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Modeling Edar expression reveals the hidden dynamics of tool signaling center patterning

The generation of patterns during development is generally viewed as a direct process. In the mouse jaw, however, the sequential patterning of molars initiates with abortive tooth signaling centers called MS and R2, thought to be vestiges of the lost rodent premolars. Moreover, the mature signaling center of the first molar (M1) is formed from the fusion of two signaling centers (R2 and early M1). Here, we report that Edar expression reveals the hidden dynamics of signalling centers patterning. First, Edar expression evidenced a hidden two-step patterning process that we modelled with a single activator-inhibitor pair: the epithelium is initially broadly activated, then activation becomes restricted in space to give rise to the signalling centers. Second, Edar expression unveils successive phases of pattern making and pattern erasing events, a phenomenon that we called a developmental palimpsest. MS is erased by a broad activation for the benefit of R2, which itself is erased before it recovers when the first molar signaling center forms. In the lower but not the upper jaw, the two neighboring signaling centers then fuse into a single elongated center. Our model recapitulated the erasure of the R2 signaling center by the wave of activation that precedes the formation of M1 signaling center, and predicted the surprising rescue of R2 in the context of an Edar mutant with reduced activation. It suggested that R2 was not intrinsically defective, but actively outcompeted by M1 formation. We confirmed this by cultivating R2 separately from the posterior tissue and showing it could then generate a tooth. Finally, by introducing chemotaxis as a secondary process of tooth germ maturation, we recapitulated the fusion of R2 and M1 in the lower jaw only, and the loss of fusion when Edar function is impaired in organ cultures. In conclusion, we have uncovered a highly indirect and dynamic nature of pattern formation in the molar field that could nevertheless be simulated with simple mathematical models. Our study argues for viewing embryonic patterns as dynamical objects rather than as fixed endpoints, where dynamics is essential to the outcome of the patterning process.

introducing chemotaxis as a secondary process of tooth germ maturation, we recapitulated the 49 fusion of R2 and M1 in the lower jaw only, and the loss of fusion when Edar function is impaired 50 in organ cultures. In conclusion, we have uncovered a highly indirect and dynamic nature of pattern 51 formation in the molar field that could nevertheless be simulated with simple mathematical 52 Introduction theoretical work of the mathematician Alan Turing (4,5,9). The generalization of this work has led to many 84 classes of reaction-diffusion (RD) mechanisms, where two (or more) molecules characterized by different 85 spatial range of action and a given topology of interaction can self-organize a stable pattern, but also exhibit 86 behaviors such as oscillations or propagating waves (4). The most iconic example is the case where a short-87 range activator that self-amplifies and activates its own long-range inhibitor, can create spots, stripes or 88 labyrinths. Recently it has been shown that many biological systems exhibit features of RD mechanisms 89 (e.g. patterning of epithelial appendages such as hair (10,11), the patterning of features such as the rugae 90 of the palate (12), digits (13) and somites (14)). This should not be taken too strictly however. Geirer and 91 Meinhardt pointed out that any process involving local self-enhancement and lateral inhibition has the 92 potential to drive spontaneous pattern formation (15). For example, color pattern formation in zebrafish 93 can be explained by RD models, but at least partly involves cell interactions rather than the diffusion of 94 biomolecules (16,17). Pattern formation can also arise from purely chemostaxis mediated self-organization. 95 When cell movement is driven by concentration gradients of chemostatic cues, positive feedbacks between 96 cell density and chemo-attractant production are known to enhance local concentration of cells and may 97 result in self-sustained aggregation. (18)(19)(20). Chemotaxis plays a prominent role in feather formation (21), 98 and this is likely also the case in most other epithelial appendages (e.g. hair (Glover et al., 2017)). 99 100 Mouse molars are a good example of repeated structures that form through sequential pattern formation. 101 Mice have only three molars per quadrant, separated from incisors by a diastema, the other mammalian 102 teeth (i.e. canine and premolars) having been lost in the evolution of mouse lineage (22). Molars develop 103 sequentially from the most anterior molar (first molar, M1) to the most posterior (third molar, M3). They 104 develop from a unique cylinder-shaped invagination of the oral epithelium, the so-called dental lamina, 105 (23-25) where tooth-specific signaling centers, called Primary Enamel Knots (PEK) are patterned. These 106 signaling centers then drive the formation of individual teeth by promoting "cap" formation; the capping 107 of the underlying condensed mesenchyme by the epithelium. Indirect evidence that activation-inhibition 108 mechanisms determine sequential formation of these signaling centers comes from the similarity of tooth 109 formation with other epithelial appendages (26), namely hair and palatal rugae, whose patterning in clearly 110 ruled by Turing-type mechanisms ((11,12)). The most direct evidence is a study by Kavanagh and 111 colleagues (27), showing that when tissue that will form M2 is separated from M1, M2 forms earlier and 112 becomes larger. PEK formation in the epithelium requires signaling from both the epithelium and the 113 mesenchyme (28), including a mechanical signal induced by mesenchyme condensation (29). 114 The sequential patterning of mouse molars in the lower jaw ( Figure 1A) involves two transient 115 signaling centers (30), that fail to drive proper cap transition, yet form morphologically 116 distinguishable buds (30,31). These buds might be the vestige of lost premolars (30,32). 117 Monitoring these signaling centers via Shh expression has shown that the signaling center called 118 MS initiates sequential patterning and then disappears (30). Subsequently, the R2 signaling center 119 forms. As it vanishes, the M1 early signaling forms posteriorly (Prochazka et al., 2010). Soon 120 after, the former R2 and M1-early signaling centers are encompassed in a giant Shh-expressing 121 signaling center (Prochazka et al., 2010;Lochovska et al., 2015). Here, we will refer to it as the 122 mature M1 signaling center. A similar situation with two abortive buds (called R1 and R2) has 123 been noticed in the upper jaw (25). Their signaling centers have not yet been characterized, 124 although they are morphologically more apparent than in the lower jaw.

