Ibrahim Cheddadi

Pierre Saramito

Benjamin Dollet

Christophe Raufaste

François Graner

Understanding and predicting viscous, elastic, plastic flows

Keywords: -47.57.Bc Foams and emulsions -83.60.La Viscoplasticity, yield stress -83.60.Df Nonlinear viscoelasticity -02.60.Cb Numerical simulation, solution of equations

. Our results demonstrate that the behaviour of soft glassy materials cannot be reduced to an intermediate between that of a solid and that of a liquid: the viscous, the elastic and the plastic contributions to the flow, as well as their couplings, must be treated simultaneously. Our approach opens the way to the realistic multidimensional prediction of complex flows encountered in geophysical, industrial and biological applications, and to the understanding of the link between structure and rheology of soft glassy systems.

Introduction

Materials such as pastes or polymer solutions display both solid-like and liquid-like behaviors; they are successfully described by visco-elastic (VE) or visco-plastic (VP) models. However, we still lack testable predictions of the timeand space-dependent flows of soft glassy materials [6,7], that are made of disordered assemblies of deformable, rearranging units [6,19]. It had been suggested that fluctuations remain relevant even at large scale, in which case detailed statistical theories of long-range correlations and avalanches would be required [20,21]. This view is challenged by recent experiments which suggest that even these materials can be treated as continuous materials described by tensorial equations [6,14]; thus in principle partial differential equations could lead to the long-awaited predictions.

Based on our experience with foams, we believe that the reason of the difficulty comes from the fact that these materials are simultaneously viscous, elastic, and plastic (VEP). Under small deformation, a foam reversibly comes Send offprint requests to: pierre.saramito@imag.fr, francois.graner@curie.fr a Present address: Laboratoire de Physique de la Matière Condensée, UMR 6622 CNRS and Univ. Nice-Sophia Antipolis, Parc Valrose, F-06108 Nice Cedex 2, France back to its shape; at large deformation, it can be irreversibly sculpted and gets a new shape; under an increasing deformation rate, it irreversibly flows, with an increasing viscous stress [START_REF] Weaire | The physics of foams[END_REF][START_REF]Collective, Les mousses -structure et dynamique[END_REF][START_REF] Höhler | [END_REF][25][26]. Existing continuous models of foam flows include for instance either a phenomenological scalar description [17,27], or a complete tensorial description of the elasticity [9] or plasticity [19] based on the micro-structure. Overall, continuous VEP models tend to successfully reproduce some experimental measurements, such as elastic and loss moduli, or compliance (see for instance refs. [START_REF] Preziosi | [END_REF]9,11,25]).

We want here to understand and predict VEP flows.

Our approach is to test whether a continuous VEP model can capture the essence and complexity of the flow properties, especially the elasticity and its advection by the flow. We thus need a VEP model as simple as possible:

with linear and isotropic coefficients, without fluctuations nor long-range interactions. We require to use only physically relevant parameters, which in principle are measurable. We need a tensorial model in a multidimensional space (what follows applies both in 2D and 3D). We need a closed system of equations: constitutive equations, specific to the material under consideration; generic conservation laws, in the spirit of hydrodynamics; and closure equations, to unify the solid and liquid descriptions.

To really test a model, and also lead to practical applications, we address the full spatial and orientational heterogeneity of a flow. We need to investigate a controlled, reproducible flow which displays significantly different V, E and P contributions. It should involve a large range of shear rates, of tensorial orientations, and of elastic deformations. It should display elastic deformations either parallel, perpendicular or at odd angles with respect to the shear rate. Most discriminant, it should actually depends on two or more dimensions of space, so that the advection of elastic stress couples the shear and normal stress components (see eqs. (8)(9)). This latter point, overlooked by most models, is crucial to test whether the elastic stress could be entirely determined by the shear rate, leading to a VP rheology (such as Bingham [START_REF] Bingham | Fluidity and plasticity[END_REF] or Herschel-Bulkley [START_REF] Herschel | [END_REF]). Recent experiments [15,16] suggest that it is not the case: in fact, the stress, the shear rate and the elastic deformation should be treated as independent variables, so that a full VEP treatment is required.

One such flow is the well documented flow around an obstacle [12] (Fig. 2). It displays a strong spatial heterogeneity, simultaneous VEP behaviors, a large range of elastic deformations, several elongation and rotation rates, and various relative orientations of the relevant tensors [19]. It enables to follow a bubble at different stages, while it stretches, then while it relaxes: thus, even in a steady flow, transient effects and relaxation times are apparent.

It is classically used as a stringent test to discriminate between different models [30] (Fig. 3).

We use foams as model systems of VEP materials. Experiments with foams or emulsions, especially in 2 dimensions, enable an easy, simultaneous visualisation of the micro-structure (bubbles or droplets, which act as tracers of velocity and deformation) and the large scale (global flow heterogeneities).

To compare mesurements from discrete experiments with continuous predictions from partial differential equations, we use the experimental tools we have developed [19]. V, E and P contributions are expressed in the same units, favoring a unified description of solid and liquid behaviors; each of them is valid in all regimes (so that e.g. elastic contributions can be measured even out of the elastic regime). These are local (in situ) measurements which link the foam structure and rheology.

Our plan is as follows. In order to make this paper self-contained, Section 2 recalls Saramito model [11]; here we write it by emphasizing how it unifies solid and liquidlike behaviours. 2 Model

Characteristics and past results of the model

We choose to use Saramito's VEP model [11] because it has a positive dissipation, thanks to the convexity of its energy function. This implies that, at least for small deformations, it obeys by construction the second principle of thermodynamics. It includes as limiting cases both the VP Bingham model [START_REF] Bingham | Fluidity and plasticity[END_REF] and the VE Oldroyd model [31],

and generalises them to VEP. It can extend to large deformations and high velocities.

It has already been used for space-and time-dependent predictions [11]. It has first been used to solve simple cases, such as steady uniaxial elongation. It has also applied to oscillatory regimes, and calculations of G ′ , G", rigidity and loss moduli.

It has then been implemented to calculate both timedependent and steady shear Couette flows, which depend on one space variable (circular [32] or planar [33] geometries). It involves two strong non-linearities, intrinsic to VEP flows, and thus independent of the model: one because the plasticity appears above a yield point (eq. 3), and the other because of the advection of elastic stress (eqs. (8)(9)). Despite these unavoidable difficulties, the model has been solved [34,35]. The resulting velocity, elasticity and plasticity fields agreed with experimental measurements.

This improved our understanding of Couette flows [35].

The localization of the velocity field results from the stress heterogeneity, so that the circular geometry by itself can induce localisation. In planar geometry, where the stress is a priori homogeneous, localisation necessarily relies on another cause of heterogeneity, such as an external friction.

Initial normal stresses can be preserved even in a steady flow, so that there are residual normal stresses which depend on initial conditions linked with the foam preparation method: the steady flow is not unique. Despite its simplicity, a Couette flow displays specific VEP features [34]. For instance, at the boundary of the localised region, the discontinuity of velocity gradient depends on the residual normal stresses, and thus on initial conditions. For all these reasons, and because the range of experimental data is limited, Couette flows have only a limited ability to discriminate between models, or between parameter values.

Notations and equations

Constitutive equations: solid mechanics

We start with constitutive equations specific of a semifluid semi-solid material (see Fig. 1). In order to emphasize the dominant role of elasticity, we express them here in terms of deformations and their rates, as is usual in the context of solid mechanics:

σ tot = -pI + 2η 1 ε + 2µε e , (1) ε
= εe + εp , (2) εp
=              1 λ |ε e | -ε Y |ε e | ε e when |ε e | > ε Y , 0 otherwise. (3)
Eq. ( 1) is a constitutive equation for the Cauchy stress σ tot , with pressure, viscous and elastic terms. Here p is the pressure, I the identity tensor; ε is the total deformation rate tensor, η 1 is the viscosity of the material apparent at small deformation (in the foam, it includes the dissipation inside the soap films); ε e is the elastic deformation tensor, µ the shear modulus.

Eq. ( 2) recalls that ε is shared between elastic εe and plastic εp contributions.

Eq. ( 3 In soft disordered materials, plasticity is related with local rearrangements. In foams, these happen when bubbles swap neighbors and are called "T1" processes [START_REF] Weaire | The physics of foams[END_REF][START_REF]Collective, Les mousses -structure et dynamique[END_REF]36]. They create a transient local deformation. Here λ is the relaxation time of the material after such a local deformation [37]. We can construct the dissipation due to plasticity, which has the dimension of an effective viscosity η 2 = λµ: it determines the loss modulus at large amplitude. The softness and deformability of the mate-rial appears in the value of ε Y , of order of unity, so that both the elastic and plastic behaviors are experimentally observable; its glassy (i.e. disordered) nature implies that λ, µ, ε Y are isotropic [6].

Conservation equations: fluid mechanics

Generic conservation equations for an isothermal flow are expressed in terms of the velocity v and its derivatives, as is usual in the context of fluid mechanics:

ρ v = div σ tot + f ext , (4)
div v = 0. (5)
Eq. ( 4) is the equation of dynamics; ρ is the density, v = ∂ t v+v.∇v, and ρ v denotes the inertia term (which we neglect below, see Section 3.3); f ext is the external force:

in the bidimensional flow experiments studied below, the friction on the top and bottom boundaries (e.g. horizontal glass or perspex plates) is approximately

f ext = -kv,
where k is a constant. We find experimentally (see Section 3.3) that k is small enough that in the present flows the effect of f ext is not measurable, so that we neglect it.

Eq. ( 5) describes the incompressibility of the flow: it applies to slow flows when the compressibility modulus is much higher than the shear modulus, as is the case in foams [START_REF] Weaire | The physics of foams[END_REF][START_REF]Collective, Les mousses -structure et dynamique[END_REF][START_REF] Höhler | [END_REF].

Closing equations: linking solid and fluid mechanics

To close the system of eqs. (1-3) and eqs. (4-5) requires to couple the deformations to the velocity.

First, the total deformation rate equals the symmetrized velocity gradient:

ε = ∇v + ∇v T 2 . (6)
Second, the time variation of the elastic deformation tensor ε e accounts for its advection by the flow velocity v.

The model should be objective, that is, the expression of the equations should remain the same for an observer who has a movement of translation or rotation with respect to the experiment. The advection of the elastic deformation tensor is thus described with a frame invariant tensorial derivative [38]:

εe = Dε e Dt . (7)
The objective derivative is [38,11]:

Dε e Dt = ∂ε e ∂t + (v • ∇)ε e + β a (ε e , ∇v), (8)
where

β a (ε e , ∇v) = ε e •W (v)-W (v)•ε e -a(D(v)•ε e +ε e •D(v)), (9)
Here W (v) = 0.5(∇v -∇v T ) is the antisymmetric part of the velocity gradient, and a ∈ [-1, 1] is the so-called "a parameter" [11,38], which effect is discussed in section 3.3.

