Elodie Cretel

Anne Pierres

Anne-Marie Benoliel

Pierre Bongrand
email: pierre.bongrand@inserm.fr

Anne- Marie Benoliel

Pierre Bongrand

Pr Pierre Bongrand

How Cells feel their environment: a focus on early dynamic events

It is now well demonstrated that cell adhesion to a foreign surface strongly influences prominent functions such as survival, proliferation, differentiation, migration or mediator release. Thus, a current challenge of major practical and theoretical interest is to understand how cells process and integrate environmental cues to determine future behaviour. The purpose of this review is to summarize some pieces of information that might serve this task. Three sequential points are discussed. First, selected examples are presented to illustrate the influence of substratum chemistry, topography and mechanical properties on nearly all aspects of cell behaviour observed during the days following adhesion. Second, we review reported evidence that long term cell behaviour is highly dependent on the alterations of cell shape and cytoskeletal organization that are often initiated during the minutes to hours following adhesion. Third, we review recently obtained information on cell membrane roughness and dynamics, as well as kinetics and mechanics of molecular interactions. This knowledge is required to understand the influence of substratum structure on cell signaling during the first minute following contact, before the appearance of detectable structural changes. It is suggested that unraveling the earliest phenomena following cell-to-substratum encounter might provide a tractable way of better understanding subsequent events. Keywords : Adhesion, cell behaviour, substratum topography, substratum rigidity, signaling Cell adhesion to foreign surfaces strongly influences nearly all functions, including proliferation, differentiation, migration or release of active mediators. These phenomena are of prominent importance for both practical and theoretical reasons. Indeed, a major challenge of tissue engineering consists of elaborating biomaterials inducing adequate response of surrounding tissues, with proper integration and inhibition of potentially harmful inflammatory or infectious processes. Also, the ultimate goal of cell biologists may well be to understand the rules followed by cells for behavioural choices. Studying the consequences of cell adhesion to well-defined controlled structures should bring major insights along this line.

During the last years, numerous investigators provided impressive information on the way cells respond to substrate properties such as molecular structure, lateral density and distribution of active sites, mechanical properties, micrometer-or nanometer-scale topography. Also, the involvement of some well-defined signalling cascades in these sensing events was convincingly demonstrated. The present challenge may well be to make sense from the huge amount of data that have been gathered. The complexity of this task may seem quite overwhelming in view of the number of molecules and genes involved in response to environmental cues. Indeed, since a limited perturbation of the cell environment may affect hundreds of important interrelated molecules, it is very difficult to obtain unambiguous proofs of an immediate relationship between a surface pattern and the triggering of a given signalling cascade in adherent cells.

The strategy we suggest to tackle with these difficulties is to analyze the phenomena occurring during the first few seconds following the encounter between a cell and a surface. Hopefully, this approach might allow us to identify surface properties liable to influence cell behaviour in a fairly immediate way. However, as will be discussed below, following this line will require to gather some insight on some cell molecular processes that remain incompletely understood at the present time. However, asking questions may be more appropriate than describing solved problems in an inaugural issue of a scientific journal. This review will include three main parts. First, we shall describe some representative examples of cell response to substratum properties. Second, we shall review some evidence supporting the concept that cell shape and cytoskeletal organization may provide a link between environmental cues and cell behavioural choices. Third, we shall describe some recent results concerning cell membrane dynamics, as a basis for cellsubstratum interaction.

SURFACE PROPERTIES KNOWN TO INFLUENCE ADHERENT CELL BEHAVIOUR

Our purpose is to illustrate basic principles with representative examples rather than presenting exhaustive reviews. Therefore, we apologize for the omission of much important work. We shall only list some surface parameters that are now recognized as important determinants of cell behaviour.

Surface chemistry.

The best known example of the importance on surface chemistry on cell behaviour may well be the need to subject plastic (polystyrene) dishes to a specific treatment to make them suitable for cell culture. This emphasizes the important of nonspecific features such as hydroxyl groups that will decrease surface hydrophobicity [START_REF] Curtis | Adhesion of cells to polystyrene surfaces[END_REF] . Another example is the longknown capacity of phagocytic cells to ingest selectively hydrophobic particles [START_REF] Capo | Nonspecific recognition in phagocytosis : ingestion of aldehyde treated erythrocytes by rat peritoneal macrophages[END_REF] . More recently, a study made at the proteomic level resulted in the identification of 21 genes of Hela Cells whose expression was substantially altered by substratum hydrophobicity after 24h adhesion [START_REF] Allen | Interaction of soft condensed materials with living cells : phenotype/transcriptome correlations for the hydrophobic effect[END_REF] . Now, while the importance of surface charge or hydrophobicity was studied for decades, it is not obvious that cells are intrinsically sensitive to these nonspecific physical-chemical properties. As recently discussed [START_REF] Vitte | Is there a predictable relationship between surface physical-chemical properties and cell behaviour at the interface ?[END_REF] , most recent evidence supports the concept that cells essentially perceive foreign surfaces through membrane receptors that are specific for well defined molecular structures. Since biomaterials become coated with adsorbed molecules within seconds following their exposure to biological media, and the conformation of adsorbed biomolecules is dependent on the physical-chemical properties of underlying surfaces, cells may detect these properties in an indirect way, through exposure of specific binding sites linked to conformational changes. Thus, fibronectin was found to support cell growth much more efficiently when it was adsorbed on hydrophilic rather than hydrophobic surfaces [START_REF] Grinnell | Fibronectin adsorption on hydrophilic and hydrophobic surfaces detected by antibody binding and analyzed during cell adhesion in serumcontaining medium[END_REF] .

Nature, density and lateral distribution of specific ligands. The most general mechanism allowing cells to respond to surfaces they have just encountered is the generation of biochemical signalling cascades following the interaction between cell membrane receptors and their specific ligands when they are exposed on the surfaces. Multiple experiments supported the general concept that the cell response is dependent on the nature of stimulated receptors. As an example, different receptors may be involved in mediating cell attachment to and spreading on a surface [START_REF] Runyan | Functionally distinct laminin receptors mediate cell adhesion and spreading : the requirement for surface galactosyltransferase in cell spreading[END_REF] . Now, in addition to the ligand species, density and distribution of binding sites may strongly influence cell behaviour. Thus, the migration behaviour of fibroblasts deposited on surfaces coated with an integrin ligand (YRGDS peptide) was markedly influenced by the spatial distribution of binding sites at the nanoscale level [START_REF] Maheswari | Cell adhesion and motility depend on nanoscale RGD clustering[END_REF] . More recently, it was reported that the spreading of rat fibroblasts on surfaces coated with RGD integrin ligands was markedly influenced by the spacing of binding sites: when the distance between binding sites was increased from 58 to 108 nm, spreading efficiency decreased with less regular progression of the cell leading edge and frequent occurrence of retraction events [START_REF] Calvacanti-Adam | Cell spreading and focal adhesion dynamics are regulated by spacing of integrin ligands[END_REF] .

Several well-demonstrated mechanisms might be responsible for these findings. First, ligand clustering may dramatically enhance cell attachment efficiency since binding strength may increase exponentially with respect to attachment valency [START_REF] Seifert | Rupture of multiple parallel molecular bonds under dynamic loading[END_REF] . Second, clustering of cell membrane molecules such as integrins may dramatically influence the triggering of signalling cascades as a consequence of interactions between intracellular molecules linked to the receptors. Thus, receptor clustering may influence signalling in a qualitative as well as a quantitative way [START_REF] Jirouskova | Ligand density dramatically affects integrin αIIbβ-mediated platelet signaling and spreading[END_REF] .

Surface topography.

It has been well demonstrated for several decades that cells deposited on substrata bearing micrometric patterns adapted their shape and orientation to the topological features of the surface, a phenomenon called "contact guidance". Thus, cells displayed marked alignment along grooves of micrometrical depth and width [START_REF] Clark | Topographical control of cell behaviour : II. Multiple grooved substrata[END_REF] . More recently, it was also shown that cells are sensitive to nanoscale topography. Thus, when fibroblasts were deposited on surfaces bearing islands of 13-nanometer height, they displayed marked enhancement of gene expression, as demonstrated with microarray technology [START_REF] Dalby | Increasing fibroblast response to materials using nanotopography : morphological and genetic measurements of cell response to 13-nm-high polymer demixed islands[END_REF] . Indeed, 584 responses were detected out of 1,718 tested genes. Also, nanoislands induced filopodium formation and cell spreading. Further work allowed the identification of molecules involved in force generation, such as myosin II, and focal contact development, such as focal adhesion kinase, in topography sensing 71 .

Additional information was obtained with different approaches. Thus, when nanoscale patterns were varied, it appeared that the adhesion of human fibroblasts was lower on ordered arrays of nanopits compared to flat surfaces or randomly distributed pits [START_REF] Curtis | Cells react to nanoscale order and symmetry in their surroundings[END_REF] . Another study might provide additional information on underlying phenomena. The activation of T lymphocytes by surfaces exposing complexes formed by cognate peptides and histocompatibility molecules (pMHC) is a process of prominent importance for the development of immune defence. When T lymphocytes were deposited on surfaces bearing pMHC freely diffusing in supported lipid layers, the addition to surfaces of nanobarriers impeding the lateral diffusion of complexes formed between T cell receptors (TCR) and pMHC resulted in marked increase of the lifetime of signal generation by peripheral TCR/pMHC clusters [START_REF] Mossman | Altered TCR signaling from geometrically repatterned immunological synapses[END_REF] . This work provided a formal proof that the presence of nanostructures on surfaces might strongly influence the development of signaling cascades.

While there is no doubt that cell behaviour is influenced by nanoscale topography, underlying mechanisms remain ill understood. The aforementioned finding that barriers as low as 50 nm might efficiently alter lateral diffusion of molecular complexes is certainly significant. Also, there is some evidence that local surface curvature might influence molecular interactions in the cell membrane [START_REF] Reynwar | Aggregation and vesiculation of membrane proteins by curvature-mediated interactions[END_REF] . Thus, substratum topography is likely to influence the in-plane movement and interactions of the proteins embedded in the cell membrane. This may drastically influence the generation of signaling cascades.

Surface stiffness.

It is now well demonstrated that the behaviour of adherent cells is markedly altered by surface mechanical properties. Thus, when fibroblasts were deposited on collagen surfaces with local variations of rigidity, they were found to migrate towards stiffer regions, a phenomenon denominated by the authors as "durotaxis" [START_REF] Lo | Cell movement is guided by the rigidity of the substrate[END_REF] . More recently, when human mesenchymal stem cells were deposited on collagen-coated surfaces of varying rigidity, cell differentiation was dramatically affected by substratum stiffness. Indeed, cells deposited on softer matrices with a Young modulus of ∼0.1 -1 kPa differentiated into neurons. Stiffer surfaces (about 10 kPa) induced muscle cell generation. Finally, cells deposited on the stiffest surfaces (25-40 kPa) turned into osteoblasts [START_REF] Engler | Matric elasticity directs stem cell lineage specification[END_REF] .

