery (indicated by arrows). Aldh1a2 is not detected after 6 hours culture (n=5/6) but is expressed in the periphery after 24hours, once segment formation occurs (n=8/9). Scale bar = 100µm. 6 W W W. N A T U R E. C O M / N A T U R E

7 a Development of Phase along Space 2π (6.28) 6.22=1.98π 6.02=1.92π 5.94=1.89π 5.97=1.90π 5.01=1.60π Phase Value (rads) π (3.14) 3.58=1.14π 2.75=0.88π Wave #-4 Wave #-3 Wave #-2 Wave #-1 Wave #+1 Wave #+2 Wave # Length (µm) b Development of Phase along Space and Projected Space 2π (6.28) Phase Value (rads) π (3.14) Length (µm) Wave #-4 Wave #-3 Wave #-2 Wave #-1 Wave #+1 Wave #+2 Wave #+3 Linear (Wave #-4) Linear (Wave #-3) Linear (Wave #-2) Linear (Wave #-1) Linear (Wave #+1) Linear (Wave #+2) Linear (Wave #+3) Supplementary Fig.7 Spatial projection of truncated phase-gradients until their completion to 2π values is based on their slopes. a, Phase kymographs for the entire cell culture assay culture, thus including both time points showing truncated mpsm (waves -4 to -1) and complete mpsm (waves +1 to +3) were used to determine the phase-gradients for each wave cycle. Waves -4 to -1 show a total phase span less than 2π, gradually increasing towards this value as time point wave +1 is approached. b, Linear regression was used to calculate the slope of the phase gradient and extrapolate to the point when they reach 2π. The projection on the length axis, shown with arrows, was used to determine the projected (phantom) mpsm length. This is the fictive length that a phase-gradient with a given slope would have in order to have a phase span (~amplitude) of 2π. 7

8 Wnt3a FGF8 FGF4 b c d e f 21h 6h a beta-catenin DAPI perk h i j k l 21h 6h g Supplementary Fig.8 Time-course analysis of Wnt- and FGF- signalling gradients in the ex vivo cell culture assay. a-f, mrna expression of Wnt- and FGF- ligands after 6h (a-c) and 24h (d-f) is detected by In-situ hybridization. In early culture stages expression of Wnt3a (a, 5/5) and FGF4 (c, 4/4) is detected throughout the culture. FGF8 (b, 7/7) is expressed in a broad domain with variable intensities. After 21h expression of Wnt3a (d, 9/12), FGF8 (e, 7/7) and FGF4 (f, 3/3) is restricted to a central domain. g-l, Wnt- and FGF-pathway effectors β-catenin and phosphorylated ERK1/2 (perk) were detected after 6h (g-i) and 21h (j-l) by double immunofluorescence. β-catenin and perk show uniform distribution at early timepoints (7/7) and show a graded distribution after 21h (7/7), from center to peripheral segments (indicated by arrows). Scale bar = 100µm. 8 W W W. N A T U R E. C O M / N A T U R E

9 Supplementary text 1 Model components We will subsequently call the phase φ(x, t) of the segmentation oscillator at position x and time t, the phase in the presumptive tail bud φ 0, and the phase shift between the two φ(x, t), with x = 0 corresponding to tail bud. Note that by definition φ(0,t) = 0. We rescale time and chose time origin so that for all cases consider, φ 0 = 1 and φ 0 (t) =t. Our experiments suggest two crucial features : Feature 1: The first important observation is that oscillation is controlled (directly or indirectly) by the phase shift between a cell within the PSM and the tail bud. In particular, when φ 2π, the clock stops. Feature 2: The second important observation directly follows from Fig. 4B. Considering the phase gradient in the ex vivo culture, our experimental data suggest it follows an exponential law in time of the φ form : = x λeαt. If we assume this gradient is approximately linear in space, we therefore get immediately, for the ex vivo culture φ(x, t) =λxe αt (1) Taking now the time derivative of equation 1, we can eliminate x to get φ t = α φ (2) Equation 2 is of particular interest as it suggests a very simple picture where clocks in individual cells slow down autonomously, with a timescale 1. This is also in line with the first observation that the clocks α stops when the phase shift φ reaches a critical value (of order 2π) : essentially, our experiments suggest that the whole process from clock slowing down to stopping of the oscillation is indeed controlled by φ. Equation 1 implies scaling. Consider the position x(t,φ ) defined by φ = φ(x(t,φ ),t ), i.e. the position in the explant of the oscillator reaching 9

