Peter S Zammit
email: peter.zammit@kcl.ac.uk

Yusuke Ono

Frederico Calbaheu

Helge Amthor

Jennifer E Morgan

Takenobu Katagiri

BMP signalling permits population expansion by preventing premature myogenic differentiation in muscle satellite cells

Keywords: Satellite cell, BMP signalling, BMP4, Noggin, Pax7, Smad, Id1, MyoD, Skeletal muscle, Proliferation, Myogenic differentiation

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Introduction

Muscle satellite cells are the resident stem cells of skeletal muscle and supply myonuclei for postnatal muscle growth, and for maintenance and repair in adult [START_REF] Zammit | All muscle satellite cells are equal, but are some more equal than others?[END_REF]. Satellite cells are located on the surface of myofibres and are mitotically quiescent in healthy adult muscle. In response to cues for routine myofibre homeostasis or hypertrophy, or the sporadic demands of muscle repair, satellite cells are activated to generate myoblasts that proliferate and eventually undergo myogenic differentiation to provide new myonuclei.

Satellite cells also self-renew, thus maintaining a population of quiescent, undifferentiated precursors available to respond to repeated demand [START_REF] Zammit | Muscle satellite cells adopt divergent fates: a mechanism for self-renewal?[END_REF][START_REF] Halevy | Pattern of Pax7 expression during myogenesis in the posthatch chicken establishes a model for satellite cell differentiation and renewal[END_REF][START_REF] Collins | Stem cell function, self-renewal, and behavioral heterogeneity of cells from the adult muscle satellite cell niche[END_REF].

Satellite cell function is controlled by various regulatory pathways, chief amongst them being Notch/Delta and Wnt signalling [START_REF] Zammit | All muscle satellite cells are equal, but are some more equal than others?[END_REF]. Many of these same regulatory networks also control embryonic myogenesis, in addition to many other processes both during development, and repair in adult. Another important network for organising embryonic and foetal myogenesis involves the Bone morphogenetic proteins (BMPs). BMPs belong to the transforming growth factor-β family and initiate signalling by binding to the transmembrane type 1 and type 2 BMP receptors. Upon BMP binding, type 1 and 2 receptors complex on the cell surface, allowing the constitutively active kinase of the type 2 receptor to transphosphorylate the type 1 receptor [START_REF] Feng | Specificity and versatility in tgf-beta signaling through Smads[END_REF]. This in turn, then phosphorylates the R-Smads -Smad1, Smad5 and Smad8 (pSmad1/5/8), which translocate to the nucleus to regulate transcription of target genes including Ids [START_REF] Hollnagel | Id genes are direct targets of bone morphogenetic protein induction in embryonic stem cells[END_REF][START_REF] Katagiri | Identification of a BMPresponsive element in Id1, the gene for inhibition of myogenesis[END_REF][START_REF] Korchynskyi | Identification and functional characterization of distinct critically important bone morphogenetic protein-specific response elements in the Id1 promoter[END_REF].

Id proteins (comprising Id1, 2, 3, 4) bind ubiquitously expressed E-proteins to form inactive heterodimers, so preventing these same E-proteins from binding with tissue specific transcription factors such as MyoD and myogenin, a step necessary for their efficient function [START_REF] Jen | Overexpression of Id protein inhibits the muscle differentiation program: in vivo association of Id with E2A proteins[END_REF]. BMP signalling can be modified in a number of ways, including via secreted antagonists such as Noggin, which bind BMPs with high affinity to interfere with interactions between BMPs and their receptors [START_REF] Feng | Specificity and versatility in tgf-beta signaling through Smads[END_REF].

Although crucial for bone and cartilage formation and repair [START_REF] Feng | Specificity and versatility in tgf-beta signaling through Smads[END_REF][START_REF] Wozney | Novel regulators of bone formation: molecular clones and activities[END_REF], BMPs also inhibit myogenic differentiation to prevent ectopic myogenesis in the lateral plate mesoderm. In the dermomyotome though, BMP signalling itself must be inhibited, to permit the onset of myogenesis [START_REF] Reshef | Regulation of dorsal somitic cell fates: BMPs and Noggin control the timing and pattern of myogenic regulator expression[END_REF][START_REF] Re'em-Kalma | Competition between noggin and bone morphogenetic protein 4 activities may regulate dorsalization during Xenopus development[END_REF][START_REF] Hirsinger | Noggin acts downstream of Wnt and Sonic Hedgehog to antagonize BMP4 in avian somite patterning[END_REF][START_REF] Amthor | The importance of timing differentiation during limb muscle development[END_REF]. BMPs not only impede myogenic differentiation in immortalised myogenic cell lines (e.g. C2) and primary myogenic cells in vitro, but also induce osteoblastic gene expression and differentiation towards the osteoblast lineage [START_REF] Wada | Generation of different fates from multipotent muscle stem cells[END_REF][START_REF] Katagiri | Bone morphogenetic protein-2 converts the differentiation pathway of C2C12 myoblasts into the osteoblast lineage[END_REF][START_REF] Asakura | Muscle satellite cells are multipotential stem cells that exhibit myogenic, osteogenic, and adipogenic differentiation[END_REF][START_REF] Dahlqvist | Functional Notch signaling is required for BMP4-induced inhibition of myogenic differentiation[END_REF].

Interestingly, although able to express early markers of osteogenic differentiation such as alkaline phosphatase, many primary myogenic cells also retain expression of proteins associated with myogenesis (e.g. Pax7 and MyoD) after exposure to BMPs for several days [START_REF] Wada | Generation of different fates from multipotent muscle stem cells[END_REF][START_REF] Asakura | Muscle satellite cells are multipotential stem cells that exhibit myogenic, osteogenic, and adipogenic differentiation[END_REF]. Since intramuscular injection of certain BMPs (e.g. BMP2 and BMP4) can lead to ectopic bone formation in vivo [START_REF] Kaplan | Skeletal metamorphosis in fibrodysplasia ossificans progressiva (FOP)[END_REF][START_REF] Lounev | Identification of progenitor cells that contribute to heterotopic skeletogenesis[END_REF], these in vitro observations were thought to provide mechanistic insight into Fibrodysplasia ossificans progressiva (FOP): a rare disorder of skeletal malformations and progressive extra-skeletal ossification in muscle. FOP is caused by a mutation in ALK-2, which renders this BMP receptor type 1 constitutively active [START_REF] Kaplan | Skeletal metamorphosis in fibrodysplasia ossificans progressiva (FOP)[END_REF], and inhibition of this mutant ALK-2 reduces ossification [START_REF] Yu | BMP type I receptor inhibition reduces heterotopic [corrected] ossification[END_REF]. However, Lounev et al. have recently shown that few (<5%) muscle progenitors actually contribute to BMP-induced heterotopic ossification in vivo [START_REF] Lounev | Identification of progenitor cells that contribute to heterotopic skeletogenesis[END_REF].

Here, we investigated the role of BMP signalling in satellite cell function during adult myogenesis.

During activation and proliferation, satellite cells robustly express BMP receptor type 1A (BMPR-1A), and pSmad1/5/8 and Smad4 are present in their nuclei, indicative of operational BMP signalling. Exogenous BMP4 can sustain satellite cell division and reduce differentiation, with BMP4 operating through BMPR-1A.

Conversely, antagonising the interaction of BMP with its receptors, or perturbing intracellular BMP signalling by either inhibiting Smad1/5/8 phosphorylation or reducing Smad5 or Smad4 levels, all induced premature differentiation. Manipulation of BMP signalling affected Id1 levels: a known inhibitor of differentiation that regulates the function of the myogenic regulatory factors (that comprise of Myf5, MyoD, myogenin and MRF4) through sequestration of E-proteins. Satellite cell progeny then up-regulate the BMP antagonist Noggin as they differentiate, and knockdown of Noggin enhanced proliferation and impeded differentiation. Since blockade of BMP signalling after muscle injury hindered regeneration, our observations show that BMP signalling is a potent regulator of routine satellite cell function in adult.

Results

BMP pathway proteins are present in activated and proliferating satellite cells

Satellite cell function can be modelled in vitro. Upon stimulation by mitogen-rich medium (termed plating, or proliferation, medium here), isolated Pax7-expressing quiescent satellite cells are activated and up-regulate MyoD. Satellite cells then proliferate, before either down-regulating Pax7, maintaining MyoD, inducing myogenin and proceeding to differentiate, or down-regulating MyoD and maintaining Pax7 to return to a quiescent-like state, modelling self-renewal [START_REF] Zammit | Muscle satellite cells adopt divergent fates: a mechanism for self-renewal?[END_REF][START_REF] Halevy | Pattern of Pax7 expression during myogenesis in the posthatch chicken establishes a model for satellite cell differentiation and renewal[END_REF].

To first determine the expression profile of BMPR-1A, we immunostained satellite cells retained in their niche on myofibres isolated from the extensor digitorum longus (EDL) muscle. BMPR-1A was undetectable in the vast majority of quiescent satellite cells on freshly isolated (T0) myofibres (Fig. 1A).

However, after culture in plating medium for 48 (T48) or 72 hours (T72), BMPR-1A became robustly expressed in activated and proliferating satellite cells (Fig. 1A). BMPR-1B was not detectable by Q-PCR (data not shown), consistent with its absence in C2C12 myoblasts [START_REF] Namiki | A kinase domain-truncated type I receptor blocks bone morphogenetic protein-2-induced signal transduction in C2C12 myoblasts[END_REF]. Intracellular BMP signalling through BMPR-1A operates via phosphorylation of the carboxyl terminal of R-Smad proteins. Immunostaining for either pSmad1/5/8 or Smad5, revealed a strong nuclear signal in activated and proliferating (T48 and T72), but not quiescent (T0), satellite cells (Fig. 1B and1C). The co-activator (Co-Smad) Smad4, which facilitates translocation of Smad1/5/8 to the nucleus and promotes their transcriptional activity [START_REF] Feng | Specificity and versatility in tgf-beta signaling through Smads[END_REF], was also present in the nuclei of activated satellite cells (Fig. 1C). Thus, the expression dynamics of BMPR-1A, pSmad1/5/8, Smad4 and Smad5 mirror each other, and indicate that BMP signalling is operating in activated and proliferating satellite cells.

Since Noggin-mediated BMP antagonism contributes to initiation of the myogenic program during embryogenesis [START_REF] Hirsinger | Noggin acts downstream of Wnt and Sonic Hedgehog to antagonize BMP4 in avian somite patterning[END_REF], we hypothesised that Noggin may also modify BMP signalling in satellite cells. Noggin was undetectable in quiescent (T0), and at low levels in activated Pax7 + , satellite cells (Fig. 2A andB). By contrast, Noggin was highly expressed by differentiating myoblasts at T72, as shown by co-immunostaining for Noggin and myogenin (Fig. 2B). Culturing satellite cells attached to a myofibre provides a useful model for studying satellite cell activation, proliferation and the initial stages of differentiation. Later events, such as myoblast fusion into large multinucleated myotubes however, are better studied in satellite cells removed from their associated myofibre and plated onto Matrigel-coated culture dishes. Such plated satellite cellderived myoblasts cultured in differentiation medium for 2 days, clearly exhibited high Noggin levels in myotubes and other Pax7 -cells (Fig. 2C andD).