134
C-In situ hybridization showing Edar expression in the isolated dental epithelium from 14.5 to 16.0 days, a 135 period corresponding to growth of the tail region ("M2 forming region") and patterning of M2 signaling 136 center in this newly grown region (at 16.0). At 14.5, Edar is restricted to the primary enamel knot of the first 137 molar (yellow on the scheme). At 15.0, Edar expression is seen in the posterior-most part of the tail, the M2-138 forming region (Gray on the scheme). Between 15.5 and 16.0, it is restricted to the primary enamel knot of 139 the second molar (M2, green on the scheme). On the scheme, the continuing expression of Edar in the 140 maturating first molar has been omitted for simplicity.

141
D-Scheme summarizing the expression pattern in C, to be compared with B. The situation in hair is similar to 142 the situation in the anterior part of the dental epithelium.

144 145
Interestingly, mutations in genes affecting various developmental pathways (FGF, Shh, Wnt, BMP and 146 Eda pathways) lead to a supernumerary tooth in front of M1, thus resembling a premolar (34). Where it 147 was specifically studied, the results were consistent with R2 signaling center being enabled to form a tooth 148 (35-39,33). The picture is thus fairly complex, especially since we lack direct evidence for the dynamics 149 of activation-inhibition mechanisms that pattern signaling centers in the dental lamina and promote tooth 150 formation. 151

152
The Eda pathway has the potential to shed light on these mechanisms. This pathway shows a consistent 153 role in activation-inhibition mechanisms throughout several epithelial appendages (hair, feather, teeth; 154 (40,41). This role has been more specifically evidenced for mouse guard hair pattern formation. The 155 receptor of the pathway, Edar, is first broadly expressed in the epidermis. Concomitantly with hair signaling 156 center patterning, it becomes upregulated in the placodal signaling center, and downregulated in the 157 neighborhood ( Figure 1B). In the absence of Edar signaling, no signaling center forms (11,42,43) , while 158 increasing signaling results in more numerous and densely packed placodes (11). Current models posit that 159 the Eda pathway is activated by Wnt, activin BA and BMP4 pathways (43,44) (11,44), but also feeds back 160 on these pathways (and others) through the transcriptional activation of their diffusing ligands and 161 inhibitors (e.g. WNT10a/b, DKK4, CCN2/CTGF, Follistatin and FGF20 (40,41) ). More recently, it 162 appeared that Eda signaling also promotes placodal fate by stimulating the centripetal migration of cells in 163 the epithelium (45,46). In teeth, the Eda pathway is dispensable for primary signaling center (PEK) 164 formation, but required for its correct sizing (46-48). Similarly, it is necessary for correct patterning of the 165 secondary signaling centers controlling cusp morphogenesis (38,47,49). Eda and Edar mutants have 166 reduced tooth size and cusp number, but intriguingly, sometimes have a small supernumerary tooth (50-167 52). In gain-of function mutations, an anterior supernumerary tooth is also found, and teeth are larger with 168 more cusps (49,53-55). 169

170
In this paper, we aimed at clarifying the temporal dynamics of signaling center formation in the dental 171 lamina. We studied the temporal dynamics of Edar gene expression, the receptor of the Eda pathway, 172 during molar pattern formation and showed it recalls the dynamics observed during hair patterning. Based 173 on these data, we built a reaction-diffusion type model of molar patterning that enables sequential signaling 174 center formation and helps reveal the exquisitely complex temporal interactions leading to the construction 175 and deconstruction of patterns in the developing molar row. Our model explains a counter-intuitive result, 176 the rescue of the abortive R2 bud in the context of reduced activation/increased inhibition of the Edar loss 177 of function mutation. Finally, we show that Edar is necessary for the formation of a fused R2-M1 signaling 178 center in the lower jaw only, possibly through a chemotactic effect. We thus showed that patterning is not 179 direct, although it follows simple mathematical rules. 180