Methods

Experimental methods

Experimental set-ups have been described in refs. [14,19,39]. Bubble distributions are monodisperse in size (area 16 mm 2 ) and disordered in geometry (shape, number of neighbors). While bubbles pass through the field of view, no rupture is observed, and coarsening is negligible.

The wet foam is prepared by direct bubbling into the 1 m long channel. At its entrance in the channel, it displays normal differences in elastic deformation, ε e xx -ε e yy (axis 2 in Fig. 7).

For the dry foam, bubbles pass first through a chamber (in which the foam drains): this chamber enables to vary the liquid fraction over more than three decades, and homogeneizes the foam while relaxing its normal differences in elastic deformation [39].

Experimental measurements treat solid and liquid behaviors with a unique set of mutually compatible tools [19]. We derive the continuous description directly from averages over almost a thousand of images of discrete measurements performed on all bubbles which can be automatically identified using image analysis, that is, which do not touch the obstacle. The entrance velocity V is measured ±2% as the average over the side of the field of view.

The texture (bubble size, elongation and packing) and its variation (bubble stretching and rearrangements) enable to measure in situ the velocity gradient (not shown), the elastic deformation ±2% and the plastic deformation rate ±7% [19]. We plot here deviatoric terms, see section 3.3.

Resolution

Eqs. ( 1)-( 5) can be solved in 2D or in 3D. Their main difficulties are intrinsinc to VEP flows, independently of the details of the model. They reduce to a set of three partial differential equations with three unknowns (ε e , v, p) and the coupled system is highly non-linear : its numerical resolution needs to be performed carefully.

Here we solve these equations in 2D with a finite element algorithm first used for a simple Couette circular geometry [35], extended here to handle more complex flow domains [34]. As in 1D, the stationary solution is obtained by solving the time-dependent problem with a second order time-splitting algorithm, already used for VE [40], generalized here to VEP; it allows to treat separatly the two main non-linearities of the equations, namely the plasticity term in eq. ( 3) and the stress transport term in the objective derivative (eq. ( 8)). Unlike in 1D [41], the stress transport term needs to be treated specifically by upwinding techniques; we chose a robust method, the discontinuous Galerkin scheme [START_REF] Lesaint | A finite element method for solving the neutron transport equation[END_REF]. In addition, the nonlinearity linked to plasticity needs a much more careful discretization than for the Couette resolution [35] to ensure a proper decreasing of the residue of the stationnary problem. The spatial discretization is performed with a mixed finite-element method as in [30]. In order to get a general method suitable for any geometry, the domain is discretized with triangles.

The calculation domain is a channel, 15 R = 22,5 cm upstream and 30 R = 45 cm downstream of the obstacle.

The mesh, made of 1100 triangles, is locally refined near the obstacle (see Fig. 5a). We start from a foam at rest and enforce the entrance velocity V . Unlike for most liquid flows, but in agreement with foam flow experiments [14],

we use slip boundary conditions. Careful tests have been performed [34] in order to ensure that the mesh is suffi-ciently refined and that discretization does not affect the results presented here. Iterations are performed (Fig. 5b) until the residue of the stationary problem is less than 10 -7 (see Fig. 5c). Calculations in 2D run in half a day on a Intel T7300 Core 2 Duo processor (2 GHz, 4 Mb cache, 32 bits). The 2D algorithm has been validated by reproducing the 1D Couette calculation [35], which runs in a few minutes.

Choice of parameters

The parameter with the main effect is ε Y : a change of ±20% suffices to visibly affects the main features of the flow. In a foam, its value is expected to depend mostly on the liquid fraction and on the dimension (2D or 3D), and possibly on the area dispersity; on the opposite, the average area and the physico-chemical properties of the foam are not expected to affect the value of ε Y .

We have tested the effect of varying the other parameters, one by one. Like in Couette flow [35], they barely affect the flow, even if varied over a large range, as we now discuss.

The value k of the friction on the plates can be determined experimentally from the overall pressure gradient ∇p across the channel length: k = ∇p/V = 9300 Pa s m -2 . If we take this value, we obtain a prediction indiscernable from k = 0. Only by taking a ten times larger friction, k = 10 5 Pa s m -2 , does the change become visible. It means that here the limit k → 0 is regular. This case is similar to cylindrical Couette geometry, but different from planar Couette geometry [35].

We choose η 1 /η 2 = 0.1 according to the Couette case [35], since it lies in the middle of a range where its exact value barely affects the flow, even up to a factor of 10.

Since the Reynolds number Re = ρV R/(η 1 + η 2 ) ≈ 5 × 10 -3 , we neglect the inertia term ρ v in Eq. ( 4).

We choose the co-rotational derivative [38], with a = 0, so that ε e is deviatoric. In that case, the term β a in eqs. The relaxation time λ is related to the Weissenberg number We = λV /R. We distinguish three velocity regimes.

At high velocity, We of order 1 or higher, the material rheology can display non-linearities in addition to those already present in the model. Since foams rupture at high velocities [39,[START_REF] Dollet | [END_REF], this regime would be easier to investigate with other materials. In the low velocity range, We greater than 10 -2 but smaller than 1, the exact value of

We does not affect the flow. This is the case for both experiments considered here, as well as for several foam flow experiments reported in the literature. Further decreasing

We over two or three decades would lead to the ultra slow range, where the fore-aft asymmetry strongly increases. This is done in very few well-controlled experiments [39] The limit We → 0 at constant ε Y is singular: it implies a divergence of the Bingham number Bi = 2ε Y /We [11]. In fact, the ultra slow regime is not quasi-static [25,27,44].

and does not exactly match quasi-static simulations [27].

Representations

Results are plotted either as maps or graphs. 4 Results

Wet foam flow

First, as a preliminary characterization, we study the flow of a wet foam (7% liquid fraction, Fig. 2a). We calculate the measurable fields: velocity v, elastic deformation ε e , plastic deformation rate εp . We use the parameters obtained in ref. [35] and rescale them to the geometry of the present set-up. We investigate separately the effect of each parameter (see Section 3.3). We check that, with a yield strain ε Y = 0.1 ± 0.02, the calculations agree well simultaneously with all available experimental data (Fig. 3 and Figs. 678). Such value of ε Y is the expected order of magnitude for a foam with this liquid fraction [19].

Other parameters have less effect. We use λ = 0.2 ± 0.1 s [14] and η 1 /η 2 = 0.1. We take k = 0 without significantly affecting the results. With a shear modulus estimated [14] around µ = 13 ± 1 N.m -2 , this translates into actual values:

σ Y = 2µε Y = 2.6 N.m -2 , η 2 = λµ = 2.6
Pa.s and η 1 = 0.1 η 2 = 0.26 Pa.s.

In such a VEP flow, elastic deformations are present in the wake of the obstacle at arbitrarily low velocity. Plasticity prevents the increase of extensional deformation, breaking the up-downstream (fore-aft) symmetry (Fig. 3).

A velocity overshoot, the so-called "negative wake", is clearly visible behind the obstacle (Fig. 3). This characteristic feature of VEP flows is barely affected even if we vary V across the low velocity regime, confirming experimental observations [14]. For instance, dividing V by 20 barely changes the overshoot (Fig. 3).

This strongly contrasts with VP flows, which are always fore-aft symmetric [30]. VE flows represent a mixed situation, where the negative wake has already been evidenced, both experimentally [45] and numerically [46,47].

In fact, it occurs for low extensional viscosity fluids and models (e.g. FENE-CR [47] but not Oldroyd-B [31]), at elongational rates large enough in comparison with the inverse relaxation time, so that the elastic deformation does not vanish. However, at the low velocity investigated here, the VE flow is completely fore-aft symmetric (Fig. 3), and even indiscernable from viscous flows, whatever the viscosity. To interpret this set of observations, it seems that the overshoot appears when the elastic deformation ceases to follow passively the total deformation rate. This can occur if there is a mechanism which saturates the value of elastic deformation, which is the case in some VE models at high velocity, and in any VEP model because of plasticity.

To summarize this first test, by adjusting only one parameter, which value has the expected order of magnitude, we can adjust both qualitative and quantitative features of all available data. We reproduce the observed negative wake and evidence the specificity of VEP flows.

Dry foam flow

Second, we turn to prediction. Since the parameter which has the most significant effect on the flow is the yield strain [14] (section 3.3), we choose to predict the flow for a twice larger value, ε Y = 0.2.

These predictions are plotted on Fig. 4a, top and Fig. 4be, lines. The overshoot on the symmetry axis (Fig. 4b) is larger and closer to the obstacle than for ε Y = 0.1: this reflects a larger effect of the elastic deformation. The elastic deformation field extends more than the velocity gradient (Fig. 4a, top), which itselfs extends more than the plastic deformation rate field (Fig. 11, top): this confirms that the three fields are physically independent quantities [19].

To check this prediction, we then perform a dry foam experiment, since decreasing the liquid fraction of a foam is expected to affect especially the yield strain. With 1.2% liquid fraction, Fig. 2b), we observe that ε e max , the maximum value of ε e measured on the experiment, is twice as much that of the wet foam. The effect of elasticity is even stronger and the agreement with our prediction even better, without adjusting any parameter. Measurements confirm the predicted spatial distribution, magnitude, direction, anisotropy of fields.

The bubble velocity (Fig. 4) passes the three most stringent tests. First, the position and magnitude of overshoot on the symmetry axis (Fig. 4b). Second, the graph along the axis 5 (Fig. 4c). And third, the exact position of the arrest points, defined in the referential of the foam, as points where v x -V = 0: close to axis 5, on y = 0 and ±5 cm, see Fig. 4a. Other axes confirm the agreement (Fig. 9).

The bubble elastic deformation too agrees remarkably well in spatial distribution, amplitude and direction (Figs.

4a,d,e and Fig. 10). Interestingly, its orientation does not directly correlate with that of streamlines, or equivalently of the velocity vectors. This confirms that the elastic deformation should be treated as a variable independent of the velocity; we have also checked (data not shown) that it does not directly correlate with the total deformation rate. Note that residual normal stresses are visible beyond the obstacle. Unlike in the Couette case, here they are reproducible and their origin is understood: they are a direct effect of the obstacle, and do not depend on the foam preparation method.

The plastic deformation rate is calculated as the total deformation rate minus the elastic deformation rate (eq.

2)

. Its predicted spatial distribution and directions agree with that of the experimental measurements, which represent the time-averaged orientation, frequency and anisotropy of the bubble rearrangements (Fig. 11).

Discussion and conclusion

To summarize, a continuous description of viscous, elastic, plastic material with physically meaningful parameters can reproduce and even successfully predict a tensorial, spatially developed flow of a disordered rearranging structure.

We analyse and interpret the effect of each parameter separately. We emphasize the dominant role of elasticity and thus identify the yield strain as the most important parameter. The flow does not reduce to VE or VP separately, so that we emphasize the specific complexity of VEP materials.

Our method opens the way to computing two-or threedimensional flows under any type or amplitude of deformation. It applies to those depending on one space variable:

for instance a flow through a channel [15,16], or during simultaneous squeezing and shearing [6]. It also applies to those depending on two space variables: for instance a flow through a hole in 2D [19,48], or in 3D with axisymmetry.

