Nicola Luminari

Christophe Airiau

Alessandro Bottaro
email: alessandro.bottaro@unige.it

Effects of porosity and inertia on the apparent permeability tensor in fibrous media

The flow in three-dimensional fibrous porous media is studied in the inertial regime by first simulating for the motion in unit, periodic cells, and then solving successive closure problems leading -after applying an intrinsic averaging procedure -to the components of the apparent permeability tensor. The parameters varied include the orientation of the driving pressure gradient, its magnitude (which permits to define a microscopic Reynolds number), and the porosity of the medium. All cases tested refer to situations for which the microscopic flow is steady. When the driving force is oriented in a direction which lies on the plane perpendicular to the fibers' axis, the results found agree with those available the literature. The fact that the medium is composed by bundles of parallel fibers favours a deviation of the mean flow towards the fibers' axis when the driving pressure gradient has even a small component along it, and this is enhanced by a decreasing porosity; this phenomenon is well quantified by the knowledge of the components of the permeability. Contrary to our initial expectations, for the over one hundred cases which we have simulated, the apparent permeability tensor remains, to a very good approximation, diagonal, a fact mainly related to the transversely isotropic nature of the medium. To obtain a complete, albeit approximate, database of the diagonal components of the apparent permeability tensor we have developed a metamodel, based on kriging interpolation, and carefully calibrated it. The resulting response surfaces can be invaluable in determining the force caused by the presence of inclusions in macroscopic simulations of the flow through bundles of fibers whose orientations and dimensions can vary in space and/or time.

Introduction

The flow through porous media is a problem of importance for several natural and technological applications. Since the original formulation by [START_REF] Darcy | Les fontaines publiques de la ville de Dijon[END_REF] , which relates the flow rate through a porous bed to the pressure drop across the bed's sides, many corrections have been made to account, for example, for viscous effects [START_REF] Brinkman | A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles[END_REF] or for the consequences of inertia [START_REF] Forchheimer | Wasserbewegung durch boden[END_REF]. All of the cited works are of empirical nature, but a volume averaging approach has been able to recover all of these formulations rigorously starting from the Navier-Stokes equations [START_REF] Whitaker | The Method of Volume Averaging[END_REF].

The theory requires the knowledge of a number of terms, most notably, in the case of an isotropic porous bed, a permeability coefficient and a Forchheimer coefficient. Initial effort s in defining these terms were based on a combination of physical reasoning and measurements, leading to expressions known as the Kozeny-Carman [START_REF] Kozeny | Über grundwasserbewegung[END_REF][START_REF] Carman | Fluid flow through granular beds[END_REF] and the Ergun [START_REF] Ergun | Fluid flow through randomly packed columns and fluidized beds[END_REF] correlations. The first coefficient provides the permeability for the laminar flow of a single-phase fluid through a packed bed of sand grains, as function of the porosity and the diameter of the grains, while the second extends Darcy's law to let the pressure drop depend on two terms, one proportional to the velocity and the second to its square, thus accounting for inertia. These approaches do not consider microstructural or geometrical features of the porous bed, which can render the permeability a tensorial quantity, and are often restricted to simple unidirectional flows. In the present work we are concerned with a transversely isotropic material composed by parallel fibers of circular cross-section, with one axis of symmetry, ( O, x 3 ); in such materials the permeability is a diagonal tensor with the component in the direction parallel to the fibers greater than those along the transverse axes. For such an arrangement we will investigate the effects of both the direction of the forcing pressure gradient and inertia. When the latter effect is present, embodied by a Reynolds number Re d , based on the mean intrinsic velocity through the medium and the fibers' diameter, exceeding an order one threshold, the permeability is no more simply defined upon geometrical properties. This new permeability, which arises from a well-defined closure problem, is then called apparent permeability .

The influence of the geometry of the solid inclusions has been addressed previously by [START_REF] Yazdchi | Microstructural effects on the permeability of periodic fibrous porous media[END_REF] for arrays of cylinders in both square and hexagonal (or staggered) patterns, with the cylinders' section which can vary in shape. The results, in the twodimensional and low Reynolds number limits, demonstrate the dependence of the permeability component along the flow direction to both the porosity and the direction of the macroscopic pressure gradient. The direction of the pressure gradient is found to have a weak effect for beds of medium-high porosity ( ε > 0.7) and a stronger dependence appears upon the geometry of the solid inclusions.