In addition to the formal demonstration that cells are highly sensitive to the substratum rigidity, important information was obtained on possibly involved mechanisms. First, it has long been found that adherent cells usually exert a pulling force on underlying substrata [START_REF] Harris | Silicone rubber substrata : a new wrinkle in the study of cell locomotion[END_REF] . Second, the force exerted by cells is dependent on the substratum resistance. This phenomenon was cleverly demonstrated by applying controlled forces to fibronectin-coated microspheres deposited on cells and held with an optical trap [START_REF] Choquet | Extracellular matrix rigidity causes strengthening of integrin-cytoskeleton linkages[END_REF] . Cells were indeed found to sense the restraining force exerted by the trap and locally increase pull. A possibly related finding is that forces were shown to stimulate focal contact development 72 . Third, using cell spread area determination to evidence rigidity sensing, Sheetz and collaborators demonstrated the involvement of some key molecules such as α V β 3 integrin and membrane-bound phosphatases in this process [START_REF] Kostic | Fibronectin rigidity response through Fyn and p130 Cas recruitment to the leading edge[END_REF] .

Several points must be clarified for full interpretation of available data. First, it is not obvious to understand which precise substratum property is sensed by cells. Indeed, while the tension of cells adhering to a surface seems correlated to the Young modulus, other substratum properties must influence cell perception. Indeed, cells probably sense the kinetics of force increase when they pull on the substratum. This clearly depends on surface viscosity as well as elasticity. Also, it should be interesting to determine whether cells are equally sensitive to resistance to pushing as well as pulling forces. Although little information is available in this respect, it is interesting to note that a force as low as a few piconewtons per µm was reported to stall lamellipodia generated by fish epithelial keratocytes [START_REF] Bohnet | Weak force stalls protrusion at the leading edge of the lamellipodium[END_REF] .

CELL-SUBTRATUM SENSING: A COMMON MECHANISM?

While there is no doubt that the behaviour of adherent cells is deeply influenced by substratum properties, there is currently no theoretical framework available to achieve a general interpretation of experimental data. In this respect, it is interesting to review several reports suggesting that cell shape might provide a link between environment and fate.

Cell shape as an integrator of environmental signals.

As recently reviewed [START_REF] Pierres | Cell fitting to adhesive surfaces : a prerequisite to firm attachment and subsequent events[END_REF] , cell spreading plays a key role in important functions such as proliferation or differentiation. Thus, human mesenchymal cells underwent osteogenic differentiation when they well spread, whereas round cells became adipocytes [START_REF] Mcbeath | Cell shape, cytoskeletal tension, and RhoA regulate stem cell lineage commitment[END_REF] . That cell shape rather than contact area and number of bound membrane receptors might be the important parameter is suggested by the finding that cell proliferation, that is often dependent on adhesion, was shown to be related to projected area, i.e. cell shape, rather than molecular adhesion area [START_REF] Chen | Geometric control of cell life and death[END_REF] .

More studies are needed to understand the link between cell shape and behaviour. As suggested above [START_REF] Reynwar | Aggregation and vesiculation of membrane proteins by curvature-mediated interactions[END_REF] , local curvature might influence interaction between membrane molecules. Another mechanism of potential importance is based on the formation of activity gradients of enzymes that might be activated by plasma membrane receptors and deactivated by cytosolic components, as supported by experimental data and theoretical modeling [START_REF] Meyers | Potential for control of signaling pathways via cell size and shape[END_REF][START_REF] Haugh | Membranebinding/modification model of signaling protein activation and analysis of its control by cell morphology[END_REF] . As another example, there is some evidence that the cytoskeleton organization might link cell shape to behaviour through a control of the small GTPase Rho [START_REF] Mammoto | Filamin links cell shape and cytoskeletal structure to Rho regulation by controlling accumulation of p190RhoGAP in lipid rafts[END_REF] . More generally, while there is ample evidence that cell cytoskeletal organization is tightly related to cell shape, there is also strong support to the hypothesis that signaling cascades are markedly influenced by cytoskeletal organization. This point is discussed below.

Cell signaling and cytoskeletal organization.

In addition to its capacity to propagate mechanical effects within cells and convert stresses into signals [START_REF] Janmey | The cytoskeleton and cell signalling : component localization and mechanical coupling[END_REF][START_REF] Zhao | Force activates smooth muscle α-actin promoter activity through the Rho signaling pathway[END_REF] , the cytoskeleton may strongly influence signaling 30 . Since signaling cascades are essentially made of sequential interactions between numerous enzymes, targets and adapters, the cytoskeleton might play a major role by promoting interactions between particular molecules [START_REF] Janmey | The cytoskeleton and cell signalling : component localization and mechanical coupling[END_REF] . A possible rationale for such a function was recently suggested on the basis of recent advances in proteomics [START_REF] Forgacs | Role of the cytoskeleton in signaling networks[END_REF] .

Forgacs and colleagues performed a mathematical analysis of the set of molecular interactions (i.e. interactome) disclosed between proteins from Saccharomyces cerevisiae. Starting from a database of 4,480 interactions between 2,115 proteins [START_REF] Uetz | A comprehensive analysis of protein-protein interactions in Saccharomyces cerevisia[END_REF] , they were able to show that cytoskeleton related proteins were endowed with a particularly high capacity to interact with molecules involved in signalling.

A FOCUS ON TRANSIENT DYNAMIC EVENTS

As illustrated by the selected examples described above, cells adhering to a foreign surface can perceive a number of features related to surface chemistry, topography or rigidity and integrate all information to select behavioural pathways. Since it is unlikely that cells view these parameters as we do, a major challenge is to understand the general mechanisms of data processing they use. A general problem is that a given cell perturbation will affect hundreds of different parameters, making it difficult to identify clearcut causal phenomena (provided they actually exist !). A possible strategy to achieve this goal might consist of identifying early phenomena determining long-term events, such as differentiation or proliferation monitored after a few days. As briefly sketched above, cell shape and cytoskeletal organization are good candidates since much evidence support the view that they are causally related to both long-term cell behaviour and substratum structure. Thus, it seems warranted to investigate the processes by which adherent substrata influence cell properties, with a special interest in shape and cytoskeletal organization. During the last years, much information was obtained on cell changes detected a few minutes or more after encounter with foreign surfaces. However, relatively little information is available on the cell response observed during the first seconds or tens of seconds following such encounters. We suggest that this study might prove rewarding, since causal relationships may be easier to detect when there is a short time interval between stimuli and responses.

A first question is to know how long it takes a cell encountering a surface to initiate a specific behavioural response. Previous studies done on cell adhesion suggest that metabolic events [START_REF] Pierres | Dynamic adhesion of CD8positive cells to antibody-coated surfaces the initial step is independent of microfilaments and intracellular domains of cell-binding molecules[END_REF] and cooperation between adhesion molecules [START_REF] Garcia | Integrinfibronectin interactions at the cell-material interface : initial integrin binding and signaling[END_REF][START_REF] Taubenberger | Revealing early steps of α2β1 integrin-mediated adhesion to collagen type I by using single-cell force spectroscopy[END_REF] are less important during the first tens of seconds after contact. Thus, it might be feasible to identify immediate consequences of cell-surface interaction by focussing on the first minutes following contact. For the sake of clarity, we shall discuss separately bulk membrane motion at interfaces, forces potentially generated by this motion, and lateral redistribution of membrane molecules at interfaces as a key determinant of signaling processes.

Bulk membrane motion at interfaces. Understanding how cells perceive foreign surfaces requires to know how the cell membrane will make contact with its environment. During the last decades, much information was obtained with at least three complementary techniques. Electron microscopy certainly provided the most accurate information. Unfortunately, the need to subject cells to fixation procedures precludes any real-time observation. Interference reflection microscopy (IRM) [START_REF] Curtis | The mechanism of adhesion of cells to glass[END_REF] also denominated as reflection interference contrast microscopy (RICM) [START_REF] Simson | Membrane bending modulus and adhesion energy of wild-type and mutant cells of Dictyostelium lacking talin or cortexillins[END_REF] allows real-time observation of the distance between a cell and a planar surface with a few nanometer accuracy, while the lateral resolution is not better than several tenths of a micrometer. The interest of this method is that no staining procedure is required. Total internal reflection microscopy (TIRF) takes advantage of evanescent waves to illuminate a region of 100-200 nm thickness adjacent to a planar glass surface. After proper labelling of the extracellular medium [START_REF] Gingell | Topography of cell-glass apposition revealed by total internal reflection fluorescence of volume markers[END_REF] or the cell membrane [START_REF] Döbereiner | Dynamic phase transitions in cell spreading[END_REF] , it is possible to achieve real-time determination of the motion of membrane along the surface. Although different cell populations may display widely different behaviour, a general trend is as follows: several minutes to hours after sedimentation on a surface, a cell may begin extending membrane protrusions parallel to the surface. They may be sheet-like lamellipodia or thin filopodia (Fig. 1). It has long been reported that well defined mediators were involved in the choice between different shapes [START_REF] Ridley | Cell migration : integrating signals from front to back[END_REF] : thus, the small GTPase Rac was reported to induce lamellipodium generation with a branched organization of actin microfilaments, while the small GTPase Cdc42 was found to initiate the extension of cylindrical filopodia shaped by a microfilament bundle. The choice between lamellipodium or filopodium formation may be influenced by substratum properties such as density of binding sites 10 , topography [START_REF] Dalby | Fibroblast reaction to island topography : changes in cytoskeleton and morphology with time[END_REF] or rigidity [START_REF] Engler | Matric elasticity directs stem cell lineage specification[END_REF] . A further point is that the cell margin was often reported to display fluctuations with periods of progress and retraction [START_REF] Döbereiner | Dynamic phase transitions in cell spreading[END_REF] . A typical period was several tens of seconds, and the reported velocity of the cell margin is of order of several tens of nanometers per second. Notably, when the density of adhesive points is high enough, this fluctuating behaviour may be replaced with a smooth progression. Now, a key point is to know how a cell can select the kind of motion it will display. At least two different mechanisms may be suggested: (i) cells might continuously form a low number of protrusions of varying morphology. Contact with the substratum might lead to reinforcement or inhibition trough a positive or negative feedback. (ii) Alternatively, the acquisition of a particular motile behaviour might be induced during an early phase of cell-substratum interaction as a consequence of some internal switch [START_REF] Pankov | A Rac switch regulates random versus directionally persistent cell migration[END_REF] .

