Tatiana Merle

Emmanuel Farge
email: efarge@curie.fr

Trans-scale mechanotransductive cascade of biochemical and biomechanical patterning in embryonic development: the light side of the force

Embryonic development is made of complex tissue shape changes and cell differentiation tissue patterning. Both types of morphogenetic processes, respectively biomechanical and biochemical in nature, were historically long considered as disconnected. Evidences of the biochemical patterning control of morphogenesis accumulated during the last 3 decades. Recently, new data revealed reversal mechanotransductive feedback demonstrating the strong coupling between embryonic biomechanical and biochemical patterning. Here we will review the findings of the emerging field of mechanotransduction in animal developmental biology and its most recent advancements. We will see how such mechanotransductive cascade of biochemical and mechanical patterning events ensures trans-scale direct cues of coregulation of the microscopic biomolecular activities with the macroscopic morphological patterning. Mechanotransduction regulates many aspects of embryonic development including efficient collective cell behaviour, distant tissues morphogenesis coordination, and the robust coordination of tissue shape morphogenesis with differentiation.

Introduction

Embryogenesis consists of the development of biomechanical morphology of tissues and biochemical cell differentiation patterns. In the developing embryo, the biomechanical and biochemical patterning were long considered as disconnected. The first was initially thought to be exclusively regulated by Newton's laws of Physics, with tissue growth as a major driving force [START_REF] Thomson | On Growth and Form[END_REF], while the second was subsequently thought to be merely induced by the biochemical cascade based on the interactions between the biomolecules produced by the genome [START_REF] Lewis | A gene complex controlling segmentation in Drosophila[END_REF][START_REF] Nusslein-Volhard | Mutations affecting segment number and polarity in Drosophila[END_REF].

The biochemical regulation of embryonic mechanical patterning by the expression of master differentiation genes was discovered in the 90's, such as mesoderm invagination controlled by the expression of the genes twist and snail in Drosophila melanogaster embryos [START_REF] Leptin | Twist and snail as positive and negative regulators during Drosophila mesoderm development[END_REF]. In addition, the antero-posterior patterning genes bicoid, nanos and torso-like are required for convergence-extension movements (C&E) at the onset of Drosophila embryo gastrulation [START_REF] Irvine | Cell intercalation during Drosophila germband extension and its regulation by pair-rule segmentation genes[END_REF]. Similarly, Xwnt-5a was suggested to be involved in C&E movements in Xenopus embryos [START_REF] Moon | Xwnt-5A: a maternal Wnt that affects morphogenetic movements after overexpression in embryos of Xenopus laevis[END_REF] and silberblick/wnt11 and wnt5 were found to be necessary for proper C&E movements in zebrafish embryos [START_REF] Heisenberg | Silberblick/Wnt11 mediates convergent extension movements during zebrafish gastrulation[END_REF].

The underlying molecular mechanisms downstream of this regulation are based on a genetic regulation of anisotropies in the intracellular concentration of a molecular motor: Myosin-II (Myo-II) [START_REF] Young | Dynamic changes in the distribution of cytoplasmic myosin during Drosophila embryogenesis[END_REF]. For instance, the Twist and Snail proteins were found to be required for the medio-apical accumulation of Myo-II in Drosophila embryos. This acto-myosin meshwork constricts mesoderm cell apexes resulting in an apical shrinkage that triggers the mesoderm epithelium invagination [START_REF] Dawes-Hoang | folded gastrulation, cell shape change and the control of myosin localization[END_REF]. These findings interestingly showed that embryo morphogenesis, that obeys the laws of Newtonian Physics, is the product of the physical conditions and of the biochemical patterns, that both introduce at a given developmental stage the internal forces that will determine the next stage.

Inversely, it has been shown in the 2000s that biochemical developmental patterns at a given stage can also be the product of the biomechanical patterns of the precedent stage through mechanotransduction processes. This introduces at given developmental stages, new biochemical signal inputs of mechanical origin. For instance, in gastrulating Drosophila embryos, mechanical strains developed by C&E induce beta-catenin (b-cat) signalling activation. This activation leads to Twist expression in the anterior endoderm, and is vitally involved in functional anterior mid-gut cell differentiation at larvae stage [START_REF] Farge | Mechanical induction of twist in the Drosophila foregut/stomodeal primordium[END_REF][START_REF] Desprat | Tissue deformation modulates twist expression to determine anterior midgut differentiation in Drosophila embryos[END_REF]. Similarly, during bone development in mouse embryos, sox9 is expressed downstream of b-cat mechanical activation in response to muscle spontaneous activity to prevent bone differentiation in joints [START_REF] Kahn | Muscle contraction is necessary to maintain joint progenitor cell fate[END_REF].

Here we will review the findings and most recent advances in the field reporting the existence of such mechanotransductive cues in animal embryonic development in vivo. We will focus on the mechanotransductive trigger of active morphogenetic biomechanical patterning, and on cell biochemical differentiation and specification patterning. We will describe the physiological functions of such coupling between the developmental biochemical patterning cascade and biomechanical shape development. Based on this trans-scale deterministic coregulation of molecular activities with macroscopic properties, we will present how long-range rapid mechanotransductive cell-cell interactions generate either collective cell behaviour or embryonic morphogenetic movement coordination. We will additionally discuss how it engenders robust coordination between biochemical and biomechanical morphogenesis during embryonic development.

The mechanical trigger of Myo-II dependent active biomechanical morphogenesis

Myo-II protein is the molecular motor driving forces in embryonic active morphogenetic movements. Its mechanosensitive behaviour was initially found in cell culture [START_REF] Effler | Mitosis-specific mechanosensing and contractile-protein redistribution control cell shape[END_REF]. In vivo, a simple micro-pipet aspiration of Drosophila embryos ectoderm increases the junctional concentration of Myo-II, suggesting a re-enforcement of junctions. Myo-II concentration also increases in ectoderm tissue in response to the C&E morphogenetic movements of gastrulation in Drosophila embryos [START_REF] Fernandez-Gonzalez | Myosin II dynamics are regulated by tension in intercalating cells[END_REF]. In addition to re-enforcing tissue resistance to deformation, tension also stabilizes Myo-II at the purse-string caused by a wound during Drosophila embryogenesis [START_REF] Kobb | Tension regulates myosin dynamics during Drosophila embryonic wound repair[END_REF]. The purse string tension increases and generates the wound closure.