The influence of the Reynolds number on the permeability and on the Forchheimer correction has been presented in a number of papers. One of the contributions most relevant here is due to [START_REF] Edwards | The influence of Reynolds number upon the apparent permeability of spatially periodic arrays of cylinders[END_REF] . These authors show that, for arrays of fibers, the apparent permeability decreases with the increase of the Reynolds number, and the rate of this decrease depends on the geometry of the array; also, the Reynolds number is found to have a stronger influence on the apparent permeability when the medium is highly porous. The results of the work by [START_REF] Edwards | The influence of Reynolds number upon the apparent permeability of spatially periodic arrays of cylinders[END_REF] agree with those by [START_REF] Zampogna | Fluid flow over and through a regular bundle of rigid fibres[END_REF] and with our own work (as shown later), all for the case of cylindrical fibers, although some issues remain on the persistence of steady solutions in the simulations by [START_REF] Edwards | The influence of Reynolds number upon the apparent permeability of spatially periodic arrays of cylinders[END_REF] in cases for which a limit cycle should have set in. A fully three-dimensional porous medium, more complex than those discussed so far, has been considered by [START_REF] Soulaine | On the use of a Darcy-Forchheimer like model for a macro-scale description of turbulence in porous media and its application to structured packings[END_REF] , confirming the decreasing trend of the apparent permeability with the Reynolds number.

Another contribution which deserves mention is that by [START_REF] Lasseux | On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media[END_REF] ; they have computed the permeability tensor for various Reynolds numbers, in a two-dimensional geometry with cylinders of square cross-section. Forcing the flow along the main symmetric directions of the fiber, [START_REF] Lasseux | On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media[END_REF] have identified different regimes:

• a creeping flow regime for 0 < Re d < 10 -3 , without Forchheimer terms; • a weak inertia regime for 10 -3 < Re d < 1 , with the Forchheimer correction quadratic in Re d ; • a strong inertia regime for 1 < Re d < 10, where the Forchheimer correction is linear with the Reynolds number; • a turbulent regime, for Re d > 10, with the Forchheimer correction again quadratic with the Reynolds number.

The boundaries between the different regimes are specific to the geometrical arrangements and to the porosities being considered; a step forward in rendering (some of) these boundaries rigorous and independent of the arrangement of the pores, through the definition of a Reynolds number which accounts for a "topological" coefficient, has been recently made by [START_REF] Pauthenet | Topological scaling for inertial transition in porous media[END_REF] . For the purposes of the present paper, we must retain that [START_REF] Lasseux | On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media[END_REF] have parametrized the Forchheimer correction with the Reynolds number, and have found that the inertial correction is orders of magnitude smaller than the Darcy's term, at least before the turbulent regime sets in. Moreover, Lasseux and co-workers have studied how a Forchheimer tensor, F , depends upon the direction of the macroscopic forcing term with respect to the orientation of the square cross-section of the fibers, for Re d up to 30. It is concluded that a deviation angle, γ , exists between the direction of the pressure gradient and that of the mean flow, because of the fibers' geometry. The inertial correction is strongly influenced by the orientation of the driving pressure gradient, and the tensor F is not symmetric (in fact the off-diagonal components are found to be inversely proportional to the diagonal terms, and symmetric with respect to rotations about the diagonal axis of the square, i.e. the direction at 45 °in the x 1x 2 plane, cf. Fig. 1 ).

The effect of variations in the forcing angle, with restrictions to angles in the x 1x 2 plane, is also examined by [START_REF] Soulaine | On the use of a Darcy-Forchheimer like model for a macro-scale description of turbulence in porous media and its application to structured packings[END_REF] with conclusions in qualitative agreement with those of both the contribution just cited and our results described further below. In all cases, the off-diagonal components of the apparent permeability tensor are small and the diagonal components display but a small variation upon rotation of the driving pressure gradient.