At the expense of longer calculations, it can even apply to flows which depend on three space variables, for instance through a twisted or branching pipe.

The model has been kept as simple as possible. On one hand, if we suppress one or another of its ingredients, our algorithm still can solve it, and the resulting solution lacks some of the experimental features. On the other hand, it can be progressively enriched by incorporating additional non-linearities, for instance related with the physico-chemistry of foams. When the velocity or the rigidity of the surfactant layer increases, shear thinning effects [26] could be introduced by an extended version of the VEP model [49] based on an Herschel-Bulkley viscoplasticity [START_REF] Herschel | [END_REF] instead of a Bingham one, as observed in [17,50]. The friction on walls too scales non-linearly with high velocity [26]. Non-linear elasticity at large deformation, although seldom reached in foams [19], can too be taken into account [9,[START_REF] Höhler | [END_REF]. At low velocity, plasticity seems to appear progressively: some bubbles begin to rearrange below the yield strain [19].

More generally, the model can be adapted specifically to any given VEP material of known properties. The value of the parameters of eqs. ( 1)-( 5) depends on the microstructure, its disorder and its physico-chemical properties.

This is where the present approach can be enriched by statistical models based on the micro-structure [7,15,16,20,21,[START_REF] Höhler | [END_REF]44,51].

  ) is a plasticity equation, which specifies εp by stating that plasticity increases above the yield strain ε Y : under a strong shear rate, the elastic deformation |ε e | can become higher than ε Y . Here |ε e | is the norm of ε e , which we define as |ε e | = [(ε e yx ) 2 + (ε e xx -ε e yy ) 2 /4] 1/2 to facilitate the comparison with a scalar shear (fixed shear direction xy with a given amplitude ε), i.e. ε e xx -ε e yy = 0 and ε e yx = ε, so that |ε e | = ε [6]; the externally measured scalar shear is then γ = 2|ε e |. Another acceptable definition would be the euclidian norm of the deviatoric elastic strain tensor [35], ([(ε e xx -ε e yy )/2]2 + (ε e xy ) 2 + (ε e yx ) 2 + [(ε e yy -ε e xx )/2] 2 ) 1/2 = [2(ε e yx ) 2 + (ε e xx -ε e yy ) 2 /2] 1/2 , which is √ 2 times larger than |ε e |. The value of ε Y is defined consistently with that of |ε e |.

( 8 - 9 )

 89 has a zero trace: hence if the trace of ε e is initially zero, it remains zero[34]. In the low velocity regime explored here, any other choice of the objective derivative, with a parameter ranging from a = -1, lower-convected derivative, to a = 1, upper convected[38], would only change the trace of ε e . Further investigations might determine the value of a by comparison with the experimental values of this trace.

Maps enable a global

  quantitative comparison between calculations and experiments: spatial distribution, magnitude, orientation of different quantities. Velocities are displayed as arrows. For clarity, they are presented in the average referential of the foam, as if a movable obstacle was displaced towards the left within a fixed foam, i.e. (v x -V )/V . Traceless tensors are represented by circles. Their two eigenvalues have same absolute value. We indicate the direction of the positive one by a thin line. The direction of the negative one, not represented, is perpendicular. The top half, in red, is the present calculation using a continuous model, performed in the half-plane assuming top-bottom symmetry. Thin lines are isovalues of the stream function. The bottom half, in blue, are experimental data obtained as averages over bubbles. Thin lines are trajectories deduced from the experimental velocity field, plotted with approximately the same density as in calculations.Graphs of components along selected axes enable detailed quantitative comparisons between calculations and experiments: position and amplitude of extrema, shape of curves, concavity, inflexion points, precision of the agreement. Axes are chosen to coincide with the measurements of ref.[14]: axis 1 is the symmetry axis, passing through the obstacle center, parallel to the flow, y = 0; axis 2 is parallel to the flow, close to the obstacle y = ±2.5 cm; axis 3 is perpendicular to the flow, passing through the obstacle center, x = 0; axis 4 is perpendicular to the flow, upstream of the obstacle, x = -2.4 cm; axis 5 is perpendicular to the flow, downstream of the obstacle, x = 2.4 cm. We do not plot components which vanish due to symmetry, such as v y or ε e xy along the symmetry axis. Red lines are predictions, blue symbols are experimental tests.

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Figures

Fig. 4 .

 4 Fig. 4. Test of dry foam flow prediction, represented as explained in section 3.4. (a) Map. Thick colored arrows: velocity field. Circles: elastic deformation tensors; the positive (resp. negative) eigenvalue corresponding to elongation (resp. compression) is represented by a line (resp. not represented). Thin lines: stream lines. Top half (red): present calculation using a continuous model. εY = 0.2, λ = 0.2 s, η1/η2 = 0.1, k = 0. Bottom half (blue): experimental data obtained as averages over bubbles shown in Fig. 2b. Scale: bar: 1 (dimensionless) for the elastic eigenvalues (circle diameter); black arrow: entrance velocity V . (b-e) Graphs of the same data along axes 1 (b,d) and 5 (c,e). Red lines are predictions, blue symbols are experimental tests. (b,c): (vx -V )/V : solid thick lines and closed squares; vy/V : dashed thin lines and open squares. (d,e): elastic normal differences (ε e xxε e yy )/2: solid thick lines and closed circles; elastic shear ε e xy : dashed thin lines and open circles.

Fig. 5 .

 5 Fig. 5. Numerical convergence of calculations (see text). (a) Mesh used for the actual calculations; for convergence tests, the results have been compared with calculations performed on a finer mesh [34]. (b) The L 2 norm of the stationnary residues of eqs. (1-5) is displayed during the calculation which led to Figs. 4 and 9-11. From top to bottom: n = 10, 200, 1000 and 3000 time iterations. Figures indicate x/R, where R is the obstacle radius. (c) The same L 2 norm of the stationnary residues is plotted versus time iterate.

Fig. 6 .

 6 Fig.6. Velocity and elastic deformation of wet foam (εY = 0.1). Red lines are calculations (same as Fig.2a), blue symbols are experimental data (from Fig.3). (top) Superimposition of velocity and elastic deformation fields, and streamlines. Scale: bar: 1 (dimensionless) for the elastic eigenvalues (circle diameter); black arrow: entrance velocity V . (bottom) Plots along axes 1-5.Same caption as Fig.4.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Elastic deformation of wet foam (same as Figs. 2a and 3): plots along axes 1-5. Same caption as Fig. 4.

Fig. 9 .

 9 Fig. 9. Velocity of dry foam (εY = 0.2). Same caption and data as Fig. 4.

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Elastic deformation of dry foam. Same caption and data as Fig. 4.

Acknowledgements We thank Y. Jiang for discussions at the beginning of this work, and Y. Bellaïche, I. Bonnet, S. Cox, L. Courbin, C. Gay, J.-M. Ghidaglia, P. Marmottant, F. Molino, C. Quilliet, H. Tabuteau for useful comments. The CNRS has funded trips within the Groupe de Recherche (GDR) "Mousses".

Appendix

Complete set of graphs

We have performed systematic calculations of all fields for several values of the parameters. They are all available upon request. Figs. 6-11 present the main ones.


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PPARβ β β β/δ δ δ δ Activation Induces Enteroendocrine L Cell GLP-1 Production

Keywords: GLP-1, PPARβ/δ, incretin, diabetes

Background and Aims: GLP-1, an intestinal incretin produced by L cells through proglucagon processing, is secreted after nutrient ingestion and acts on endocrine pancreas β-cells to enhance insulin secretion. PPARβ/δ is a nuclear receptor which improves glucose homeostasis and pancreas islet function in diabetic animal models. Here, we investigated whether PPARβ/δ activation regulates L cell GLP-1 production. Methods: Proglucagon regulation and GLP-1 release were evaluated in murine GLUTag and human NCI-H716 L cells and in vivo using wild-type, PPARβ/δ-null and ob/ob C57Bl/6 mice treated with the PPARβ/δ synthetic agonists GW501516 or GW0742. Results: PPARβ/δ activation increased proglucagon expression and enhanced glucose-and bile acid-induced GLP-1 release by intestinal L cells in vitro and ex vivo in human jejunum. In vivo treatment with GW0742 increased proglucagon mRNA in the small intestine in wild-type, but not in PPARβ/δ-deficient mice. Treatment of wild-type and ob/ob mice with GW501516 enhanced the increase in plasma GLP-1 after an oral glucose load and improved glucose tolerance. Concomitantly, proglucagon and GLP-1 receptor mRNA increased in the small intestine and pancreas respectively. Finally, PPARβ/δ agonists activate the proglucagon gene transcription by interfering with the β-catenin/TCF-4 pathway. Conclusions: Our data show that PPARβ/δ activation potentiates GLP-1 production by the small intestine. Pharmacological targeting of PPARβ/δ is a promising approach in the treatment of type 2 diabetes, especially in combination with DPP-4 inhibitors.

Introduction

Glucagon-like peptides (GLPs) and glucagon are hormones encoded by the same gene, proglucagon, each having different physiological activities 1 . Due to alternative posttranslational processing, glucagon is predominantly produced in endocrine pancreatic α-cells by Prohormone Convertase (PC)-2 whereas GLPs are predominantly produced in the intestine and also the brain after cleavage by PC-1/3 2,-4 . Glucagon stimulates hepatic glucose production to maintain blood glucose levels upon fasting [START_REF] Jiang | Glucagon and regulation of glucose metabolism[END_REF] and inhibits insulin gene expression in pancreatic β-cells [START_REF] Hussain | Glucagon stimulates expression of the inducible cAMP early repressor and suppresses insulin gene expression in pancreatic beta-cells[END_REF] . Intestinal L cell GLP-1 is primarily produced during the postprandial state to promote β-cell insulin secretion to decrease α-cell glucagon secretion [START_REF] Baggio | Biology of incretins: GLP-1 and GIP[END_REF] .

The role of GLP-1 in the metabolic response to glucose ingestion has been established by several studies both in human and animal models. Acute administration of a GLP-1 antagonist leads to increased blood glucose levels in humans [START_REF] Edwards | Glucagon-like peptide 1 has a physiological role in the control of postprandial glucose in humans: studies with the antagonist exendin 9-39[END_REF] , whereas GLP-1 receptor gene disruption in mice results in glucose intolerance 9,-11 . The insulinotropic activity of GLP-1 is preserved in type 2 diabetes 12,-14 , but is reduced compared to healthy subjects [START_REF] Toft-Nielsen | Determinants of the impaired secretion of glucagon-like peptide-1 in type 2 diabetic patients[END_REF][START_REF] Nauck | Reduced incretin effect in type 2 (non-insulin-dependent) diabetes[END_REF] . Treatment of diabetic patients with GLP-1 increases meal-stimulated insulin levels and suppresses postprandial hyperglycaemia without causing hypoglycaemia [START_REF] Elahi | The insulinotropic actions of glucose-dependent insulinotropic polypeptide (GIP) and glucagon-like peptide-1 (7-37) in normal and diabetic subjects[END_REF][START_REF] Meier | Normalization of glucose concentrations and deceleration of gastric emptying after solid meals during intravenous glucagon-like peptide 1 in patients with type 2 diabetes[END_REF] . However, circulating GLP-1 has a very short half-life due to inactivation by the enzyme dipeptidyl peptidase IV (DPP-4). The potential of GLP-1 for the treatment of diabetes has led to the development of long-acting DPPIVresistant GLP-1 analogs and orally bioavailable DPP-4 inhibitors, both approaches having proven efficacy in lowering blood glucose levels and HbA 1c in patients with type 2 diabetes [START_REF] Toft-Nielsen | Determinants of the impaired secretion of glucagon-like peptide-1 in type 2 diabetic patients[END_REF][START_REF] Klonoff | Exenatide effects on diabetes, obesity, cardiovascular risk factors and hepatic biomarkers in patients with type 2 diabetes treated for at least 3 years[END_REF] .