182
Edar regulation highlights activation in the growing dental epithelium 183 To get insights into molar row patterning, we examined the regulation of the Edar gene ( Figure 1B). 184 Because the early period of molar row patterning is complicated by the presence of vestigial signaling 185 centers, we first focused on the patterning of the second molar. Since Edar is exclusively expressed in the 186 epithelium, we performed in situ hybridization on mandibular epithelium that has been dissociated from 187 the mesenchyme, thus providing a 3D view of Edar expression. At 14.5 dpc, Edar expression is restricted 188 to the primary signaling center (PEK) of the first molar, and no expression is seen in the second molar field, 189 looking like a "tail" (Figure 1C). At 15.0 dpc, the "tail" has elongated and Edar expression is upregulated 190 in the posterior most part of it. By late 15.5 dpc, it starts restricting to the M2 primary enamel knot, just 191 before M2 cap transition occurs (at 16.0 dpc). The restriction was concomitant with Shh expression starting 192 in M2 PEK (data not shown and (30)). 193 This dynamic of an initial broad upregulation of Edar followed by its restriction to a signaling center is 194 reminiscent of what happens during hair patterning (compare Figure 1B and 1D). It suggests that the 195 decision to form a tooth signaling center in the growing molar field proceeds in two phases. First, the whole 196 dental epithelium is activated. This activation results in broad Edar expression, so far the only gene to show 197 such an expression pattern marking the epithelium competent to form a tooth. Second, activation gets 198 restricted spatially and gives rise to a signaling center. This results in the focused expression of Edar, and 199 many other genes known as primary enamel knot genes (e.g. Shh). In this view, Edar expression is a read-200 out of activation levels in the molar field: where it is high enough, Edar is expressed. To further formalize 201 these ideas, we built a mathematical model of activation in the dental epithelium, as followed by Edar 202 expression pattern. 203 204 Activation dynamics can be modeled by a transition from a bistable system to a Turing system in a 205

growing domain 206
Activation (monitored by Edar expression) can switch between different states: from no activation (that is 207 no Edar expression) to broad activation (broad Edar expression) and from broad activation to spatially-208 restricted activation (focused Edar expression) suggestive of Turing mechanisms in the dental lamina. 209 From a mathematical point of view, these complex behaviors can be modeled as two regimes of the same 210 reaction-diffusion system (Figure 2A). The first regime, named throughout this paper the bistable regime, 211 describes solutions connecting two constant, stable states (respectively, a high activation state and a low 212 (no) activation state). This corresponds to the anterior part of the tissue with transient up-regulation of 213 Edar. Second, the so-called Turing regime, differs from the previous one by the modification of one 214 parameter (auto-inhibition strength). It is characterized by stable heterogeneous patterns which emerge 215 from homogeneous patterns (e.g. spots). Thus the same system of equations describing the interaction of 216 an activator and its inhibitor models waves of activation in the dental lamina as well as its restriction to the 217 signaling center. A simple change in one reaction parameter could switch the system from the bistable 218 region to the Turing regime. Because the restricted expression is only seen in the developmentally advanced 219 parts (i.e. the anterior part) and follows broad activation, we impose that the activator has a positive 220 feedback on tissue maturation, resulting in the switch to Turing regime. Biologically, this means that upon 221 the broad wave of activation, new gene products have been produced that modify the activation-inhibition

237
Newly grown parts of the dental epithelium get activated on a periodic basis (bright yellow anterior domains, 238 snapshot 2, figure 1C 15.0-15.5 dpc), and upon maturation, produce a Turing peak (snapshot 3, Figure 1C    The tissue grows from its anterior end, and the newly produced tissue matures exponentially in time (albeit 250 at a slow rate). Maturation is stimulated in presence of the activator, with a certain time delay. In zones 251 where maturation reaches a threshold value, the system parameters irreversibly switch from a bistable to a 252 Turing regime ( Figure 2B, snapshots 2 to 3). 253 Before this switch can occur, such simple system first needs to reach high levels of broad activation in the 254 newly grown part ( Figure 2B, snapshot 2). Based on the literature, epithelia-mesenchyme interactions may 255 play an important part in this broad activation (see Discussion), but the mechanism is largely unknown. 256 Here, we simply assumed an extrinsic component representing the interaction with the mesenchyme. Below 257 a certain threshold of activation, it will act to increase the concentration of the activator. Above a certain 258 threshold, it will feedback negatively on it. This introduces an oscillatory behavior at the anterior end of 259 the domain. Interestingly, although we do not explicitly put this in the model, the oscillatory behavior of 260 the mesenchyme is spatially coupled to the domain growth: the transition from "no activation" to 261 "activation" is promoted by the positive feedback from the mesenchyme, but will only happen when the 262 domain has grown enough to escape from the influence of the inhibitor from the Turing peak ( Figure 2B, 263 snapshot 1 to snapshot 2). 264 In summary, our theoretical model based on Edar expression involves activation-inhibition mechanisms in 265 the dental epithelium, coupled with periodic activation of the growing dental epithelium. following the wave: R2 recovers and the early M1 signaling center is newly formed (although soon later, 282 a single domain is formed see Fig 3C, 14.0 dpc, this will be discussed in detail below). We call this 283 phenomenon a developmental palimpsest, because a palimpsest is a manuscript page that has been scraped 284 or washed off to be used again for a novel text: here, a first Turing pattern (a first text) is erased by the 285 Edar expression wave (text scraping) and a new pattern is formed (novel text).