As already anticipated, this work investigates how the direction of the macroscopic pressure gradient, the porosity and the Reynolds number can modify the Darcy and Forchheimer closures arising from a volume-averaged model of a fibrous porous medium. We will consider a three-dimensional unit cell for the microscopic model (such a unit cell is sometimes denoted REV, for Representative Elementary Volume), with a generic forcing whose direction is defined by two Euler angles. Given the formidable space of parameters, some representative results are first shown and discussed. Response surfaces in the space of parameters are then identified by the use of a metamodel based on kriging interpolation. For the sake of space, only the first diagonal component of the apparent permeability tensor is discussed in detail in the paper; however, all components have been computed. They represent an extremely useful data base which we are now in the process of using in macroscopic simulations of flows through bundles of fibers of varying orientation and density.

The volume-averaged Navier-Stokes (VANS) method

A brief description of the method

The system under investigation consists of an incompressible Newtonian fluid which flows through a rigid porous medium. In the following, the subscript β is used to indicate the fluid phase while σ is adopted for the solid phase. The governing equations valid at the microscale for the flow through the pores are:

∂ v β ∂t + v β • ∇ v β = - 1 ρ β ∇ p β + ν β ∇ 2 v β + f , (1) ∇ • v β = 0 , (2)
where v β , p β , ρ β and ν β stand, respectively, for the velocity, the pressure, the density and the kinematic viscosity of the fluid. The right-hand side term, f , is a force (per unit mass) which drives the fluid motion and can be interpreted as the macroscopic pressure gradient acting on the system. The concept of REV of the porous medium is classically introduced in the framework of the VANS approach. An example of REV is depicted on Fig. 1 , together with relevant notations (volume shape and size, indication of the fact that the normal unit vector is directed from the fluid to the solid phase, centroid x of the REV). The REV represents the domain over which the microscopic problem is solved; its size is defined so as to capture all the microscopic features of the flow. As a rule of thumb, the REV is the smallest fluid domain over which periodic boundary conditions can be applied.

In the computational domain, any flow variable ψ can be decomposed into an intrinsic average part < ψ > β plus a perturbation ˜ ψ , as:

ψ = ψ β + ˜ ψ .
The intrinsic average is defined with an integration carried out only on the fluid phase [START_REF] Whitaker | The Method of Volume Averaging[END_REF]:

ψ β β = 1 V β V β ψ β (x ) dV β .
(3)

Applying such an operator to Eqs. ( 1) and ( 2) , and following [START_REF] Whitaker | The Forchheimer equation: a theoretical development[END_REF] we have:

∂ v β β ∂t + v β β • ∇ v β β = - 1 ρ β ∇ p β β + ν β ∇ 2 v β β + f + + 1 V β A βσ - ˜ p β ρ β I + ν β ∇ ˜ v β • n βσ dA, (4) ∇ • v β β = 0 , ( 5
)
upon neglecting in Eq. ( 4) the sub-REV-scale dispersion term

(linked to ˜ v β ˜ v β β
) which is often small in porous media flows [START_REF] Breugem | The influence of wall permeability on laminar and turbulent flows[END_REF]. The surface integral term in Eq. ( 4) represents the drag (per unit mass) due to surface forces at the fluid-solid interface of the medium. It is called the Darcy-Forchheimer microscale force, F m . The equations are however often to be solved at the macroscale, so that a macroscale force model, F M , must be used to replace F m in the governing equation. Such a model is often based on a permeability tensor, K , and a Forchheimer tensor, F , and reads:

F M = -ν β εK -1 ( I + F ) < v β > β ,
(6) so that the system is closed by imposing

F m = F M . ( 7
)
The drag force F m computed by direct numerical simulations (DNS) with account of all individual pores will be later compared to the model based on the permeability and Forchheimer tensors (whose equations are given below). This is a useful exercise to demonstrate consistency of the approach and accuracy of the numerical simulations; it does nothing else since, as briefly described below, to derive the Forchheimer tensor the microscopic velocity field must be known anyhow. Nonetheless, knowledge of the behaviour of these tensors (or, equivalently, of the related apparent permeability) might prove both useful and instructive, in particular should one wish to extend the range of applicability of the model to cases for which the microscopic solution is not available. The core of the VANS approach consists in the identification of the permeability and Forchheimer tensors. This problem, referred to as the closure problem, is discussed at length by [START_REF] Whitaker | Flow in porous media i: a theoretical derivation of Darcy's law[END_REF][START_REF] Whitaker | The Forchheimer equation: a theoretical development[END_REF] . He derives two partial differential equation systems, the first valid in the zero Reynolds number limit (system (8) below), while the second applies when inertial terms are not negligible (system (10) ).