However, an alternative approach may be to increase endogenous GLP-1 production through modulation of proglucagon gene transcription in enteroendocrine L cells.

The proglucagon promoter contains several transcriptional control elements localized in the 2.5-kb 5'-upstream sequence of the transcriptional initiation site [START_REF] Jin | The proglucagon gene upstream enhancer contains positive and negative domains important for tissue-specific proglucagon gene transcription[END_REF] . A number of transcription factors control proglucagon gene expression in a tissue-specific manner allowing physiologically appropriate regulation of the production of the different active peptides [START_REF] Holst | Glucagon and glucagon-like peptides 1 and 2[END_REF] . The transcription factor Pax6 activates proglucagon gene transcription in α-cells by binding the G1 response element in the proglucagon promoter [START_REF] Morel | The upstream promoter element of the glucagon gene, G1, confers pancreatic alpha cell-specific expression[END_REF][START_REF] Philippe | Islet-specific proteins interact with the insulin-response element of the glucagon gene[END_REF] . Disruption of the murine Pax6 gene not only markedly disrupts islet development, but also selectively eliminates enteroendocrine cell populations in the intestine including the subpopulation of GLP-1-producing cells, indicating the importance of Pax6 also in the control of intestinal proglucagon gene expression [START_REF] Hill | Essential requirement for Pax6 in control of enteroendocrine proglucagon gene transcription[END_REF] . By contrast, the G2 response element of the proglucagon promoter appears involved in the tissue-specific regulation of proglucagon transcription since protein kinase C activation enhances transcription in α-cells via this site [START_REF] Philippe | Glucagon gene transcription in an islet cell line is regulated via a protein kinase C-activated pathway[END_REF] , without influencing intestinal expression [START_REF] Drucker | Proglucagon gene expression is regulated by a cyclic AMP-dependent pathway in rat intestine[END_REF] .

Moreover, insulin and GSK-3β inhibitors, such as lithium, stimulate G2 response element activity specifically in enteroendocrine L cells through the β-catenin/TCF-4 signaling pathway resulting in increased GLP-1 production [START_REF] Ni | Transcriptional activation of the proglucagon gene by lithium and betacatenin in intestinal endocrine L cells[END_REF][START_REF] Yi | TCF-4 mediates cell type-specific regulation of proglucagon gene expression by beta-catenin and glycogen synthase kinase-3beta[END_REF] .

The Peroxisome Proliferator-Activated Receptors (PPARs) are ligand-activated transcription factors belonging to the nuclear receptor superfamily which regulate fatty acid, cholesterol and carbohydrate metabolism [START_REF] Wan | Activation of PPARdelta up-regulates fatty acid oxidation and energy uncoupling genes of mitochondria and reduces palmitate-induced apoptosis in pancreatic betacells[END_REF] . Whereas PPARα and PPARγ are the targets of the fibrates and glitazones respectively, the PPARβ/δ isoform appears an interesting pharmacological target for the treatment of disorders associated with the metabolic syndrome [START_REF] Gross | PPAR agonists: multimodal drugs for the treatment of type-2 diabetes[END_REF][START_REF] Coll | Peroxisome proliferator-activated receptor (PPAR) beta/delta: a new potential therapeutic target for the treatment of metabolic syndrome[END_REF] . PPARβ/δ-deficient mice challenged with a high-fat diet exhibit obesity [START_REF] Wang | Peroxisome-proliferator-activated receptor delta activates fat metabolism to prevent obesity[END_REF] and glucose-intolerance [START_REF] Lee | PPARdelta regulates glucose metabolism and insulin sensitivity[END_REF] . Furthermore, in vivo PPARβ/δ activation reduces body weight and lipid accumulation in adipose tissue and improves insulin sensitivity through increased skeletal muscle fatty acid oxidation [START_REF] Wang | Peroxisome-proliferator-activated receptor delta activates fat metabolism to prevent obesity[END_REF][START_REF] Lee | PPARdelta regulates glucose metabolism and insulin sensitivity[END_REF] .

In this study, we examined the hypothesis that PPARβ/δ activation enhances GLP-1 production by L cells, which would contribute to the improvement of glucose tolerance in vivo.

Our data identify intestinal tissue-specific regulation of proglucagon gene expression as a new mechanism of action for PPARβ/δ in glucose metabolism thus expanding its interest as target for the treatment of type 2 diabetes.

Materials and Methods

Chemicals and reagents. See Supplementary Material.

In Vitro Studies

Cell culture and treatment. The mouse GLUTag L cell model was kindly provided by D.J.Drucker (University of Toronto, Canada). The human NCI-H716 L cell, murine α-TC1/9 α-cells and rat INS1-E β-cells were purchased from ATCC ® (CCL-251™ and CRL-2350™ respectively). For details, see Supplementary Material.

Transient transfection assays. GLUTag L cells (14x10 4 cells/cm 2 ) were transfected for 12h using Lipofectamine TM 2000 reagent (Invitrogen) in serum free culture medium. For details, see Supplementary Material.

Gene reporter assays. See Supplementary Material.

GLP-1 secretion assays. GLUTag L cells were starved for 30min in glucose-free Krebs-phosphate-buffered medium (120mM NaCl, 5mM KCl, 0.25mM MgCl 2 , 0.5mM CaCl 2 and 2.2mM NaHCO 3 , pH7.2) supplemented with diprotin-A 100µM and BSA 0.1%. Cells were subsequently stimulated for 30min with Krebs buffer in the presence or not of glucose 5mM or a BA mix (50µM each; LCA and DCA). The cell supernatants were centrifuged at 500 x g at 4°C for 5min and GLP-1 measured with Elisa Kit (EGL P-35K, Millipore) using Mithras technology (Berthold).

Western blot analysis. See Supplementary Material.

Ex vivo Studies

Human islet isolation. Human islet isolation and culture conditions were previously described [START_REF] Popescu | The nuclear receptor FXR is expressed in pancreatic beta-cells and protects human islets from lipotoxicity[END_REF][START_REF] Lalloyer | Peroxisome proliferator-activated receptor alpha improves pancreatic adaptation to insulin resistance in obese mice and reduces lipotoxicity in human islets[END_REF] .

Human jejunum tissue isolation. Fresh human jejunum tissue was obtained with informed consent from obese patients undergoing gastric bypass surgery (gastrojejunal derivation) and enrolled in the ABOS study (ClinGov NCT01129297). For details, see Supplementary Material.

Animal Models and Experimental Protocols

Results

Glucose is a Positive Modulator of Proglucagon and Prohormone Convertase-1/3 (PC-1/3) Gene Expression in GLUTag L Cells.

Glucose is the primary stimulus for GLP-1 secretion by enteroendocrine L cells.

However, the transcriptional impact of glucose on proglucagon expression has not yet been reported. Time course analysis of gene expression in GLUTag L cells shows that proglucagon (Figure1A) and PC-1/3 (Figure1B) mRNA levels significantly increased upon incubation with glucose compared to lactate. By contrast, PC-2 mRNA was not regulated by glucose within 12h of incubation, while both glucose and lactate tended to increase its expression after 24h (Figure1C). Expression of GPBAR1 (also known as TGR5), the Gs-coupled membrane receptor that binds BAs to increase GLP-1 secretion 36,-38 , was detected in GLUTag L cells.

GPBAR1 mRNA was not regulated by glucose (Figure1D). The expression of the three PPAR isoforms was analysed in GLUTag L cells. PPARβ/δ and PPARγ were expressed at high levels (Ct=24 and Ct=26 respectively), while PPARα was barely detectable (Ct≥34). Glucose had no major effect on PPARβ/δ mRNA expression (Figure1E). By contrast, PPARγ mRNA levels were significantly lowered by glucose (Figure1F).

PPARβ β β β/δ δ δ δ Activation in GLUTag L Cells Increases Proglucagon Expression and

Improves Glucose-and BA-Induced GLP-1 Release.

Since PPARβ/δ and PPARγ are expressed in GLUTag L cells, it was next examined whether activation of these PPARs may regulate GLP-1 production. Surprisingly, PPARγ activation with rosiglitazone transiently decreased proglucagon mRNA levels (supplementary FigureS1). By contrast, treatment with the PPARβ/δ agonist GW501516 increased proglucagon mRNA in a time-dependent manner, reaching >2-fold increase after 24h (Figure2A). Treatment of GLUTag L cells with GW0742, another PPARβ/δ agonist, also significantly increased proglucagon mRNA levels (supplementary FigureS2-A). Incubation of GLUTag L cells for 24h with increasing doses of GW501516 resulted in >4-fold elevated proglucagon mRNA levels at the highest tested dose (Figure2B). To test whether this effect is mediated via PPARβ/δ, GLUTag L cells were transfected with PPARβ/δ siRNA before incubation for 24h with GW501516. PPARβ/δ knockdown abolished the induction of proglucagon mRNA levels by GW501516 (Figure2C). Western blot analysis showed that proglucagon protein levels increased upon treatment for 24h with GW501516, an effect blunted by PPARβ/δ-silencing (Figure2D). PC-1/3 mRNA levels were not influenced by PPARβ/δ agonist treatment (data not shown).

Next, it was determined whether the increased proglucagon expression due to PPARβ/δ activation influences GLP-1 production according to the protocol described in To determine how PPARβ/δ regulates proglucagon transcription, the -350-bp proglucagon-Luc reporter gene was transfected together with the PPARβ/δ expression vector in GLUTag L cells which were subsequently treated with PPARβ/δ agonists. Treatment of transfected GLUTag L cells with GW501516 or GW0742 resulted in an activation of proglucagon promoter activity, an effect enhanced by PPARβ/δ co-transfection (Figure4A).