BMP regulates the balance between proliferation and differentiation in satellite cells

We next examined the effects of stimulating BMP signalling, and used administration of recombinant BMP4 to achieve this, since it is present in serum [START_REF] Kodaira | Purification and identification of a BMP-like factor from bovine serum[END_REF]. Addition of recombinant BMP4 protein (100 ng/ml) to EDL satellite cells retained in their niche on the myofibre, significantly increased the number of Pax7 + MyoD + satellite cells, but decreased the number of cells with the differentiating Pax7 -MyoD + phenotype (Fig. 3A). Inhibition of differentiation by BMP4 was confirmed by co-immunostaining for myogenin and Pax7 (Fig. 3B). Plating medium contains chick embryo extract and horse serum, so is rich in growth factors, and contains BMP4 [START_REF] Kodaira | Purification and identification of a BMP-like factor from bovine serum[END_REF]. Under these culture conditions, satellite cell proliferation is likely to be near maximal, hence addition of BMP4 did not further increase cell division (Fig. 3A andB). Since satellite cells retained expression of genes associated with myogenesis, these short exposures to BMP4 did not induce any obvious loss of myogenic identity.

To interfere with BMP binding to its receptors, we added recombinant Noggin protein (50 ng/ml) or soluble BMPR-1A fragments (sBMPR-1Af) (200 ng/ml) to cultures of satellite cells associated with a myofibre for 72 hours. Exposure to either Noggin or sBMPR-1Af resulted in a significant reduction in the mean total satellite cell number per myofibre, particularly of cells with the Pax7 + MyoD + phenotype (Fig. 3A).

Numbers of differentiating Pax7 -MyoD + cells were already increased at T48, and became the predominant phenotype at T72, signifying a precocious onset of differentiation (Fig. 3A). There were also significantly fewer self-renewing Pax7 + MyoD -cells present at T72 (Fig. 3A). Co-immunostaining for Pax7 and myogenin confirmed that both Noggin and sBMPR-1Af enhanced differentiation, revealing a significant increase in cells with the Pax7 -myogenin + phenotype (Fig. 3B). These effects were not restricted to satellite cells from somite-derived muscles, since those from the branchiomeric masseter muscle, which has a different embryonic origin [START_REF] Ono | Muscle satellite cells are a functionally heterogeneous population in both somite-derived and branchiomeric muscles[END_REF], also exhibited fewer differentiating Pax7 -MyoD + cells in the presence of BMP4, but increased differentiation when exposed to Noggin (Fig. 3C).

Culture in proliferation medium does not readily support differentiation of plated satellite cells.

Addition of recombinant BMP4 resulted in a higher proportion of cells co-immunostained for Pax7 and MyoD (Fig. 3D). The differentiation index under these culture conditions however, is already so low that any inhibitory effect of BMP4 on formation of myosin heavy chain (MyHC) containing myocytes was not significant (Fig. 3E, quantified in F). Addition of either Noggin or sBMPR-1Af however, triggered a robust differentiation, as shown by both the reduced proportion of plated cells with Pax7 (Fig. 3D), and the marked increase in myotubes containing MyHC (Fig. 3E andF). The proportion of satellite cell progeny with the self-renewing Pax7 + MyoD -phenotype was also significantly increased, from effectively zero in control cultures, to ~0.7% after exposure to either Noggin or sBMPR-1Af (Fig. 3D).

Finally, we determined the effects of exogenous BMP4 on satellite cells following serum deprivation, which normally promotes myogenin expression, cell cycle exit, and fusion into myotubes. However, addition of recombinant BMP4 (200 ng/ml) to differentiation medium led to a significant drop in the proportion of cells containing myogenin (Fig. 3G,quantified in H) and fusing into myotubes (Fig. 3G,quantified in I).

However, MyoD remained readily detectable in the nuclei of these cells after exposure to BMP4 for 2 days (Fig. 3G). Importantly, BMP4 caused a 1.46±0.2-fold increase in total cell number, indicating that BMP4 not only inhibits differentiation, but also stimulates satellite cell proliferation, despite serum depletion (Fig. 3J).

BMP operates through BMPR-1A to prevent precocious differentiation

To determine whether BMP signalling was transduced through BMPR-1A in activated satellite cells, we used siRNA-mediated knockdown [START_REF] Ono | Presenilin-1 acts via Id1 to regulate the function of muscle satellite cells in a gamma-secretase-independent manner[END_REF], which reduced the mean total number of satellite cells per myofibre when assayed at T72 (Fig. 4A, quantified in B). In particular, BMPR-1A knockdown significantly decreased the number of both Pax7 + MyoD + cells and Pax7 + MyoD -self-renewed cells, but not the number of Pax7 -MyoD + differentiating cells, which became the predominant phenotype (Fig. 4B). The efficient reduction in BMPR-1A levels by siRNA was confirmed by Western blot analysis of plated satellite cells, which also revealed decreased Pax7 levels, but an increase in the amount of MyHC (Fig. 4C).

Immunostaining (Fig. 4D, quantified E-G) confirmed that reducing BMPR-1A led to a reduction in the proportion of Pax7 + cells (Fig. 4E), which was accompanied by a significant reduction in Ki67 + proliferating cells (Fig. 4D andF). In contrast, there was an increase in the proportion of MyHC + differentiated cells after BMPR-1A knockdown (Fig. 4D andG).

Blockade of intracellular BMP signalling promotes myogenic differentiation

We next analysed the pSmad1/5/8 signalling cascade [START_REF] Feng | Specificity and versatility in tgf-beta signaling through Smads[END_REF]. The protein kinase inhibitor Dorsomorphin is a specific inhibitor of Smad1/5/8 phosphorylation by BMP receptors [START_REF] Yu | Dorsomorphin inhibits BMP signals required for embryogenesis and iron metabolism[END_REF]. Dorsomorphin (1-5 μM) was not toxic to plated satellite cells, as shown using the Tunel assay (data not shown). Exposure of cultured myofibres to Dorsomorphin significantly decreased the mean number of associated Pax7 + MyoD + satellite cells, and so Pax7 -MyoD + differentiating cells became the main phenotype (Fig. 5A). Dorsomorphin also induced myogenic differentiation in a dose dependent (0.1-1.0 μM) manner in plated satellite cells maintained in proliferation medium (Fig. 5B). Additionally, we also targeted Smad signalling using siRNAmediated knockdown of Smad5 or Smad4, which effectively reduced the levels of the translated protein (Fig. 5C andD). Co-immunostaining for Ki67 and MyHC revealed that Smad5 or Smad4 knockdown significantly reduced cell proliferation, and induced precocious differentiation (Fig. 5E,quantified in F/G).

BMP signalling regulates the ability of MyoD to activate its transcriptional targets

To understand how interference with BMP signalling promotes myogenic differentiation, we investigated Id1, a downstream target of BMPs (6-8) and a negative controller of MyoD and myogenin [START_REF] Jen | Overexpression of Id protein inhibits the muscle differentiation program: in vivo association of Id with E2A proteins[END_REF][START_REF] Benezra | The protein Id: a negative regulator of helix-loop-helix DNA binding proteins[END_REF]. We recently showed that Id1 is highly expressed in proliferating satellite cells, before being downregulated during myogenic differentiation [START_REF] Ono | Presenilin-1 acts via Id1 to regulate the function of muscle satellite cells in a gamma-secretase-independent manner[END_REF]. Immunostaining showed that exposure to Dorsomorphin to inhibit Smad1/5/8 phosphorylation, reduced Id1 levels in both satellite cells retained in their niche on the myofibre (Fig. 6A), and in plated satellite cells (Fig. 6B).

Muscle creatine kinase (CKM) is a well-characterised gene and provides a useful tool to quantify both the ability of MyoD to transactivate its target genes, and myogenic differentiation [START_REF] Benezra | The protein Id: a negative regulator of helix-loop-helix DNA binding proteins[END_REF]. Therefore, we measured the activity of a CKM-luciferase construct (CKM-LUCpCS2) in response to both exogenous Id1, and perturbation of BMP signalling (Fig. 6C). As expected, CKM-LUCpCS2 activity was significantly decreased after transfection with Id1 (Fig. 6C). Importantly, transfection of a plasmid encoding a constitutively active BMPR-1A (caBMPR-1A) that phosphorylates Smad1/5/8 in the absence of BMP ligands [START_REF] Akiyama | Constitutively active BMP type I receptors transduce BMP-2 signals without the ligand in C2C12 myoblasts[END_REF], also significantly reduced CKM-LUCpCS2 activity (Fig. 6C). In contrast, inhibition of Smad1/5/8 phosphorylation with Dorsomorphin significantly increased the activity of CKM-LUCpCS2, but Id1 was able to completely reverse this Dorsomorphin-mediated increase (Fig. 6C). Finally, caBMPR-1A also reduced CKM-LUCpCS2 activity back to control levels, despite the presence of Dorsomorphin (Fig. 6C).

Thus, BMP signalling in satellite cells acts to limit the transcriptional activation of genes controlled by MyoD and enhanced during differentiation.

Noggin antagonises BMPs to facilitate myogenic differentiation

Noggin is up-regulated as satellite cells differentiate (Fig. 2). To examine the role of Noggin, we first performed siRNA-mediated knockdown in plated satellite cells, and immunostaining showed that it worked efficiently (Fig. 7A). Plated satellite cells transfected with Noggin siRNA and co-immunostained for Ki67 and MyHC (Fig. 7B, quantified C-E) had a significantly increased proportion of proliferating Ki67 + satellite cells (Fig. 7C). However, the proportion of differentiating MyHC + cells and the fusion index were both markedly reduced, with only myocytes or small myotubes present (Fig. 7B, D andE). Consistent with these observations, Western blot analysis revealed a slight decrease in myogenin levels (Fig. 7F). We also found that the fusion defect induced by Noggin knockdown could be rescued by Dorsomorphin-mediated blockade of intracellular BMP signalling (Fig. 7G, quantified H), indicating that endogenous Noggin from differentiating cells is required for proper myotube formation, through suppression of BMP signalling.

BMP signalling is operative during muscle regeneration in vivo

Finally, we evaluated whether BMP signalling was also operative in satellite cells in regenerating muscle in vivo. Cardiotoxin was used to induce damage in the gastrocnemius muscle of adult mice. Three days later, the regenerating muscles were removed, cryosectioned and immunostained for pSmad1/5/8, together with MyoD to identify satellite cell-derived myoblasts: 97.4 ± 2.9% (n= 50 cells from each of 3 mice) of MyoD + cells contained pSmad1/5/8 (Fig. 8A). Undamaged areas of the muscle periphery provided the control, and no pSmad1/5/8 immunostaining was associated with myofibres in these non-regenerating regions (data not shown).