In the first system of equations a three-component vector d and a 3 × 3 tensor D are introduced. This system can be divided into three separate independent problems which resemble a forced Stokes problem where each component of d and the corresponding row of D play, respectively, the role of a pressure and a velocity field. Together with the periodic boundary conditions, the problem reads:

⎧ ⎪ ⎨ ⎪ ⎩ 0 = -∇d + ∇ 2 D + I , ∇ • D = 0 , D = 0 on A βσ , d (x + i ) = d (x ) , D (x + i ) = D (x ) i = 1 , 2 , 3 . ( 8
)
The permeability tensor is found by applying the intrinsic average on the D tensor and multiplying by the porosity ε

= V β V , i.e. K = ε < D > β .
In the Stokes regime, it is

F M = -ν β εK -1 < v β > β . ( 9
)
The second closure problem differs from the first only for the presence of a linearised convective term in which the microscopic velocity obtained from the DNS, v β , is used as an input. This of course implies knowledge of the microscopic velocity field. A Oseen-like approximation which relaxes this constraint has been proposed by [START_REF] Zampogna | Fluid flow over and through a regular bundle of rigid fibres[END_REF] .

The new unknowns are a vector and a tensor called, respectively, m and M , with the same meanings of d and D . The system reads:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 ν β v β • ∇M = -∇m + ∇ 2 M + I , ∇ • M = 0 , M = 0 on A βσ , m (x + i ) = m (x ) , M (x + i ) = M (x ) i = 1 , 2 , 3 . ( 10
)
The average of the tensor M multiplied by the porosity is

the apparent permeability , H = ε M β . When inertia is important
Eq. ( 6) can be written as

F M = -ν β εH -1 < v β > β , ( 11
)
as shown by [START_REF] Whitaker | The Forchheimer equation: a theoretical development[END_REF] . Two remarks are in order at this point. First, the equations in the closure problem (10) are time-independent because the microscopic velocity v β is a solution of a stationary DNS. Thus, the Reynolds number should be sufficiently small for unsteady effects not to be present. Should the wake behind a solid inclusion display regular or irregular temporal oscillations, the equations of system (10) may be used, as an approximation, by replacing the instantaneous velocity in the REV with its time-averaged distribution. This case is however not of present concern. Secondly, the closure problems reflect the structure of the solution of the two systems ( 8) and (10) . In particular, the solution of (8) depends only on the geometry of the porous medium so that the permeability tensor K is symmetric. This is not the case for H , because of the effect of the microscopic velocity amplitude and direction. Clearly, the solution of system (8) tends to that of (10) when with v β the amplitude of the intrinsic velocity and d the diameter of the fiber.

Re d = v β β d ν β → 0 ,

Validation and setup

In this section the numerical methodology, the parameters, the setup and the validation for some reference cases are given.

Computational domain

The geometry used for the base REV is shown in Fig. 2 : a cylindrical inclusion is present at the centre of the REV and four quarters of cylinders are situated at the corners. The lateral length of the cubic envelop is , which is used as length scale for the microscopic problem; the diameter d of the cylinders is adapted as a function of the desired porosity ε, ratio between the fluid volume over the total REV volume ( 3 ). The forcing term f of the DNS is a vector whose direction is defined by two Euler angles, with rotations of the form: θ e 3 + φ e 2 I (cf. Fig. 2 ). Its amplitude is set a priori and is connected to the Reynolds number, Re d which results from the calculations once the mean velocity is evaluated.