Although it is not yet feasible to chose between aforementioned hypotheses, it seems reasonable to investigate the early phenomena following cell-to-surface encounter and preceding the morphological changes associated to spreading. Thus, it seems desirable to achieve a quantitative description of the motion and mechanical properties of the surface of an isolated cell in order to predict the consequences of interaction with a surface of known structure. While it has long been shown with electron microscopy that cell membranes are studded with numerous cylindrical or sheet-like protrusions appearing as folds of the plasma membrane, less information is available on the kinetic and mechanical properties of the membrane. The typical thickness of these protrusions or microvilli is about 0.1 µm, and length may range between a few tenths of micrometers and several micrometers. Since this value is far higher than the length of typical adhesion receptors, it is not surprising that the initial interaction between cells and surfaces involves the tip of microvilli [START_REF] Jones | Rapid modification of the morphology of cell contact sites during the aggregation of limpet haemocytes[END_REF] . Now, there remains to understand the dynamics of the cell surface immediately before adhesion. Recently, microscopic studies based on IRM/RICM suggested that the membranes of phagocytes approaching adhesive surfaces displayed fluctuations of higher than 1 Hz frequency and several nanometer amplitude [START_REF] Zidovska | Brownian motion of nucleated cell envelopes impedes adhesion[END_REF][START_REF] Pierres | How cells tiptoe on adhesive surfaces before sticking[END_REF] . A typical map of cell topography and dynamics near a surface is shown on Fig. 2. Unfortunately, the lateral resolution of IRM/RICM may be insufficient to yield accurate information on the motion of individual microvilli. Also, the mechanical properties of these surface protrusions remain poorly understood. In two sets of experiments based on micropipette and biomembrane force probe, it was shown [START_REF] Shao | Static and dynamic lengths of neutrophil microvilli[END_REF][START_REF] Evans | Nano-to Microscale dynamics of P-selectin detachment from leukocyte interfaces. I. Membrane separation from the cytoskeleton[END_REF] that blood neutrophil microvilli could withstand a pulling force of about several tens of pN before separation between the membrane and underlying cytoskeleton and formation of a lipid tether. Clearly, more information is required to help us determine the kinetics of cell-to-substratum initial contacts together with the intensity of generated forces. This knowledge is important since local molecular organization and signal generation are expected to be strongly influenced by these parameters.

Forces between cell and substratum.

A basic question we must address is to know what force a cell membrane will perceive when approaching a foreign surface. While numerous nonspecific interactions such as electrodynamic or electrostatic forces are likely to occur, it seems acceptable to focus on two dominant phenomena: steric repulsion and specific ligand-receptor interactons [START_REF] Vitte | Is there a predictable relationship between surface physical-chemical properties and cell behaviour at the interface ?[END_REF] .

Steric repulsion.

As previously reviewed [START_REF] Robert | Glycocalyx Regulation of Cell Adhesion[END_REF] , it is well established that essentially all living cells are coated with a polysaccharide-rich layer of widely varying thickness, ranging between a few tenths of a micrometer and several micrometers. This is called the glycocalyx. This highly hydrophilic layer will impede close approach between the plasma membrane and a nearby surface. Therefore, it is usually considered as antiadhesive, although in some cases the outermost carbohydrate group may bind to lectin-like receptors exposed on adjacent surfaces. The glycocalyx may involve huge polysaccharides or proteoglycans with a molecular weight higher than 1,000,000 dalton. Also, particularly on white blood cells, it includes large mucin-like molecules that have been well identify. The most important examples may be leukosialin (CD43) and CD45.

Clearly, it would be desirable to know the distance dependence of repulsion generated by the glycocalyx. This is difficult in view of the heterogeneity of glycocalyx components. However, a major point that emerged nearly a decade ago [START_REF] Patel | P-selectin must extend a sufficient length from the plasma membrane to mediate rolling of neutrophils[END_REF][START_REF] Sabri | Influence of surface charges on cell adhesion : difference between static and dynamic conditions[END_REF] is that this repulsion exhibits a strong time-dependent decay that may be due (i) to an internal reorganization of repulsive chains (this has not been well demonstrated todate) and (ii) to an egress of repulsive molecules from contact areas [START_REF] Soler | Leukosialin (CD43) behavior during adhesion of human monocytic THP-1 cells to red blood cells[END_REF][START_REF] Leupin | Exclusion of CD45 from the T-cell receptor signaling area in antigenstimulated T lymphocytes[END_REF] . As will be discussed in the next section, this point is of paramount importance since it may strongly influence the outcome of cell-surface interaction.

Molecular attractive bonds.

As recently reviewed [START_REF] Pierres | Dissecting individual ligandreceptor bonds with a laminar flow chamber[END_REF][START_REF] Robert | What is the biological relevance of the specific bond properties revealed by single molecule studies ?[END_REF] , the formation and dissociation of bonds between surface-attached molecules was subjected to considerable scrutiny during the last decade. A thorough description of these phenomena would not fall into the scope of the present paper and we shall only summarize essential conclusions.

A few years ago, it seemed reasonable to consider that the outcome of an interaction between two surfaces bearing cognate ligand and receptor molecules could be satisfactorily described by two parameters: -The rate of bond dissociation k off (F) as a function of force exerted on the bond. In many circumstances, it appeared that k off (F) followed so-called Bell's law:

k off (F) = k 0 exp(F/F 0 ) (1)
Many experiments performed at the single molecule level with different tools such as laminar flow chambers, atomic force microscopes, biomembrane force probes or optical traps yielded for parameter F 0 values usually ranging between several piconewtons and several tens of piconewtons. This is the order of magnitude of the force that can be exerted by a bond linking two surfaces subjected by a disruptive force.

-The rate of bond formation k on when surfaces are at binding distance. This parameter proved much more difficult to measure, and even to define, than the rate of bond dissociation, and new methods might bring substantial progress in the near future [START_REF] Chen | Monitoring receptor-ligand interactions between surfaces by thermal fluctuations[END_REF] . A major problem is that the probability of bond formation between two surfaces bearing ligands and receptors is proportional to the number of receptor-ligand couples that are close enough to interact. Since the height of membrane asperities is often much larger than the length of typical adhesion receptors, the number of interacting molecules is strongly dependent on the details of membrane-to-surface alignment. Indeed, surface roughness was shown to change binding frequencies by nearly two orders of magnitude [START_REF] Williams | Quantifying the impact of membrane microtopology on effective twodimensional affinity[END_REF] .

Recently, another difficulty was recognized. Dissecting individual ligandreceptor couples made more and more obvious the concept that bond formation is a multiphasic process involving numerous intermediate binding states [START_REF] Pierres | Dissecting streptavidinbiotin interaction with a laminar flow chamber[END_REF][START_REF] Pincet | The solution to the streptavidin-biotin paradox : the influence of history on the strength of single molecular bonds[END_REF][START_REF] Marshall | Force history dependence of receptor-ligand dissociation[END_REF] . This means that bond formation may not be viewed as an all-or-none phenomenon, and the force that can be sustained by a newly formed bond is highly dependent on its history. Thus, it was recently found that adhesion molecules such as cadherins could form associations of widely different strength, with a spontaneous lifetime ranging between at least a few milliseconds and several seconds [START_REF] Pierres | Dissecting subsecond cadherin bound states reveals an efficient way for cells to achieve ultrafast probing of their environment[END_REF] .

Thus, when a cell membrane is close to a ligand-bearing surfaces, the frequency of bond formation and the force exerted by newly formed bonds on the membranes is dependent on complex binding properties that could be understood and measured only very recently. Clearly, this new information must be incorporated in a theoretical framework aimed at explaining how cell membranes perceive the presence of a potentially adhesive surface.

Signaling in contact zones: importance of lateral reorganization of membrane molecules. Clearly, the basic problem addressed in this review is to understand which signaling cascades will be generated by membrane-tosurface interactions.

In view of the above discussion, mechanical forces exerted on the cell membrane may generate signaling cascades through several mechanisms. Indeed, it has long been shown that membrane tension may activate calcium channels through direct interaction with lipid bilayers [START_REF] Kung | A possible unifying principle for mechanosensation[END_REF] . Also, it recently became clear that some adhesion molecules such as integrins are flexible machines liable to display large deformations resulting in exposition of new antigenic sites [START_REF] Luo | Structural basis of integrin regulation and signaling[END_REF] . Clearly, this process might result in formation of docking sites for a variety of signaling molecules. Thus, it is not surprising that mechanical forces exerted on cells were often found to generate multiple biochemical processes such as calcium rise [START_REF] Horoyan | Localization of calcium and microfilament changes in mechanically stressed cells[END_REF] or phosphorylation [START_REF] Matthews | Cellular adaptation to mechanical stress : role of integrins, rho, cytoskeletal tension and mechanosensitive ion channes[END_REF] .

However, the main mechanism responsible for signal generation as a consequence of membrane-to-substratum interaction may well be the lateral segregation of membrane molecules. Indeed, due to the huge number of potential interactions between cell molecules [START_REF] Uetz | A comprehensive analysis of protein-protein interactions in Saccharomyces cerevisia[END_REF] , generating encounters between enzymes and potential targets may be sufficient to initiate a biochemical cascade. Thus, integrin clustering is likely to play an important role in signal generation after integrin engagement [START_REF] Giancotti | Integrin signaling[END_REF] . Also, some evidence suggests that the mere passage of T lymphocyte receptors in a small phosphatasefree zone might increase phosphorylation of activating sites and recruitment of kinases [START_REF] Burroughs | Ligand detection and discrimination by spatial relocalization : a kinase-phosphatase segregation model of TCR activation[END_REF] .

As a consequence, several different mechanisms might play a role in the perception of an adhesive substratum by a cell: i) Clusters of binding sites for membrane receptors might result in receptor clustering.

ii) The rearrangement of mobile repulsive molecules might result in phase separation and additional segregation of membrane molecules [START_REF] Bruinsma | Adhesive switching of membranes : experiment and theory[END_REF] . iii) Modulation of membrane molecule diffusion by topographic structures [START_REF] Mossman | Altered TCR signaling from geometrically repatterned immunological synapses[END_REF] might further alter the formation of molecular complexes.

Thus, available evidence suggests potential mechanisms for signal generation during the earliest phase of interaction between a cell and a foreign surface.

CONCLUSION AND PERSPECTIVE.