Regenerating gastrocnemius muscle was also injected with either 50 μl of 10 μM Dorsomorphin (Fig. 8B-D) or 50 μl of 100 μg/ml sBMPR-1Af (Fig. 8E-G) 1 and 3 days after cardiotoxin-induced injury. The contralateral regenerating muscle served as the control, and was injected with 50 μl of vehicle. Regenerating muscles were removed on day 8 and immunostained for MyHC and collagen type 1 (Fig. 8B andE).

Administration of Dorsomorphin resulted in a significant ~20% decrease in the mean perimeter of centrally nucleated regenerating myofibres compared to controls, while the reduction was nearer ~40% with sBMPR-1Af (Fig. 8C andF). A significant ~1.5-fold increase in collagen type 1 deposition following either Dorsomorphin or sBMPR-1Af administration was also observed (Fig. 8D andG).

Discussion

Here we investigated the role of BMP signalling in normal adult myogenesis. We found that exogenous BMP4 was able to sustain proliferation in satellite cells and inhibit differentiation. Conversely, blocking interaction of BMP with its receptors, down-regulating BMPR-1A or perturbing intracellular BMP signal-mediation, all induced rapid differentiation. Thus, BMP signalling initially serves to allow expansion of the satellite cell pool by stimulating proliferation and preventing precocious myogenic differentiation (Fig.

9).

The source of BMPs that act on satellite cells in vivo remains to be determined, but myogenic cells can express and secrete BMP4 (29, 30), with higher levels produced by myoblasts from dystrophic muscle [START_REF] Sterrenburg | Gene expression profiling highlights defective myogenesis in DMD patients and a possible role for bone morphogenetic protein 4[END_REF]. Additionally, muscle damage may also expose satellite cells to BMP4 in serum [START_REF] Kodaira | Purification and identification of a BMP-like factor from bovine serum[END_REF], and/or from cells such as macrophages, since microglia/macrophages produce BMPs after CNS damage [START_REF] Xiao | Bone morphogenetic proteins mediate cellular response and, together with Noggin, regulate astrocyte differentiation after spinal cord injury[END_REF]. While prolonged exposure of myogenic cells to high levels of BMPs can induce expression of various osteogenic genes (16), we found that satellite cells exposed to BMP4 for the shorter periods likely encountered during muscle repair/regeneration in vivo, did not obviously compromise their myogenic identity.

Satellite cells respond to BMPs because they express BMPR-1A, which was undetectable on most quiescent satellite cells, but up-regulated as cells were activated. An early event in satellite cell activation is induction of MyoD, which can bind in the promoter of Bmpr-1a to enhance its expression, and MyoD overexpression leads to BMPR-1A induction [START_REF] Liu | Myoblast sensitivity and fibroblast insensitivity to osteogenic conversion by BMP-2 correlates with the expression of Bmpr-1a[END_REF]. Thus up-regulation of BMPR-1A is part of the response of satellite cells to serum stimulation, and siRNA-mediated knockdown of BMPR-1A induced premature differentiation, showing that BMP signalling operates through BMPR-1A. Consistent with our observations, a dominant-negative BMPR-1A mutant enhanced myogenic differentiation in C2 cells maintained in highserum medium, but a dominant-negative BMPR-1B mutant did not [START_REF] Kodaira | Purification and identification of a BMP-like factor from bovine serum[END_REF]. By contrast, we found that a constitutively active BMPR-1A (28) reduced the activity of a CKM reporter that is normally up-regulated during myogenic differentiation.

BMPs are known to control the ability of MyoD to transactivate its target genes by Smad-mediated regulation of Id proteins (6-8), negative regulators of MyoD function and terminal differentiation [START_REF] Jen | Overexpression of Id protein inhibits the muscle differentiation program: in vivo association of Id with E2A proteins[END_REF][START_REF] Benezra | The protein Id: a negative regulator of helix-loop-helix DNA binding proteins[END_REF].

Proliferating satellite cell-derived myoblasts contain both MyoD and Id1, with Id1 presumably curtailing MyoD function to prevent premature differentiation. Id1 levels are then dramatically reduced as satellite cells undergo differentiation [START_REF] Ono | Presenilin-1 acts via Id1 to regulate the function of muscle satellite cells in a gamma-secretase-independent manner[END_REF]. Consistent with these observations, while Id1 and Id3 are undetectable in muscle lysates from uninjured muscle, they are up-regulated during muscle regeneration, peaking around day 3 [START_REF] Clever | Inefficient Skeletal Muscle Repair in Inhibitor of Differentiation (Id) Knockout Mice Suggests a Crucial Role for BMP Signaling During Adult Muscle Regeneration[END_REF]. In satellite cells, we found that inhibiting BMP signalling reduced Id1 levels, promoted differentiation and increased CKM promoter activity. Delta/Notch signalling permits satellite cell expansion and prevents precocious differentiation [START_REF] Conboy | The regulation of Notch signaling controls satellite cell activation and cell fate determination in postnatal myogenesis[END_REF].

Notch signalling is required for BMP4 to impede myogenic differentiation in C2 cells: inhibitors of γsecretase prevent cleavage of Notch to release the active intracellular domain, and can partially reverse the BMP-induced differentiation block [START_REF] Dahlqvist | Functional Notch signaling is required for BMP4-induced inhibition of myogenic differentiation[END_REF]. The catalytic subunit of the γ-secretase complex is presenilin-1, and we recently showed that presenilin-1 can inhibit satellite cell differentiation via both γ-secretase-dependent and γ-secretase-independent mechanisms [START_REF] Ono | Presenilin-1 acts via Id1 to regulate the function of muscle satellite cells in a gamma-secretase-independent manner[END_REF]. The γ-secretase-dependent effects of presenilin-1 on impeding differentiation are likely through controlling Notch signalling. Crucially however, the γ-secretase (Notch)-independent presenilin-1 mechanism operates by augmenting Pax7 and controlling Id1 levels [START_REF] Ono | Presenilin-1 acts via Id1 to regulate the function of muscle satellite cells in a gamma-secretase-independent manner[END_REF], and intriguingly, Id1, Id2 and Id3 are direct transcriptional targets of Pax7 [START_REF] Kumar | Id3 is a direct transcriptional target of Pax7 in quiescent satellite cells[END_REF]. This γ-secretase-independent mechanism is unlikely to work through BMP signalling though, since there is no difference in pSmad1/5/8 levels between presenilin-1-null or control embryonic fibroblasts transfected with MyoD, even though only the MyoD-expressing presenilin-1-null cells readily differentiate in high serum medium (unpublished observations). Furthermore, over-expression of Wnt3a in C2 cells is capable of overcoming the BMP2mediated block of differentiation by down-regulating Id1 [START_REF] Nakashima | Cross-talk between Wnt and bone morphogenetic protein 2 (BMP-2) signaling in differentiation pathway of C2C12 myoblasts[END_REF]. Thus γ-secretase-independent presenilin-1 signalling and the BMP pathway both appear to separately converge on Ids to regulate satellite cell function.

Antagonism of BMP signalling by Noggin is employed during development to protect pre-myogenic cells, and expansion of the Noggin expression domain promotes ectopic myogenesis [START_REF] Reshef | Regulation of dorsal somitic cell fates: BMPs and Noggin control the timing and pattern of myogenic regulator expression[END_REF][START_REF] Re'em-Kalma | Competition between noggin and bone morphogenetic protein 4 activities may regulate dorsalization during Xenopus development[END_REF]. We found that Noggin-mediated interference with BMP signalling was also redeployed during adult myogenesis. Noggin was up-regulated as satellite cells differentiated, and knocking-down Noggin resulted in more cells proliferating, while fewer differentiated. Since Noggin is a secreted signalling molecule, it may also act in a paracrine manner to reduce BMP signalling in proliferating cells to potentiate and augment timely differentiation. Noggin expression in C2 cells is up-regulated by exogenous BMP2 [START_REF] Takayama | RNA interference for noggin enhances the biological activity of bone morphogenetic proteins in vivo and in vitro[END_REF] and administration of Noggin to regenerating muscle reduces both pSmad1/5/8 and Id1 levels [START_REF] Clever | Inefficient Skeletal Muscle Repair in Inhibitor of Differentiation (Id) Knockout Mice Suggests a Crucial Role for BMP Signaling During Adult Muscle Regeneration[END_REF]. In addition, rhabdomyosarcoma cells express myogenic genes including MyoD and myogenin but form myotubes poorly, and have relatively high levels of BMP4 and lower expression of Noggin, than primary myoblasts [START_REF] Goldstein | FGFR1 over-expression in primary rhabdomyosarcoma tumors is associated with hypomethylation of a 5' CpG island and abnormal expression of the AKT1, NOG, and BMP4 genes[END_REF]. Furthermore, specification of myogenic cells is unaffected in Noggin-null embryos, but differentiation is clearly perturbed, with meagre myotube formation during foetal myogenesis [START_REF] Tylzanowski | The Noggin null mouse phenotype is strain dependent and haploinsufficiency leads to skeletal defects[END_REF], again suggesting a common role for Noggin in developmental and adult myogenesis.

We observed that inhibition of BMP signalling during muscle regeneration resulted in smaller regenerated myofibres. Similarly, Id1 -/+ Id3 -/-mice also exhibit a muscle regeneration defect, with the crosssectional area of regenerated myofibres only ~50% of wild-type [START_REF] Clever | Inefficient Skeletal Muscle Repair in Inhibitor of Differentiation (Id) Knockout Mice Suggests a Crucial Role for BMP Signaling During Adult Muscle Regeneration[END_REF]. These mice also had fewer proliferating satellite cells during regeneration [START_REF] Clever | Inefficient Skeletal Muscle Repair in Inhibitor of Differentiation (Id) Knockout Mice Suggests a Crucial Role for BMP Signaling During Adult Muscle Regeneration[END_REF].

In conclusion, BMP signalling is part of the program that regulates routine satellite cell function.

BMPs stimulate proliferation and prevent precocious differentiation, to allow amplification of satellite cellderived myoblasts. Noggin is then up-regulated as satellite cell progeny differentiate, to antagonise BMP signalling and facilitate coordinated differentiation (Fig. 9).