Numerical setup

The numerical simulations of the full Navier-Stokes Eqs. ( 1) -(2) in the REV have been carried out with the open-source code Open-FOAM [START_REF] Weller | A tensorial approach to computational continuum mechanics using object-oriented techniques[END_REF], based on a finite volume discretization with a colocated arrangement for the unknowns. The standard solver icoFoam (incompressible Navier-Stokes) has been modified in order to include a constant pressure gradient acting as a forcing term f in Eq. ( 1) . The coupling between the velocity and the pressure equations is based on the pressure implicit split operator referred to as the PISO algorithm. The time derivative term is discretized using the second order backward Euler scheme and all the spatial terms use a second-order central difference stencil based on Gauss finite volume approach. The velocity system is solved with a preconditioned bi-conjugate gradient iterative solver with the tolerance on the velocity residuals set to 10 -8 , associated to a diagonal incomplete lower upper pre-conditioner. The pressure equation is solved with a geometric-algebraic multigrid algorithm associated to a Gauss-Seidel smoother and the tolerance on the pressure residuals is taken equal to 10 -6 . Periodic boundary conditions are applied to all fields (velocity and pressure) on all fluid boundaries along the three directions, and the no-slip condition is imposed on the surface of the solid inclusions. The time step t is automatically determined to ensure that the maximum Courant number, Co , respects the condition:

Co = v β t/ s 1 / 2 , in which v β is the
modulus of the local velocity on the REV and s is the local grid spacing in the direction of v β . Co is basically the ratio between the fluid speed and the velocity to propagate information through the mesh and the condition Co < 1/2 is found to be sufficient to have a stable solver. The convergence of the solutions is assessed on the basis of a Richardson extrapolation which employs the results obtained on successively refined grids (with up to 1.5 million cells in the finer grid used). The mesh convergence analysis is described in Appendix A .

Validation on two different configurations

The results published in the literature by [START_REF] Zampogna | Fluid flow over and through a regular bundle of rigid fibres[END_REF] and [START_REF] Yazdchi | Microstructural effects on the permeability of periodic fibrous porous media[END_REF] are now used to validate both the methodology and our choice of the computational parameters. In the cited papers, three-dimensional computations of the permeability components in different cells geometries are presented.

Fig. 3 displays the comparison for a cell with a square arrangement of the fibers; here the permeability is evaluated along the two principal directions, x 1 and x 3 . To quantify the accuracy of the results we introduce a relative error on the permeability components, defined as

e r K ii = | K ii present -K ii literature | K ii literature
and reported in Table 1 . Good agreement is found with the published results: the relatives errors in all the cases tested is always below 5%. Fig. 4 shows a similar comparison for a staggered arrangement of the inclusions in the unit cell. In this case the section of the cell is rectangular. The agreement for the only permeability component available in the literature is again satisfactory. Finally, to check the correct implementation of the closure model ( 10) it is important to verify the equality (7) between the amplitude F M of the macroscopic force and its microscopic counterpart obtained through an integration of the DNS fields over the solid boundaries of the inclusions in the REV. Fig. 5 shows a plot of the relative error between these two forces, i.e.

|| F M -F m || || F m ||
, as function of the Reynolds number. We consider the successful comparison displayed in Fig. 5 as the conclusive demonstration of the validity of the approach described here. We have nonetheless carried out the same verification displayed in Fig. 5 for most of the simulations described in the following, to our satisfaction.

Tests with larger REV's

Since the REV is the unit cell within the porous medium over which average quantities of the VANS are computed, it is important to choose its dimensions appropriately in the inertial regime for, if the REV is too small, it might be easy to miss crucial features of the wakes. For example, to predict the critical Reynolds number, Re c , of the first Hopf bifurcation, a REV containing at least three solid inclusions in the direction of the mean pressure gradient is necessary in the configuration used by [START_REF] Agnaou | From steady to unsteady laminar flow in model porous structures: an investigation of the first Hopf bifurcation[END_REF] . Among the results reported, it is found that, for a fixed REV size, the error committed in the evaluation of the critical Reynolds number increases with the porosity. This same error is considerably reduced when the mean pressure gradient angle is θ = 45 • . Thus, the choice of the number of inclusions in a REV is a task not to be overlooked, and the final choice must account for the porosity, the direction of the pressure gradient and the microscopic Reynolds number.

Here, the influence of the numbers of inclusions present in a REV is assessed by focussing only on the velocity components after averaging over the REV. The unit cubic cell of side is used as reference: starting from this, two additional REV's are built, as shown in Fig. 6 . The first one is doubled in both the x 1 and x 2 directions and the case tested numerically is characterised by θ = 0 , φ = 0 For both these test cases, no appreciable differences, neither in the mean velocity nor in the forces on the fibers, have been observed, with relative errors on the mean velocity with respect to the reference case which remain below 2%. We take this as sufficient evidence to use, in the following, only the reference cubic REV of side equal to , with the understanding that only cases with a microscopic Reynolds number up to around 100 can be considered.