The generalisation of such mechanically induced reenforcement of junctions and of tissue integrity in response to embryonic morphogenetic movements was recently observed in early zebrafish embryos. In mutants with central domain-defective a-catenin, a domain known to be mechanosensitive in cell culture, the mechanical strains due to gastrulation led to C&E and embryo elongation defects [START_REF] Han | alphaE-catenin-dependent mechanotransduction is essential for proper convergent extension in zebrafish[END_REF] (Figure 1A). In cell culture, such domains mechanically open to effectors such as Vinculin leading to the active re-enforcement of adherens junctions [START_REF] Yao | Force-dependent conformational switch of alpha-catenin controls vinculin binding[END_REF]. A similar molecular process was thus proposed to be at work to ensure normal C&E morphogenesis in response to the strains generated by the earliest embryonic morphogenetic movements [START_REF] Han | alphaE-catenin-dependent mechanotransduction is essential for proper convergent extension in zebrafish[END_REF]. A mechanosensitive maintenance of adherens junctions has also been demonstrated to preserve tissue integrity during Drosophila mesoderm invagination and during C-elegans embryo morphogenesis [START_REF] Weng | Myosin-dependent remodeling of adherens junctions protects junctions from Snail-dependent disassembly[END_REF][START_REF] Zhang | A tension-induced mechanotransduction pathway promotes epithelial morphogenesis[END_REF]. Interestingly, the Yap/Taz mechanosensitive signalling pathway [START_REF] Dupont | Role of YAP/TAZ in mechanotransduction[END_REF] has also been found to be required for the maintenance of the overall shape of the zebrafish embryo. In this case, the Rho GTPase activating protein ARHGAP18 preserves the embryonic tissue stiffness against gravity-induced deformations in a Yap/Taz dependent process [START_REF] Porazinski | YAP is essential for tissue tension to ensure vertebrate 3D body shape[END_REF].

Yet, Myo-II mechanosensitivity is not only involved in the tissue resistance to endogenous morphogenetic movements. It also triggers morphogenetic movements in response to the mechanical strains developed by the morphogenetic movements of the precedent stage. Indeed, the Fog-dependent medio-apical accumulation of Myo-II -that leads to the epithelial mesodermal cells apex constrictions and to mesoderm invagination in Drosophila embryos -has been found to be mechanotransductively induced by the Snail-dependent cell apex pulsations that just precede.

First indicated by mechanical perturbations through indenting the embryo [START_REF] Pouille | Mechanical signals trigger myosin II redistribution and mesoderm invagination in Drosophila embryos[END_REF], this was demonstrated by the quantitative rescue of sna-dependent pulsations in snail mutants. A network of pulsating micromagnets was applied to snail mutant embryos injected with magnetic liposomes [START_REF] Mitrossilis | Mechanotransductive cascade of Myo-II-dependent mesoderm and endoderm invaginations in embryo gastrulation[END_REF]. Such physiological mechanical rescue, of pure physical origin, quantitatively mimicked the Snaildependent mechanical strains, and Fog-dependently rescued both the medio-apical accumulation of Myo-II and mesoderm invagination (Figure 1B).

As an underlying molecular mechanism, the sna-dependent apex pulsations were found to mechanically inhibit Fog endocytosis and to enhance Fog exocytosis, presumably by membrane flattening and tension increase, thereby triggering the Rho downstream pathway leading to medio-apical accumulation of Myo-II [START_REF] Pouille | Mechanical signals trigger myosin II redistribution and mesoderm invagination in Drosophila embryos[END_REF]. Interestingly, the existence of a defect of endocytosis in the mesoderm versus the ectoderm was recently corroborated in early gastrulating Drosophila embryos [START_REF] Jha | Quantitative control of GPCR organization and signaling by endocytosis in epithelial morphogenesis[END_REF].

In addition to mechanotransductive cues, the shape of the embryo was found to determine the shape of the actomyosin meshwork [START_REF] Chanet | Actomyosin meshwork mechanosensing enables tissue shape to orient cell force[END_REF].

Furthermore, the mechanical strains developed by mesoderm invagination stretches the posterior pole of the embryo, in which Fog is also expressed. As a consequence, this leads to the Fog-dependent mechanotransductive activation of apical Myo-II accumulation and to the initiation of the posterior endoderm invagination [START_REF] Mitrossilis | Mechanotransductive cascade of Myo-II-dependent mesoderm and endoderm invaginations in embryo gastrulation[END_REF]. This process, in turn, mechanically participates in the morphogenetic movement of C&E by anteroposterior cell elongation [START_REF] Lye | Mechanical coupling between endoderm invagination and axis extension in Drosophila[END_REF][START_REF] Collinet | Local and tissue-scale forces drive oriented junction growth during tissue extension[END_REF]. And this C&E morphogenetic movement also positively feedbacks into apical Myo-II accumulation in the posterior endoderm and robustly participate together with mesoderm invagination in the initiation of the posterior endoderm invagination [START_REF] Mitrossilis | Mechanotransductive cascade of Myo-II-dependent mesoderm and endoderm invaginations in embryo gastrulation[END_REF]. Thus, the mechanosensitive behaviour of Myo-II is at the node of an active self-induced biomechanical cascade of morphogenetic movements. Mechanotransductive feedback processes are also involved in organogenesis. The mechanical strains developed by contractile apoptotic cells were found to enhance junctional apicobasal Myo-II concentration in neighbouring cells, thereby leading to tissue folding in Drosophila leg joint formation [START_REF] Monier | Apico-basal forces exerted by apoptotic cells drive epithelium folding[END_REF]. Cells can also stock geometrical surrounding information via biochemical cues to control their own geometrical properties. In the Drosophila pupal epithelium, through the recruitment of Mud proteins, tricellular junctions (TCJs) orient cell division via the astral microtubules by pulling on the mitotic spindle [29 ]. Mechanotransduction is also involved in plant development.