Material and methods

Isolation and culture of primary satellite cells and myoblasts

Adult (8-12 weeks old) C57BL10 mice were killed by cervical dislocation, and the EDL (or masseter for Fig. 3B only) muscle isolated and digested in collagenase type1 as previously described [START_REF] Ono | Presenilin-1 acts via Id1 to regulate the function of muscle satellite cells in a gamma-secretase-independent manner[END_REF]. Most satellite cells are mitotically quiescent in healthy adult muscle, so must first be activated and will then enter the cell cycle (2). To induce satellite cell activation in vitro, myofibers and associated satellite cells were cultured in plating medium [DMEM supplemented with L-glutamine or GlutaMax (Sigma), 10% horse serum (PAA Laboratories), 0.5% chicken embryonic extract (ICN Flow) and 1% penicillin-streptomycin (Sigma)] at 37°C in 5% CO 2 , as described previously [START_REF] Ono | Muscle satellite cells are a functionally heterogeneous population in both somite-derived and branchiomeric muscles[END_REF]. To prepare plated satellite cells, either cells were removed from myofibers by enzymatic treatment with 0.125% trypsin-EDTA (Sigma) for 10 min at 37 o C, or allowed to migrate from isolated myofibres cultured on Matrigel. Satellite cells were then re-plated on Matrigel at equal density and maintained in proliferation medium (DMEM supplemented with L-glutamine or GlutaMAX, 20-30% foetal bovine serum, 1% chicken embryo extract, 10 ng/ml bFGF and 1% penicillin-streptomycin). Satellite cells were treated with either 100 ng/ml recombinant BMP4 protein, (Peprotech), 50 ng/ml recombinant Noggin protein (Peprotech), 200 ng/ml sBMPR-1Af (R&D systems) or 0.1-3 μM Dorsomorphin (also known as Compound C; Calbiochem) administered to the culture medium. In other experiments, satellite cells were switched from proliferation medium to differentiation medium (DMEM with L-glutamine or GlutaMax and supplemented with 2% horse serum and 1% penicillin-streptomycin) and cultured for 48 hours with or without BMP4 (200 ng/ml). Myofibres and associated satellite cells or plated satellite cells were fixed with warm 4% paraformaldehyde (PFA).

Antibodies used in the study

Antibodies were obtained from the following sources. Rabbit anti-BMPR-1A from Orbigen and a gift from Professor B. Kay [START_REF] Kariyawasam | Basal expression of bone morphogenetic protein receptor is reduced in mild asthma[END_REF]; rat anti-Ki67 from DAKO; goat anti-Smad5, mouse anti-Smad4, mouse anti-MyoD and rabbit anti-MyoD antibodies were obtained from Santa Cruz (Santa Cruz); goat anti-Noggin antibody was obtained from R&D systems; mouse anti-MyHC (MF20), anti-Pax7, anti-myogenin (F5D) and anti-β-tubulin antibodies (E7) were supplied by the DSHB (Iowa City, IA); rabbit anti-pSmad1/5/8 antibody was from Cell Signalling Technology (Beverly, MA); goat anti-collagen type 1 antibody was acquired from Southern Biotech (Birmingham, AL). Mounting medium containing 4,6-diamidino-2-phenylindole (DAPI) to counterstain all nuclei and the M.O.M kit were both purchased from Vector Laboratories.

Immunoblot analysis

Immunoblot analysis was performed as previously described [START_REF] Ono | Presenilin-1 acts via Id1 to regulate the function of muscle satellite cells in a gamma-secretase-independent manner[END_REF]. Anti-BMPR-1A, anti-MyHC, anti-Pax7, anti-Myogenin or anti-β-tubulin antibodies were applied in PBS containing 0.1% Tween20 and incubated at 4°C overnight. HRP-conjugated secondary antibodies were used for visualisation by chemiluminescence (Fujifilm).

Immunostaining

Immunocytochemistry was as previously described [START_REF] Ono | Presenilin-1 acts via Id1 to regulate the function of muscle satellite cells in a gamma-secretase-independent manner[END_REF]. Anti-BMPR-1A, anti-Ki67, anti-Smad5, anti-Smad4, anti-MyoD, anti-Noggin, anti-MyHC, anti-Pax7, anti-myogenin, anti-Id1, or anti-pSmad1/5/8 antibodies were applied at 4°C overnight. For co-immunostaining sections, frozen regenerating muscle cross-sections were fixed with 4% PFA, blocked using the M.O.M kit and incubated with either anti-pSmad1/5/8 and anti-MyoD antibodies, or with anti-MyHC and anti-collagen type 1 antibodies. Immunostained myofibres, plated cells or cryosections were viewed on a Zeiss Axiophot 200M using Plan-Neofluar lenses, or on a Nikon C1si confocal using Plan-Fluor lenses. Digital images were acquired with a Zeiss AxioCam HRm Charge-Coupled Device using AxioVision software version 4.4. Images were optimised globally and assembled into figures using Adobe Photoshop. The Click-iT Tunel Alexa fluor imaging assay (Invitrogen) was used as per the manufacturer's instructions on satellite cells cultured for 24 hours in proliferation medium containing either 1µM or 5µM Dorsomorphin (n= 3 mice).

CKM luciferase reporter Assay

Plated satellite cells were co-transfected with the CKM firefly Luciferase reporter plasmid (CKM-LUCpCS2) and either pcDNA3-Cre-expressing control vector, pcDNA3-mId1-expressing vector [START_REF] Benezra | The protein Id: a negative regulator of helix-loop-helix DNA binding proteins[END_REF] or pcDEF3-constitutively active BMPR-1A-expressing vector (7), using Lipofectamine LTX according to the manufacturer's instructions.

The Renilla luciferase plasmid pRL (Promega), driven by a minimal tk promoter, was included as an internal control. Dorsomorphin was added to a final concentration of 5 μM. 48 hours post-transfection, the dual-Luciferase reporter Assay (Promega) was performed according to the manufacturer's instructions and measured on an Anthos Lucy3 (JENCONS-PLS).

siRNA-mediated gene knockdown

The transfection of siRNA (Invitrogen) into primary satellite cell-derived myoblasts was performed using Lipofectamine 2000 reagent (Invitrogen) or Lipofectamine RNAiMAX (Invitrogen) as previously described [START_REF] Ono | Presenilin-1 acts via Id1 to regulate the function of muscle satellite cells in a gamma-secretase-independent manner[END_REF].

Transfection of siRNA into single myofibers was carried out 12-16 hours after isolation with Lipofectamine RNAiMAX (Invitrogen) [START_REF] Ono | Presenilin-1 acts via Id1 to regulate the function of muscle satellite cells in a gamma-secretase-independent manner[END_REF]. All samples were examined 1-3 days after the transfection. The following siRNA sequences were used: BMPR-1A siRNA: 5'-GAAGUGCUGGAUGAAAGCCUGAAUA-3', Smad4 siRNA: 5'-GGAGAGACGUUUAAGGUCCCUUCAA-3', Smad5 siRNA: 5'-CCUGGGAUUGUUGUCAAAUGUUAAU-3' and Noggin siRNA: 5'-CGACCCGGGCUUUAUGGCUACUUCG-3'. Control siRNAs were supplied by Invitrogen.

Muscle regeneration studies

Animal husbandry, breeding and experimental procedures were passed by the Ethical Review Process Committee of King's College London, and were carried out in accordance with British law under the provisions of the Animals (Scientific Procedures) Act 1986. To assay pSmad1/5/8 expression, the gastrocnemius muscle of 10week old male C57BL10 mice was injected with 50 μl of 10 μM cardiotoxin (Sigma) using a 29 G 1/2 insulin syringe. Muscles were removed 3 days later and immediately frozen in isopentane cooled in liquid nitrogen, and stored at -80°C before being cryosectioned and immunostained for pSmad1/5/8 and MyoD. To assay regenerative capacity in the presence of perturbed BMP signalling, the belly of both the left and right gastrocnemius muscles of 7-9 week old male C57BL10 mice were injected with 25 μl of 10 μM cardiotoxin (Sigma). 1 and 3 days after cardiotoxin injection, 50 μl of 10 μM Dorsomorphin/DMSO dissolved in saline or 50 μl of 100 μg/ml sBMPR-1Af dissolved in saline, were injected intramuscularly into the right gastrocnemius muscle at the same site as the cardiotoxin. Control solutions (same concentration of DMSO dissolved in saline for Dorsomorphin or BSA dissolved in saline for sBMPR-1Af) were injected into the left muscle as a control. Muscles were removed 8 days after cardiotoxin injection and immediately frozen in isopentane cooled in liquid nitrogen and stored at -80 o C.

Transverse sections of muscle were cut with a cryostat and co-immunostained for MyHC and collagen type 1.

Measurement of centrally nucleated regenerating myofibres and the area of endomysial fibrosis was performed using Image J software (developed by NIH and available for free download at http://rsb.info.nih.gov/ij).

Statistic analysis

Data are presented as mean±standard deviation (SD) or mean±standard error (SEM). Immunostained satellite cells from at least ten myofibres were scored from each of at least three mice for each experiment. Significant differences between test conditions and controls were determined using Student t-test, where p values of < 0.05 were considered to be statistically significant.

Titles and legends to figures

Fig. 1 .

 1 Fig. 1. BMPR-1A, pSmad 1/5/8, Smad4 and Smad 5 are up-regulated during satellite cell activation Isolated EDL myofibres with their associated satellite cells were either immediately fixed (T0) or cultured in plating medium for either 48 hours (T48) or 72 hours (T72) before fixation and immunostaining. (A) BMPR-1A was undetectable on the majority of Pax7 + quiescent cells at T0, but robustly expressed in both Pax7 + and Pax7 -cells at both T48 and T72. (B) Quiescent Pax7 + satellite cells (T0) did not contain pSmad1/5/8, whereas pSmad1/5/8 became readily detectable in the nuclei of proliferating Pax7 + cells at T48, and both Pax7 + and Pax7 -cells at T72. (C) In accordance with the dynamics of pSmad1/5/8 expression, Smad5 levels were very low/absent in quiescent satellite cells at T0, but up-regulated in both proliferating Pax7 + and MyoD + cells at T48, as was Smad4. Arrows indicate the same satellite cell at each time point to aid comparison. Representative images from at least three independent experiments are shown. Scale bar equals 30 μm.

Fig. 2 .

 2 Fig. 2. Noggin is highly expressed in satellite cell progeny committing to myogenic differentiation Isolated EDL myofibres with their associated satellite cells were either immediately fixed (T0) or cultured in plating medium for 72 hours (T72) before fixation and immunostaining. (A) Noggin was not expressed in quiescent Pax7 + satellite cells (arrows) at T0. (B) After culture, Noggin was highly expressed in both Pax7 - and Myogenin + cells committed to myogenic differentiation (arrowheads) at T72, but at much lower levels in Pax7+ cells (arrows). (C and D) Plated satellite cell-derived myoblasts were cultured in differentiation medium for 2 days and immunostained for Noggin and Pax7. (C) Pax7 -differentiating cells (arrowheads) and (D) myotubes both expressed high levels of Noggin protein, while Noggin levels in Pax7 + cells were low (arrows). Representative images from at least three independent experiments are shown. Scale bar equals 30 μm.