As summarized in the first part of this review, there is now ample evidence that cells adhering to a surface integrate several properties including chemistry, micrometer-and nanometer-scale topography, and mechanical properties to determine future behaviour. However, relating substratum properties to alterations of the expression of hundreds of genes as a consequence of the perturbation of a complex network of biochemical reactions seems a formidable task.

As indicated in the second part of this review, a possible way of simplifying this challenge may be provided by the frequent observation that important aspects of cell behaviour observed days or weeks after interaction with a surface are tightly related to modifications of cell shape and cytoskeletal organization that may be observed several minutes or days after adhesion. Since much progress was recently achieved in unraveling the mechanisms of cell spreading on a surface, it seems warranted to look for a better understanding of the relationship between substratum properties and cell shape. This is still a most difficult goal since even during the minutes an hours following cell adhesion a huge number of signaling cascades may be triggered.

As briefly sketched in the third part of this review, a possible way of progressing further might consist of investigating the earliest steps of cell-to-substratum interaction. Indeed, relating substratum structure to the phenomena occurring during the first seconds following contact might be conceptually easier than linking this structure to delayed events. The main question is to determine which parameters a cell is really probing. Thus, while it is well accepted that substratum rigidity strongly influences cell behaviour, the very stimulus responsible for cell response is not well understood. Indeed, if cells are sensitive to tension, there remains to understand how the tension generated by a cell is related to substrate resistance to force (is elasticity, or viscosity, or a combination of both the parameter to consider ?). Are the substratum resistance to pulling or pushing forces of similar importance? A logical way of addressing this problem is to try to relate substratum structure to signal generation, since the perception of a given environmental cue may be considered as equivalent to the signal it will generate. A requirement to approach this goal is to obtain a detailed figure of cell spontaneous motion in the vicinity of a potentially adhesive surface. Much progress was recently done in this domain.

Therefore, it is hoped that the suggested research line might be rewarding. However, a point of caution may be useful: while most studies were done on cells deposited on a 2-dimensional surfaces, it must be kept in mind that in many cases a 3dimensional environment should be more relevant physiologically [START_REF] Cukierman | Taking cell-matrix adhesions to the third dimension[END_REF][START_REF] Mao | Stimulatory effects of a three-dimensional microenvironment on cell-mediated fibronectin fibrillogenesis[END_REF][START_REF] Jiang | Cell-matrix entanglement and mechanical anchorage of fibroblasts in three-dimensional collagen matrices[END_REF] . Despite this restriction, the exquisitely accurate pieces of information that can be obtained on cells interactions with surfaces should strongly increase our understanding of the way cells perceive their environment in the near future.

Figure 1 .

 1 Figure 1. Studying the morphology of cell-tosubstratum contact extension with interference reflection microscopy. Human T lymphocytes were deposited on glass surfaces coated with nonactivating anti-HLA antibodies (A, C, E) or activating anti-CD3 antibodies (B, D, F). Cell morphology was monitored with standard microscopical observation (A, B) and interference reflection 15 minutes (C, D) or 30 minutes (E, F) after deposition. Clearly, contact extension was mediated by lamellipodia or filopodia depending of substratum structure. Bar length is 2 µm.

Figure 2 .

 2 Figure 2. Three-dimensional reconstruction of cell surface morphology and dynamics near an adhesive surface. Human monocytic THP-1 cells were deposited on fibronectin-coated surfaces and observed with interference-reflection microscopy. The shape of the cell membrane is shown as a coded-colour map (A) or a 3-D drawing (B) together with the amplitude of spontaneous membrane fluctuations. Bar length is 2 µm. See ref 44 for more details.


Joffrey Wolf

Dr Agnès Labande
email: labande@lcc-toulouse.fr

Michael Natella

Jean-Claude Daran

Rinaldo Poli

Aryl Grignard Cross-Coupling of Aryl Chlorides Catalysed by New, Highly Active Phosphine/Imidazolium Nickel(II) Complexes

Keywords: Nickel, Phosphine, N-heterocyclic carbene, C-C coupling, Grignard

Two new zwitterionic Nickel(II) complexes, bearing phosphine/imidazolium ligands, have been prepared. Their catalytic activity, and the activity of an analogous Ni(II) complex, previously described by us, has been evaluated with a range of aryl chlorides, an aryl bromide and arylmagnesium halides. The catalytic activity is related to the length of the tether between the carbene and phosphine moieties. Thus the catalysts possessing a 6-membered metallacycle are more active than that having a 7-membered metallacycle. Changing from a mesityl (Mes) to a 2,6-diisopropylphenyl (DIP) on the imidazole ring does not strongly influence the activity. The three complexes show moderate to very good activities with most substrates, and enhanced selectivities compared to previously published Ni(II)/N-heterocyclic carbene systems. The low activity observed with 4-chlorobenzotrifluoride seems to be related to the presence of a phosphine on the ligands.

Introduction

The development of very active, yet robust catalysts for application to organic synthesis is still a challenge. Complexes bearing N-heterocyclic carbenes are actively studied nowadays, because of their high thermal stability, and because metal-NHC bonds have high dissociation energies [1][2][3]. On the other hand, phosphine complexes of nickel(II) are very active in the C-C coupling reaction between aryl halides and aryl Grignard reagents (Kumada-Corriu reaction), but it is not rare to observe some ligand dissociation [4][5][6][7][8][9][10][11]. Consequently, mixed phosphine/N-heterocyclic carbene nickel complexes should be good candidates as catalysts for C-C coupling reactions [12][13][14][15][16][17]. To date, only two Ni(II)/N-heterocyclic carbene systems have been successfully tested in the reaction of aryl chlorides with aryl Grignard reagents [18].

We have recently described the preparation of two zwitterionic nickel(II) complexes, bearing a phosphine/imidazolium ligand (1a and 1d) [19]. These complexes proved very active as precatalysts for the coupling of phenylmagnesium chloride and 4-chloroanisole, giving an essentially total conversion of the starting aryl chloride after 1 h at room temperature, with only 3 mol% of precatalyst.

Based on these results, we wanted to further explore the field and study the scope and limitations of our systems in the Kumada-Corriu reaction. In the present paper, we describe the preparation and characterisation of two new members of this family of nickel(II) phosphine-imidazolium complexes, and the evaluation of their catalytic activity with a range of aryl chlorides, one aryl bromide and arylmagnesium halides.

Experimental

General comments

All reactions were carried out under a dry argon atmosphere using Schlenk glassware and vacuum line techniques. Solvents for syntheses were dried and degassed by standard methods before use. Elemental analyses were carried out by the analytical service of the Laboratoire de Chimie de Coordination in Toulouse. 1 H data were recorded on Bruker AC200, AM-250 and AV-500 spectrometers, operating at 200, 250 and 500 MHz, respectively. 13 C{H, P}and 31 P{H} NMR data were recorded on the Bruker AV-500 instrument, operating at 125.8 and 202.5 Hz respectively. The spectra were referenced internally using the signal from the residual protiosolvent ( 1 H) or the solvent signals ( 13 C), and externally using 85% H3PO4 for 31 P. Mass spectra were obtained from acetonitrile or methanol solutions on a TSQ7000 instrument from ThermoElectron (electrospray ionisation or chemical ionisation), and from DMSO or DMF solutions on a Nermag R10-10 instrument (FAB). GC chromatograms were recorded on a Fisons 8000 Series GC equipped with a SPB-5 capillary column and the products were identified by comparison with authentic samples. Commercial chemicals were from Acros, Aldrich, Alfa Aesar, Avocado or Fluka and used as received. N- (2,4,6-trimethylphenyl)imidazole 2a and N-(2,6diisopropylphenyl)imidazole 2b were prepared as described by Arduengo [20] and Johnson [21]. Compound 1a was synthesised as previously described [19].

Synthesis of ligands 1-(2-hydroxyethyl)-3-(2,6-diisopropylphenyl)-imidazolium bromide 3b

2-Bromoethanol (2.5 mL, 35 mmol) was added to a toluene solution (150 mL) of 2b (6.2 g, 27.2 mmol). The mixture was stirred at 120°C for 20 h. During this time a yellow-orange oil separated out in the flask. The mixture was cooled to room temperature and the toluene phase was removed. The oil was extracted with CH2Cl2 (20 mL), the resulting solution was slowly precipitated in ether (400 mL), and the pale yellow solid which formed was filtered and dried in vacuo. Yield: 6.83 g (72 %). 1

1-(3-hydroxy-n-propyl)-3-(2,4,6-trimethylphenyl)imidazolium bromide 3c

3-Bromo-1-propanol (2.2 mL, 24.0 mmol) was added to a toluene solution (60 mL) of 2a (3.72 g, 20.0 mmol). The mixture was stirred at 120°C for 15 h. During this time a cream solid precipitated in the flask. The suspension was cooled to room temperature, the solid was decanted, washed with toluene (2 x 15 mL) and dried in vacuo. A white solid was obtained. Yield: 4.47 g (69 %). 1 H NMR (, 250 MHz, CDCl3): 9.64 (s, 1H, N-CH-N), 7.97 (t, J = 1.5 Hz, 1H, MesN-CH=C), 7.16 (t, J = 1.5 Hz, 1H, AlkN-CH=C), 6.99 (s, 2H, CH (Mes)), 4.85 (t, J = 4.9 Hz, 2H, N-CH2), 4.03 (t, J = 4.8 Hz, 2H, CH2-O), 3.54 (br, 1H, OH), 2.33 (s, 3H, p-was dried (MgSO4), filtered and the solvent was removed in vacuo to give a pale yellow solid. Yield: 3.37g (87 %). 1

1-(2-diphenylphosphinoethyl)-3-(2,6-diisopropylphenyl)imidazolium bromide 5b

KPPh2, freshly made from t-BuOK (337 mg, 3.00 mmol) and HPPh2 (0.54 mL, 3.15 mmol) in DMSO (8mL), was added to a DMSO solution (8 mL) of 4b (1.19 g, 2.86 mmol).

The solution was allowed to stir for 2 h at room temperature. The solvent was then removed under vacuum. Methanol (5 mL) was added to quench the excess KPPh2, then removed under vacuum. Dichloromethane (10 mL) was added and the resulting mixture was filtered and diethyl ether (100 mL) was added. The resulting precipitate was separated by filtration and dried in vacuo. The product was obtained as a white, air-sensitive solid. Yield: 1.37 g (92 %). Hz, 2H, N-CH2), 2.84 (t, J = 6.8 Hz, 2H, CH2-P), 2.25 (h, J = 6.8 Hz, 2H, CH-(CH3)2), 1.17 (d, J = 6.6 Hz, 6H, CH-(CH3)2), 1.07 (d, J = 6.9 Hz, 6H, CH-(CH3)2); 31 P NMR (, 81 MHz, CDCl3): -20.77 (CH2-PPh2) (data consistent with those found in the literature [13]).