Meristem shaping and coordinated cell growth are regulated by mechano-sensitive microtubule reorganisation via Katanin [START_REF] Uyttewaal | Mechanical stress acts via katanin to amplify differences in growth rate between adjacent cells in Arabidopsis[END_REF][START_REF] Verger | A tension-adhesion feedback loop in plant epidermis[END_REF].

An active biomechanical patterning cascade is thus at work during embryonic development, adding to the wellknown biochemical inductive cascade of patterning gene expression.

Moreover, as we will see, the mechanical strains developed by the biomechanical cascade can also participate, in turn, to determine the biochemical patterning cascade.

The mechanical regulation of biochemical patterning

Myo-II is by far not the only biomolecule whose behaviour is mechanosensitive. Transcription or co-transcription factor activation can also be strain dependent. For instance, the transcription factor NF-KB translocates into the nucleus in response to mechanical deformation in endothelial cells in culture leading to PDGF growth factor expression [START_REF] Hay | Activation of NF-kappaB nuclear transcription factor by flow in human endothelial cells[END_REF]. In zebrafish, laminar shear stress generated by blood flow induces Yap nuclear translocation. This transcription factor then stimulates endothelial cell division and contributes to vessel maintenance [START_REF] Nakajima | Flow-dependent endothelial YAP regulation contributes to vessel maintenance[END_REF]. And Smad1 is one of the first transcription factors shown to be mechanotransductively involved in cell differentiation by triggering myoblast-osteoblast transdifferentiation [START_REF] Rauch | C2C12 myoblast/ osteoblast transdifferentiation steps enhanced by epigenetic inhibition of BMP2 endocytosis[END_REF].

In vivo, the b-cat protein is responsible for the mechanical induction of the twist mesodermal gene expression in the early drosophila ectoderm submitted to external uni-axial global deformation [START_REF] Farge | Mechanical induction of twist in the Drosophila foregut/stomodeal primordium[END_REF]. The mechanosensitivity of b-cat was found to vitally trigger the mechanical induction of Twist expression in the anterior endoderm. b-Cat activates the downstream differentiation of the anterior midgut following its mechanically induced partial release from the cell junctions and its transfer from the cytoplasm to the nucleus in response to C&E compression during gastrulation. To perturb mechanical strains in epithelium and inhibit the anterior endoderm compression within the wild-type genetic background (WT), compressing tissues were ablated with a two photons laser. To rescue physiological compressions in the WT, a magnetic field gradient was used to micromanipulate the neighbouring tissues loaded with super-paramagnetic nano-particles [START_REF] Desprat | Tissue deformation modulates twist expression to determine anterior midgut differentiation in Drosophila embryos[END_REF].

The b-cat mechanosensitivity is also an important mechanism to stimulate and trigger mesodermal genes expression in the Drosophila (Twist) and zebrafish (brachyury) embryos mesoderm, in response to its invagination and to epiboly initiation, respectively. Under mechanical tensions, Src42A phosphorylates the Y654-b-cat major site of interaction with E-cadherins (E-cad) leading to the release of a pool of b-cat from the junctions to the cytoplasm and the nucleus [START_REF] Brunet | Evolutionary conservation of early mesoderm specification by mechanotransduction in Bilateria[END_REF]. Mechanical induction might also be involved in brachyury expression during Nematostella gastrulation [START_REF] Pukhlyakova | beta-Catenindependent mechanotransduction dates back to the common ancestor of Cnidaria and Bilateria[END_REF].

Interestingly in pathological contexts, b-cat mechanical activation, leading to the expression of tumorous target genes like cyclin D1, myc and zeb-1 in strained healthy tissues, is also involved in tumor progression due to skin and breast tumours stiffness and colon tumor growth pressure in vivo [START_REF] Fernandez-Sanchez | Mechanical induction of the tumorigenic beta-catenin pathway by tumour growth pressure[END_REF][START_REF] Samuel | Actomyosin-mediated cellular tension drives increased tissue stiffness and beta-catenin activation to induce epidermal hyperplasia and tumor growth[END_REF]. We also can notice that treatments activating the b-cat pathway with mechanical vibrations in adipocytes, repressing the adipogenic Glycogene Synthase, are developed to fight against obesity [START_REF] Shoham | Mechanotransduction in adipocytes[END_REF].

b-cat mechanosensitivity was recently found to be physiologically involved in several distinct developmental processes. In particular, it was proposed to initiate feather follicle differentiation during chicken development. Indeed, in response to epidermis buckling under Myo-II contractile tension, b-cat becomes phosphorylated on Y654 and translocates into the nucleus of bent cell domains. When Myo-II contractile activity was blocked with blebbistatin, no tissue buckling was observed. As a result, b-cat activation and the expression of its target gene bmp2, known to be upstream of follicle cell differentiation, were inhibited [START_REF] Shyer | Emergent cellular self-organization and mechanosensation initiate follicle pattern in the avian skin[END_REF] (Figure 2A).

In addition to b-cat, phosphorylation of the mechanosensitive Yap factor was also recently found to regulate the earliest differentiation of inner-cell mass of pre-implantation mouse embryos, during cell engulfment morphogenesis. This follows the first asymmetric division after fertilization, and was proposed to be the consequence of the high acto-myosin contraction of the cell engulfed by the less contractile one [START_REF] Maitre | Asymmetric division of contractile domains couples cell positioning and fate specification[END_REF]. The inhibition of the Myo-II contractile activity with blebblistatin prevents cell engulfment. As a result, in a doublet, both cells express inner cell-like markers, that is high levels of cytoplasmic phosphorylated-Yap and low levels of Cdx2 expression (Figure 2B).

We saw above how internal morphogenetic forces can regulate tissue developmental biochemical patterning. Interestingly, mechanical strains that are external to tissues can also influence biochemical development.

For instance, mechanical constraints applied by the uterus on the growing mouse embryo lead to the mechanical induction of Cer1 expression in its distal endoderm. This protein is involved in the determination of the antero-posterior axis of mouse embryonic development [START_REF] Hiramatsu | External mechanical cues trigger the establishment of the anteriorposterior axis in early mouse embryos[END_REF].