Since lithium activates the -350-bp proglucagon promoter in GLUTag L cells via the βcatenin/TCF-4 signaling pathway [START_REF] Ni | Transcriptional activation of the proglucagon gene by lithium and betacatenin in intestinal endocrine L cells[END_REF][START_REF] Yi | TCF-4 mediates cell type-specific regulation of proglucagon gene expression by beta-catenin and glycogen synthase kinase-3beta[END_REF] and since PPARβ/δ activation has been shown to influence the β-catenin/TCF-4 signaling pathway [START_REF] Korinek | Two members of the Tcf family implicated in Wnt/beta-catenin signaling during embryogenesis in the mouse[END_REF] , it was determined whether the PPARβ/δinduction of proglucagon gene transcription in enteroendocrine L cells is mediated through the β-catenin-TCF/Lef signaling pathway. siRNA knockdown of TCF-4 abolished the GW501516induced increase of proglucagon mRNA levels in GLUTag cells (Figure4B). Moreover, siRNA knockdown of β-catenin in GLUTag L cells did not alter basal proglucagon mRNA levels, but significantly reduced GW501516-induced proglucagon mRNA expression. In addition, overexpression of the constitutively active mutant β-catenin-S33Y significantly increased basal and GW501516-induced proglucagon mRNA expression (Figure4C). Therefore, PPARβ/δ transcriptionally regulates enteroendrine L cell proglucagon expression through stimulation of the β-catenin/TCF-4 pathway (Figure4D).

Activation of PPARβ β β β/δ δ δ δ Increases Intestinal Proglucagon mRNA Expression and GLP-1 Secretion in Response to an Oral Glucose Load in Mice.

To determine whether PPARβ/δ activation regulates proglucagon gene expression in vivo, C57Bl/6 mice were treated with synthetic PPARβ/δ agonists. GW0742 treatment resulted in a significant increase of proglucagon mRNA levels in the different parts of the small intestine (Figure5 and supplementary FigureS3 Top). Measured as a marker of PPARβ/δ activity and a well-characterized PPARβ/δ target gene [START_REF] Kim | PPARbeta/delta selectively induces differentiation and inhibits cell proliferation[END_REF][START_REF] Chawla | PPARdelta is a very low-density lipoprotein sensor in macrophages[END_REF] , PLN2 (perilipin2, also called ADRP)

intestinal mRNA levels increased upon GW0742 treatment measured (supplementary FigureS3 Bottom). This effect was mediated via PPARβ/δ, since proglucagon mRNA was not increased upon GW0742-treatment of PPARβ/δ-deficient mice (Figure5). A similar, increase of proglucagon mRNA levels was observed in the jejunal mucosa upon GW501516 treatment (Figure6A, left panel). However, proglucagon mRNA levels were not regulated in the pancreas To investigate whether PPARβ/δ activation enhances the incretin effect in vivo, treated and control wild-type mice were challenged with a glucose load. GW501516-treated mice displayed an increase of plasma GLP-1 (Figure6B) and insulin concentrations (Figure6C)

15min after the oral glucose gavage, associated with an improved glucose tolerance (Figure6D). Indeed, the iAUC glucose was reduced by 25% (P<.05) after PPARβ/δ activation (Figure6D, inset). These results suggest that PPARβ/δ activation may improve glucose homeostasis, at least in part, through enhancing the GLP-1 pathway.

Activation of PPARβ β β β/δ δ δ δ Enhances Intestinal GLP-1 Release, Restores Pancreas

Responsiveness to Insulin Secretion and Reduces Postprandial Glycemia in ob/ob

Mice.

The pathophysiological relevance of the enhanced oral glucose-induced GLP-1 secretion upon PPARβ/δ activation was examined in insulin-resistant obese mice. Treatment of ob/ob mice with GW501516 resulted in a significant increase of proglucagon, as well as PLN2 mRNA levels (Figure7A). In line, plasma GLP-1 increased 15min after oral, but not intraperitoneal, glucose loading in GW501516-treated ob/ob mice (Figure7B), evidencing the contribution of the incretin axis. Moreover, fasting plasma insulin was lower in GW501516treated compared to control ob/ob mice (Figure7C), likely a reflection of the previously reported peripheral insulin-sensitizing effects of PPARβ/δ activation [START_REF] Wang | Peroxisome-proliferator-activated receptor delta activates fat metabolism to prevent obesity[END_REF][START_REF] Lee | PPARdelta regulates glucose metabolism and insulin sensitivity[END_REF] . However, the capacity of β-cells to secrete insulin upon glucose challenge was also improved in GW501516-treated ob/ob mice. A significant increase of plasma insulin was observed in GW501516-treated ob/ob mice 15min after glucose tolerance test, which was more pronounced upon oral versus intraperitoneal glucose loading, thus illustrating the enhanced incretin effect after PPARβ/δ activation. By contrast, no significant changes in plasma insulin were found in control ob/ob mice likely due to the pronounced hyperinsulinemic insulin-resistant phenotype (Figure7C).

Finally, glucose excursion in the oral glucose tolerance test was improved in GW501516treated ob/ob mice (Figure7D) with a significant reduction of iAUC glucose compared to untreated ob/ob mice (Figure 7D, inset).

Discussion

Our results show that PPARβ/δ activation positively regulates enteroendrine L cell GLP-1 production and enhances its response to glucose and BAs. Proglucagon mRNA levels and GLP-1 production were markedly increased in PPARβ/δ agonist-treated mice contributing to a significant improvement of glucose homeostasis. Likewise, activation of mouse and human enteroendocrine L cells in vitro or ex vivo with PPARβ/δ agonists significantly increased proglucagon mRNA levels and enhanced glucose-and BA-induced GLP-1 secretion.

As described by Drucker et al. [START_REF] Brubaker | Regulation of glucagon-like peptide-1 synthesis and secretion in the GLUTag enteroendocrine cell line[END_REF][START_REF] Drucker | Activation of proglucagon gene transcription by protein kinase-A in a novel mouse enteroendocrine cell line[END_REF] , the murine GLUTag L cell line is a useful model for the analysis of the molecular determinants of enteroendocrine gene expression. We thus used this model and studied the transcriptional regulation of genes whose products participate in GLP-1 production. Our data show that glucose is not only a primary stimulus of GLP-1

release, but it is also a positive transcriptional modulator of proglucagon and PC-1/3 mRNA levels in GLUTag L cells. Moreover, we found that GLUTag L cells express PPARγ and

PPARβ/δ at high levels, whereas PPARα is not expressed. A significant decrease of PPARγ was observed in glucose-stimulated GLUTag L cells, whereas no major change of PPARβ/δ expression was observed. Interestingly, proglucagon mRNA levels were significantly decreased in GLUTag L cells treated with the PPARγ agonist rosiglitazone. Additionally, in wild-type mice treated with rosiglitazone, a significant decrease of proglucagon mRNA levels in small intestine was also observed (data not shown). This observation is consistent with results obtained in pancreatic α-cells in which proglucagon gene transcription is also decreased by rosiglitazone treatment through interfering with the Pax6/G1 enhancer element pathway [START_REF] Schinner | Repression of glucagon gene transcription by peroxisome proliferator-activated receptor gamma through inhibition of Pax6 transcriptional activity[END_REF][START_REF] Krätzner | A peroxisome proliferator-activated receptor gamma-retinoid X receptor heterodimer physically interacts with the transcriptional activator PAX6 to inhibit glucagon gene transcription[END_REF] . We thus speculate that the negative rosiglitazone effect on proglucagon gene expression in enteroendocrine L cell may be due to inhibition of the Pax6 pathway, which has an important role in intestinal proglucagon gene expression [START_REF] Hill | Essential requirement for Pax6 in control of enteroendocrine proglucagon gene transcription[END_REF] . In line with these observations, it is reasonable to speculate that PPARγ does not enhance GLP-1 biosynthesis.

Since previous reports have demonstrated that GLP-1 receptor agonists inhibit β-cell apoptosis under conditions of glucolipotoxicity [START_REF] Hui | Glucagon-like peptide-1 inhibits apoptosis of insulin-secreting cells via a cyclic 5'-adenosine monophosphate-dependent protein kinase A-and a phosphatidylinositol 3-kinase-dependent pathway[END_REF][START_REF] Buteau | Glucagon-like peptide-1 prevents beta cell glucolipotoxicity[END_REF] and that PPARβ/δ activation also prevents palmitate-induced β-cell apoptosis [START_REF] Wan | Activation of PPARdelta up-regulates fatty acid oxidation and energy uncoupling genes of mitochondria and reduces palmitate-induced apoptosis in pancreatic betacells[END_REF] , we speculated that the protective effect of PPARβ/δ on βcell apoptosis also may be indirectly modulated via GLP-1. Interestingly, GLP-1 receptor levels were induced in the pancreas of GW501516-treated mice which may potentiate β-cell GLP-1 responsiveness. Our data demonstrate that PPARβ/δ activation enhances GLP-1 secretion both in response to an oral glucose load in vivo and in presence of GLP-1 secretagogues (glucose and BAs) in enteroendocrine L cells in vitro and ex vivo. These effects correlated with higher proglucagon mRNA in both L cells and the small intestine, whereas in pancreas, proglucagon mRNA remained unaffected. The increase of proglucagon mRNA levels in L cells in vitro by PPARβ/δ activation was confirmed by western blot analysis. Furthermore, these effects were absent both in vivo using PPARβ/δ-deficient mice, and in vitro using siRNA PPARβ/δ in GLUTag L cells. These observations collectively demonstrate that the observed effects of GW501516 and GW0742 are mediated via PPARβ/δ. More importantly, GW501516activated human NCI-H716 L cells and human intestinal tissue displayed a significant increase of proglucagon mRNA levels supporting the notion that PPARβ/δ activation of this pathway is also relevant to humans. Together, our results demonstrate that PPARβ/δ activation likely potentiates GLP-1 action through the increase of both enteroendocrine L cell GLP-1 production and pancreatic GLP-1 receptor expression.

Further, it has been established that lithium, a GSK-3β inhibitor, and insulin increase proglucagon expression through stimulation of the β-catenin-TCF/Lef signaling pathway only in enteroendocrine GLP-1-producting L cells, but not in pancreatic glucagon-producing αcells [START_REF] Ni | Transcriptional activation of the proglucagon gene by lithium and betacatenin in intestinal endocrine L cells[END_REF][START_REF] Yi | TCF-4 mediates cell type-specific regulation of proglucagon gene expression by beta-catenin and glycogen synthase kinase-3beta[END_REF] . This is due to the fact that the transcription factor TCF-4 is expressed in small intestine epithelium [START_REF] Korinek | Two members of the Tcf family implicated in Wnt/beta-catenin signaling during embryogenesis in the mouse[END_REF] , but not in islet α-cells [START_REF] Yi | TCF-4 mediates cell type-specific regulation of proglucagon gene expression by beta-catenin and glycogen synthase kinase-3beta[END_REF]

. Han et al. recently demonstrated that the

PPARβ/δ agonist GW501516 stimulates the Wnt-β-catenin-TCF/Lef signaling pathway in a human cholangiocarcinoma cell line model [START_REF] Han | Regulation of Wnt/beta-catenin pathway by cPLA2alpha and PPARdelta[END_REF] . The role of the β-catenin/TCF-4 signaling pathway was thus investigated in the response of enteroendocrine L cells to PPARβ/δ activation. Our data strongly suggest that the mechanism through which PPARβ/δ activates proglucagon promoter activity involves the β-catenin/TCF-4 pathway. GW501516 induction of proglucagon gene expression was abolished by TCF-4 knockdown and significantly decreased by β-catenin knockdown. However, an increase of proglucagon mRNA levels was observed in GLUTag L cell overexpressing constitutively β-catenin which was significantly much more important after PPARβ/δ activation. This result is consistent with a previous report showing that G2 element activity of the proglucagon was increased in β-catenin-S33Y transfected GLUTag L cells, whereas in the InR1-G9 α-cell line, G2 response element activity remained unaffected [START_REF] Yi | TCF-4 mediates cell type-specific regulation of proglucagon gene expression by beta-catenin and glycogen synthase kinase-3beta[END_REF] . This mechanism is consistent with the observation that PPARβ/δ activation does not influence proglucagon gene expression in pancreatic α-cells.