Fig. 3 .

 3 Fig. 3. Inhibition of BMP signalling promotes myogenic differentiation in satellite cellsIsolated myofibres associated with satellite cells, or plated satellite cell-derived myoblasts, were cultured in the presence of recombinant BMP4 protein (100 ng/ml), recombinant Noggin protein (50 ng/ml) or sBMPR-1Af (200 ng/ml). (A) EDL myofibres were cultured in plating medium supplemented with BMP4, Noggin, or

Fig. 4 .

 4 Fig. 4. BMP signalling operates through BMPR-1AWe used siRNA-mediated protein knockdown to explore if BMP signalling operates through BMPR-1A. (A and B) Control (Cont si) or BMPR-1A (BMPR-1Asi) siRNA were transfected into satellite cells retained in their niche on isolated EDL myofibers and cultured for ~2.5 days before fixation and co-immunostaining for Pax7 and MyoD. BMPR-1A knockdown reduced the mean total number of cells and especially the number of Pax7 + MyoD + cells. (C-G) Control siRNA or BMPR-1A siRNA duplexes were also transfected into plated satellite cell-derived myoblasts. (C) Immunoblot analysis clearly demonstrated efficient knockdown of

Fig. 5 .

 5 Fig.5. Inhibiting intracellular BMP signal mediation promotes myogenic differentiation. To inhibit BMP signal mediators, we used either Dorsomorphin, an inhibitor of Smad1/5/8 phosphorylation, or siRNA to knockdown the R-Smad -Smad5 or Co-Smad -Smad4, levels. (A) EDL myofibres and their associated satellite cells were cultured with or without 1 μM Dorsomorphin and the plating medium changed daily. Dorsomorphin markedly decreased the mean total number of cells per fibre, and particularly the number of cells with the Pax7 + MyoD + phenotype. (B) Satellite cell-derived myoblasts were plated at high cell density (80-90% confluency) and cultured in proliferation medium for 2 days with or without Dorsomorphin. Immunostaining for MyHC showed that Dorsomorphin promoted myogenic differentiation and fusion in a dose-dependent manner. (C and D) Efficient siRNA-mediated knockdown of Smad5 or Smad4 was confirmed by immunostaining of plated satellite cells 24 hours after transfection. (E-G) Coimmunostaining for MyHC and Ki67 after Smad5 or Smad4 silencing for 3 days in proliferation medium, was sufficient to promote cell cycle exit and induce precocious differentiation, characterised by a decreased proportion of cells with Ki67 + expression and an increase in MyHC + cells and myotubes. Data is mean±SEM (A), or mean±SD (F and G) from at least three independent experiments. An asterisk indicates that data are significantly different from control conditions (p<0.05). Scale bar equals 50 μm.

Fig. 6 .

 6 Fig. 6. BMP signalling regulates the MyoD-controlled CKM promoter Id1 is a target gene of BMP signalling, and a negative regulator of MyoD activity. (A) Exposure to 3 μM Dorsomorphin for 6 hours to inhibit BMP signalling resulted in a down-regulation of Id1 expression in satellite cells associated with a myofibre. (B) Similarly, immunostaining of plated satellite cells after exposure to 3 μM Dorsomorphin for 18 hours in proliferation medium also resulted in down-regulation of Id1 protein. Scale bar equals 20 μm. (C) In order to investigate the ability of MyoD to activate its transcriptional targets, plated satellite cell-derived myoblasts were transfected with a CKM (muscle creatine kinase) Luciferase reporter plasmid (CKM-LUCpCS2), whose activity is regulated by MyoD, together with a Renilla luciferase plasmid pRL driven by a minimal tk promoter, as an internal control. Plated satellites cells were co-transfected with either Cre-expressing control plasmid or Id1-expressing plasmid or constitutively active BMPR-1A (caBMPR-1A)-expressing plasmid, together with CKM-LUCpCS2. After transfection, the cells were cultured in proliferation medium with or without 5 µM Dorsomorphin for 24 hours before reporter assay. Id1 or caBMPR-1A reduced CKM-LUCpCS2 activity. By contrast, inhibition of BMP signalling with Dorsomorphin had the opposite effect, and increased CKM-LUCpCS2 activity. Both Id1 and caBMPR-1Awere able to reverse this increase in CKM-LUCpCS2 activity produced by exposure to Dorsomophin. Data is represented as mean±SD from at least three independent experiments. An asterisk denotes that data are significantly different from control conditions, while a cross indicates that data is different from the level of CKM-LUCpCS2 activity in the presence of Dorsomorphin (p<0.05), using Student's T-test.

Fig. 7 .

 7 Fig. 7. Reducing Noggin levels enhances satellite cell proliferation but inhibits differentiationTo further understand the role of Noggin in myogenic progression, we used siRNA-mediated knockdown of Noggin protein. (A) Efficient Noggin knockdown in plated satellite cell-derived myoblasts was confirmed by immunostaining for Noggin 3 days after siRNA (Noggin si) transfection. (B-E) The effects of Noggin knockdown were analysed by co-immunostaining for MyHC and Ki67, which revealed an increase in myoblast proliferation, but a reduction in differentiation and fusion into large myotubes. (F) Western blotting also showed that Noggin siRNA reduced myogenin levels in plated satellite cells. (G and H) The effect of Dorsomorphin (1 µM) treatment to inhibit BMP signalling on Noggin siRNA transfected cells was analysed by immunostaining for MyHC. Dorsomorphin reversed the fusion defect induced by Noggin knocked-down.

Fig. 8 .

 8 Fig. 8. BMP signalling operates during normal muscle regeneration in vivo. Regeneration was induced in mouse gastrocnemius muscles by injection of cardiotoxin. (A) Satellite cellderived myoblasts, identified by MyoD expression, also expressed pSmad1/5/8 (arrows) in regenerating muscle on day 3 of regeneration. (B-G) To examine the effects of blockade of BMP signalling, 50 μl of 10

Fig. 9 .

 9 Fig. 9. Role of BMP signalling during myogenic progression in satellite cellsDuring muscle repair and regeneration, quiescent Pax7 + satellite cells are activated to co-express Pax7 and MyoD. Satellite cell-derived myoblasts proliferate extensively before many then down-regulate Pax7 and differentiate to either repair damaged muscle fibres, or fuse together to generate new myofibres. Other satellite cells maintain Pax7 expression but lose MyoD, and self-renew to maintain a viable stem cell

Acknowledgements

We would like to thank Dr. Stephen J. Tapscott for providing the CKM-LUCpCS2 reporter plasmid; Dr. Robert Benezra for generously sharing pcDNA3-mId1 construct; Dr. Naohiro Hashimoto for helpful discussions and Paul Knopp for much help. We gratefully acknowledge colleagues who shared antibodies, including through the Developmental Studies Hybridoma Bank developed under the auspices of the NICHD and maintained by the University of Iowa. Y.O. was funded by the Muscular Dystrophy Campaign (RA3/737). F.C. is sponsored by the Association of International Cancer Research (07-0151). J.E.M. is backed by a Wellcome Trust university award. The laboratory of P.S.Z. is also supported by The Wellcome Trust, The Medical Research Council, the MYORES Network of Excellence (contract 511978) from the European Commission 6th Framework Programme and the collaborative Project OPTISTEM (Grant Agreement number: 223098) from the European Commission 7th Framework Programme.

Conflict of interest

Authors declare no conflict of interest. compartment. Normal BMP signalling via BMPR-1A and Smad1/5/8 phosphorylation is required during muscle repair/regeneration to allow the satellite cell-derived myoblast population to expand, by preventing precocious differentiation. BMP signalling operates through controlling Id levels, a negative regulator of MyoD and myogenic differentiation. As satellite cell progeny differentiate, Noggin is up-regulated to antagonise the BMP signal to facilitate the differentiation programme.


In Mitteilungen des SFB 586 "Differenz und Integration", N° 6, Halle, 2004, 57-69 1 Neither Segmentary, nor Centralized: the Sociopolitical Organisation of a Nomadic Society (Tuaregs) beyond Categories Hélène Claudot-Hawad

The patterns of the Tuareg sociopolitical organisation combine several principles which are generally deemed incompatible in current anthropological theories. How can one capture realities which defeat classical analytical categories? The aim of this paper will be to describe and analyse the workings of the Tuareg political system in the early XXth century, its various manifestations and transformations during the anticolonial war.

The "Segmentary theory" -in the different ways it developed from its premises by [START_REF] Durkheim | De la division du travail social[END_REF] who spoke of "agrégats semblables entre eux, semblables aux anneaux de l'annelé", to Fortes and Evans-Pritchard (1940) who systematised it, to authors like [START_REF] Gellner | Saints of the Atlas[END_REF] and [START_REF] Hart | Dadda' Atta and His Forty Grandsons: The Socio-political Organization of the Ait' Atta od Southern Morocco[END_REF] who focused on North -Africa, contains the general idea of the equality or the equilibrium between the different segments that form the group or the society as a whole. This type of social organisation, i. e. the segmentary lineage system, has often been opposed to hierarchical and centralized structures. This opposition has also been expressed as a gap between a kinship age and a political age of the human societies. This evolutionist vision is highly present in the early French literature concerning Africa and the Sahara, and continues to be influential today.

This explains why the question of the political structure of the Tuaregs was never really addressed during the colonial period. From the start, the Berber-speaking nomadic or seminomadic groups of Tuaregs were said to be anything but the embodiment of "civilisation".

And this all the more so as the Tuaregs had been particularly hostile to the first French incursions into their country and had dared, in 1881, to slaughter the mission led by Flatters.

As a result, all the colonial writings present the Tuaregs as a mixture of aggressive and 1 Edited by Bernhard Streck (Hg.): Segmentation und Komplementarität. Organisatorische, ökonomische und kulturelle Aspekte der Interaktion von Nomaden und Sesshaften. Beiträge der Kolloquia am 25.10.2002und 27.06.2003. Orientwissenschaftliche Hefte 14) pugnacious tribes, always waging wars with one another, ever ready to vent their instinctive tendencies, that is to steal, despoil the weaker ones of their belongings, in short to plunder.

The theories of Berberic anarchy and the endemic uncontrolled character of the nomadic peoples merged and were both applied to the Tuareg case by observers of the first part of the XXth century 2 . Within this perspective, Captain Benhazera for instance explains in his book about the Kel Ahaggar published in 1908 ( 122) that "the writing of the history of the Tuaregs would amount to constantly telling stories of raids", raids against their own tribes and against their neighbours: "Ecrire l'histoire des Touaregs se ramènerait à raconter une série de razzias continuelles entre eux et leurs voisins (les gens de l'Aïr, les Berabich, les Aoullimminden, d'une part, les Chamba de l'autre." And the author concludes : "It is impossible to make any sense out of this" ("On ne s'y reconnaît plus").