In late stages of zebrafish embryos development, the oscillatory shear stress induced by the blood flow stimulates the expression of the endocardial flow-responsive klf2a gene. This gene was found to control the fibronectin synthesis that initiates cardiac valve development through the canonical Wnt/b-cat pathway [START_REF] Steed | klf2a couples mechanotransduction and zebrafish valve morphogenesis through fibronectin synthesis[END_REF][START_REF] Goddard | Hemodynamic forces sculpt developing heart valves through a KLF2-WNT9B paracrine signaling axis[END_REF].

In mouse embryos, the oscillatory shear stress of the lymphatic fluid enhances the b-cat signalling pathway in endothelial cells, and the expression of target genes, such as FOXC2, that regulates lymphatic vasculature development [START_REF] Cha | Mechanotransduction activates canonical Wnt/beta-catenin signaling to promote lymphatic vascular patterning and the development of lymphatic and lymphovenous valves[END_REF].

The integrative physiological functions of trans-scale mechanical and mechanotransductive cues in development

Predicting the macroscopic behaviours of living systems based on their molecular activities through an integrative reductionist approach from one scale to another remains an active field of biology. The challenge lies in untangling the complex interactions through different scales. In contrast, mechanical strains consist of a very simple cue, as it crosses all scales, thereby directly coupling the molecular microscopic with the macroscopic patterning scale. Indeed, as we saw, the Myo-II activity, which is biochemically patterned, generates mechanical strains leading to macroscopic biomechanical morphogenetic movements. These morphogenetic movements in turn mechanotransductively modulate the activity of Myo-II at the molecular level, which triggers other active morphogenetic movements. Macroscopic strains additionally regulate the biochemical activities leading to (co-)transcription factor nuclear translocation at the molecular level, thereby modulating developmental gene expression.

In other words, in embryonic patterning, the mechanosensitivity of the biochemical and active biomechanical components allows mechanical cues to be privileged mode of direct interaction between the biochemically active and reactive microscopic molecular regulators of embryogenesis and the macroscopic morphological phenotype of the embryo [START_REF] Chan | Coordination of morphogenesis and cell-fate specification in development[END_REF][START_REF] Fernandez-Sanchez | Mechanical induction in embryonic development and tumor growth integrative cues through molecular to multicellular interplay and evolutionary perspectives[END_REF].

Among the physiological functions of such mechanobiochemical cascade that underlie embryogenesis, we can notice integrative rapid and long-range cell-cell interactions. For instance, in the Drosophila embryo, line tensions of acto-myosin cables robustly maintain differentiation boundaries across tissues [START_REF] Monier | An actomyosin-based barrier inhibits cell mixing at compartmental boundaries in Drosophila embryos[END_REF][START_REF] Aliee | Physical mechanisms shaping the Drosophila dorsoventral compartment boundary[END_REF]. Also, collective cell apex constrictions, coordinated across the mesoderm tissue by rapid and long-range mechanotransductive cell-cell interactions, are required for tissue invagination at gastrulation [START_REF] Mitrossilis | Mechanotransductive cascade of Myo-II-dependent mesoderm and endoderm invaginations in embryo gastrulation[END_REF].

Physiological functions also reveal a robust coordination between biomechanical and biochemical patterning. For instance, the mechanical induction of Twist expression ensures that only the internalized tissue at gastrulation will maintain Twist expression, and will in fine differentiate into mesodermal internal organs [START_REF] Brunet | Evolutionary conservation of early mesoderm specification by mechanotransduction in Bilateria[END_REF]. Similarly in early mouse embryos only the not-ingulfed cells express the Cxd2 gene responsible for the trophectoderm identity, and will not develop as an embryonic tissue [START_REF] Maitre | Asymmetric division of contractile domains couples cell positioning and fate specification[END_REF].

Such mechanical cues were additionally found, and proposed, to mediate long-range coordination of biomechanical and biochemical patterning, throughout the whole embryo. For instance, ventral mesoderm invagination triggers the initiation of posterior endoderm invagination within the 2 min time scale, thereby coordinating mesoderm with endoderm invagination at gastrulation [START_REF] Mitrossilis | Mechanotransductive cascade of Myo-II-dependent mesoderm and endoderm invaginations in embryo gastrulation[END_REF]. In addition, germ-band extension mediates both the anterior endoderm specification -through b-cat activation -and the posterior endoderm invagination -through the Fog/ MyoII pathway -, thereby synchronising two processes 100 cells (500 mm) apart [START_REF] Desprat | Tissue deformation modulates twist expression to determine anterior midgut differentiation in Drosophila embryos[END_REF][START_REF] Mitrossilis | Mechanotransductive cascade of Myo-II-dependent mesoderm and endoderm invaginations in embryo gastrulation[END_REF]. These effects may be of particular importance under conditions in which the dynamics and complex topology of the embryonic tissues prevent the establishment of the long-range morphogen gradients that are efficient at earliest stages where cells are arranged in simpler, static geometrical patterns.

And indeed, integrating both the macroscopic biomechanical and microscopic biochemical patterns through such reciprocal trans-scale mechanical and mechanotransductive interaction, explains embryogenetic processes as important as gastrulation, initial tissue differentiation, maintenance of patterning boundaries, coordination of morphogenetic movements and the coordination of biomechanical/biochemical patterning (Figure 3). Epithelial tricellular junctions act as interphase cell shape sensors to orient mitosis. Nature 2016, 530:495-498. InDrosophila pupal epithelium, cells round up during mitosis and therefore loose their interphase shape anisotropy. In these conditions, Bosveld et al. asked how cells can achieve the Herwig's rule that stipulates that cells will divide along their long-axis direction. They showed that tricellular junctions (TCJs) recruit the Dynein-associated protein Mud to exert forces on the mitotic spindle and control its position during mitosis. Using laser ablation, they could show that Mud pulls forces via the astral microtubules. Remarkably, the predictions of the associated Mud-model were significantly better that the former Shape-model ones. By showing how TCJs play a major role in orienting cell divisions in Drosophila embryo epithelium, this study describes how cells can stock geometrical surrounding information via biochemical cues and control their own geometrical properties.