1-(3-diphenylphosphinopropyl)-3-(2,4,6-trimethylphenyl)imidazolium bromide 5c

KPPh2, freshly made from t-BuOK (152 mg, 1.35 mmol) and HPPh2 (0.25 mL, 1.42 mmol) in DMSO (3mL), was added to a DMSO solution (2 mL) of 4c (500 mg, 1.29 mmol).

The solution was allowed to stir for 15 h at room temperature. The solvent was then removed under vacuum. Methanol (5 mL) was added to quench the excess KPPh2, then removed under vacuum. Dichloromethane (10 mL) was added and the resulting mixture filtered. The filtrate was concentrated (ca. 1.5 mL) and diethyl ether (20 mL) was added. The resulting precipitate was separated by filtration and dried in vacuo. The product was obtained as a white, airsensitive solid. Yield: 500 mg (79%). 1 MHz, CDCl3): -14.1 (s, CH2-PPh2) (data consistent with those found in the literature [13]).

3. Synthesis of Nickel(II) complexes: 1-(2-diphenylphosphinoethyl)-3-(2,6-diisopropylphenyl)imidazolium nickel tribromide 1b

NiBr2(DME) (410 mg, 1.33 mmol) was added to a solution of 5b (700 mg, 1.34 mmol) in THF (20 mL). The mixture was stirred at room temperature for 30 min to give a green solution. The solvent was removed under vacuum and CHCl3 (20 mL) was added. The mixture was filtered and the solvent removed to give a green solid. Yield: 953 mg (97 %). 1

1-(3-diphenylphosphinopropyl)-3-(2,4,6-trimethylphenyl)imidazolium nickel tribromide 1c

NiBr2(MeCN)2 (312 mg, 1.01 mmol) was added to a suspension of 5c (435 mg, 0.88 mmol) in THF (10 mL). The mixture was stirred at room temperature for 1 h. The precipitate was filtered, the green solid/oil obtained was washed with THF (2 x 5 mL), CHCl3 (2 x 10 mL) and dried under vacuum to give a green solid. Yield: 612 mg (97 %). Green crystals were obtained by diffusion of diethyl ether into an acetonitrile solution. 1

Standard procedure for Kumada-Corriu reactions:

The aryl halide (1 mmol) and diethyleneglycol-di-n-butylether (30 μL as internal standard) were mixed with 1a, 1b or 1c (3 mol % based on aryl halide) in THF (1 mL). The arylmagnesium halide (1.5 mmol) was slowly added to the mixture. Reactions were kept at room temperature (between 23-25°C) for 18 h and quenched with a minimum of ethanol. In order to remove the catalyst and the magnesium salt, mixtures were filtered on silica with ether as solvent. The resulting mixtures were analysed by gas chromatography.

X-ray analyses:

A single crystal was mounted under inert perfluoropolyether at the tip of a glass fibre and cooled in the cryostream of an Oxford-Diffraction XCALIBUR CCD diffractometer. Data were collected using the monochromatic MoK radiation (= 0.71073). The Final unit cell parameters were obtained by the least-squares refinement of a large number of selected reflections. Only statistical fluctuations were observed in the intensity monitors over the course of the data collection.

The structure was solved by direct methods (SIR97 [START_REF] Altomare | SIR97 a program for automatic solution of crystal structures by direct methods[END_REF]) and refined by least-squares procedures on F 2 with the SHELXL-97 program [START_REF] Sheldrick | SHELX97, Programs for Crystal Structure Analysis (Release 97-2)[END_REF] using the integrated system WINGX(1.63) [24]. Hydrogen atoms attached to carbon atoms were introduced at calculated positions and treated as riding on their parent atoms [d(CH)= 0.96-0.98 Å] with a displacement parameter equal to 1.2 (C6H5, CH2, OH) or 1.5 (CH3) times that of the parent atom. Owing to the poor quality of the crystal, only reflections with theta below 23.3° were used in the refinement procedure. The Molecular view was realised with the help of ORTEP-III [25] . Crystal data and refinement parameters are shown in Table 1.

Table 1.

Crystallographic data (excluding structure factors) have been deposited with the Cambridge Crystallographic Data Centre as supplementary publication no. CCDC 299074.

Copies of the data can be obtained free of charge on application to the Director, CCDC, 12 Union Road, Cambridge CB2 1EZ, UK (fax: (+44) 1223-336-033; e-mail: deposit@ccdc.cam.ac.uk).

Results and discussion

Following our recent discovery that the nickel(II) complexes 1a and 1d are excellent precatalysts for the coupling of phenylmagnesium chloride and 4-chloroanisole [19], it was of interest to see whether small structural changes in the phosphine-imidazolium ligand could affect the properties or catalytic activity. For this purpose, we considered two modifications:

(i) replacement of the ortho substituents on the aryl group of the imidazolium moiety by bulkier isopropyl substituents; (ii) an increased length of the tether from 2 to 3 carbons, thus forming a 7-membered metallacycle instead of a 6-membered one in the putative phosphinocarbene complex (Figure 1).

Ligands 5b and 5c were prepared by a natural extension of the procedure previously described for ligand 5a [19]. Thus, 2-bromoethanol reacted with N-(2,6diisopropylphenyl)imidazole to give imidazolium salt 3b in good yield. Analogously, the reaction of 3-bromopropanol with N-mesitylimidazole gave imidazolium salt 3c in good yield (in this case, the procedure described by Fürstner [26] was slightly modified and allowed us to improve the yield from 28% to 69%). Bromination of 3b and 3c with PBr3 gave compounds 4b and 4c. Finally, nucleophilic substitution with KPPh2 gave the expected imidazolium salts 5b and 5c, with Br -as the anion, in good to excellent yields. Zwitterionic nickel(II) complexes 1b and 1c were obtained, like the previously reported 1a [19], in excellent yields by stirring a mixture of the desired ligand and NiBr2L2 for 1 h in THF at room temperature (Scheme 2). Like 1a, the two new complexes are green, paramagnetic compounds, with a distorted tetrahedral geometry. They have been characterized by NMR, elemental analysis and mass spectrometry, and by an X-Ray diffraction study for 1c (Figure 2). The paramagnetism is indicated by the large paramagnetic shift of certain resonances in the 1 H NMR spectrum. Unfortunately, we did not manage to isolate the corresponding carbenic Ni(II) complexes, obtained by deprotonation of the imidazolium moiety in the presence of LiN(SiMe3)2. However, the deprotonation of 1a-c induced a colour change to yellow-brown, consistent with the formation of diamagnetic, square-planar complexes. An NMR monitoring of the deprotonation of 1a confirmed the formation of nickel phosphine-carbene complexes. Indeed, the 31 P NMR spectrum shows two sharp, very close signals at 3.28 ppm (major) and 2.99 ppm (minor), in addition to a sharp signal at -18.3 ppm which could correspond to the free phosphine. In the 13 C NMR spectrum, two small doublets were observed in the typical region of coordinated NHC's at respectively 168.3 ppm (JPC = 108.2 Hz) and 159.85 ppm (JPC = 35.2 Hz). A selective 31 P decoupling experiment showed that these two doublets are coupled to the phosphorus nuclei resonating at 3.28 and/or 2.99 ppm, but not to the one at -18.3 ppm. The presence of two doublets, with very different JPC coupling constants, could be explained by the formation of two carbenic complexes (Figure 1): a monomer, bearing the chelating carbene-phosphine ligand (cis coupling), and a dimer, with the carbene and the phosphine being in trans on the nickel. Moreover, we did not observe the characteristic 1 H NMR signal at ca. 10.5 ppm corresponding to the acidic proton of the imidazolium moiety. The 31 P NMR spectrum of the deprotonation product of 1c shows a similar pattern, with two close singlets at -5.2 and -5.5 ppm, and a very small signal at -15.0 ppm.

A view of compound 1c is illustrated in Figure 2. The Ni atom is tetrahedrally coordinated by the phosphorus atom of the phosphine group and by three bromine atoms. The Ni-P distance is comparable with the value found in [NiBr3(PPh3)] -(C24H20As) + (2.323(1) Å ) [27]. The Br atoms are rotationally staggered with respect to the phenyl and the imidazolium groups. The three Br-Ni-Br angles show a significant deviation from their mean, 112.8(1)°, with two values greater than the ideal tetrahedral angle. The mesityl group is found roughly orthogonal to the imidazolium ring with a dihedral angle of 88.2(4)°, as previously observed in [1][2]4,imidazolium-3-{CH2C(t-Bu)=N(i-Pr)}]bromide [28]. The internal bond lengths and angles of the imidazolium ring are unexceptional and lie within the range expected [29]. It is interesting to note that there are three weak intermolecular C-H…Br (1) hydrogen bond interactions (Table 2), the shortest one being with the acidic imidazolium proton.

The new complexes, together with the previously reported complex 1a, were tested as precatalysts for the coupling reaction between a variety of aryl chlorides and Grignard reagents, extending the previous study of the coupling of 4-chloroanisole with phenylmagnesium chloride catalyzed by 1a [19]. The corresponding trichloride complex 1d was not further investigated, because our previous study showed that the activity does not depend on the nature of the halide ligand. As we have previously shown [19], the catalytic results are independent on whether the Grignard reagent or an external base (e.g. LiN(SiMe3)2) is used to deprotonate the precatalysts. Therefore, we consider that a chelating phosphine-carbene coordination mode is present in the catalyst as indicated by the NMR results described above. The results are given in Table 3. A more detailed analysis of the product distribution yields the results shown in Tables 4 and5. Some general trends emerge from this work. Complex 1a, bearing a mesityl group on the imidazolium moiety, is a bit more active than 1b. Both complexes, with the notable exceptions of the couplings involving the CF3-substituted aryl chloride substrate (runs 2 and 8), also show an equal or slightly better activity than the Ni(acac)2/IPr system [18], as can be seen from the kinetic studies (Figures 3 and4). Indeed, in the reaction of 4-chloroanisole with phenylmagnesium chloride, the 4-chloroanisole conversion reached 95% with 1a and 82 % with 1b after 1 h (Figure 3). On the other hand, complex 1b was a bit more selective than 1a and gave less by-products (Tables 4 and5) [START_REF]the reaction of a Grignard reagent with 1-chloro-2,6-dimethylbenzene was carried out but only gave 5 to 10[END_REF]. Both are more selective than the Ni(acac)2/IPr system.