Previous studies suggested that PPARβ/δ activation is similar to PPARγ activation in their effects on improving insulin sensitivity [START_REF] Lee | PPARdelta regulates glucose metabolism and insulin sensitivity[END_REF][START_REF] Winzell | Improved insulin sensitivity and islet function after PPARdelta activation in diabetic db/db mice[END_REF][START_REF] Tonelli | Mechanisms of early insulin-sensitizing effects of thiazolidinediones in type 2 diabetes[END_REF] . PPARβ/δ agonist treatment of diabetic db/db, ob/ob or high fat-fed mice reduces weight gain, increases skeletal muscle fatty acid oxidation and decreases plasma triglycerides [START_REF] Wang | Peroxisome-proliferator-activated receptor delta activates fat metabolism to prevent obesity[END_REF][START_REF] Schuler | PGC1alpha expression is controlled in skeletal muscles by PPARbeta, whose ablation results in fiber-type switching, obesity, and type 2 diabetes[END_REF][START_REF] Wang | Regulation of muscle fiber type and running endurance by PPARdelta[END_REF] . In line with these previous data, our results show that GW501516-treated ob/ob mice also displayed a 10% decrease of body weight in association with a reduction of epididymal adipocyte tissue mass (2.7±0.1 vs.

3.1±0.1g ; P<.05) and fasting glycemia (167.8±11.5 vs. 205.8±7.1mg/dl ; P<.05). Moreover, GW501516 treatment of ob/ob mice significantly decreased basal plasma insulin levels and restored insulin secretion after a glucose load. In addition to the peripheral effects of PPARβ/δ treatment, our results also show that GW501516 treatment of ob/ob mice leads to improved glucose homeostasis through stimulation of the GLP-1 signaling pathway, as demonstrated by the increase of intestinal proglucagon expression associated with a higher GLP-1 release in response to oral glucose. As a result, a more pronounced increase in plasma insulin levels is observed when glucose is administered orally.

In conclusion, our study identifies a new role for PPARβ/δ as a positive regulator of GLP-1 signaling by increasing both proglucagon and GLP-1 receptor gene expression in enteroendocrine GLP-1-producing L cells and pancreas, respectively. Together, pharmacological targeting of the GLP-1 pathway by PPARβ/δ agonists may prove to be a promising approach for the treatment of type 2 diabetes, especially in combination with DPP-4 inhibitors.

Cyclophilin

F GCA TAC GGG TCC TGG CAT CTT GTC C 100 nM R ATG GTG ATC TTC TTG CTG GTC TTG C F. forward primer; R. reverse primer.

Figure2E. 4

 4 Figure2E. Treatment for 24h with GW501516 enhanced GLP-1 secretion both in basal and

(

  Figure6A, central panel), nor in αTC-1/9 cells (supplementary FigureS4-A) or isolated human islets (supplementary FigureS4-B), while PLN2 or CTP-1α, analysed as markers of PPARβ/δ activity, were induced. These data indicate that the proglucagon gene regulation by PPARβ/δ activation is restricted to the intestine. By contrast, GLP-1 receptor mRNA levels increased both in the pancreas of GW501516-treated mice (Figure6A, right panel) and in isolated human islets (supplementary FigureS4-B). This up-regulation may enhance the response to the increased intestinal GLP-1 production. By contrast, treatment of rat INS1-E pancreatic β-cells with GW51516 did not influence RGLP-1 mRNA, whereas CPT-1 mRNA increased (FigureS4, panel C). This suggests that the effects of PPARβ/δ activation on RGLP-1 gene regulation in the pancreas are likely due to indirect mechanisms requiring paracrine signaling.

Figure 1 .

 1 Figure 1. Glucose increases proglucagon and prohormone convertase 1/3 and decreases PPARγ γ γ γ mRNA levels in GLUTag L cells. GLUTag L cells were 12h-starved in glucose-free medium and then stimulated with glucose (5mM) or lactate (10mM) as control for

Figure 2 .

 2 Figure 2. PPARβ β β β/δ δ δ δ activation with GW501516 increases proglucagon mRNA levels and enhances glucose-and BA-induced GLP-1 release in GLUTag L cells. Proglucagon

Figure 3 .

 3 Figure 3. PPARβ β β β/δ δ δ δ activation increases proglucagon expression and enhances glucoseand BA-induced GLP-1 release in human enteroendocrine L cells. Differentiated NCI-

Figure 4 .

 4 Figure 4. PPARβ β β β/δ δ δ δ Activation Induces Proglucagon Promoter Activity through the β β β β-catenin/TCF-4 Signaling Pathway in GLUTag L cells. (A) Cells were co-transfected with -

Figure 5 .

 5 Figure 5. The increase of intestinal proglucagon mRNA levels by GW0742 is PPARβ β β β/δ δ δ δ dependent. Wild-type (PPARβ/δ +/+ ) and PPARβ/δ-deficient (PPARβ/δ -/-) male mice (n=4-5

Figure 6 .

 6 Figure 6. In vivo PPARβ β β β/δ δ δ δ activation with GW501516 increases intestinal proglucagon mRNA levels, enhances GLP-1 secretion and improves glucose homeostasis. Wild-type

Figure 7 .

 7 Figure 7. PPARβ β β β/δ δ δ δ activation enhances GLP-1 secretion and restores insulin secretion in ob/ob mice. Obese (ob/ob) C57BL/6 male mice (8-10 animals per group) were treated or

Oral and intra-peritoneal glucose tolerance tests (OGTT, IPGTT). Oral

	PPARβ β β β/δ δ δ δ-deficient and ob/ob mice. 6-8 week-old male PPARβ/δ-deficient 35 , obese
	ob/ob and wild-type mice (Charles River laboratories), fed a chow diet (A03; UAR, France)
	and maintained in a temperature-controlled room (22°C) on a 12h light-dark cycle, were
	treated by gavage with vehicle or the PPARβ/δ synthetic agonists (GW0742 or GW501516) at
	the indicated doses and times. Plasma and several tissues including different intestinal
	sections (duodenum, jejunum, ileum, and colon) and pancreas were collected after 6h fasting.
	All tissues were snap-frozen in liquid nitrogen and stored at -80°C until RNA isolation. Animal
	care and experimental procedures were performed according to approved institutional
	guidelines (# CEEA 02/2008).
	(OGTT) and
	intra-peritoneal (IPGTT) glucose tolerance tests were performed on overnight fasted mice.
	Sitagliptin was given orally at a dose of 25 mg/kg 45min before the glucose challenge (3g/kg
	of body weight). For details, see Supplementary Material.

RNA isolation and quantification by real time-quantitative PCR (RT-QPCR). RNA

	was extracted from cells and intestinal mucosa using the Trizol ® reagent (Life Technologies,
	Gaithersburg, MD, USA). Mouse and human intestinal tissues and pancreas RNA were
	extracted using the guanidinium thiocyanate (GSCN)/phenol/chloroform (Sigma). RNA
	extracts were treated with DNase I to eliminate contaminating genomic DNA. For details on
	RNA quantification, see Supplementary Material.
	Statistical analysis
	Statistical analyses were performed by using the unpaired Student's t test. All data are
	expressed as mean±SE. Statistically significant differences between treatments are reported
	to DMSO (control) and CMC (vehicle) for in vitro and in vivo experiments respectively. P
	values < .05 were considered statistically significant.

Table S1 .

 S1 Oligonucleotide sequences used for Real Time PCR analysis

					final concentration
	species	gene name	type a	sequence (5'to 3')	L cells	small intestine
				α-cells or Islets
	Mouse	Proglucagon	F ATT GCC AAA CGT CAT GAT GA R GGC GAC TTC TTC TGG GAA GT	100 nM	500 nM
		PPARβ/δ	F ACT TGG CGT GGC GCC TGC R AGC GGT GTG GGT ATG CGC A	100 nM	300 nM
		PPARγ	F CCG TGA TGG AAG ACC ACT CG R AGG CCT GTT GTA GAG CTG GGT C	100 nM	300 nM
		PPARα	F ATC GCG TAC GGC AAT GGC TTT A R TAA TTT GCT GTG GAG ATC GGC CTG	300 nM	300 nM
		GPBAR1	F AAC GCT ACA TGG CAG TGT TG R GGA GGC CAT AAC TTC CAG GTA GA	300 nM	300 nM
		PLN2	F ATC CAG AAG CTG GAG CCA CAA A R CCA CAT CCT TCG CCC CAG TTA		100 nM
		RGLP-1	F CTG AAG CGA TGA CCA AGG CGG R CGG AGT GTG AAG AGT CTA AGC G	300 nM
	Human	Proglucagon	F GCA CAT TCA CCA GTG ACT ACA GCA R TGG CAG CTT GGC CTT CCA AAT A	100 nM	500 nM
		RGLP-1	F TCC TCC TCG GCT TCA GAC ACC TGC A R CCA CTT CAG GGC TGC GTC CTT GAT G	300 nM
		CPT-1α	F TCT GGC TGG ACA TGC AGT TGG C R TAT CGT GGT GGT GGG CGT GAT G	100 nM
	Rat	RGLP-1	F ACA TCC ACT TGA GGG CAG CG R CAG ACA CTT GCA CTG CAC CAG G	300 nM
		CPT-1	F TCC AAA CAT CAC TGC CCA AG R GAA TTG TGG CTG GCA CAC TG	300 nM
	Mouse/Human	TFIIB	F ACC AGC CGT TTG GAT GCT C R CCC ACA TCA ATA ACC CGG TC	100 nM	100 nM

Acknowledgements

The authors gratefully acknowledge D.J.Drucker and F.Gribble for providing the GLUTag L cells and A.Billin and T.Willson for providing GW0742. We appreciate the generous gifts of -350bp-proglucagon-Luc plasmid from W.Knepel and pEGFP-β-catenin-S33Y plasmid from D.Bershadsky.