Figure 1 (

 1 Figure 1

  Figure 2

  Figure 3

  Markova O, Guirao B, Martin C, Wang Z, Pierre A, Balakireva M, Gaugue I, Ainslie A, Christophorou N et al.:

Acknowledgements

The 'Mechanics and Genetics of Embryonic and Tumor Development' lab is currently funded by the FRM (Grant # DEQ20150331702), the ANR (Grant # 16CE14002801), and the PSL-IRIS-OCAV (Grant # ANR1-10-IDEX-0001-02 PSL). Tatiana Merle is funded by the ED-PIF (doctoral funding 1767/2014) and the FRM (Grant # FDT20170436944).

Conflict of interest statement

Nothing declared.


Saint-Venant Christos E Papoutsellis
email: christos.papoutsellis@cerema.fr

Marissa L Yates
email: marissa.yates-michelin@cerema.fr

Bruno Simon
email: simon@irphe.univ-mrs.fr

Michel Benoit
email: benoit@irphe.univ-mrs.fr

FULLY NONLINEAR MODELING OF NEARSHORE WAVE PROPAGATION INCLUDING THE EFFECTS OF WAVE BREAKING

INTRODUCTION

Nearshore wave modeling over spatial scales of several kilometers requires balancing the fine-scale modeling of physical processes with the model's accuracy and efficiency. In this work, a fully nonlinear potential flow model is proposed as a compromise between simplified linear, weakly nonlinear or weakly dispersive models and direct CFD approaches.

The core of present approach is the use of a series representation for the velocity potential. This series contains prescribed vertical functions and allows the determination of the velocity potential in terms of unknown horizontal functions.

The resulting dimensionally reduced model retains the structure of the Hamiltonian water wave system Zakharov (1968), [START_REF] Craig | Numerical Simulation of Gravity Waves[END_REF], avoiding the solution of the Laplace problem for the potential. Instead, a numerically convenient linear system of horizontal equations needs to be solved at each step in the temporal evolution. No simplifications concerning the deformation of the physical boundaries are introduced, apart from the typical requirement of a smooth, non-overturning free surface and seabed. The main limitation of this formulation is its inability to account for wave breaking. The treatment of this process is the subject of the present work. Two different techniques are implemented in the present model. Simulation results are compared to laboratory measurements for two test cases: (1) shoaling and breaking of regular waves over a barred bathymetry [START_REF] Beji | Experimental investigation of wave propagation over a bar[END_REF] and (2) shoaling and breaking of regular waves on a plane beach [START_REF] Ting | Observation of undertow and turbulence in a laboratory surf zone[END_REF].

MODEL

By considering an inviscid fluid and assuming that the flow is incompressible and irrotational, the wave motion is described in terms of the velocity potential and the free surface elevation. The kinematic and dynamic free surface boundary conditions are expressed as a system of evolution equations for the free surface elevation and the free surface velocity potential, which requires calculating the normal fluid velocity, or the Dirichlet-to-Neumann (DtN) operator, at each time step [START_REF] Zakharov | Stability of periodic waves of finite amplitude on the surface of a deep fluid[END_REF], [START_REF] Craig | Numerical Simulation of Gravity Waves[END_REF]. The computation of the DtN operator is performed by using the Coupled-Mode method in the vertical direction Athanassoulis & Papoutsellis (2017). The corresponding model, called Hamiltonian Coupled-Mode system (HCMS), has been successfully validated with comparisons to analytical solutions, laboratory experiments and other computational approaches for non-breaking waves, see e.g. Papoutsellis & Athanassoulis (2017), [START_REF] Papoutsellis | Implementation of a fully nonlinear Hamiltonian Coupled-Mode Theory, and application to the interaction of solitary waves with varying bathymetry and vertical walls[END_REF]. Here, HCMS is extended to take into account the effects of depth-induced wave breaking.

When using potential flow theory, wave breaking cannot be modelled directly. The effects of wave breaking can however be included as a mechanism of wave energy dissipation that is activated by appropriate wave breaking criteria. Here, a criterion on the velocity of the free surface elevation, 𝜕𝜕 𝑡𝑡 𝜂𝜂, is considered. A wave is assumed to start breaking when this velocity exceeds a specified value, proportional to the shallow water velocity: 𝜕𝜕 𝑡𝑡 𝜂𝜂 > 𝛾𝛾�𝑔𝑔ℎ, where ℎ is the local water depth and 𝛾𝛾 is a constant depending on the configuration. A smaller threshold value is used for the termination of the breaking process Kennedy, Chen, Kirby, & Dalrymple (2000). Once a breaking wave is detected, a dissipative term is triggered in the dynamic free surface boundary condition and is applied over a specified spatial region of the breaking wave. Two different methods are implemented and tested here: (1) an artificial pressure term proposed by [START_REF] Guignard | Modeling of wave shoaling in a 2D-NWT using a spilling breaker model[END_REF], acting in a region around the breaking crest and (2) an eddy viscosity term proposed by Kurnia & van Groesen (2014), acting on the front face of the breaking wave.

RESULTS

The first test case evaluates the details of wave breaking over a barred bathymetric profile studied by [START_REF] Beji | Experimental investigation of wave propagation over a bar[END_REF] At station #2, located in the shoaling region and close to the initial breaking point, the two wave breaking methods and the experimental results are nearly indistinguishable. Slight differences appear at stations #4 and #8, located in and after the breaking region, respectively.

The second test case investigates the shoaling and breaking of regular waves on a plane beach with slope 1/35, following the experiments of Ting & Kirby (1994) (TK94). For the spilling breaker case presented here, with incident wave height 0.0125 m and period T = 2.0 s over an offshore depth of 0.4 m, the models reproduce well the transition from symmetric to asymmetric waves with steeper wave faces and gentler rear slopes, and more peaked crests and wider troughs. The wave envelope (maximum and minimum free surface positions, Figure 2) and wave skewness and kurtosis (not shown here) are predicted well up to the breaking point. After the breaking point, the results obtained using the pressure-type dissipation term show slightly better agreement with the measurements of the maximum free surface elevation.