302
Snapshot 3 shows recovery of the R2 peak, together with the newly formed M1 peak (early 14.0 dpc in 303 panel C). The movie corresponding to the snapshots is available as supplementary material 3.

305 306
In our model, we could reproduce this behavior by imposing to the immature dental epithelium a bistable 307 regime with inactive and active stable states. This is a generic reaction-diffusion mechanism for wave 308 propagation. A representative simulation of this modified model is shown in Figure 3D (see also the movie 309 as supplementary material 3). The active state is more stable, so that once activation is primed in the anterior 310 part, a wave activates progressively the inactive area ( Figure 3D, snapshot 2). It is important to notice that, 311 before wave initiation, the immature area is maintained naturally in the (less) stable inactive state under 312 the influence of the Turing initial peak. The wave can even propagate into the mature tissue and erase the 313 previously formed Turing pattern ( Figure 3D, snapshot 1 and 2). Then, as a consequence of tissue 314 maturation subsequent to wave activation, two activation peaks are formed by a secondary Turing 315 patterning ( Figure 3D, snapshot 3). This would correspond to R2 recovery and the newly formed M1 316 signaling center. For this palimpsest to occur, the wave that initiates in the immature (bistable) domain 317 should interact with the stable pattern in the mature Turing domain. Both the activation wave and the Turing 318 peak feature stability. As such, understanding their interaction is far from trivial. Several conditions must 319 be fulfilled in order to observe a palimpsest in the numerical tests, which are reviewed in Supplementary 320 Mat S3. In particular, we found that auto-inhibition, if increased in the bistable regime, strengthened the 321 wave and favored the palimpsest. We also found that it is sensitive to the temporal dynamics. It requires a 322 suitable synchronization between domain growth, anterior activation, wave speed, and maturation rate. Most of the numerous mutants where a premolar-like tooth forms, supposedly from R2, have larger or 327 simply normally sized molar teeth. The teeth of loss of function mutants for the Eda pathway are poorly 328 grown, yet a premolar-like tooth can form. A rescue of R2 is counter-intuitive in such context, therefore, 329 we decided to re-examine one of these mutants (Edar DownlessJ , abreviated Edar DlJ ) in the light of Edar 330 dynamics and the present model. 331 First, we looked at the dynamics of Edar expression in Edar DlJ mutant, to check if R2 was indeed rescued 332 in this mutant. This mutant encodes for a defective Edar protein due to a single amino-acid change, but the 333 gene is still transcribed. In contrast with the mutant epidermis, in which Edar regulation is lost (Edar 334 expression stays at uniform low levels in the epidermis) and hair fails to form (56), we still observe Edar 335 restriction to tooth signaling centers ( Figure 4A), consistent with teeth being formed. However, the 336 dynamics of activation-inhibition mechanisms was modified in this mutant. We observed high variability 337 between embryonic tooth rows (including left/right), in line with the high phenotypic variability seen in 338 adults with losses of function for the Eda pathway (teeth rows with 2 or 3 or in rare cases 4 teeth) (50-52). 339 In the lower jaw ( Figure 4A), we did not find obvious differences early in 12.5 dpc Edar DlJ-/embryos as 340 compared with Edar DlJ+/+ embryos, all of them exhibiting restriction of Edar to MS signaling center that is 341 also stained by Shh expression (not shown). In both cases, wild type and mutant, Edar expression is next 342 found in the whole dental lamina (13.0 dpc, Figure 4A). However, no restriction was observed in Edar DlJ-343 /-13.5 dpc embryos as normally seen in their wild type counterpart (FVB background). Homogenous Edar 344 expression was still observed in most 14.0 dpc Edar DlJ-/embryos at a time when homogenous Edar 345 expression is again observed in Edar DlJ+/+ embryos. From 14.5 dpc, a restriction to a signaling center was 346 observed in most embryos (here named T1 PEK, with or without expression in the "tail"), while others still 347 display more or less continuous Edar expression. We noticed that this signaling center in the mutant is 348 found more posteriorly in the jaw than is the R2 signaling center ( Figure 4B). At 15.0 dpc, we see either 349 one signaling center with Edar expression in the tail or two signaling centers (named T1 and T2). Possibly 350 the later case is due to approximately simultaneous patterning of two signaling centers from a dental    consistent here with T1 signaling center being patterned further away from the anterior end of the dental 374 epithelium. Therefore, we analyzed numerically the effect of stronger inhibition (by decreasing auto-375 inhibition rate). Interestingly, this was sufficient to recover qualitative behaviors consistent with the data: 376 1-T1 and T2 are formed later than R2 and M1 (red dashed line). 2-T1 is displaced anteriorly (blue dashed 377 line) 3 -the wave no longer destabilizes T1: a Turing pattern is formed that is not subjected to a palimpsest. 378 These results suggest two things. First, that a new wave of activation associated with the formation of the 379 next signaling center will naturally destabilize any pre-existing signaling center, if inhibition from this pre-380 existing center is weak enough. Second, because we do not impose any difference between signaling 381 centers, this shows that R2 is not intrinsically defective, but actively outcompeted by the activation wave 382 associated with M1 formation. This is a major output of our modeling effort but it is in contradiction with 383 the previous hypothesis, in which the R2 signaling center is considered to be intrinsically defective. 384 Therefore, we decided to directly test this hypothesis. If the anterior part of the dental epithelium is intrinsically defective for tooth formation, it should not be 391 able to give rise to a fully develop tooth when removed from the early M1 signaling center. On the contrary, 392 if the anterior part is not intrinsically defective, but normally outcompeted by the M1 as we suggest, a tooth 393 should be able to develop from it when removed from M1 influence. To test this, we cut the anterior part 394 (R2 part) from the anterior part (early M1 signaling center and anterior tail). As expected from our 395 predictions, the anterior part developed into a fully growing tooth. Remarkably, the timing of development 396 is advanced compared to the anterior part by 1 day, in accordance with the R2 signaling center having been 397 patterned earlier than the M1 one ( Figure 5). 398 Taken together, these results are consistent with a model in which the R2 region is fully competent for 399 tooth formation, but is actively outcompeted by the forming M1, resulting in the developmental palimpsest 400 effect described. Left: The developing molar region was dissected from 14.3 dpc embryos and the anterior most part corresponding 405 to R2 signaling center was separated from the rest, including the early M1 signaling center and the anterior "tail".