Grant Support

This work was supported by EU Grants AMI-DIAB (Fonds de Compétitivité des Entreprises, convention n°06 2 90 6169) and Hepadip (LSHM-CT-2005-018734) w ith the collaboration of Université Lille 2, Région Nord/Pas-de-Calais, the FEDER and French State. We acknowledge grant support from the 'Nouvelle Société Française d'Athérosclérose' (to M. Daoudi). ; CMC, Carboxy-Methyl-Cellulose; DCA, DeoxyCholic Acid; DPP-4, DiPeptidyl Peptidase-4; FBS, Fetal Bovine Serum; GLP, Glucagon-Like Peptide; GPBAR1,

Supplementary Material

Chemicals and reagents. The DPP-4 inhibitors diprotin A and sitagliptin were purchased from Sigma and MSD respectively. BSA, Deoxycholic acid (DCA) and lithocholic acid (LCA) was purchased from Sigma-aldrich. BSA was dissolved in demineralised water.

DCA and LCA were dissolved in DMSO. The synthetic PPARβ/δ agonist GW501516 was a generous gift of Genfit (Parc Eurasanté, Loos F-59120, France). The synthetic PPARβ/δ agonist GW0742 was synthesized by GlaxoSmithKline (Research Triangle Park, NC). GW501516 and GW0742 were dissolved in dimethylsulfoxide (DMSO, Sigma-aldrich) and 0.5% Carboxy-Methyl-Cellulose (CMC, low viscosity, Sigma) for in vitro and in vivo experiments respectively. These PPARβ/δ agonists pharmacologically activate both murine and human PPARβ/δ 1 with similar and potent EC 50 values in the nanomolar range for PPARβ/δ activity and with >1000-fold selectivity for PPARβ/δ over the other PPAR receptor subtypes 1,2 .

RNA quantification by real-time PCR.

Total RNA (1-2 µg) was reverse transcribed using the SuperScript First-strand Synthesis System for Reverse transcription (Life Technologies, Gaithersburg, MD, USA). Reverse transcription reactions without reverse transcriptase were performed as negative controls for subsequent PCR reactions. Reverse transcribed cDNAs were quantified by SYBR ® green based real-time PCR using specific oligonucleotides (listed in Supplemental Table S1) on an Mx3000 apparatus (Stratagene, La Jolla, CA) after 1/5 and 1/2.5 dilution of RT products for the in vitro and in vivo samples respectively. mRNA levels were normalized to TFIIB or cyclophilin as internal control.

In vitro studies

Cell culture and treatment. GLUTag L cells were grown and maintained in DMEM + GlutaMAX TM -1 with glucose 1 g/l (Cat. No. 21885-025, Invitrogen) supplemented with 10% FBS, glutamine 2mM, penicilin 10 4 IU/ml and streptomycin 10 mg/ml. For the experiments, Western blot analysis. Total cellular extracts from GLUTag and NCI-H716 L cells were obtained after treatment with SDS-lysis buffer (Nonidet P401%, Tris/HCl (pH8.0) 20mM, glycerol 10%, NaCl 150mM, and EDTA 1mM) supplemented with protease inhibitor cocktail (Complete TM Mini, Roche Diagnostics). Cellular proteins (50µg) were separated on 12% SDS-PAGE and transferred to nitrocellulose membrane (Hybond TM , ECL TM Nitrocellulose membrane; Amersham Biosciences). After blocking non-specific binding sites with 5% BSA in TBST (20mM Tris HCl, 137mM NaCl, pH7.6, 0.1% Tween 20), the blots were probed with monoclonal antibodies raised against proglucagon (11E2, sc-80603) and TFIIB (sc-225) as control (Santa Cruz Biotechnology, INC.) Horseradish peroxidase-conjugated secondary antibodies were used for detection using the enhanced chemiluminescence detection kit (ECL, Amersham Biosciences) and autoradiography (Camera Gbox, Ozyme).

In vivo Models and Experimental Protocols

Oral and Intra-Peritoneal Glucose Tolerance Tests (OGTT, IPGTT). Blood samples were collected from the tail vein at 0, 15, 30, 45, 60 and 90min and glycemia was measured using a Glucometer (Accu-Check active®, Roche Applied Science). Retro-orbital blood samples were collected in diprotin A (100µM) 15min after the glucose load following immediate centrifugation at 4°C. Plasma Insulin and GLP-1 were measured using Rat insulin Elisa kit (Mercodia, Uppsala, Sweden) and Glucagon-Like peptide-1 (Active) Elisa kit (EG LP-35K, Millipore) respectively.

Human jejunum tissue isolation. The pieces of human jejunum tissue freshly isolated (about 2 cm 2 ) were cultured for 24h in RPMI + GlutaMAX TM -1 with glucose 1 g/l (Cat.

No. 61870-010, Invitrogen) supplemented with 10% FBS, penicilin 10 4 IU/ml and streptomycin 10 mg/ml in presence of PPARβ/δ agonist GW501516 (1µM) or DMSO (control).

Supplementary Data


Some Considerations on Routing In Particular and Lossy Environments

Keywords: Réseaux de capteurs, RPL, Routage, Diffusion, Multicast

Le mémorandum présente une sélection d'observations et d'expériences acquises lors de la réalisation d'une implémentation prototype de RPL et lors d'une évaluation de l'applicabilité de ce protocole à différents déploiements spécifiques dans le monde réel.

Introduction

RPL -the "Routing Protocol for Low Power and Lossy Networks" (RPL) [START_REF] Winter | RPL: IPv6 Routing Protocol for Low power and Lossy Networks[END_REF] -is a proposal for an IPv6 routing protocol for Low-power Lossy Networks (LLNs), by the ROLL Working Group in the Internet Engineering Task Force (IETF). The basic construct in RPL is a DODAG -a destination oriented directed acyclic graph, rooted in a "controller".

Traffic inside the LLN flows along this DODAG, either upward (towards the "controller") or downward. In RPL, upward routes, having the controller as destination (either by way of explicitly addressing the destination, or by using the controller as "gateway"), are provided by the DODAG construction mechanism: each LLN router selects a set of parents, on a path towards the controller, as well as a preferred parent. Once a router is part of a DODAG (i.e. has selected parents) will emit DODAG Information Object (DIO) messages, using link-local multicasting, indicating its respective rank in the DODAG (i.e. its position -distance according to some metric(s), in the simplest form hopcount -with respect to the root).

Routes for any destination inside the LLN, other than the controller, are provided by these destinations generating Destination Advertisement Objects (DAOs).

RPL Data Traffic Flows

RPL makes a-priori assumptions of traffic patterns: sensor-to-controller traffic (multipoint-to-point) is predominant, controller-to-sensor traffic (point-tomultipoint) is rare and sensor-to-sensor traffic is somewhat esoteric.

An LLN router in RPL will select a "preferred parent", to serve as a default route towards the controller (or prefixes advertised by the controller). Thus, RPL provides "upward routes" or "multipoint-to-point routes" from the sensors towards the controller.

An LLN router in RPL, which wishes to act as a destination for traffic ("downward routes" or "point-to-multipoint"), will issue DAOs upwards in the DODAG towards the controller, describing which prefixes belong to, and can be reached via, that LLN router.

Sensor-to-sensor routes are supported by having the source sensor transmit, via its default route, to the controller, which will send the data packet downward towards the destination sensor.

Why This Is A Critical Point

The data traffic characteristics assumed by RPL does not represent a universal distribution of traffic patterns in LLNs, for example:

• There are scenarios in which sensor-to-sensor traffic is assumed a more common occurrence, such as [START_REF] Martocci | Building Automation Routing Requirements in Low Power and Lossy Networks[END_REF].

• There are scenarios, in which all traffic is bi-directional, such as for example in case sensor devices in the LLN are, in majority, "actively read": a request is issued by the controller to a specific sensor, and the sensor value is expected returned.

For the former, all sensor-to-sensor routes transit the root, possibly causing congestion in the wireless spectrum near the root, as well as draining energy from the intermediate routers on an unnecessarily long path.

For the latter, all LLN routers are required to generate DAOs, which generates a considerable control traffic overhead.

DAO Mechanism

Two distinct "modes of operation" for the downward mechanism are specified in RPL:

(i) In storing mode, each router is assumed to maintain routes to all LLN routers in its sub-DODAG, i.e. routers that are "deeper down" in the DAG. DAOs propagate from the routers towards the controller, where each intermediate router adds its reverse routing stack to the DAO message (aggregating routes where possible).

(ii) In non-storing mode, only the controller stores routes to all LLN routers in the network. Each LLN router unicasts DAOs to the controller, which then calculates routes to all destinations by "piecing together" the information collected from DAO messages (which contain the destination prefix of the LLN router and addresses of the parents through which it is reachable). In nonstoring mode, downward traffic is sent by way of source.

Why This Is A Critical Point

In non-storing mode, source-routes increase the L3 header length -with small MTUs, this may lead to increased fragmentation, thus successful delivery of an IP packet depends on successful delivery of possibly more fragments -a single fragment lost renders otherwise successfully delivered fragments of the IP packet lost. LLNs are, generally, characterized by higher loss-rates and smaller MTUs [START_REF]ROLL Charter[END_REF]. In addition to possible fragmentation, the maximum length of the source routing header [START_REF] Hui | An IPv6 Routing Header for Source Routes with RPL[END_REF] is limited to 255 octets at maximum. As each IPv6 address has a length of 16 octets, not more than 15 hops from the source to the destination are possible. Even with address compression, such as specified in [START_REF] Montenegro | Transmission of IPv6 Packets over IEEE 802.15.4 Networks[END_REF], the maximum path length may not exceed 127 hops. This excludes scenarios with long "chain-like" topologies, such as traffic lights along a street.

In storing mode, each LLN router has to store routes for its sub-DODAG. This implies that, for LLN routers near the controller, the required storage is only bounded by the number of paths to all other LLN routers in the network. As RPL targets constrained devices with little memory, but in networks consisting of thousands of routers, the storing capacity on these LLN routers may not be sufficient.

Bidirectionality Hypothesis

Parents (and the preferred parent) are selected based on receipt of DIOs, without verification of the ability for a LLN router to successfully communicate with the parenti.e. without any bidirectionality check of links. However, the basic use of links is for "upward" routes, i.e. for the LLN router to use a parent INRIA (the preferred parent) as relay towards the DODAG controller -in the opposite direction of the one in which the DIO was received.

Why This Is A Critical Point

Unidirectional links are no rare occurrence, such as is known from wireless multihop networks. If an LLN router receives a DIO on such a unidirectional link, and selects the originator of the DIO as parent, that would be a bad choice: unicast traffic in the upward direction would be lost. If the router had verified the bidirectionality of links, it might have selected a better parent, to which it has a bidirectional link.

5 Why NUD is NOT a Solution [START_REF] Winter | RPL: IPv6 Routing Protocol for Low power and Lossy Networks[END_REF] suggests using Neighbor Unreachability Detection (NUD) [START_REF] Narten | Neighbor Discovery for IP version 6 (IPv6)[END_REF] to detect and recover from the situation of unidirectional links between a LLN router and its (preferred) parent(s). When a tries (and fails) to actually use b for forwarding traffic, NUD is supposed engaged to detect and prompt corrective action, e.g. by way of selecting an alternative preferred parent.