10 phase φ at time t, and let us call φ 0 the phase in the tail bud. We can then easily compute x(t +2π, φ +2π). We have from Equation 1 and φ = φ 0 + λx(t,φ )e αt = t + λx(t,φ )e αt (3) φ +2π = t +2π + λx(t +2π, φ +2π)e α(t +2π) (4) so that we immediately get x(t +2π, φ +2π) =e 2απ x(t,φ ) (5) This is scaling : over one cycle of oscillation, we recover the same phase profile at time t +2π than at time t, but on a spatial zone shrunk of a factor e 2πα. More generally, it is not difficult to show that if we assume that the phase gradient is shrunk by a factor α in space within interval dt, then phase equation φ t =1+αx φ x holds. A general solution of equation 6 is (6) φ(x, t) =t + f(xe αt ) (7) where f is an arbitrary function, fixed by the initial conditions (φ(x, 0) = f(x)). Equation 1 is equation 7 with f(x) =x. Provided f is invertible, same reasoning as in equations 3 to 5 can be followed to explain scaling. 2 Integrative kinematic scaling model So far, we have explained how the observed phase dynamics naturally explains scaling, relying purely on kinematic, e.g. cell-autonomous oscillator behavior. However, it is known that PSM cells are coupled [1] and additionally, it is reasonable to assume that growth further impacts on this coupling. Thus, in order to built a complete and realistic model, both cell-coupling as well as growth are taken into account (term ɛ, see below). Equation 2 is a phase equation [2, 3, 4]. It can be taken literally, i.e. we can imagine that some (still unknown) mechanism is able to (indirectly) compute the actual phase shift between the tail bud and any given cells. An equation including the two features described previously for a single oscillator would then be : t φ = (1 ɛ + α φ)θ( φ) with Θ( φ) = φ 6 φ 6 + φ 6 (8) The term Θ( φ) is a phenomenological term which accounts for Feature 1 and imposes the clock stops when phase shift reaches φ 2π. 10

11 1 and imposes the clock stops when phase shift reaches φ 2π. The term α is the Feature 2, the exponential slowing down of the clock. Finally, term ɛ accounts for a change of intrinsic frequency of the clock in response to external cues. Again, while the molecular mechanism accounting for ɛ remains to be determined, we assume this term depends on tissue property such as growth rate and cell-to-cell coupling. Term ɛ is not necessary to account for scaling per se, but in effect, enables scaling to actively occur upon changes in growth rate (In the absence of ɛ, scaling under different growth rates would still occur but would merely be a passive consequence of a difference in laying down a kinematic oscillator array). The simulation presented in Fig. 4C is therefore done with ɛ>0 while the embryo is growing, and ɛ = 0 when growth is stopped. The analytic solution of 8 will be presented elsewhere. Here, we present an intuitive interpretation that can account for scaling. As long as the tail bud grows, the posterior most part of the PSM at time t +2π is essentially identical to the same part of the PSM at time t, and therefore periodic segmentation ensues. It can be shown analytically that during growth, the phase profile is then exponential in space. For individual cells, phase varies exponentially in time as expected from equation 2. These two exponentials (spatial and temporal) then combine within the PSM and compensate to produce a pattern of constant size. Changing ɛ essentially breaks up this compensation between the spatial phase exponential and the temporal phase exponential by, roughly speaking, linearizing the spatial phase profile, akin to Eq. 1. As a consequence, scaling actively occurs. 3 Hypotheses of biological explanations Equation 8 suggests that two oscillators coexist in the PSM, the prediction would then be that φ 0 stays synchronized in the whole PSM and within the tail bud while the oscillator φ would slowly phase-shift. Some unknown molecular machinery would evaluate phase shift between the two oscillators and act back on oscillator φ. It is known indeed that several oscillators coexist in the PSM [5, 6, 7], and based on our previous findings, the possibility that their phase-shift varies along the PSM exists. However, dynamic and quantitative measurements of various oscillators (linked to Notch, Wnt and Fgf-singaling oscillations) are required to test this prediction. Another possible molecular explanation involves the assumption that a morphogen acting locally controls the frequency of the oscillation. The complete analytical and numerical description of this case will be developed elsewhere. Since cells are moving linearly in time along the PSM, exponential degradation of this morphogen would result into an exponential spatial gradient in the embryo, thus converting temporal into spatial information. Similar to above, dynamic and quantitative measurements of morphogen gradients are needed to directly test this possibility. 11

12 References [1] Horikawa, K., Ishimatsu, K., Yoshimoto, E., Kondo, S., and Takeda, H. Noise-resistant and synchronized oscillation of the segmentation clock. Nature 441(7094), , June (2006). [2] Goodwin, B. C. and Cohen, M. H. A phase-shift model for the spatial and temporal organization of developing systems. J Theor Biol 25, (1969). [3] Kopell, N. and Howard, L. N. Horizontal bands in the belousov reaction. Science 180(4091), , June (1973). [4] Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence. Dover Publications, Inc, (1984). [5] Aulehla, A., Wehrle, C., Brand-Saberi, B., Kemler, R., Gossler, A., Kanzler, B., and Herrmann, B. G. Wnt3a plays a major role in the segmentation clock controlling somitogenesis. Developmental Cell 4(3), , March (2003). [6] Niwa, Y., Masamizu, Y., Liu, T., Nakayama, R., Deng, C.-X., and Kageyama, R. The initiation and propagation of Hes7 oscillation are cooperatively regulated by Fgf and notch signaling in the somite segmentation clock. Developmental Cell 13(2), , August (2007). [7] Dequeant, M.-L., Glynn, E., Gaudenz, K., Wahl, M., Chen, J., Mushegian, A., and Pourquié, O. A complex oscillating network of signaling genes underlies the mouse segmentation clock. Science 314(5805), (2006). 12