Microscopic solutions

In this section, some local microscopic fields computed with direct numerical simulations are shown, together with components of the intermediate tensor M coming from the numerical solution of the closure equations (10) .

In Another interesting point emerges by inspection of Fig. 8 where two off-diagonal components of M are shown for two porosity values; the first image (left frame) represents a plane flow in the Stokes regime while the second is the plane cut of a threedimensional solution in the inertial regime. Positive and negative values of the microscopic fields can be seen in both images but, once averaging is applied over the REV, the resulting permeability component is very close to zero (in fact, exactly equal to zero in the Stokes case). This same features occurs for all off-diagonal terms in all cases examined, so that, within the current range of Reynolds numbers, the apparent permeability tensor is, to a good approximation, diagonal 1 A three-dimensional case is shown in Fig. 9 , where all the nonzero terms of the M tensor are plotted for a porous structure with 1 In fact, there are always at least two orders of magnitude differences between the diagonal and the off-diagonal components. While the latter should not, in prin-ε = 0 . 6 . The components shown are M 11 , M 22 , M 33 , M 12 and M 21 , while M i 3 and M 3 j are not plotted because they are identically zero to machine accuracy. Distinct features are visible in each image; in particular, in the last frame the M 33 microscopic component displays a low wavelength structure along the cylinder's axis. Increasing the dimensions of the REV along x 3 does not alter such a structure, i.e. the 3 domain chosen with its periodic boundary conditions does not filter out significant high wave-numbers of the flow. We further note that the tensor M is not symmetric in this case since each off-diagonal component represents the solution of the closure problem in a specific direction (first index of the field) and the forcing term acts orthogonally to it (second index of the field). Once averaged over the REV it is found that H 12 H 21 very close to zero. ciple, we will attention here only on the dominant terms of the permeability tensor.

The apparent permeability tensor

In this section the variations of the diagonal components of the permeability tensor H are discussed as function of the direction of the mean forcing, the Reynolds number and the porosity. As stated previously, the Reynolds number Re d ranges from 0 to approximately 100 in order to capture phenomena associated with inertia; the cases considered never lead to unsteady signals. The porosity parameter ε is set to either 0.4 (low porosity), 0.6 (medium) or 0.8 (high).

The forcing direction is defined by the Euler angles and all the configurations considered in this section are summarized in Table 2 ; the choice has been made to explore a reasonably large range of parameters, with both two-dimensional and three-dimensional flows characterized by symmetric and asymmetric patterns.

Let us briefly recall the methodology. First, a DNS is carried out to compute the microscopic flow. Then the closure problem is solved for the tensor M . Finally, each component of the apparent permeability H is obtained by averaging ( Eq. ( 3) ). The results are collected in Figs. 10,11 and 12 , showing the variation of the diagonal components of H .

In the left column of each figure we focus on the low-Re d regime (0 < Re d < 2), while in the right column the effect of inertia can be assessed. As expected, when Re d is small the apparent permeability is quasi-Reynolds-number-independent (and can be approximated well by the true permeability). As the Reynolds number increases above a few units, inertial effects grow in importance yielding typically a monotonic decrease of all components of H , aside from case indexed 5 (φ = 90 • ) for which the flow remains aligned with the cylinder's axis. In case 5 the microscopic flow solution is invariant with x 3 and does not change with Re d in the range considered, so that H is a constant tensor. When the porosity is large all components show a similar behaviour irrespective of the forcing angle (except, clearly, case 5).