On the other hand, the social hierarchy of the Tuaregs -which made a clear distinction between the nobility, the men of religion, the tributaries (often called by the French observers "vassals"), the artisans and the slaves -was often equated with European feudal models. Thus, the Tuaregs were also rejected as archaic men rooted in the past.

In the 1960's, during the "Independence" years, the Tuaregs were divided into five different nation-states: Libya, Algeria, Niger, Mali, and the then Upper-Volta. But if we accept the rationale of the Colonial vision, that is the absence of any social or political unity among the Tuaregs, then this partition would not hold to be true and it would only add administrative divisions to disorganized groups that were already isolated.

Nowadays in the press or in various scientific journals comments regarding the Tuareg situation in the 1990's -dealing with the movements of rebellion and the emergence of armed fronts in Mali and Niger -are not very different from the colonial representation of the Tuareg world. They allege that the revolt sprang from their isolation and their basic incapacity to really adapt to the modern world, the workings of which they cannot understand. These acts would be their last attempt to perpetuate the raids of yore, their last feudal reflex. Also, the repression waged against the Tuaregs is explained as "a backlash against their former enslavers" 3 . Finally, it is also imagined that the Tuaregs have only recently discovered the concepts of "nation", "revolution" or "territory" borrowed from modern states by westerneducated Tuaregs who are thought to have been at the root of the present agitation 4 . 2 See on this point Claudot -Hawad 1990. 3 For a detailed analysis about this literature, see Claudot-Hawad / Hawad 1996, 13-36; Claudot-Hawad 2000a. 4 See for example [START_REF] Bourgeot | Les vertus de la démocratie[END_REF] In such a context, it is indeed difficult to formulate the original question concerning the political organisation of the Tuaregs. Yet, this is a constantly recurring theme of the ethnohistorical study of the Tuaregs during the colonial period. On the one hand, colonial documents saw no coherence in the facts which they reported about the Tuaregs: their approach to the war led by Kawsen (Kaosen, Kaocen) against the French invasion in 1915-18 is one obvious example. On the other hand, the Tuareg versions of the same event stressed a very strong internal coherence based on their political organisation. Therefore, it is not surprising for any scholarly attention to infer that this political order was not built at the level of encampments or tribes as the French colonial postulate of Berberic anarchy wants it to be, but that it existed at the level of much larger political formations binding different confederations (tegéhé, tighmawin) and different tuareg political poles (each including also various "ethnic" and linguistic groups in their political definition).

To refer to the whole as well as to its various constitutive parts, the word temust is used.

Temust n imajaghen means "The society or the nation of the Tuaregs", a translation that is always a matter of passionate semantic polemic within the academic context which shares with the colonial perspective the idea that the Tuaregs never thought of themselves as an organised political entity and therefore cannot be considered as a community, a people or a nation.

The analysis of Tuareg political conceptions and their institutional aspects offers another insight into these realities. At the beginning of the twentieth century, the Tuareg world appears structured into five great confederations: in the North-East the Kel Ajjer; in the North-West,the Kel Ahaggar, in the South-East the Kel Aïr, in the South-West the Kel Tademmekkat; and between these powerful political poles there was a new formation: the Tagaraygarayt meaning "the middle" or the "go-between", to be taken in a political rather than a geographical sense.

The Tuaregs use several registers to express their political relationships.

Kinship

The first register favours kinship concepts to determine the connection between the various units within the same genealogy. When the context is matrilinear, as it is for the Ahaggar, when two groups are said to be equal, they are presented as being "the children of two sisters".

On the other hand, "the children of a sister" will be set in opposition to "the children of a brother" to mean that the former are the rightful owners of power, titles, goods, etc. while the latter are excluded from these rights and possessions. When there is a change in leadership, the genealogy is remodelled accordingly. For example, in the Kel Ahaggar confederation, the female ancestor called Tin-Hinan that is "the one (woman) of the tents", is sometimes presented as the mother of the three noble tribes between which chieftancy passed round until the seventeenth century: Kel Ghela, Taytoq and Tégéhé Mellet. Other times she is presented as the mother only of the Kel Ghela, of those who had toppled the other tribes.

If at one point the right to command is confiscated by a group which keeps out the other aspiring parties, the latter will be considered either as the descent of a younger sister, or excluded from the genealogy.

Concurrently, the relationships between suzerains and tributaries can be interpreted as the relationship between older and younger branches. Finally, the relationships of negative exchange -that is honor's pillage according to very strict rules (Claudot-Hawad 1987) -that the Kel Ahaggar practiced for instance in the second part of the XIXth century with regard to the wealthy Iwellemmeden of the south-west -can also be translated into the language of kinship through the relationship between a nephew (ag elet ma, "son of the daughter of the mother") and his maternal uncle (agna, litteraly ag ma: "son of the mother"). The matrilinear rule -which established the transmission of some undivided goods, of rights and of power between uncle and nephew -allows also this nephew to "pillage" whatever he needs from the belongings of his uncle. Thus the relationships of pillages made according to the rules of honour (as opposed to other types of aggressive actions) that exist between these various confederations are incorporated into the social norm.

Among the Tuaregs, there are also some very interesting metaphorical ways of expressing political and social relationships.

The body

First, the society can be viewed metaphorically as a body, each part representing in its turn some kind of body in miniature, built according to the same structure as the bigger one.

The smallest social unit: aghiwen, "encampment", is placed within tawsit, a term meaning both "tribe" and "wrist"; this unit itself finds its place within a confederation of tribes called ettebel in the Ahaggar: ettebel refers to the commanding drum, while in the Aïr, it is called taghma and means also "thigh". At last, this unit is included in a tégéhé, a federation of confederations, a term meaning "hips". The gathering of "hips" makes up temust n imajaghen, that is the Tuareg society at large, and this notion is associated with the anatomic image of the "chest", which is the seat of identity (called temusa that has the same linguistic root as temust).

The tent

If the various levels of social and political articulations are expressed through body metaphors, the whole society is seen also as a tent. Each stake which makes up the frame of the tent is also perceived within a narrower perspective as a complete unity, built upon the same pattern as the bigger unit. In addition, the word ehan, which refers to the "home" (here a tent made of leather or mats), is used to define social units from the smaller unit -the nuclear family -to the larger one -the society at large and even the whole universe.

The tent represents a shelter and this image connotes other notions as well. For this shelter to exist, it needs a foundation, that is a space upon which it can be built. If this shelter is to last long, its foundations must constantly be strengthened. In other words, this means that the attributes that are necessary to establish a human group, whatever its size, are at the same time the territory and the resources used to "feed" it.

Thus, the sociological metaphor of the tent, in a very graphic and effective way, establishes a vital link between organized social units and access to the land and goods that are necessary for the survival of the group and all its constitutive parts.

In legal terms, these vital goods (called akh iddaren, the "living milk") can be analysed as a matrimony that can neither be divided up nor alienated. Access to them is given to the members of the tent and is always temporary. These rights concern material possessions (herds, land, slaves, etc.) or symbolic ones (chieftaincy, suzerainty, status) that imply various dues.

Political models

But how is this shelter, representing the greater society tent, to be built? There are two different types of architecture. The first one is called titek, a term which refers to the screens made of matting and leather that help separate the tent into different parts. In the figurative sense, titek expresses the movement that pushes away and the gesture that rejects. Thus, this evocative image defines the hierarchy in which each social category is included according to its functions, and in which only a part of society has the political power. In this context, the same actions have different meanings depending upon the status of the protagonists. If the warrior's code of honour, for example, is very strict for the noblemen, it is not the same for the other members of society who, up to a point, are not considered as being completely "responsible" for their behaviour.

The architectonic metaphor is further extended: it equates the suzerains and the central stake (tamenkayt) which supports the leather roof of the tent, surrounded by the side stakes which represent the dependants. The distribution of the tribes within the political territory of the federations reproduces the frame of the nomadic leather tent that is very widely used in the Ajjer, the Ahaggar and the Tadmekkat. This frame conveys the prominent roles of the leaders: not only their protective function but also its counterpart, that is their domination over the "weaker ones" who must pay them their tribute. Thus, all the goods that are necessary to the community are controlled and even distributed by the chief, amenukal, who, for example, manages the organisation of the territory, the distribution of the grazing areas and the sharing of the vassals' dues. Within this context, the role of the chief -and at least of a fraction of the lineage he represents -is associated with power and the advantages that go with it.

This system based on hierarchy is contrasted with another model defined in philosophical discourse as igagan meaning "vertebra", "vaults" and "arches". This refers to another architectural concept which eliminates all separations inside the tent as well as the central stake. Igagan are the wooden arches supported by the lateral stakes that make up the frame on which the roof of the shelter rests. This model corresponds to the tent made of mats which is used in the Aïr.

The Aïr went even further in this egalitarian conception of society by which all the stakes of the tent have similar functions. This model is made concrete in the political system of the ighollan. The various tribes that compose this organisation are placed on equal footing and they have given up tribute and status, even if in some other fields, such as history or culture, they sometimes "recall" the noble or vassal origin of their members.

One can easily understand that this system leaves no room for a chief who would be inclined to hold the power or supremacy over the others.

Yet, one must note that it rests on two complementary institutions: the assembly and the chiefcum-arbiter, the latter incarnated by the man whom the French called the Agadez "Sultan" 5 .4

The Kel Ahaggar

The Kel Ahaggar correspond to the first model of political organisation.

The occupation of their territory is thus conveyed by the central place the leaders occupy while 5 For further discussion of the invention by the French colonial administration of such a political function among the Tuaregs, see Claudot-Hawad (1996) 2000a, chap. II. their clients and dependants are placed all around it like the lateral stakes of the tent.

At the beginning of the XXth century, the Kel Ahaggar constituted a tégéhé, that is, according to the body metaphor, the "hips" where the three confederations of tribes called ettebel, commanding drum, met: they were the Kel Ghela, Taytoq and Tegéhé Mellet. Each confederation was composed of different tribes ("wrists") with non-egalitarian status: the suzerains (ihaggaren) and the tributaries (imghad).

Power was transmitted matrilinearly. Those who were entitled to rule were called kel ettebel, "those of the drum". The chief of the whole confederation of the Kel Ahaggar was chosen solely from the Kel Ghela tribe and took the title of amenukal in contrast with the other chiefs of equivalent units (Tegéhé Mellet and Taytoq), called amghar, meaning "old man, wiseman" which applies to all men after a certain age.

The chief was elected by all the respresentatives of the tribes, including the dependants. In fact, the chief's power was limited and mitigated by a consensus from the subjects. In the absence of coercive power which was the case, his arbitrage was only accepted in a fair trial.

All in all, chieftancy among the Tuaregs assigned as many duties as there were rights. The duty of "protection" was bound up both with economy and war. The collective assets belonging to each political unit had therefore to be distributed among those in need in case of famine or epizootic disease.