406
The following day, a signaling center was recovered in the anterior part and formed a tooth germ (6 days cultivation).

407
After two days, a signaling center formed in the anterior part and formed a similarly sized tooth as the anterior part.

408
In the control experiment (right), the second molar does not form before the third day.

The formation of a large signaling center depends on Edar activity 412
We then focused on another feature of the mouse dental row: the incorporation of R2 into M1 which is 413 hypothesized to play a crucial role for the formation of the anterior part of M1, both during development 414 and evolution (30,33). This corresponds to another curious behavior of Edar expression dynamics: the 415 fusion of R2 and early M1 signaling centers soon after recovering from the palimpsest ( Figure 3C). 416 We examined the 13.5-14.5 dpc period, which corresponds to M1 PEK formation, in detail. To do so we 417 followed in parallel the dynamics of Edar expression, Shh expression (a recognized marker of tooth 418 signaling centers) and Wnt pathway activity (monitored by the TOPGAL reporter) ( Figure 6A). In late 13.5 419 dpc/ early 14.0 dpc embryos, Shh expression and TOPGAL X-gal staining reveals that the M1 signaling 420 center starts to form. Some faint Shh expression is occasionally seen in R2, while X-gal staining persists 421 there, presumably in part due to B-galactosidase long half-life. At this stage, Edar is also focused in R2 422 and M1 signaling centers, yet low expression can also be seen around. In slightly older embryos, robust 423

457 458
Because Edar expression prefigures the anterior expansion of Shh expression domain and Edar has been 459 shown to regulate Shh (42,57,58), we wanted to test if Edar signaling is necessary for anterior expansion 460 and the formation of a large M1 PEK. To specifically test this, we dissected 13.0 dpc lower molar regions, 461 when R2 has already formed, and cultured them for 48h with or without an interfering antibody, so that we 462 knock-down Edar signaling in the next period of M1 PEK formation. We then visualized Shh expression 463 on isolated epithelia. In untreated samples, we occasionally observed a large M1 PEK similar to in vivo 464 samples ( Figure 6B, state 0), but most often it was split in two spots (corresponding to R2 and an extended 465 early M1 expression) bridged by a narrow domain of Shh expression ( Figure 6B, state 1). This can be 466 explained by the fact that the dissection process could change the activation-inhibition balance in favor of 467 inhibition (as proposed in (59)), which according to the predictions of our model, should favor R2 468 persistence. In treated samples, we mostly recover a small, very posterior signaling center, which 469 corresponds to early M1 Shh expression ( Figure 6B, state 2 and 3). Moreover, the dental epithelium is 470 morphologically different, showing a bud, followed by a small cap and a "tail". Thus, in the absence of 471 Edar signaling, Shh expression is lost in the R2 region and only a small PEK forms, which is equivalent in 472 size and position to the early M1 signaling center, and which drives cap transition there. Taken together, 473 our results show that Edar signaling is essential for the formation of a large PEK that encompass R2 and 474 early M1 signaling centers. 475 476