Differences start appearing at ε = 0 . 6 ; the two cases with φ = 0 • (index 1 and 2) behave similarly, and so do the two cases indexed 3 and 4 (with φ = 45 • ). This seems to suggest a weaker effect of θ on the permeability components. For even smaller poros- ity ( ε = 0 . 4 ), the blockage effect caused by the inclusions produces the unexpected behaviour displayed in Fig. 12 . When the flow is purely two-dimensional (cases 1 and 2), variations in the Reynolds number affect H significantly; when a pressure gradient along x 3 is present the strong packing of the fibers constrain the fluid to flow prevalently along the fibers' axis, and the apparent permeability is almost Re d -independent. When assessing variations in H jj for this case, attention should also be paid to the fact that the permeability is now at least one order of magnitude smaller than in the previous cases so that variations of the diagonal components shown in Fig. 12 are tiny in absolute terms. This is related to the fact that the inverse of the permeability plays the role of a drag coefficient in the macroscopic expression of the force (cf. equation ( 6) ). In other Applying the intrinsic average operator to the off-diagonal components of the tensor M results in terms that are negligible with respect to their diagonal counterparts, and these results are true for all the parameters considered. This means that there is a very weak coupling between the principal directions of the fiber. The directional decoupling and the diagonal property of the apparent permeability tensor have also been computationally demonstrated on a completely different REV geometry by [START_REF] Soulaine | On the use of a Darcy-Forchheimer like model for a macro-scale description of turbulence in porous media and its application to structured packings[END_REF] . Conversely, [START_REF] Lasseux | On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media[END_REF] have carried out a two-dimensional study with fibers of square cross-section, finding that the off-diagonal terms are non-negligible and only about one order of magnitude smaller than the diagonal components. This result is a consequence of the non-rotationally-invariant geometry considered. The present work and the two articles just cited suggest that the diagonal property of the tensor H is more closely associated to the geometry of the porous material than to the flow regime, at least in the range of Re d considered.

A metamodel for H

Sections 4 and 5 have shown how the apparent permeability depends on the two Euler angles, the Reynolds number and the porosity. The space of parameters is formidable and the results found so far are not sufficient to treat, for example, cases characterized by multiple inclusions' sizes and orientations in different regions of the domain, or cases involving a poroelastic medium, with temporally and spatially varying porosity, flow direction and local Reynolds number. The complete solution of the closure problem for a single set of parameters takes approximately 4 CPU hours on our two-processor Intel(r) IVYBRIDGE 2.8Ghz, each with 10 cores and 64 GB of RAM, so that a complete parametric study is, to say the least, unpractical. In view of this, the construction of a metamodel capable to provide a full characterisation of the permeability as a function of all parameters is a worthy endeavor. We have tested several surrogate models, before eventually settling on the kriging approach [START_REF] Kleijnen | Regression and kriging metamodels with their experimental designs in simulation: a review[END_REF] described in Appendix B .

The metamodel provides a scalar function (for each term of the H tensor) defined in a four-dimensional space. In each of the following figures two parameters are fixed and the response surface is displayed as function of the remaining two, focussing on the H 11 component. The other diagonal components of the apparent permeability tensor behave in a similar fashion and will not be shown for brevity. All the results of the metamodel are, however, available on: https://github.com/appanacca/porous _ solvers _ OF.git .

In Fig. 13 the angle φ is fixed to zero, and the isolines display H 11 as function of the angle θ and of the Reynolds number, Re d , for three values of the porosity. The white square symbols indicate the samples used to build the metamodel. The maximum value of each surface is always found for Re d equal to zero and H 11 typically decreases with Re d , when the porosity is sufficiently large. As seen previously, for a porosity approximately greater or equal to 0.6 the variation of the apparent permeability with the angle θ is weak in this two-dimensional configuration. For the lowest porosity studied (left frame) the permeability has very small values and the isolines display an irregular behaviour; this is a feature common to all plots relative to the smaller value of ε, signaling that it is probably necessary, in this specific case, to insert additional sample points in building the response surfaces.

In Fig. 14 the parameter θ is set to 0 °and the response surface is displayed in the Re d -φ plane. As already indicated, the results confirm that an increase of the Reynolds number is generally associated to a decrease of the first diagonal component of the appar- ent permeability tensor. However, the H 11 variations with respect to φ are more pronounced than those found with respect to θ and are due to a real three-dimensionalization of the flow. This conclusion remains to be verified in the lower porosity case (left frame) where the variations are very tiny and more irregular.

In Fig. 15 the Reynolds number is set to the inertial range value of 40 and the response surface is displayed in the θφ plane.