In most cases, complex 1c showed a poorer activity than 1a or 1b, as can be seen from Table 3 and Figure 3. It seems that the formation of a 7-membered cycle slows the reaction down. For several substrates, complex 1c also shows a reduced selectivity, even relative to the Ni(acac)2/IPr system. One noticeable exception was observed with 4-chlorobenzotrifluoride, where conversions were low, with up to 71% starting material left for entry 2 (Table 4) and 66% for entry 8 (Table 5). A similar behaviour was reported for a monophosphine-based system, namely Ni(acac)2/P(t-Bu)3 (up to 53% starting material left), whereas the use of a simple imidazolium salt gave excellent conversions under the same conditions (Table 4, entry 2) [18]. A GC monitoring of this reaction provided insight as to the possible cause of this reduced activity.

The rate of conversion is initially high but dramatically drops after ca. 2 h. This result clearly indicates that the catalyst is rather active at the early stages of the reaction but dies as the reaction progresses. This stability problem appears closely related to the simultaneous presence of CF3 groups and a phosphine in the ligand system.

Finally, the presence of ortho substituents relative to the chlorine atom had a dramatic effect on conversion with 1c, but a much less pronounced effect with 1b. A curious effect, for which we cannot offer an obvious rationalization, is the significantly lower activity of 1a relative to 1b for entry 11, whereas about the same activity of the two catalysts is observed for entry 5.

Conclusion

Zwitterionic complexes of Ni(II), bearing phosphine/imidazolium ligands, have been prepared and show high activities and selectivities for the coupling of aryl chlorides with aryl Grignard reagents at room temperature. We have observed that small structural changes, such as increasing the length of the ligand tether from 2 to 3 carbons, could have a dramatic effect on the catalytic activity. Very high conversions have been achieved, except in the case of 4chlorobenzotrifluoride, where the presence of a phosphine moiety in the catalyst seems to hamper the reaction. We are now studying the coordination chemistry of these ligands on other metals, and the catalytic properties of the resulting complexes will be tested in due course. (i) x-1/2,y,-z+3/2 ; (ii) -x+2,y+1/2,-z+3/2 Table 3. Cross-coupling reactions of aryl chlorides with arylmagnesium chloride or bromide. GC yield (%) of heterocoupling product 8. [a][ b] Entry R 1 E R 2 X Cat. 1a [c] Cat. 1b [c] Cat. 1c [c] IPr/Ni(acac)2 [a] For the meaning of R 1 , R 2 , E and X and for the nature of the possible reaction products, see Scheme 3.

[b] Conditions: 1.0 eq aryl halide, 1.5 eq aryl Grignard, 3 mol% 1a-c, THF, 25°C, t = 18 h.

[c] GC yield using diethyleneglycol-di-n-butylether as the internal standard. The relative yields in percent are normalized to the reagent in defect (6): 6+8+9+10 = 100.

[d] Ref. [18].

  H NMR (, 500 MHz, CDCl3): 9.50 (s, 1H, N-CH-N), 8.19 (s, 1H, (DIP)N-CH=C), 7.38 (t, J = 7.9 Hz, 1H, p-CH (DIP)), 7.15 (d, J = 7.9 Hz, 2H, m-CH (DIP)), 7.09 (s, 1H, AlkN-CH=C), 4.73 (t, J = 4.8 Hz, 2H, N-CH2), 3.96 (br, 1H, OH), 3.84 (t, J = 4.8 Hz, 2H, CH2-OH), 2.15 (h, J = 6.8 Hz, 2H, CH-(CH3)2), 1.01 (d of d, J = 6.8 Hz, 12H, CH-(CH3)2); 13 C NMR (, 125.8 MHz, CDCl3): 145.35 (o-C (DIP)), 137.63 (N-CH-N), 131.75 (p-CH (DIP)), 130.00 (N-C (DIP)), 124.50 (m-CH (DIP)), 124.31 ((DIP)N-CH=), 123.73 (AlkN-CH=), 60.17 (CH2-OH), 52.01 (CH2-N), 28.43 (CH-(CH3)2), 24.33 (CH-(CH3)2), 24.00 (CH-(CH3)2). Mp 169-170°C. Anal. cal. for C17H25BrN2O; C: 57.79, H: 7.13, N: 7.93%; found: C: 57.54, H: 6.90, N: 7.79%. MS (ESI) m/z, (%): 273.15, (100) [C17H25N2O + ]; 79, (100) [Br -].

1 H

 1 NMR (, 200 MHz, CDCl3): 10.20 (s, 1H, N-CH-N), 7.85 (s, 1H, MesN-CH=C), 7.46 (t, J = 7.8 Hz, 1H, p-CH (DIP)), 7.39 (m, 4H, o-CH (C6H5)), 7.30 (m, 6H, m/p-CH (C6H5)), 7.23 (d, J = 8 Hz, 2H, m-CH (DIP)), 7.07 (s, 1H, AlkN-CH=C), 4.84 (d of t, J = 12.4 Hz -J = 6.9

  H NMR (, 200 MHz, CDCl3): 10.35 (t, J = 1.7 Hz, 1H, N-CH-N), 7.75 (t, J = 1.8 Hz, 1H, MesN-CH=C), 7.36 (t, J = 7.8 Hz, 1H, o-CH (C6H5)), 7.28 (m, 6H, m/p-CH (C6H5)), 7.16 (t, J = 1.7 Hz, 1H, AlkN-CH=C), 6.92 (s, 2H, CH (Mes)), 4.76 (d of t, J = 25.6 Hz -J = 6.5 Hz, 2H, N-CH2), 2.97 (d of t, J = 23 Hz -J = 6.5 Hz, 2H, CH2-P), 2.27 (s, 3H, p-CH3), 2.63 (m, 2H, C-CH2-C), 1.98 (s, 6H, o-CH3); 31 P NMR (, 81

Figure 1 .Figure 2 .

 12 Figure 1. Proposed structures of Ni(II) carbenic complexes from 1a: monomer (cis

(TIFF file)Figure 3 .

 3 Figure 3. Conversion of 4-chloroanisole vs time with 1a-c (Table3, entry 4).

Figure 4 .

 4 Figure 4. Conversion of 4-chlorotoluene vs time with 1a-c (Table 3, entry 10) and

Figure 5 .Scheme 1 .

 51 Figure 5. Conversion of 4-chlorobenzotrifluoride vs time with 1a.

Table 4 .

 4 Detailed results of cross-coupling reactions with phenylmagnesium chloride.[a]

	Entry	R	E	Cat. 1a	Cat. 1b	Cat. 1c	IPr/Ni(acac)2
	1	H	N	100/0/0/0/6	100/0/0/0/17	100/0/0/0/21	100/0/0/0/4
	2	4-CF3	C	37/63/0/0/30	53/47/0/0/30	29/71/0/0/27	96/4/0/0/4
	3	4-CH3	C	86/2/5/7/17	96/2/1/1/10	63/32/4/1/15	81/7/3/9/11
	4	4-OCH3	C	95/4/0/1/19	92/6/1/1/22	80/18/1/1/19	71/19/1/9/16
	5	2-CH3	C	78/10/12/0/24	82/11/7/0/20	23/67/10/0/22	73/9/10/8/15
	6	bromomesitylene C	4/64/32/0/27	3/71/26/0/23	7/53/40/0/26	5/80/15/0/12

[a]

The product distribution is given as 8/6/9/10/11; the amount of 10 was approximated by comparison of the peak integrations.

Table 5 .

 5 Detailed results of cross-coupling reactions with 4-anisylmagnesium bromide.[a]

	Entry	R	E	Cat. 1a	Cat. 1b	Cat. 1c	IPr/Ni(acac)2
	7	H	N	100/0/0/0/7	100/0/0/0/6	100/0/0/0/10	100/0/0/0/1
	8	4-CF3	C	36/63/1/0/17	33/66/1/0/15	42/57/1/0/16	100/0/0/0/8
	9	H	C	99/0/0/1/4	98/1/0/1/6	76/23/0/1/4	93/3/3/1/16
	10	4-CH3	C	92/7/0/1/5	87/11/1/0/3	89/10/1/0/5	88/5/4/3/15
	11	2-CH3	C	48/45/6/1/6	87/9/3/1/5	12/83/5/0/5	77/20/0/3/10
	[a] The product distribution is given as 8/6/9/10/11; the amount of 10 was approximated by comparison of the
	peak integrations.

  Scheme 3. Reagents and products for the catalytic Kumada-Corriu reaction (for the results, see Tables3, 4 and 5).

	R 1	E	Cl	+	XMg			R 2	THF, rt, 18h	R 1	E	R 2
		6							7		8
		R 1 by-products:	E	R 1	E	E	R 1	R 2	R 2
							9				10	11
						(i) or (ii)				(iii)	(iv)
			4,6-Me	3 C 6 H 2		3a: R=2,4,6-Me	3 C 6 H 2	4a-c	5a-c
	2b: R=2,6-i-Pr 2 C 6 H 3			n=1
									3b: R=2,6-i-Pr	2 C 6 H	3
										n=1
									3c: R=2,4,6-Me	3 C 6 H 2
										n=2
											n	N
				Br						+
								(i)	or	(ii)	N
							Ar				Ph	2 P
	Ph	2 P		N	+	N					Ar
			n
											Ni
											Br
											Br
											Br
			5a-c
											1a:	Ar	=	2,4,6-Me	3 C 6 H	2
											n	=	1
											1b:	Ar	=	2,6-i-Pr 2 C 6 H	3
											n	=	1
											1c:	Ar	=	2,4,6-Me	3 C 6 H	2
											n	=	2

Scheme 2. Reagents and conditions: (i) NiBr2(DME), THF, rt, 30 min to 1h (1a,b); (ii) NiBr2(MeCN)2, THF, rt, 1h (1c).

Acknowledgements

We are grateful to the CNRS for support of this work and to the "Fonds Social Européen" for a Ph.D. fellowship to JW.

Tables and figures:

N


Jorge Villagra
email: jvillagr@ing.uc3m.es

Brigitte D'andréa-Novel
email: brigitte.dandrea-novel@ensmp.fr

Michel Fliess
email: michel.fliess@polytechnique.edu

Hugues Mounier
email: hugues.mounier@ief.u-psud.fr

Robust grey-box closed-loop stop-and-go control

This paper presents a robust stop-and-go control law, especially well adapted to car following scenarios in urban environments. Since many vehicle/road interaction factors (road slope, rolling resistance, aerodynamic forces) are very poorly known and measurements are quite noisy, a robust strategy is proposed within an algebraic framework. On the one hand, noisy signals will be processed in order to obtain accurate derivatives, and thereafter, variable estimates. On the other hand, a grey-box closed-loop control will be implemented to compensate all kind of unmodeled dynamics or parameter uncertainties.