The hierarchy

A reading that singles out the integrative pattern of kinship shows every unit, emerging both at the tribe level and the tribes confederation level, as an ensemble of relatives organised matrilinearly. As a result, the Kel Ghela, Taytoq and Tégehé Mellet are sometimes defined as the respective descendants of three sisters. This equality in theory, however, expressed through the parental-lineage vision of the society, is counterbalanced by the political categories which on the contrary emphazise hierarchy by differentiating the nobles (ihaggaren), the tributaries (imghad), the religious (ineslimen, i. e.

the "muslims"), the artisans (inaden) and the slaves (iklan).

Similarly, a deeper investigation into the genealogical Tuareg data shows that the "egalitarian" representation of clans -each of which is viewed as a group of uterine relatives with a common ancestor and who, by definition, should attain power and rights transmitted matrilinearly -is some kind of mystification. Indeed, the unilinear form of rules of social order, i. e. filiation, inheritance, the transmission of power, are juxtaposed by another principle, that of endogamy allowing for marriage between the members of the same lineage. In other words, a man who holds the power through his mother, i. e. who is a "son of the drum" (ag ettebel), can marry the daughter of his mother's sister -a classificatory "sister" -who will hand down chieftancy to her descendants. Such a marriage will result in mixing the categories of "son" and "nephew" in the next generation and will have apparently transmitted power patrilinearly, whereas it is because power is passed down by his mother that the son of a chief will succeed his father.

Thus, endogamy combined with matrilinear determination of political rights offers the ideal way to model and reduce the group of potential successors and opens the way to hierarchy.

Among the Kel Ghela, the matrilinear nucleus in power for five generations corresponds to thirty four per cent of the tribe6 . This is a privileged endogamous group in which the marriage with a classificatory sister will be sought to keep the power within the lineage: those who rule set store by this type of marriage, they are also however ready to establish new marriage bonds outside their confederation in order to widen their political zones of influence.

A society in movement

What is most striking among the Tuaregs, in their political theory as well as their cosmogony, is their dynamic vision of the order of the world, the universe or society. Each state is seen as only one step leading to another step that will, exactly like in the nomadic cycle, eventually leads back to the initial steps7 .

According to this theory, slaves are bound one day to become free men, tributaries to become suzerains, and suzerains to go back to where they had started. Social movement and changes are always underlying the social order that will always be temporary. Some social rules illustrate this mobility in practice: a slave can change masters, a dependent tribe can choose another suzerain, thereby depriving worthless masters of their power. Also, when a slave has acquired the Tuareg culture, he must be given his freedom.

So, two contradictory orientations influence the itinerary of each tent. The first one is to "put on weight" through the development of its various branches (through its descendents, the extension of its allies and clients). The second orientation is the tendency to become completely independent from the mother tent: that was for instance the case of the Kel Ahaggar when they became independent from the Kel Ajjer in the XVIIIth century.

The evolution of political power

In conclusion, let us stress that what the Tuaregs privileged politically was a confederal model.

In this schema, the power of a chief was never coercive; it was based on the community's wishes as expressed through consensus so that in fact the role of the chief was no more than that of an arbiter. It was the arbiter-chief's responsibility to maintain the balance between the various equal units as well as the partnership between the different social categories. In the early twentieth century, the political decisions of the five great confederations and the relations between them could be coordinated by the correspondent assembly summoned each time it was deemed necessary (for wars, conflicts, peace agreements, etc.).

With the growing encroachment of the colonial forces within the Tuareg country, the attitude to be adopted on the political scene regarding the French occupation became of paramount importance. It was around this new problem that competition for power and rivalry between political candidates emerged.

In the Ahaggar, after the Tuaregs' total defeat in Tit (1902), the party for peace with the occupying power, led by Musa ag Amastan (Kel Ghela) gained the upperhand while the party for resistance joined the Tuareg forces who fought on the side of the Ajjer or in the Fezzan, and then in 1916 with Kawsen in the Aïr.

A great number of tributary groups (imghad), such as the Dag Ghali, joined the resistance: this, once more, underlines their relative independence from the nominal chief of the confederation.

Because of the intervention of the colonial forces and their prevailing logic, the very nature of Tuareg power was transformed. Legitimacy was no longer a matter of being elected or being chosen by one's subjects but rather a question of subservience to the colonial power. Many small local chiefs, with the help of the colonial military forces, became arbitrary and despotic rulers (Claudot -Hawad 1990-Hawad , 1993b-Hawad , 1993c)).

Henceforth, the mobility that characterized social hierarchy came to a stop. The links established between the various confederations were interrupted as well as the elaborate networks of political, social, economic and cultural relationships woven with the neighbouring societies. The political and economic space of the Tuaregs shrank, was shattered and became petrified.

To conclude, the words pronounced in tamajaght language at the end of 1989 by Ghayshena

For a detailed analysis, see[START_REF] Claudot-Hawad | Parenté touarègue et informatique[END_REF] id. 1987.

About this philosophy and its relations with the political and social institutional order, seeClaudot-Hawad 1993d; id. 2000a.


Afaf Bouharguane
email: bouharg@math.univ-montp2.fr

On the instability of a nonlocal conservation law

Keywords: Fractional anti-diffusive equation, instability, finite difference schemes. Mathematics Subject Classification: 35L65, 45K05, 35G25, 35C07, 35B35, 65M06

. We prove that constant solutions of Fowler's equation are non-linearly unstable. We also illustrate this fact using a finite difference scheme.

Introduction

Partial Differential Equations with nonlocal or fractional operators are widely used to model scientific problems in mechanics, physics, signal processing and other subjects, see for example [START_REF] Azerad | Simultaneous denoising and enhancement of signals by a fractal conservation law[END_REF] and references therein. We consider in this paper a nonlocal conservation law which appears in the formation and dynamics of sand structures such as dunes and ripples [START_REF] Fowler | Dunes and drumlins[END_REF][START_REF] Lagree | Stability of an erodible bed in various shear flows[END_REF]. Namely, Fowler ([7, 8, 9]) introduced the following equation

∂ t u(t, x) + ∂ x u 2 2 (t, x) + I[u(t, •)](x) -∂ 2 xx u(t, x) = 0 t ∈ (0, T ), x ∈ R, u(0, x) = u 0 (x) x ∈ R, (1)
where u = u(t, x) represents the dune height, u 0 ∈ L 2 (R) is an initial condition, T is any given positive time and I is a nonlocal operator defined as follows: for any Schwartz function ϕ ∈ S(R) and any x ∈ R,

I[ϕ](x) := +∞ 0 |ζ| -1 3 ϕ ′′ (x -ζ)dζ. (2)
Equation ( 1) is valid for a river flow over an erodible bottom u(t, x) with slow variation. The nonlocal term appears after a subtle modeling of the basal shear stress. This operator appears also in the work of Kouakou & Lagrée [START_REF] Kouakou | Evolution of a model dune in a shear flow[END_REF][START_REF] Lagree | Stability of an erodible bed in various shear flows[END_REF].

The nonlocal term I can be seen as a fractional power of order 2/3 of the Laplacian with the bad sign. Indeed, it has been proved [1]

F I[ϕ] -ϕ ′′ (ξ) = ψ I (ξ)Fϕ(ξ) (3)
where

ψ I (ξ) = 4π 2 ξ 2 -a I |ξ| 4 3 + i b I ξ|ξ| 1 3 , (4)
with a I , b I positive constants and F denotes the Fourier transform. One simple way to establish this fact is the derivation of a new formula for the operator I [1]

I[ϕ](x) = C I 0 -∞ ϕ(x + z) -ϕ(x) -ϕ ′ (x)z |z| 7/3 dz, (5)
with C I = 4 9 . The operator I[u] is a weighted mean of second derivatives of u with the bad sign and has therefore an anti-diffusive effect and creates instabilities which are controlled by the diffusive operator -∂ 2 xx . This remarkable feature enabled to apply this model for signal processing. Indeed, the diffusion is used to reduce the noise whereas the nonlocal anti-diffusion is used to enhance the contrast [START_REF] Azerad | Simultaneous denoising and enhancement of signals by a fractal conservation law[END_REF].

Remark 1. For causal functions (i.e. ϕ(x) = 0 for x < 0), this operator is up to multiplicative constant, the Riemann-Liouville fractional derivative operator which is defined as follows [START_REF] Podlubny | An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF] 1 Γ(2/3)

+∞ 0 ϕ ′′ (x -ξ) |ξ| 1/3 dξ = d -2/3 dx -2/3 ϕ ′′ (x) = d 4/3 dx 4/3 ϕ(x), (6)
where Γ denotes the Euler function.

Recently, some results regarding this equation have been obtained, namely, existence of travellingwaves u φ (t, x) = φ(xct) where φ ∈ C 1 b (R) and c ∈ R represents wave velocity, the global wellposedness for L 2 -initial data, the failure of the maximum principle, the local-in-time well-posedness in a subspace of C 1 b [START_REF] Alibaud | A non-monotone nonlocal conservation law for dune morphodynamics[END_REF][START_REF] Alvarez-Samaniego | Existence of travelling-wave and local well-posedness of the Fowler equation[END_REF] and the global well-posedness in a L 2 -neighbourhood of C 1 b , solving namely for u = u φ + v, with v ∈ L 2 (R) [START_REF] Bouharguane | Global well-posedness and instability of a nonlocal conservation law[END_REF]. For this purpose, the following Cauchy problem solved by the perturbation v has been considered:

∂ t v(t, x) + ∂ x ( v 2 2 + u φ v)(t, x) + I[v(t, •)](x) -∂ 2 xx v(t, x) = 0 t ∈ (0, T ), x ∈ R, v(0, x) = v 0 (x) x ∈ R, (7)
where v 0 ∈ L 2 (R) is an initial perturbation and T is any given positive time.

Let us note also that any constant is solution of the Fowler equation. We shall prove below that these solutions are unstable. To prove that a solution u φ is unstable when φ is constant, we introduce the notion of mild solution (see Definition 1) based on Duhamel's formula [START_REF] Fowler | Evolution equations for dunes and drumlins[END_REF]. We also give some numerical results that illustrate this fact.

The remaining of this paper is organized as follows: in the next section, we define the notion of mild solution and we give some results. Section 3 contains the proof of the instability. We introduce in section 4 an explicit finite difference scheme for which we give numerical simulations to illustrate the theory of the previous section.