CONCLUSION

The test cases presented here show the ability of the proposed model to simulate well the nonlinear processes of wave propagation and transformation over a barred beach and a mildly inclined beach. Breaking waves are detected by using a criterion on the velocity of the free surface elevation. In conjunction with this criterion, two wave-breaking techniques that successfully take into account the effects of breaking through wave energy dissipation, were evaluated and compared to experimental measurements of spilling and plunging breaking waves. Ongoing investigation and modification of the above approaches indicates that the simulation results can be improved by carefully selecting the spatial extent of the activation of the implemented wave breaking terms. Both methods can be extended to model domains with two horizontal dimensions. Investigation of the sensitivity of the above approaches on the choice of the involved free parameters will also be discussed and reported during the Conference and in the final paper.

  (BB93); see Figure 1, bottom. The incident wave height and period are 0.054 m and 2.5 s respectively, over a depth of 0.4 m, and the breaking type is classified as plunging. The processes of shoaling, breaking and harmonic generation are reproduced well by both of the considered methods. Comparisons between the computed and measured free surface elevation are shown in Figure 1.

Figure 1 .

 1 Figure 1. Comparison of simulated and free surface elevation at stations 2,4, and 8 (top three panels), and bathymetric profile (bottom panel) for the plunging wave experiment of BB93.

Figure 2 .

 2 Figure 2. Comparison of simulated and measured wave envelopes along the wave channel for a spilling wave experiment of TK94.


Goulven Eynard
email: goulven.eynard@enst-bretagne.fr

Christophe Laot
email: christophe.laot@enst-bretagne.fr

Non Data Aided Timing Recovery Algorithm for Digital Underwater Communications

Synchronization is a critical operation in an underwater acoustic data communication receiver. This paper proposes a comparison of two digital NDA (non data aided) timing recovery schemes, referenced in literature as the Gardner, and the Oerder and Meyr algorithms. We consider the context of a single QPSK carrier continuous transmission, where a timing shift has to be estimated continuously in order to track the optimum sample time. Simulations processed on real data taken from sea trials, collected by the GESMA in collaboration with SERCEL and ENST Bretagne, reveal that a large Doppler shift or a large jitter on the timing shift estimation can introduce cycle slips in the clock synchronizer, which generate a large burst of errors in the data receiver. These perturbations can dramatically affect the global performance of the transmission system.

I. INTRODUCTION

Synchronization is a critical operation in an underwater acoustic data communication receiver. In fact, without a sufficiently accurate knowledge of the synchronization parameters, the task of retreiving the symbol sequence from the received signal can be highly time and power consuming for the receiver. This is particulary the case in a digital underwater acoustic link, where the channel can be subject to severe time-space variability, strong multipath and Doppler effect that make the estimation of the synchronization parameters very difficult in practice. The particular case of the Doppler carrier phase tracking with Doppler compensation is relevant and is well explored in [START_REF] Freitag | Efficient Equalizer Update Algorithms for Acoustic Communication Channels of Varying complexity[END_REF] and [START_REF] Johnson | Improved Doppler Tracking and Correction for Underwater Acoustic Communications[END_REF]. This paper adresses the problem of NDA (Non Data Aided) timing recovery. Timing recovery is often the first synchronization operation processed by the digital receiver and so is a vital part of any synchronous receiver. We consider the context of a single carrier QPSK continuous transmission, where a timing shift has to be estimated continuously in order to track the optimum sample time. We also suppose that the received signal is sampled at a frequency sufficiently high so that interpolation is not necessary. The first recovery scheme is an error-tracking synchronizer, based on the tracking of the average zero-crossing instant of the received signal using a digital locked loop (DLL). The fact that this adaptive algorithm provides an estimation of the timing shift each symbol time allows a high sensitivity to strong variations of the timing estimate. On the other hand, the feedback structure can introduce cycle-slips and hang-up problems [START_REF] Meyr | Digital Communication Receivers: Synchronization, Channel Estimation, and Signal Processing[END_REF]. The second recovery scheme, commonly referred as the Oerder and Meyr algorithm, uses the cyclostationnary property of the received signal to recover the timing shift. This algorithm uses a feedforward structure: the timing error estimation is processed directly from the received signal sampled at a constant frequency. This type of algorithm is particulary adapted in short burst mode, where it is sufficient to make a single estimation for each burst of data. We consider here the situation of a continuous transmission: we divide the received data into blocks of length L symbols and we successively estimate the timing shift for each block. In order to track variations of the timing shift caused for instance by a relative motion between the emitter and the receiver, the successive estimates have to be unwrapped by means of a post-processing structure [START_REF] Oerder | Digital Filter and Square Timing Recovery[END_REF], [START_REF] Meyr | Digital Communication Receivers: Synchronization, Channel Estimation, and Signal Processing[END_REF], [START_REF] Mengali | Synchronization Techniques for Digital Receivers[END_REF]. This algorithm supposes that the timing shift has no variation during the transmission of an observation interval. The major drawback of this algorithm comes from variations of the timing shift that are not taken into account. The main contribution of this paper is the study of these two timing recovery algorithms applied to the digital underwater transmission context. We observe that a large Doppler shift or a large jitter on the timing shift estimation can introduce the deletion of symbols in the clock synchronizer, generating large bursts of errors. This pertubation, known in literature as a cycle slip, can dramatically affect the global performance of the transmission system. The paper is organized as follows. In Section II the baseband model is introduced. We present succesively in Section III the Gardner algorithm, and in Section IV the Oerder and Meyr algorithm. In Section V, the two algorithms adapted to the single input, multiple output context is presented. Finally, the performance of the two algorithms are compared using data from sea trials in Section VI.