Coupling chemotaxis to the Turing system reproduces biological variability in signaling center fusion 477
Recent studies have pointed out that chemotaxis may plays a role in the formation of tooth and hair placodes 478 (10,45,46) and the Eda pathway activated centripetal migration in the placodal epithelium (45). We noticed 479 that the TOPGAL stainings tend to contract in the antero-posterior direction as the M1 signaling center 480 matures and the distance between R2 and early M1 signaling center decreases (compare 14.0 and 14.5 dpc 481 samples in figure 6A). This suggested us that cell movements may take part in the formation of the large 482 signaling center. 483 To evaluate this possibility, we incorporated cell motion through chemotaxis in a simple Turing system 484 producing two peaks, thus starting with the situation when R2 and M1 signaling center co-exist (Figure 6 Figure 6E). This is consistent with the intuitive idea that fusion requires sufficiently long range 493 communication between the two Turing peaks, and thus a feedback on the long-range diffusing species (in 494 our case the inhibitor). Reducing chemotaxis efficiency resulted in the lack of fusion, consistent with our 495 experiments reducing Edar signaling, and presumably, chemotaxis ( Figure 6D). 496 Nonetheless, adding chemotaxis to our system does not necessarily result in the fusion into a single spot. 497 In our in silico simulations, we observed that the transition between fusion and the absence of fusion 498 depends on various parameters. The reason is that the activator-chemoattractant has a direct positive effect 499 on the inhibitor, but an indirect negative effect on the same inhibitor by means of cell recruitment. These 500 ambivalent effects make chemotaxis able to compensate the segregation due to Turing patterning in some 501 situations, or reinforce it in other situations. For example, we observed that chemotaxis can favor pattern 502 formation where the Turing system fails to produce a pattern alone. This suggests us that chemotaxis may 503 be part of the normal formation of tooth signaling centers, even when they stay separated. 504 In line with this, we noticed that there is no fusion in the upper jaw ( Figure 6F), where the distance between 505 signaling centers is initially larger by about 30% (Figure 6F and G). A small increase of the domain size, 506 as seen between lower and upper jaw, increased the Turing wavelength and was sufficient to abolish the 507 fusion under the same chemotaxis efficiency ( Figure 6H, a 15% increase is sufficient with our parameters 508 setting). 509 In conclusion, the non-trivial interaction between chemotaxis and a Turing system appears to be a plausible 510 mechanism to explain the variability in the dynamics of tooth signaling centers in lower jaw, upper jaw 511 and various mutants, including Edar loss of function. 512

516
In this study, we have revealed the highly complex and dynamic behavior of signaling centers responsible 517 for tooth patterning in the mouse jaw. Patterning is usually seen as a directional process, rather than a 518 dynamic process that could take circuitous routes. However, we show that patterning of the first molar 519 involves what we called a developmental palimpsest, where patterns are established, erased or remodeled 520 to give rise to new patterns. Using a mathematical approach, we show that these behaviors (pattern erasing, 521 recovery, rescue, fusion) despite seeming to be complex can be be produced by the activity of simple 522 mechanisms (a Turing pair with two regimes, as well as chemotaxis). 523 524