I. INTRODUCTION A. Generalities

Adaptive cruise control (ACC) and stop-and-go control systems have been deeply studied in recent years [START_REF] Vahidi | Research Advances in Intelligent Collision Avoidance and Adaptive Cruise Control[END_REF]. Let us recall that while ACC automatically accelerates or decelerates the vehicle to keep a quasi-constant target velocity and headway distance, stop-and-go deals with the vehicle circulating in towns with frequent and sometimes hard stops and accelerations. Both situations present completely different comfort and safety constraints, and therefore, in most of the reported works, ACC and stop-and-go problems are treated separately. Some approaches ( [START_REF] Persson | Stop & Go Controller for Adaptive Cruise Control[END_REF], [START_REF] Yi | A Driver-Adaptive Stop-and-Go Cruise Control Strategy[END_REF]) have tried to reproduce human behavior in order to achieve a 'comfort-based' control. Unfortunately, this kind of strategy may not necessarily lead to safe operation (see e.g. [START_REF] Vahidi | Research Advances in Intelligent Collision Avoidance and Adaptive Cruise Control[END_REF]). Besides, external factors such as road characteristics, weather conditions, and traffic load should be taken into account in a robust and safe control system.

Furthermore, an accepted comfort criteria is to guarantee bounded longitudinal accelerations and jerks. Using this idea, many authors (e.g. [START_REF] Brackstone | Car-following: A historical review[END_REF], [START_REF] Chien | Automatic vehicle-following[END_REF], [START_REF] Hiraoka | Modeling of driver following behavior based on minimum-jerk theory[END_REF]) have modeled interdistance using different types of time polynomials, whose coefficients are obtained respecting safety acceleration and jerk constraints.

In general, these approaches produce acceptable results in an ACC scenario. However, during a sudden deceleration of the preceding car, the vehicles present a large transitory relative velocity and the actual inter-distance decreases abruptly.

Hence, this dynamical scenario would not be suitably represented by static polynomial models, but by some kind of inter-distance dynamic model.

In [START_REF] Martinez | A Safe Longitudinal Control for Adaptive Cruise Control and Stop-and-Go Scenarios[END_REF], the authors proposed a nonlinear reference model taking into account safe and comfort specification in an intuitive way. In addition, the model is combined with a simple feedback loop used to compensate unmodeled dynamics and external disturbances1 . However, this work makes two assumptions that are never met in real situations:

• the inter-distance and the velocity of the leader vehicle are perfectly measured from suitable sensors; • the reference acceleration generated by the dynamic inter-distance model is instantaneously applied to the following vehicle. Our contribution consists in elaborating the engine/brake torque to produce the expected reference acceleration of the follower vehicle, that is, when taking into account measurement noises as well as unmodeled dynamics, such as road inclination, aerodynamic forces or rolling resistance. To achieve this task, a unified approach on estimation and control has been used.

An algebraic framework is proposed to deal with filtering, estimating derivatives, and finally, model free control design.r

It is important to point out that these filters, differentiators and estimators are not of asymptotic nature, and do not require any statistical knowledge of the corrupting noises. This original way of treating conventional problems can be viewed as a change of paradigm in many control and signal processing aspects (cf. [START_REF] Fliess | Non-linear estimation is easy[END_REF]).

Finally, in order to minimize the loss of performances due to uncertain road parameters, a model-free control philosophy will be used, which will be adapted by including specific well-known dynamics, in a kind of grey-box model control.

B. Outline of the paper

The general control scheme will be presented in Section II. In the third Section, the algebraic setting for model-free control will be introduced. Section IV will be devoted to recall the vehicle dynamics and the feedforward control, where a longitudinal acceleration and a consequent torque is generated under ideal circumstances. This section also shows the motivation for the choice of algebraic techniques introduced in Section III. Noise and parameter robustness will be tackled with a grey-box closed-loop control approach in Section IV. Simulation results will show a very good compromise between performance and robustness. Finally, the conclusion and some future work will be drawn in Section V.

II. CONTROL SCHEME

Figure 1 graphically summarizes the whole control scheme. The stop-and-go system uses radar information and CAN 2 bus accessible data to generate, via a dynamic model and a reliable follower velocity estimation (see e.g. [START_REF] Villagra | Estimation of longitudinal and lateral vehicle velocities: an algebraic approach[END_REF]), the desired acceleration. This model provides a safe and comfortable reference inter-distance between the leader and the following vehicle. A reference longitudinal acceleration is then generated as a feedforward control. Since this model is based on corrupted measures and not always valid assumptions, a feedback term is introduced. This closed-loop will not only behave as a typical PID controller, but it will also estimate linear or non-linear unmodeled dynamics (road slope, wind, rolling resistance) in order to anticipate the controller action. The resulting control will provide an acceleration as close as possible to the desired one.

III. ALGEBRAIC SETTING FOR MODEL-FREE CONTROL

A. Numerical differentiation

Start with a polynomial time function

x N (t) = ∑ N ν=0 x (ν) (0) t ν ν! ∈ R[t], t ≥ 0, of degree N.
The usual notations of operational calculus (see, e.g., [START_REF] Yosida | Operational Calculus: A Theory of Hyperfunctions[END_REF]) yield

X N (s) = N ∑ ν=0 x (ν) (0) s ν+1
Multiply both sides by positive powers of d ds . The quantities x (ν) (0), ν = 0, 1, . . . , N, which are linearly identifiable, satisfy the following triangular system of linear equations:

d α s N+1 X N ds α = d α ds α N ∑ ν=0 x (ν) (0)s N-ν 0 ≤ α ≤ N -1
(1) Multiplying both sides of Eq. ( 1) by s -N , N > N, permit to get rid of time derivatives, i.e., of

s µ d ι X N ds ι , µ = 1, . . . , N, 0 ≤ ι ≤ N.
Consider now an analytic time function, defined by the power series x(t) = ∑ ∞ ν=0 x (ν) (0) t ν ν! , which is assumed to be convergent around t = 0. Approximate x(t) by the truncated 2 Controller Area Network Taylor expansion x N (t) = ∑ N ν=0 x (ν) (0) t ν ν! of order N. Good estimates of the derivatives are obtained by the same calculations as above.

Remark 3.1: A most elegant and powerful algorithmic procedure for obtaining a corresponding numerical differentiator is provided in [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF]. It will be exploited in the sequel. 3Take a finite-dimensional SISO system E(t, y, ẏ, . . . , y (ι) , u, u, . . . , u (κ) ) = 0 which is linear or not, where E is a sufficiently smooth function of its arguments. Assume that for some integer n,

B. Model-free control

0 < n ≤ ι, ∂ E ∂ y (n) ≡ 0.
The implicit function theorem yields then locally

y (n) = E(t, y, ẏ, . . . , y (n-1) , y (n+1) , . . . , y (ι) , u, u, . . . , u (κ) )
This equation becomes by setting E = F + αu:

y (n) = F + αu (2)
where

• α ∈ R is a non-physical constant parameter, which is chosen by the engineer in such a way that F and αu are of the same magnitude, • F is determined thanks to the knowledge of u, α, and of the estimate of y (n) . Remark 3.2: A system might only be partially unknown as in Sect. IV-E. It is straightforward to adapt the previous method in this case.

In all the known examples until today, n was chosen to be equal to 1 or 2 in Eq. ( 2). If n = 2, the desired behavior is obtained via the intelligent PID controller, which is of the form

u = 1 α ÿ * -F + K P e + K I edt + K D de dt (3)
where

• y * is a reference trajectory • e = yy * is the tracking error,

• K P , K I , K D ∈ R are suitable gains, the tuning of which is quite straightforward.

IV. HIGH-LEVEL LOOP CONTROL The feedforward high level control will be briefly recalled to point out its main features and the fundamental limitations that have been addressed in the present work:

• The closed-loop control is not at all robust to radar noisy measurements.

• When sudden accelerations/decelerations are needed, the corresponding open-loop engine/brake torques may be hard to compute. After a brief introduction to vehicle longitudinal dynamics, the next subsections will detail how each issue has been addressed, under the algebraic framework presented in Section III.

A. Vehicle longitudinal dynamics

A force balance along the vehicle longitudinal axis (cf. [START_REF] Rajamani | Vehicle Dynamics And Control[END_REF]) yields

M ẍ = F x f + F x r -F a -R x f -R x r -mg sinθ
where F x f , F x r are respectively the front and rear longitudinal tire forces, R x f and R x r the front and rear tire forces due to rolling resistance, θ the angle of inclination of the road, and F a is the longitudinal aerodynamic drag force.

The rolling resistance forces are often modeled as a timevarying linear function of normal forces on each tire, i.e. R x = kF z , with k the rolling resistance coefficient.

The aerodynamic forces can be written (e.g. [START_REF] Kiencke | Automotive Control Systems[END_REF]) as

F a = 1 2 ρC d A F (V x + V wind ) 2
whith ρ being the mass density of air, C d the aerodynamic drag coefficient, A F the frontal area of the vehicle (the projected area of the vehicle in the direction of travel) and V x , V wind respectively the longitudinal vehicle and wind velocities. Finally, Pacejka model [START_REF] Pacejka | The magic formula tyre model[END_REF] is used for longitudinal tire/road interaction forces F x . They depend on many factors, but essentially on longitudinal slip and normal forces. These normal forces will be computed as realistically as possible within a 10 d.o.f vehicle model (6 d.o.f. of the vehicle center of gravity and one supplementary d.o.f. on each wheel).

B. Feedforward control

A reference model proposed by [START_REF] Martinez | A Safe Longitudinal Control for Adaptive Cruise Control and Stop-and-Go Scenarios[END_REF] will act as a feedforward term into the longitudinal high level control law. The basis of this model will be sketched in the next lines.

The inter-distance reference model describes a virtual vehicle dynamics which is positioned at a distance d r (the reference distance) from the leader vehicle. The reference model dynamics is given by dr = ẍlẍr f [START_REF] Fliess | Non-linear estimation is easy[END_REF] where ẍl is the leader vehicle acceleration and

ẍr f = u r (d r , ḋr ) (5)
is a nonlinear function of the inter-distance and of its time derivative.

Introducing d d 0d r in [START_REF] Hedrick | Longitudinal Vehicle Controller Design for IVHS Systems[END_REF], where d 0 is the safe nominal inter-distance, the control problem is then to find a suitable control when d 0:

u r = u 2 ( d, ḋ), d 0
such that all the solutions of the dynamics (4) fulfill the following comfort and safety constraints:

• d r d c , with d c the minimal inter-distance.

• ẍr B max , where B max is the maximum attainable longitudinal acceleration.

• ... x r
J max , with J max a bound on the driver desired jerk.