II. BASEBAND MODEL

We consider the timing recovery for digital data transmission with linear modulation schemes. The received, filtered and sampled baseband signal from the input channel can be written as:

r(kT e ) = +∞ n=-∞ a n g(kT e -nT s -τ [k]T e ) + w(kT e ), (1)
where the transmitted symbols are represented by {a n } (QPSK modulation) and assumed to be mutually uncorrelated. g(t) includes the transmitting and the receiving filter of the communication system as well as the channel impulse response. T e Fig. 1: The feedback structure (digital locked loop) denotes the sampling period. Let T s be the symbol duration. We suppose that T s = N T e , where N is the oversampling factor. In the specific case of an underwater acoustic communication, the relatively low data rate allows us to take a large value for the oversampling factor N , so that no interpolation is necessary. τ [k] is the unknown varying timing shift. w(t) is the channel noise which is assumed to be white and gaussian. First, the digital receiver operates the timing recovery. Then, data are estimated with the SA-DFE (self adaptive decision feedback equalizer), which is presented in [START_REF] Labat | Adaptive Decision Feedback Equalization: Can You Skip The Training Period ?[END_REF] for the SISO (single input, single output) case, and in [START_REF] Labat | Blind adaptive Multiple-Input Decision Feedback Equalizer with a Self-Optimized Configuration[END_REF] and [START_REF] Trubuil | Improved AUV autonomy provided by an underwater acoustic link[END_REF] for the SIMO (single input, multiple output) case.

III. GARDNER TIMING RECOVERY ALGORITHM

At the heart of the feedback structure (or digital locked loop) is a TED (timing error detector), which serves to extract timing error information from the received signal. The error signal e[k] at the output of the TED is filtered through the loop filter. When the DLL is in lock, e[k] is nearly proportionnal to the difference between τ [k] and the timing estimate τ [k]. The NCO (numerical controlled oscillator) tends to lock thec local clock onto the incoming signal using the timing error information provided by the loop filter. The Gardner TED [START_REF] Gardner | A BPSK/QPSK Timing-Error Detector for Sampled Receivers[END_REF] uses the zero-crossing instant of the received signal as an information to perform timing-recovery. Defining:

T [k] = k T s + τ [k] T e ,
the output of the Gardner timing error detector can be expressed as:

e[k] = ℜ r T [k -1] -r T [k] r * T [k] -Ts/2 . (2)
The error signal is then filtered throught a first-order filter loop. The transfert function loop filter employed here is:

F (z) = 1 -λ 1 -λz -1 , (3)
where the forgetting factor λ is taken at the value: λ = 0.99. This factor has been chosen as the best heuristic trade-off between the acquisition time and the filtering of the noise.

The NCO provides at its output the corrected sampling signal. Further details of this algorithm can be found in [START_REF] Gardner | A BPSK/QPSK Timing-Error Detector for Sampled Receivers[END_REF].

IV. OERDER AND MEYR ALGORITHM

A. Feedforward Structure

The main difference of the feedforward structure compared to the feedback structure is that the estimation of the timing error is obtained directly from the received signal r(t) sampled at a constant frequency f e = 1/T e , with T s = N T e . No information previously computed is used when the sampling operation is processed. Also we call L the length of the observation interval in symbol periods and r[m] = {r(kT e )} mLN ≤k≤(m+1)LN -1 the observation interval itself. We note that each observation interval r[m] is composed of N L samples. The squared sequence of the observation interval contains a spectral component at 1/T s that can be used for the estimation of the parameter τ :

x[m] = {x(kT e )} mLN ≤k≤(m+1)LN -1 , (4)
where

x(kT e ) = r(kT e ) 2 .
The spectral component is extracted by computing the complex Fourier coefficient:

X m = (m+1)LN -1 k=mLN x(kT e )e -j2πk/N . (5)
The argument of the Fourier transform gives the modulo N of the timing estimate within a factor -N/2π:

τ [m] = -N/2π Arg(X m ). (6)
Further details can be found in [START_REF] Oerder | Digital Filter and Square Timing Recovery[END_REF] about this estimator.

Successive estimations are needed to track fluctuations of the synchronization parameter. Feedforward estimation involves dividing the received signal into observation intervals that are short enough to make the approximation that the timing shift is constant. We notice that the resulting feedforward estimates at the output of the

B. Unwrapping Algorithm

In this section, we give details on how to accomplish the unwrapping operation. Let SAW N (x) be the sawtooth function with period N , defined on the interval -

N 2 ≤ x ≤ N 2 as: SAW N ( ∆τ [m]) = ∆τ [m] if | ∆τ [m]| ≤ N 2 . ( 7
)
The sawtooth function is a useful function to unwrap the data.

Having a look on Figure 3, we notice that the expression of the sawtooth function can be expressed from the modulo N function:

SAW N ( ∆τ [m]) = M OD N ( ∆τ [m]) if M OD N ( ∆τ [m]) ≤ N/2, M OD N ( ∆τ [m]) -N else. (8)
A more useful expression is given by the equation:

SAW N ( ∆τ [m]) = M OD N ( ∆τ [m] -N/2) -N/2. ( 9
)
Fig. 3: Plot of the Modulo and the Sawtooth functions, supposing N the period of the two functions.

Having expressed the sawtooth function, we are now able to express τ [m], supposing the knowledge of only τ

[m -1] and τ [m]. Let ∆ τ [m] = τ [m] -τ [m -1].
We have:

τ [m] = τ [m -1] + ∆ τ [m]. Let ∆τ [m] = τ [m] -τ [m -1].
Using the expression of SAW N ( ∆τ [m]), it can be proven that:

∆ τ [m] = SAW N ( ∆τ [m]). ( 10
)
The expression of τ [m] is finally obtained:

τ [m] = τ [m -1] + SAW N ( ∆τ [m]), (11)
which is illustrated by the unwrapping structure of Figure 4.

Fig. 4: The feedforward timing estimator with the unwrapping structure

V. SINGLE INPUT MULTIPLE OUTPUT TIMING RECOVERY

To exploit spatial diversity, timing estimation is processed on M channels independantly for both algorithms. Then, an average of the M timing estimates is used for sampling synchronously the M sensors. Figure 5 and 6 describe the timing recovery schemes for respectively the Oerder and Meyr and for the Gardner algorithm.