Duhamel formula and some results

Definition 1. Let u φ be a constant, T > 0 and

v 0 ∈ L 2 (R). We say that v ∈ L ∞ ((0, T ); L 2 (R)) is a mild solution to (7) if for any t ∈ (0, T ), v(t, x) = K(t, •) * v 0 (x) - t 0 ∂ x K(t -s, •) * v 2 2 (s, •)(x) ds (8)
where

K(t, x) = F -1 e -tφ I (•) (x) is the kernel of the operator I -∂ 2 xx + u φ ∂ x and φ I (ξ) = 4π 2 ξ 2 - a I |ξ| 4 3 + i b I ξ|ξ| 1 3 + i 2πu φ ξ.
The expression [START_REF] Fowler | Evolution equations for dunes and drumlins[END_REF] is the Duhamel formula and is obtained using the spatial Fourier transform. The use of this formula allows to prove the local-in-time existence with the help of the contracting fixed point theorem. The global existence is obtained thanks to L 2 a priori estimate. We refer to [START_REF] Alibaud | A non-monotone nonlocal conservation law for dune morphodynamics[END_REF][START_REF] Bouharguane | Global well-posedness and instability of a nonlocal conservation law[END_REF] for the proof.

Lemma 1. Let t > 0. Then, ∂ x K(t, •) ∈ L 2 (R) and satisfies ||∂ x K(t, •)|| L 2 (R) ≤ C t -3/4 + e αt , ( 9
)
where C is a positive constant independent of t.

Proof. By Plancherel formula, we have

||∂ x K(t, •)|| 2 L 2 (R) = ||F (∂ x K(t, •)) || 2 L 2 (R) = ||ξ → 2iπξe -tφ I (ξ) || 2 L 2 (R) , = 2 +∞ 0 4π 2 ξ 2 e -2t(4π 2 ξ 2 -a I ξ 4/3 ) dξ.
Let ξ 0 > 0 such that for all ξ > ξ 0 ,

4π 2 ξ 2 -a I ξ 4/3 ≥ ξ 2 .
Then, if we denote by α = -min Re(φ I ) (see Figure 2), we have

||∂ x K(t, •)|| 2 L 2 (R) ≤ 8π 2 ξ 0 0 ξ 2 e -2t(4π 2 ξ 2 -a I ξ 4/3 ) dξ + 8π 2 +∞ ξ 0 ξ 2 e -2t(4π 2 ξ 2 -a I ξ 4/3 ) dξ, ≤ 8π 2 ξ 0 0 ξ 2 e 2tα dξ + 8π 2 +∞ 0 ξ 2 e -2tξ 2 dξ, ≤ 8π 2 ξ 3 0 3 e 2tα + 4π 2 √ 2 t -3/2 +∞ 0 ξ 2 e -ξ 2 dξ, ≤ C t -3/2 + e 2αt ,
which completes the proof of this lemma.

Remark 2. Using again Plancherel formula, we have for any initial data v 0 with values in L 2 (R), the following L 2 -estimate [START_REF] Alibaud | A non-monotone nonlocal conservation law for dune morphodynamics[END_REF]:

||v(t, •)|| L 2 (R) ≤ e αt ||v 0 || L 2 (R) .

Instability

In this section, we give the proof of the instability.

Let us first note that a fundamental property of the kernel K is the non-positivity [START_REF] Alibaud | A non-monotone nonlocal conservation law for dune morphodynamics[END_REF], see Figure 1. This feature enabled to prove the failure of the maximum principle. We use again this property to show that the constant solutions of the Fowler equation are unstable. Proof. We adapt the strategy used by De Bouard in [START_REF] De | Instability of stationary bubbles[END_REF], making it more elementary in our setting. We denote by S(t) the linear semigroup associated with equation [START_REF] Fowler | Dunes and drumlins[END_REF] i.e. S(t)w = K(t, •) * w. Hence, we have

T (t)v 0 := v(t, •) = S(t)v 0 - 1 2 t 0 ∂ x K(t -s, •) * v 2 (s, •) ds, (10)
where T denotes the nonlinear semigroup associated with equation [START_REF] Fowler | Dunes and drumlins[END_REF]. First, by Young inequality, Lemma 1 and Remark 2, we have

||T (t)v 0 -S(t)v 0 || L 2 (R) ≤ 1 2 t 0 ||∂ x K(t -s, •)|| L 2 (R) ||v(s, •)|| 2 L 2 (R) ds, ≤ C 2 t 0 (t -s) -3/4 + e α(t-s) e 2αs ||v 0 || 2 L 2 (R) ds, ≤ b(t)||v 0 || 2 L 2 (R) , (11)
where b(t) = C 2 e 2αt (4t 1/4 + 1 α e αt ) > 0 for all t ∈ (0, T ). Hence, we obtain for t 0 > 0 fixed,

||T (t 0 )v 0 -S(t 0 )v 0 || ≤ b 0 ||v 0 || 2 L 2 (R) , (12)
where b 0 = b(t 0 ). Assume now that the constant u φ is a stable solution of the Fowler equation i.e,

∀ε > 0, ∃η > 0, ||v 0 || < η ⇒ ||v(t, •)|| L 2 (R) < ε,
for all positive t. Let then

0 < ε < e αt 0 -1 16b 0 ,
and let η > 0 be such that

||v 0 || L 2 (R) < η ⇒ ||v(t, •)|| ≤ ε, for all positive t.
Let N be an integer large enough such that

e αt 0 N ≥ e αt 0 -1 4ηb 0 . ( 13
)
Let us now consider the following initial data: v 0 = δw 0 , where

F(w 0 ) = 1 √ d-c 1 [c,d] ,
with 0 < c < d satisfying (see Figure 2):

• Re (φ I ) (c) < Re (φ I ) (d) = -β < 0, • for all ξ ∈ [c, d], Re (φ I ) (ξ) ≤ -β, • β = α -γ where 0 < γ < ln(2)
N t 0 . And δ is defined as

δ = e -αN t 0 e αt 0 -1 4b 0 . ( 14
)
The shape of w 0 is given by

w 0 (x) = 1 √ d -c 2 sin( d-c 2 x) x e -i d+c 2 x if x = 0, d -c otherwise.
It follows from ( 13) and ( 14) that ||v 0 || L 2 (R) = δ ≤ η, and thus for this initial data, we get that

||v(t, •)|| L 2 (R) ≤ ε for all t > 0. For n ∈ {0, • • • , N }, we have v(nt 0 ) = S(nt 0 )v 0 + n-1 k=0 S ((n -1 -k)t 0 ) [v((k + 1)t 0 ) -S(t 0 )v(kt 0 )] , = S(nt 0 )v 0 + n-1 k=0 S ((n -1 -k)t 0 ) [T (t 0 )v(kt 0 ) -S(t 0 )v(kt 0 )] .
Hence, we get using [START_REF] Podlubny | An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF] and the L 2 -estimate given in Remark 2

||v(nt 0 ) -S(nt 0 )v 0 || L 2 (R) ≤ n-1 k=0 e αt 0 (n-k-1) ||v((k + 1)t 0 ) -S(t 0 )v(kt 0 )|| L 2 (R) , ≤ n-1 k=0 e αt 0 (n-k-1) b 0 ||v(kt 0 )|| 2 L 2 (R) , ≤ n-1 k=0 e αt 0 (n-k-1) b 0 e 2αkt 0 ||v 0 || 2 L 2 (R) , = δ 2 b 0 e αt 0 (n-1) e αnt 0 -1 e αt 0 -1 , ≤ δ 4 e αt 0 n .
Moreover, by Plancherel formula, we have

||S(t)v 0 || 2 L 2 (R) = ||K(t, •) * v 0 || 2 L 2 (R) = ||F(K(t, •) * v 0 )|| 2 L 2 (R) , = R |F(K(t, •))(ξ)F(v 0 )(ξ)| 2 dξ, = d c δ 2 d -c e -2t Re(φ I )(ξ) dξ, ≥ e 2βt ||v 0 || 2 L 2 (R) .
We finally infer that

||v(N t 0 )|| L 2 (R) ≥ ||S(N t 0 )v 0 || L 2 (R) -||v(N t 0 ) -S(N t 0 )v 0 || L 2 (R) , ≥ δe βN t 0 - δ 4 e αt 0 N , ≥ e αt 0 -1 16b 0 > ε,
which gives us a contradiction and completes the proof of this theorem.

Remark 3. We can give a physical interpretation of this result: a flat profile u φ = constant is unstable under the morphodynamics described by the Fowler model.

Numerical simulations

The spatial discretization is given by a set of points x j ; j = 1, ..., N and the discretization in time is represented by a sequence of times t 0 = 0 < ... < t n < ... < T . For the sake of simplicity we will assume constant step size δx and δt in space and time, respectively. The discrete solution at a point will be represented by u n j ≈ u(t n , x j ). The schemes consist in computing approximate values u n j of solution to (1) on [nδt, (n + 1)δt[×[jδx, (j + 1)δx[ for n ∈ N and j ∈ N thanks to the following relation:

u n+1 j -u n j δt + 1 2 (u n j ) 2 -(u n j-1 ) 2 δx - u n j+1 -2u n j + u n j-1 δx 2 + I δx [u n ] j = 0, (15)
where I δx is the discretization of the nonlocal term I. We use a basic quadrature rule to approximate I given by ( 2)

I δx [ϕ] j = δx -4/3 +∞ l=1 l -1/3 (ϕ j-l+1 -2ϕ j-l + ϕ j-l-1 ) . (16)
Let us note that for compactly supported initial datum the sum is a finite sum. In this section, we want to simulate the instability stated previously. To this aim we must be careful to distinguish between the instability which stems from the model and numerical instabilities, which could be caused by a careless discretization.

The numerical stability ensures that the difference between the approximate solution and the exact solution remains bounded for increasing t for δx, δt given.

As noticed, the Fowler model amplifies slowly the low frequencies whereas the high frequencies are quickly dampened. Thus, the classical notion of A-stability (strong stability) and C-stability are not suitable nor desirable. A new definition of stability for this model has been considered in [START_REF] Azerad | Finite difference schemes for a nonlocal conservation law[END_REF] and numerical stability criteria have been obtained.

In the following numerical test, we have to take care to choose δx and δt accurately following the stability condition, see [START_REF] Azerad | Finite difference schemes for a nonlocal conservation law[END_REF]. We expose in Figure 3(a) the evolution of an initial flat bottom disturbed by a small bump for different times. As we can see the perturbation creates small ripples that grow with time without bound. This result proves that the perturbed solution goes away from the non-perturbed solution. This illustrates the instability of constant solutions for the Fowler model [START_REF] Alibaud | A non-monotone nonlocal conservation law for dune morphodynamics[END_REF]

Figure 1 :Theorem 1 .

 11 Figure 1: Evolution of the kernel K for t = 0.1 and t = 0.5 s

Figure 2 :

 2 Figure 2: Behaviour of Re (φ I )

Figure 3 (

 3 b) shows the logplot for the L 2 -norm evolution of this perturbed bottom, which confirms the validity of the estimate given in Remark 2.

Figure 3 :

 3 Figure 3: Evolution of a flat bottom disturbed by a small bump.

Acknowledgements

We thank Pascal Azerad and Clément Gallo for helpful comments. The author is supported by the ANR MATHOCEAN ANR-08-BLAN-0301-02