The authors of [START_REF] Martinez | A Safe Longitudinal Control for Adaptive Cruise Control and Stop-and-Go Scenarios[END_REF] propose to use a nonlinear damper/spring model u 2 = -c| d| ḋ, which can be introduced in the dynamics equation ( 4) to give:

d = -c| d| ḋ -ẍl .
The previous equation may be analytically integrated and expressed backwards in terms of d r as follows, assuming that ẋl (0) = 0: 5), the feedforward control law is then obtained applying 4ẍr

ḋr = c 2 (d 0 -d r ) 2 + ẋl (t) -β , β = ẋr f (0) + c 2 d 0 -d r (0) 2 . (6) From (
f = u r = -c|d 0 -d r | ḋr (7)
where the inter-distance evolution comes from the numerical integration of (6). Remark 4.1: In practice, the leader velocity is not measured and we have to construct an estimator, using for example techniques developed in section III.

C. Closed-loop control

Some kind of feedback control must be introduced in order to avoid errors induced by measurement noises. A standard PID compensation leads to extremely noisy perturbed results when a derivative term is used, and to instability or important tracking errors when it is not.

In order to avoid this kind of problem, a PD compensator5 has been implemented, where inter-distance and its timederivatives are obtained using Sect. III-A. The signal can be locally approximated by a linear polynomial (N = 1). Thus, d(t) = d 0 + d 1 t, t 0, d 0 , d 1 ∈ R. In the first case, an estimator for d 0 is sought; in the second one, d 1 will be estimated. If we take for instance N = 3 and N = 2, the estimators can be respectively written as follows :

d = d0 = 2 T 2 T 0 (2T -3τ)d(τ)dτ d = d1 = -3! T 3 T 0 (T -2τ)d(τ)dτ (8)
Figure 2 shows6 the difference between applying two discrete PD controller with a different low-pass filter and an algebraic PD controller. Inter-distance and jerks evolution with different closed-loop controllers: a discrete PD with low-pass filter of cut-off frequency equal to 100 Hz (top), another one with cut-off frequency equal to 5 Hz (middle) and an algebraic PD (bottom). The P = 0.7 and D = 0.4 parameters are identical in all cases.

D. Reference torque generation

The wheel rotation dynamics can be written as follows

I ω = -rF x + τ e -τ b ( 9
)
where I is the rotation inertia moment, ω the wheel angular velocity, r is the tire radius, τ e the applied engine torque, and τ b the brake torque, both of them applied to the wheel center.

A commonly used assumption ( [START_REF] Hedrick | Longitudinal Vehicle Controller Design for IVHS Systems[END_REF], [START_REF] Nouvelière | Experimental vehicle longitudinal control using a second order sliding mode technique[END_REF], [START_REF] Rajamani | Vehicle Dynamics And Control[END_REF]) consists in considering rolling without slipping, i.e. V x = R g rω, where R g is the gear ratio. However, in a stop-and-go context, where fast responses to sudden decelerations are required, this is not an acceptable hypothesis.

If a generalized wheel torque τ eb = τ e -τ b is considered, it is straightforward to see its dependence on tire/road interaction forces. Therefore, a realistic estimation of this generalized torque from equation ( 9) turns out to be quite hard.

The sum of the 4 wheels rotation dynamics equations and of the vehicle dynamic longitudinal equation Mγ

x = ∑ 4 i=1 F x i yields τ g = 4τ eb = I 4 ∑ i=1 ωi + rMγ x . ( 10
)
The main inconvenient for such an estimator is that a good numerical differentiator for ωi is needed. An equivalent algebraic estimator to [START_REF] Join | Étude préliminaire d'une commande sans modèle pour papillon de moteur[END_REF] will then be used to compute ωi .

Finally, figure 3 compares inter-distances between openloop generated torque under no slipping assumption and open-loop torque with our dynamic estimation approach. A remarkable improvement can be obtained when this new strategy is used in demanding situations. Indeed, results shown in figure 3 are obtained with longitudinal accelerations up to 5.5 ms -2 , which are rarely found in an ACC context.

E. Grey-box feedback control

The procedure described in Sect. III-B can here be applied in a particular way. Since some specific dynamics are very well known, it is worth to integrate them in our predictive scheme. Thus, the design parameter α corresponds here to well-known quantities. Recall the local input-output model introduced in Eq. ( 2) and compare it with the reordered torque expression [START_REF] Liubakka | Adaptive automotive speed control[END_REF]

γ x = 1 Mr τ g -I 4 ∑ i=1 ωi + G(t) , (11)
G(t) = r F a -R x f -R x r -Mg sin θ
If rolling without turning is considered ( Vx = γ x ), the next equation can then be written:

Vx = F(t) + αu(t) + β (t) (12)
where

F = G Mr , α = 1 Mr and β = - I ∑ 4 i=1 ωi Mr and u = τ g is the control variable.
The goal is to obtain an accurate closed-loop estimation of F. Following the theoretical ideas described in Sect. III, the procedure consists, first of all, in rewriting [START_REF] Martinez | A Safe Longitudinal Control for Adaptive Cruise Control and Stop-and-Go Scenarios[END_REF] in the operational domain, with the assumption F = F 0 in the estimation time window,

sV x -V 0 = F 0 s + ατ g (s) + β (s) (13)
and then applying the operator d ds with the aim of eliminating the initial condition V 0 V x + s dV x ds = -F 0 s 2 + α dτ g ds + dβ ds Finally, s -ν , with ν = 2 is applied in order to eliminate any non causal term

F 0 s 4 = - 1 s 2 V x - 1 s dV x ds + α 1 s 2 dτ g ds + 1 s 2
dβ ds which, expressed backwards in the time domain, yields When the complete strategy is tested on a quite demanding scenario 7 , the inter-distance model reference is pretty well tracked (see figure 4), the follower acceleration remains under the comfort constraints, and consequently, the jerk bounds are also guaranteed (cf. [START_REF] Martinez | A Safe Longitudinal Control for Adaptive Cruise Control and Stop-and-Go Scenarios[END_REF]). Furthermore, the generalized torque applied to the vehicle seems very robust to noise perturbations.

F 0 = 3! T 3 T 0 ((-T + 2t)V x (t) -(T -t)t(ατ g (t) + β (t)))dt (
However, the most important source of uncertainty comes from road conditions. Thus, if rolling resistance, aerodynamic efforts and a sloped road are introduced, the results are slightly different. Figure 5a shows that even if the interdistance trends are already very well respected, a variable bias cannot be annihilated with the "standard" control. The grey-box control strategy has been applied in order to obtain more robust results. The dashed line in figure 5a represents the tracking performance when the estimator F of global disturbances is introduced. A considerable improvement (almost 400%) is obtained when the global effects of disturbances are estimated via equation [START_REF] Pacejka | The magic formula tyre model[END_REF]. It can be appreciated from figure 5b that road slope, rolling resistance and aerodynamic forces are pretty well estimated in an overall term F = F 0 . Note that aerodynamic forces are not very significant when compared with the road slope. However, big wind gusts can appear at high speeds. In this case, a reliable and fast estimator should applied. Figure 6 shows the behavior of control law proposed in [START_REF] Persson | Stop & Go Controller for Adaptive Cruise Control[END_REF] when severe wind gusts longitudinally knock the car. It can be appreciated that our control is much more robust when F is estimated. Moreover, the time window estimation size T e can be used as a tuning parameter for safety or comfort. When tracking performance is more important than comfort (sudden changes in acceleration), a small window will be used. If important jerks are not desired, a bigger window estimation will be more appropriate.

V. CONCLUDING REMARKS

A grey-box closed-loop stop-and-go control for vehicles has been presented. Its main feature is its ability to deal with usual disturbances (wind, road slope, rolling resistance...) which are not easily measurable. It can be seen that, as expected, our method leads to a closed-loop robust behavior with respect to noises and unmodeled dynamics.

The next step will be to develop a low-level control including the engine and brake dynamics. An algebraic approach is under study to generate the physical control variables: the throttle angle (see already [START_REF] Join | Étude préliminaire d'une commande sans modèle pour papillon de moteur[END_REF]) and brake pressure. Furthermore, the whole algorithm is being adapted to real vehicles and will be soon presented.

Fig. 1 .

 1 Fig. 1. General Stop-and-go control scheme.

  Fig. 2.Inter-distance and jerks evolution with different closed-loop controllers: a discrete PD with low-pass filter of cut-off frequency equal to 100 Hz (top), another one with cut-off frequency equal to 5 Hz (middle) and an algebraic PD (bottom). The P = 0.7 and D = 0.4 parameters are identical in all cases.

2 =320 2 =123Fig. 3 .

 223 Fig.3. Comparison between torque generation under no slipping assumption, and with expression[START_REF] Liubakka | Adaptive automotive speed control[END_REF].

14 ) 4 ∑Fig. 4 .

 1444 Fig.4. Inter-distance, velocity, acceleration and generalized torque for highly demanding scenario (up to 5 ms -1 ).

Fig. 5 .

 5 Fig. 5. (a) Inter-distance evolution with and without F estimation. (b)Comparison between real F and its estimate F 0 . Aerodynamic (F a ), road slope (F s ) and rolling resistance (F r ) terms are also depicted.

Fig. 6 .

 6 Fig. 6. (a) Wind velocity. (b) Inter-distance evolution with different sliding windows for F estimation. (c) Longitudinal acceleration with different estimation time windows. (d) F estimation

Examples of adaptive car following controllers dealing with this problem can be found in works by[START_REF] Liubakka | Adaptive automotive speed control[END_REF],[START_REF] Ioannou | Throttle and Brake Control Systems for Automatic Vehicle Following[END_REF] or[START_REF] Youcef-Toumi | Application of time delay control to an intelligent cruise control system[END_REF].

See[START_REF] Fliess | Commande sans modèle et commande à modèle restreint[END_REF] for more details, for numerous computer simulations, and for references on already existing applications.

Note that the parameter c is an algebraic function of safe and comfort parameters d c , V max , B max and J max (cf.[START_REF] Martinez | A Safe Longitudinal Control for Adaptive Cruise Control and Stop-and-Go Scenarios[END_REF]).

The integral term is not used in order to avoid an unstable behavior of the system (see[START_REF] Martinez | A Safe Longitudinal Control for Adaptive Cruise Control and Stop-and-Go Scenarios[END_REF]).

Since the inter-distance reference trajectory depends on the closed-loop behavior, it is difficult to exactly obtain the same testbed for the 3 cases presented in Figure2.

Several heavy accelerations/decelerations are applied to the vehicle on a flat road, where neither rolling resistance nor aerodynamic forces are considered.