VI. COMPARISON OF THE ALGORITHMS ON REAL DATA

The evaluation of timing recovery algorithms is performed from the database collected by the GESMA in collaboration with SERCEL and ENST Bretagne during series of trials in the Atlantic ocean. At the receiver, the signal is demodulated using a free running oscillator. Then, timing recovery is processed on the baseband signal. Since the data rate is relatively low, we are able to choose an oversampling factor N sufficiently high so that interpolation is not necessary. Then, we jointly process equalization and phase synchronization using the unsupervised SA-DFE (Self-Adaptive Decision Feedback Equalizer), presented in [START_REF] Labat | Adaptive Decision Feedback Equalization: Can You Skip The Training Period ?[END_REF] and [START_REF] Labat | Blind adaptive Multiple-Input Decision Feedback Equalizer with a Self-Optimized Configuration[END_REF] for the multi-sensors case. To exploit spatial diversity, M = 4 sensors are used at the receiver. For each simulation, we plot the evolution of the timing shift for both timing algorithms and of the estimated Mean Square Error (MSE) of the signal observed after the SA-DFE equalizer. Let the estimated MSE be:

e M SE[k] = β e M SE[k-1]+(1-β) d[k] -y[k] 2 , ( 12
)
where β = 0.99, y[k] is the output of the SA-DFE equalizer and d[k] is the decision made on the estimated symbol y[k]. The estimated timing shift gives a reliable information on the Doppler shift present on the transmission. Also, it provides information on the jitter of the timing shift, due for instance to ISI (Intersymbol Interference) present on the channel. Moroever, a cycle slip is particularly recognizable in the evolution of the timing shift as a jump of length ≥ T s , where T s is the symbol duration. The estimated MSE provides good indications on the performance of the global transmission system and on channel time-variations. We first consider the case of a communication between an AUV (Autonomous Underwater Vehicle) and its surface base station. The relative motion between the emitting and the receiving structures introduces a Doppler shift that can affect severely the tracking of the timing estimation. The transmission rate is 14kbp/s, the modulation type is QPSK with carrier frequency at 35 kHz. At the receiver, the signal is sampled at T e = T s /N , where T s = 1.4 10 -4 s and N = 20.

We observe in Figure 8 that the Doppler shift is very large on this transmission (800 T e = 40 T s over 30000 transmitted symbols: the Doppler shift in timing recovery is 0.13% of the symbol rate). It can be seen from Figure 8, that the large Doppler shift present on this channel causes two consecutive cycle slips in a relative small period of time (approximatively 500 T s ) when Gardner algorithm is used. Both cycle slips have an immediate impact on the estimated MSE of the signal observed after the SA-DFE equalizer, as depicted in Figure 8. We choose L = 50 so that the Oerder an Meyr algorithm can track the fast evolutions of the timing shift. A larger value for L can be chosen: in [START_REF] Oerder | Digital Filter and Square Timing Recovery[END_REF], it is recommended to choose the length of the observation interval so that the variation of the timing shift τ [m] is smaller than T s /2. Hence, for a Doppler of 0, 13%, we can take a maximum value L = 375. However, to have a reliable continuous evaluation of the timing shift present on the channel and to face for example a brutal burst of noise, a heuristic value for the maximum variation of the timing shift on the observation interval could be T s /4 or T s /8. Therefore, the best value for L would be between 50 and 100 for this transmission. We now consider the case of a static communication between the emitter and the receiver structure. The transmission rate is 8.75kbp/s, the modulation type is QPSK with carrier frequency at 35 kHz. At the receiver, the signal is sampled at T e = T s /32, where T s = 2.2 10 -4 s.

During the transmission, a strong impulsive noise occurs. We choose various values for parameter L (L = 30 , 50) in order to erase this impulsive noise. As far as the length L of the observation interval is sufficient, we observe that the jitter of the timing-shift is considerably reduced. This observation is confirmed by the result in [START_REF] Oerder | Digital Filter and Square Timing Recovery[END_REF], where it is shown that the mean square error of the estimate τ [m] is inversely proportionnal to the parameter L when L is large. For the clarity of Figure 9 and 10, results obtained for L > 50 are not reported, since the performance are essentially the same as for L = 50. However, we notice that results obtained with L = 30 are far worse than results obtained with the Gardner algorithm. This can be explained by the fact that the effect of the impulsive noise can be attenuated if the observation interval is sufficiently large. Concerning the Gardner algorithm, we observe that the cycleslip cannot be avoided by modifying the parameter of the loop filter λ. The cycle-slip in the tracking of the timing estimate is immediately repercuted in the estimated MSE of the SA-DFE in Figure 10. Results obtained in these two examples show that the second algorithm is particulary well-fitted to underwater acoustic communications if the parameter L is well chosen.

VII. CONCLUSION

The comparison of two well-known timing recovery algorithm is described here. A particular focus has been made on the unwrapping technique. Simulation on real data reveals that the Oerder and Meyr algorithm with a parameter L well chosen is more robust to cycles slips due to severe Doppler shift and burst of noise than the Gardner algorithm. The two transmissions reveal also that it exists a trade-off for the value of L where the Oerder and Meyr algorithm is able to face whether a large Doppler spread or a large burst of noise. Simulations conducted on various situations reveal that the value L = 50 or L = 100 appears to be a good trade off in our case to face large frequency offset and impulsive noise.

Fig. 2 :

 2 Fig. 2: Block diagram of the Oerder and Meyr estimator

  Oerder and Meyr estimator (Figure 3) are restricted to the basic interval: -N ≤ τ [m] ≤ N . In fact, the feedforward estimates can be considered as estimates of the synchronization parameters reduced modulo the interval [-N ; N ]. We want to remove this modulo N operation to track the timing shift τ [m]. This problem is solved by unwrapping the feedforward estimates.

Fig. 5 :

 5 Fig. 5: Single Input -Multiple Output timing recovery scheme for the Oerder and Meyr Algorithm.

Fig. 6 :

 6 Fig. 6: Single Input -Multiple Output timing recovery scheme for the Gardner Algorithm.

Fig. 7 :

 7 Fig. 7: Comparison of the evolution of the timing shift in time for the Oerder and Meyr, and the Gardner Algorithm.

Fig. 8 :Fig. 9 :

 89 Fig. 8: Estimated MSE for the Oerder and Meyr and the Gardner Algorithm.

Fig. 10 :

 10 Fig. 10: Estimated MSE for the Oerder and Meyr and the Gardner Algorithm.