From similarities in Edar expression dynamics to differences between hair and tooth patterning 525
In this study, we have revealed the highly dynamic expression of Edar in the developing molar row. This 526 dynamic is superficially similar to that seen during hair patterning. This is not surprising since hair and 527 tooth patterning share many common features (Mikkola, 2007;Biggs and Mikkola, 2014), making their 528 comparison highly instructive. Below, we compare these two systems in light of our results. 529 In teeth, as in hair, Edar expression becomes restricted to the signaling center as it is patterned. We have 530 noticed however two substantial differences: i) In skin, the initial basal levels of Edar are upregulated in 531 the placode and downregulated in its vicinity. This is thought to be pivotal for placode patterning, where 532 Eda signaling is necessary to stabilize and refine an otherwise labile Wnt-dependent placode prepattern 533 (11,43). In the molar field, Edar expression in the dental lamina reaches levels pretty similar to restricted 534 expression in the signaling center, suggesting that Edar is actively stimulated in both cases. This 535 upregulation may rely on ActivinbA (stimulating Edar expression in tooth cultures, (Laurikkala et al. 2001) 536 and the Wnt pathway (which plays a central role in tooth formation, and is involved in Edar basal 537 expression in hair (11,43)). Downregulation may rely on the BMP pathway, as in hair ((11,43)). ii) We 538 show that this regulation still occurs in the Edar mutant, a major difference with hair, for which the 539 regulation does not occur and uniform basal levels of Edar expression are maintained in the absence of 540 Eda signaling (11). Self-activation of the pathway thus plays a more minor role, if any, in teeth. 541 We believe that these differences on Edar regulation may reflect differences in the balance of the different 542 processes participating in hair and tooth formation. For the formation of hair placodes, Turing-like 543 mechanisms establish a noisy pre-pattern, with local sources of FGF signaling. Mesenchyme condensation 544 towards these sources then refines and reinforces the pattern (Glover 2017). The mesenchyme is also able 545 of autonomous self-organisation, but this is masked by the pre-pattern imposed by the epithelium. 546 The formation of tooth signaling centers seems to rely on a different equilibrium between the two tissues. 547 The formation of a PEK is highly dependent on mesenchyme condensation, as seen in bud-arrested tooth 548 In this study, we assume that the complex spatio-temporal changes in Edar expression highlight waves of 556 activation in the dental epithelium. Each of these waves resumes with the patterning of a signaling center 557 and they are reiterated upon posterior growth of the dental lamina. We note that this growth zone could be 558 the Sox2 positive region shown in (23). We built a reaction-diffusion mathematical model to describe this 559 behavior. In this macroscopic model, molecules are treated as a continuum, and set on a 1-dimensional 560 space to model the antero-posterior dimension of molar row formation. We also chose to consider only 2 561 types of Turing in-phase molecules, corresponding to an activator and an inhibitor. This is of course a high 562 level of abstraction. Tooth genetics has revealed many molecules from the epithelium or the mesenchyme 563 that could participate in the activation-inhibition mechanisms (with both in-phase and out-of-phase 564 patterns), but it was not our purpose here to identify these molecules. Instead, our modeling effort aimed 565 at providing a theoretical framework for sequential tooth formation. Moreover, although our model 566 explicitly aims to describe activation in the epithelium (Edar dynamics), this does not mean that the 567 activator-inhibitor couple in our model should be seen as an abstraction for Turing reactions in the 568 epithelium only. We do not rule out that our model could synthesize the activation-inhibition reactions 569 arising from epithelia-mesenchymal interactions and giving rise to the Turing pattern. 570 We focused on qualitative insights and assessed robustness of pattern formation and developmental 571 palimpsest in our model. We found a suitable model parametrization, and tested its sensitivity with respect 572 to patterning (see Supp Mat for details). Although the results are generally robust enough to moderate 573 parameter changes (10%-50%), it is interesting that the developmental palimpsest can be abolished in 574 many ways, changing auto-inhibition but also temporal dynamics and synchronicity between events. This 575 is consistent with the marked tendency of molar row development towards supplementary molar 576 formation: it can happen in mutants from many different pathways, and moreover it often occurs without 577 major changes in other aspects of tooth development. 578 Our model explicitly assumes that the mesenchyme is responsible for periodic activation priming the newly 579 grown epithelium. This dependence is consistent with a body of evidence showing that mesenchyme 580 activity is necessary for the induction of primary enamel knot formation and sequential tooth formation 581 (61,28,27). We also know that mesenchyme activity depends on the msx1-Bmp4 feedback loop (62-64), 582 which is itself dependent on a mechanical signal provided by mesenchyme condensation (Mammoto et al. 583 )(29). When this loop is defective, sequential tooth formation can stop at different stages from no tooth 584 forming, only one, or only two instead of three (65,63). It can also simply stall until adequate levels of 585 Bmp4 signaling are reached, as seen in the barx1 mutant (66). The mesenchymal Bmp4 signal is part of a 586 Wnt-Bmp regulatory network whose integration drives signaling center formation (28). It is also known 587 that the mesenchyme produces ActivinbA, a potent inducer of both tooth formation (Kavanagh et al. 2007) 588 and Edar expression. In the absence of further knowledge about how the mesenchyme could prime the 589 waves of activation observed in the epithelium, we introduced in our model an extrinsic component 590 representing the interaction with the mesenchyme, and chose a parsimonious way to provide it an 591 oscillatory behavior. For this, we assumed that the mesenchyme activity is stimulated by the activator and 592 feedbacks on it in the newly grown area. Below a certain threshold, it will act to increase the concentration 593 of the activator. Above a certain threshold, it will act to decrease the concentration of the activator. 594 Another interesting feature of our model is that inhibition from the Turing spot locks the bistable system of 595 the newly grown epithelium in the "no activation" state. This means that, in the absence of a wave of 596 activation triggered by the mesenchyme, sequential addition will stop, in contrast with a standard Turing 597 system in a growing field. This is consistent with mutants in the bmp4-msx1 axis where sequential addition 598 resumes after M1 or M2 formation. Experimental approaches will be needed to determine the mechanisms 599 enabling periodicity in our system, and to implement this in the model.  . (14)). Even cells isolated from the presomitic mesoderm exhibit oscillations (67). However, 607 whether such a bona fide molecular oscillator will be found in the tooth system remains an open question. 