NUD is based upon observing if a data packet is making forward progress towards the destination, either by way of indicators from upper-layer protocols (such as TCP) 1 or -failing that -by unicast probing by way of transmitting a unicast Neighbor Solicitation message and expecting that a solicited Neighbor Advertisement message be returned.

Why This Is A Critical Point

A LLN router may receive, transiently, a DIO from a router, much closer (in terms of rank) to the controller than any other router from which a DIO has been received. Some, especially wireless, link layers may exhibit different transmission characteristics between multicast and unicast transmissions2 , leading to a (multicast) DIO being received from farther away than a unicast transmission can reach. DIOs are sent (downward) using link-local multicast, whereas the traffic flowing in the opposite direction (upward) is unicast. Thus, a received (multicast) DIO may not be indicative of useful unicast connectivity -yet, RPL might cause this LLN router to select this attractive router as its preferred parent. This may happen both at initialization or at any time during the LLN lifetime, as RPL allows attachment to a "better parent" at any time.

A DODAG so constructed may appear stable and converged until such time that unicast traffic is to be sent and, thus, NUD invoked. Detecting only at that point that unicast connectivity is not maintained, and causing local (and possibly global) repairs exactly at that time, may lead to traffic not being deliverable.

Also, absent all LLN routers consistently advertising their reachability through DAO messages, a protocol requiring bi-directional flows between the communicating devices, such as TCP, will be unable to operate.

Finally, upon having been notified by NUD that the "next hop" is unreachable, a LLN router must discard the preferred parent and select another preferred parent -hoping that this time, the preferred parent is actually reachable. Also, if NUD indicates "no forward progress" based on an upper-layer protocol, there is no guarantee that the problem stems exclusively from the preferred parent being unreachable. Indeed, it may be a problem father ahead, possibly outside the LLN, thus changing preferred parent will do nothing to alleviate the situation.

RPL Implementability and Complexity

RPL is designed to operate on "LLN routers [...] with constraints on processing power, memory, and energy (battery power)" [START_REF] Winter | RPL: IPv6 Routing Protocol for Low power and Lossy Networks[END_REF]. However, the 159 pages long specification 3 , describes complex mechanisms (e.g. the upwards and downward data flows, a security solution, manageability of LLN routers, auxiliary functions for autoconfiguration of LLN routers, etc.), and provides no less than 9 message types, and 10 different message options.

To give one example, the ContikiRPL implementation 4 , which does not provide non-storing mode or any security features, consumes about 50 KByte of memory. Sensor hardware, such as MSP430 sensor platforms, does not contain much more memory than that, i.e. there may not be much space left to deploy any application on the LLN router.

Why This Is A Critical Point

Since RPL is designed to be the routing protocol for LLNs, which covers all the diverse applications requirements listed in [START_REF] Martocci | Building Automation Routing Requirements in Low Power and Lossy Networks[END_REF][START_REF] Pister | Industrial Routing Requirements in Low-Power and Lossy Networks[END_REF][START_REF] Brandt | Home Automation Routing Requirements in Low-Power and Lossy Networks[END_REF][START_REF] Dohler | Routing Requirements for Urban Low-Power and Lossy Networks[END_REF], it is possible that (i) due to limited memory capacity of the LLN routers, and (ii) due to expensive development cost of the routing protocol implementation, many RPL implementations will only support a partial set of features from the specification, leading to non-interoperable implementations.

As RPL is targeted for a "Standards Track" publication, interoperable implementations are desired. Already during the IPSO Interoperability Workshop in 2010 was it observed that several implementations were not interoperable, as they only implemented one mode of operation or the other (i.e. storing or non-storing mode).

RPL Underspecification

While [START_REF] Winter | RPL: IPv6 Routing Protocol for Low power and Lossy Networks[END_REF] is verbose in many parts, as described in section 6, some mechanisms are underspecified.

While for DIOs, the Trickle timer specifies an efficient and easy-to-understand timing for message transmission, the timing of DAO transmission is not explicit. 3 Plus additional specifications for routing headers [START_REF] Hui | An IPv6 Routing Header for Source Routes with RPL[END_REF], Trickle timer [START_REF] Levis | The Trickle Algorithm[END_REF], routing metrics [START_REF] Vasseur | Routing Metrics used for Path Calculation in Low Power and Lossy Networks[END_REF] and objective function [START_REF] Thubert | RPL Objective Function 0[END_REF]. 4 http://www.sics.se/contiki

INRIA

As each DAO may have a limited lifetime, one "best guess" for implementers would be to send DAO periodically, just before the life-time of the previous DAO expires. Since DAOs may be lost, another "best guess" would be to send several DAOs shortly one after the other in order to increase probability that at least one DAO is successfully received. The same underspecification applies for DAO-ACK messages: optionally, on reception of a DAO, an LLN router may acknowledge successful reception by sending back a DAO-ACK. Again, timing of the DAO-ACK messages is not specified by RPL.

Why This Is A Critical Point

By not specifying details about message transmission intervals and required actions when receiving DAO and DAO-ACKs, implementations may exhibit a bad performance if not carefully implemented. Some examples are:

1. If DAO messages are not sent in due time before the previous DAO expires (or if the DAO is lost during transmission), the routing entry will expire before it is renewed, leading to a possible data traffic loss.

2. RPL does not specify to use jitter [START_REF] Clausen | Jitter Considerations in Mobile Ad Hoc Networks (MANETs)[END_REF] (i.e. small random delay for message transmissions). If DAOs are sent periodically, adjacent routers may transmit DAO messages at the same time, leading to link layer collisions.

3. In non-storing mode, the "piece-wise calculation" of routes to a destination from which a DAO has been received, relies on previous reception of DAOs from intermediate routers along the path. If not all of these DAOs from intermediate routers have been received, route calculation is not possible, and DAO-ACKs or data traffic cannot be sent to that destination.

Other examples of underspecification include the local repair mechanism, which may lead to loops and thus data traffic loss, if not carefully implemented: a router discovering that all its parents are unreachable, may -according to the RPL specification -"detach" from the DODAG, i.e. increase its own rank to infinity. It may then "poison" its sub-DODAG by advertising its infinite rank in its DIOs. If, however, the router receives a DIO before it transmits the "poisoned" DIO, it may attach to its own sub-DODAG, creating a loop. If, instead, it had waited some time before processing DIOs again, chances are it would have succeeded in poisoning its sub-DODAG and thus avoided the loop.

Position

While RPL provides support for all of multipoint-to-point, point-to-multipoint and sensor-to-sensor traffic, its strength clearly is in providing connectivity for multipoint-to-point flows. Modulo the issues presented regarding bi-directionality of links and the possibility of loops, the DODAG formation mechanism is elegant, efficient, and relatively well understood. The DIO message generation/processing rules and the trickle timers [START_REF] Levis | The Trickle Algorithm[END_REF], necessary for this DODAG formation, are relatively straight-forward to implement as well -are actually not very complex. The state required in each router is also minimal and bounded -down to a single entry in the routing table (the preferred parent). Such pure multipoint-to-point traffic flows are not rare either: data-acquisition networks, where sensors on their own impulse communicate their readings to a controller are important, e.g. in various environmental monitoring or data acquisition scenarios.

As elegant as the support for pure multipoint-to-point traffic flows (i.e. pure upward routes) by way of DODAG construction is, the support for other traffic flows appears less so: the DAO mechanism, supporting downward routes, is what enables bi-directional traffic flows (e.g. for active reading of sensors) and sensorto-sensor flows by way of dog-leg-routing through the controller. This mechanism appears as a complicated addition to the elegant DODAG mechanism. It is, actually, also underspecified. Examples of such "underspecification" include the proper behavior with respect to DAO-ACKs and DAO retransmissions. The strength of the DODAG mechanism is that, by way of trickle timers, DIO emissions automatically taper off as the network becomes stable. For DAOs, a "best guess" -as this is not specified -is for these to be periodic. Similarly, RPL specifies two incompatible modes for such downward routing: storing mode, wherein all LLN routers are expected to have "unbounded" memory (or, at least, enough to store complete routing tables), and non-storing mode necessitating sourcerouting thus possibly more fragmentation and higher probability of IP packets being lost. Both of these appear to be challenging in Low-power Lossy Networks with resource-constrained devices.

An example of an active reading deployment is "smart metering", where a utility company wishes to interrogate individual consumption of its clients: each reading is initiated by the controller requesting the meter reading and the meter replying with the current consumption. Typically, a utility company will wish to read "several thousand meters" -but that over the course of a day (or more). Thus, all downward routes need not be immediately available -nor permanently maintained -but need be discoverable when needed.

Protocols so discovering routes on demand exist, such as AODV [START_REF] Perkins | Ad hoc On-Demand Distance Vector (AODV) Routing[END_REF] and a proposed simplification for LLNs entitled LOAD [START_REF] Kim | 6LoW-PAN Ad Hoc On-Demand Distance Vector Routing (LOAD)[END_REF]; neither of which requires source-routing, however both may require more state than a single routing table entry in each router. It should be noted that the RPL storing-mode, which also eliminates the need for source-routing, likewise does not have a bounded state. In an active reading deployment, however, this state requirement may, in the case of a protocol such as LOAD or AODV, be reasonably managed by the controller by way of spacing readings appropriately so as to keep the number of active concurrent routes (and, so, the state required in each router) below a threshold supported by each LLN router.

While evaluating the complexity of a protocol based on the complexity of its specification isn't entirely fair, it is still indicative that while the RPL specification counts 159 pages, and depends on several other specifications (trickle, metrics, 6lowpan routing header, etc.), LOAD counts 17 pages and AODV (published as RFC) 37 pages. One possible solution to the problem of complexity could be to "modularize" RPL, i.e., extract different modules (such as for upward data traffic, storing mode, non-storing mode) into separate specifications with a common framework that allows implementations to select which modules to implement, and to define a mechanism to assure that two interoperating implementations provide the same modules, or at least stipulate which modules a given implementation contains. A consequence of the above could be to suggest that a possible LLN routing protocol solution would be modular, consisting of:

• the DODAG formation mechanism from RPL, specifically the DIO and Trickle components;

• an on-demand route discovery mechanism, for example derived from LOAD or AODV.

With the ability to combine these, as needed, for a given deployment/scenario, it might provide a both elegant, efficient and simple routing solution.

INRIAA

  legitimate question, from the position of active reading, is therefore what the extra complexity of implementing RPL brings. A further legitimate question to ask is what resulting protocol would have emerged, had the design-basis been active reading rather than data acquisition.

RR n°7540

Though not called out in[START_REF] Narten | Neighbor Discovery for IP version 6 (IPv6)[END_REF], also from lower-layer protocols (such as Link Layer ACKs)

Such is the case for some implementations of IEEE 802.11b, where multicast/broadcast transmissions are sent at much lower bit-rates than are unicast. IEEE 802.11b is, of course, not suggested as a viable interface for LLNs, but serves to illustrate that such asymmetric designs exist.RR n°7540