608 We note that tissue-scale oscillations have been observed in limbs, whose development share similarities 609 with that of epithelial appendages including teeth (68). We also envision other possibilities relying on tissue 610 properties rather than cell properties, for example emerging from the cross-talk between the epithelium and 611 the mesenchyme (as suggested above). will be interesting to test if the palimpsest can be obtained with such a model. 627 Our study also shares superficial similarities with another system showing sequential patterning: feather 628 patterning. In this system, a priming wave of activation is observed in the epithelium, giving rise to a stripe 629 in the chick embryo's back, which is then broken into a spot pattern giving rise to individual feathers. to the breaking of the stripe into spots. These models also converge conceptually. Stripe formation in the 636 feather model, and Edar activation wave in the tooth model, mainly rely on positive feedback. Spot 637 formation and signaling center formation both rely on the introduction of a sharper negative feedback. We 638 take this as an indication that this sequence of activation might be a general property of epithelial 639 appendages (feathers, hair, teeth), that can be captured by very different, non-exhaustive models. We also 640 want to stress that our model is meant to recapitulate activation/long-range inhibition mechanisms, rather 641 than specifically reaction-diffusion mechanisms, and we do not exclude that the biological mechanisms it 642 captures are based on chemotaxy, as in the Painter model. The present data and our simple model suggest that these complex behaviors are the fruit of rather simple 653 but highly dynamic interactions in the growing tooth field. 654 As viewed from Edar expression, the pattern constituted by MS, and later R2 signaling centers is erased to 655 give raise to a second wave of patterning, materialized by a broad Edar expression in the dental epithelium 656 at respectively 12.5-13.0 dpc and 13.5-14.0 dpc. This was recapitulated in the model by enabling the 657 bistable domain to form a traveling wave, that can destabilizes a previously formed signaling center, if 658 inhibition in the later is not too strong. Aside from recapitulating Edar expression, the travelling wave has 659 more profound implications. Indeed, it implies a first paradigm shift that vestigial buds are not committed 660 to abort as usually thought (for example due to their proximity with the diastema thought to serve as a 661 source of inhibitors (70,71), or through the expression of specific molecules (72)), but rather (or on top of 662 that) that they are actively competed by the next round of activation as the dental epithelium grows. Such 663 a balance explains why the anterior part of the molar row is very sensitive to genetic perturbations (with 664 many genetic conditions exhibiting a supplementary tooth there) and environmental perturbations (i.e. 665 tooth culture in this study). It also explains why even conditions that produce a more inhibitory context 666 than the wild type can produce such supplementary tooth. Indeed, our model predicts that if inhibition is 667 increased (e.g. auto-inhibition is decreased), like it is commonly assumed in the Eda pathway mutants, then 668 the Turing pattern remains (with a slightly longer wavelength), but the traveling wave is almost 669 immediately suppressed. This is exactly what we document in Edar DlJ mutants for the R2 signaling center: 670 it forms more posteriorly, and we see no traveling wave that would erase it. Rather, it persists to form a 671 tooth bud. Our cultivation of anterior parts of the molar field, corresponding to R2 signaling center, also 672 show that it has the potential to fully form a tooth, but is actively competed by the M1 signaling center in 673 the wild type situation. Consistent with our results, Li et al. reported that FGF8 application could rescue 674 tooth germ development in the mouse diastema only when it was separated from the molar and incisor buds 675 (73). In conclusion, our results extend the prevailing model (that of Kavanagh et al.), where inhibition 676 between forming teeth is unidirectional (from M1 to M2, to M3), by showing that inhibition can be 677 bidirectional and subtly dependent on the temporal dynamics of the system. 678 In the wild type, following broad Edar activation at 13.5-14.0, a new pattern of Edar restriction forms that 679 is markedly different between the lower and upper jaw. In the lower jaw, independent R2 signaling center 680 and early M1 signaling center are transiently seen (very transiently with Edar expression, for a longer time 681 with Shh expression or TOPGAL activity) but they are rapidly included in a single elongated signaling 682 center. In the upper jaw, Edar restricts to R2 signaling center and M1 signaling center and remains as such. 683 Therefore, the palimpsest is observed in both jaws, enabling the co-occurrence of two signaling centers, 684 but only in the lower jaw does some additional mechanism enable their fusion into a large signaling center. 685 Here, we introduced chemotaxy to the model, because chemotaxy has been evidenced in hair placode 686 formation, both at the level of the epithelium (45,46) and the mesenchyme (10,21). This was sufficient to 687 recapitulate a number of interesting features: 1) chemotaxy changes the reaction-diffusion so that the 688 system first makes two peaks that later fuse into a single, larger peak: this is reminiscent of the large M1 689 signaling center 2) this behavior is sensitive to the distance between the initial peaks, and a 15% increase 690 was sufficient to impede fusion. The measured 30% difference between the R2-M1 distance in the lower 691 jaw, where fusion occurs, and the R2-M1 distance in the upper jaw, where fusion does not occur, may thus 692 be sufficient to explain difference in fate in the two jaws. 3) finally, reducing chemotaxy was sufficient to 693 impede fusion. This may explain why in our culture system, inhibition of Edar activity impedes fusion 694 although the distance between R2 and M1 does not seem drastically changed. 695 Interestingly, we show that chemotaxy plays an ambivalent role in our model. Depending on the conditions, 696 it acts in favor or against the Turing pattern, or it is relatively neutral. We propose that this ambivalent role 697 contributes to explain the versatility of this biological system in regard to genetical and environmental (e.g. An important lesson from the tooth system is that patterning events may be less straightforward 703 than usually thought, and patterns may be dynamically drawn and erased or refined during 704 embryogenesis. In other words, developmental palimpsests may be a common feature. One 705 reason for this is historical: systems are the product of evolution, and as pointed out by F. Jacob, 706 evolution proceeds as a tinkerer, not as an engineer (74): extant patterning mechanisms are 707 modified versions of ancestral mechanisms, and not purposely designed from scratch. Our study 708 is consistent with recent studies on the fine-scale temporal dynamics of gap gene patterns in 709 dipterans (e.g. a progressive anterior shift of the gap genes pattern). As shown here for the Edar 710 mutant, incorporating this dynamics into models provided a better explanation for mutant 711 phenotypes (8,75,76). Moreover, this curious dynamics is also likely the vestige of an ancestral 712 mode of segmentation (75,77). In summary, we believe these two systems illustrate that temporal 713 dynamics of developmental systems needs to be studied, and moreover to be studied in the light 714 of evolution, to fully explain how the system reacts to perturbations. Indeed, embryonic patterns 715 can be highly dynamic and thus dynamics can be essential to the outcome of the patterning 716 process.