For the two highest porosity values, 0.6 and 0.8, the results confirm that H 11 has a much stronger dependence on φ than on θ , suggesting that the real test of permeability models must include three-dimensional effects. As seen earlier, the behaviour of the permeability when the porosity is low (left frame in the figure) is not intuitive, with a significant effect of the angle φ and a minor influence of θ . Again this occurs from the constraint provided to the flow by the inclusions, and from the occurrence of a large deviation γ in these cases (see Appendix B for the definition of the deviation angle γ ).

The response surface is shown in the Re d -ε plane of Fig. 16 for three sets of θφ angles. Here significant effect of the porosity with respect to the Reynolds number observable. In fact the surface gradient is almost aligned with the porosity direction, i.e. a quasi-Reynolds independence is demonstrated in this plane, and the apparent permeability can change by one order of magnitude in the range of the analysed porosity. Some relatively small Reynolds number effects are visible for ε equal to 0.8, when the wake of the flow has more space to develop in the inertial regime. In the central figure the flow is aligned with the direction of the fibers and, as expected, it shows practically no dependence with respect to the Reynolds number.

The response surface analysis has confirmed the qualitative trends which had been reached earlier on the basis of a few selected flow cases, yielding at the same time much more detailed information on the behavior of the apparent permeability with the parameters of the problem. The data base which has been built will be used in future work which will focus, via the VANS approach, on configurations for which neither the porosity nor the local Reynolds number are constant in space or time.

Concluding remarks

The components of the permeability tensor are essential ingredients for any solution of flow through anisotropic porous media. When the flow through the pores resents of significant acceleration effects, the permeability must be modified (it is then called appar-ent ) by the presence of a second tensor, the Forchheimer tensor F , defined by

F = KH -1 -I .
The permeability, K , and the apparent permeability, H , can be formally deduced by two closure problems which have been briefly recalled in Section 2 . The real obstacle to the solution of the problem for H is the need to know the microscopic velocity fields through the pores. We have solved for such fields in a unit cell (the REV), varying the forcing amplitude and direction, treating over one hundred different cases of flows through arrangements of parallel fibers. From this, we have thus been able to solve the linear system (10) for all the unknown elements of the intermediate tensor M , from which, through averaging, we obtain the apparent permeability. Such a tensor is indispensable to evaluate accurately the drag force caused by the presence of the fibers, for a macroscopic solution of the flow on the basis of Eqs. ( 4) and ( 5) when inertial effects are present.

It has been found that the apparent permeability tensor is strongly diagonally dominant for whatever forcing direction and porosity, provided the microscopic Reynolds number remains below a value approximately equal to 100; this results -which is a direct consequence of the transverse isotropy of the material which has been considered here -can be used to compute H rapidly, approximating it as a diagonal tensor.

Finally, a metamodel has been used to produce results so as to cover the whole space of parameters, and this has allowed the construction of a complete data base. The elements of the data base provide a robust approximation to the "true" apparent permeability values as discussed in Appendix B (see in particular B.2 and B.3 ). This data base is now being used in simulations of poroelastic media based on the VANS approach.

B2. Kriging interpolation method

The kriging approach is a linear interpolation/extrapolation method that aims to build a predictor field based on a set of observations ( x i , y ( x i )), for i = 1 , . . . , n .

The predictor ˆ f (x ) is a sum of a trend function t ( x ) and a Gaussian process error model e ( x ):

ˆ f (x ) = t(x ) + e (x ) .

(B.3)

The aim of the error model is to make adjustments on the trend function so that, for any point of the sampling the predictor is exactly equal to the sample, i.e. ˆ f ( x i ) = y ( x i ) . This property represents one of the main qualities of this approach. In addition, when the model parameters are conveniently set, the trend function and the covariance model can take into account both smooth and steep variations in the data set. The trend function defined here is based on a second order least-square regression, with the coefficients found from the solution of the associated linear system. The Gaussian process error model has zero-mean and its covariance between two generic data-points, x i and x j , is written as Cov (y ( x i ) , y ( x j )) = σ 2 r( x i , x j ) .

The coefficient σ is an amplitude parameter and r ( x i , x j ) is a correlation function, based on the Matérn covariance model, that reads: r( x i , x j ) = parameters it is found that the choice λ = (1 . 2 , 1 , 1 , 1) yields acceptable results; in particular, the weight along θ is mildly larger than in the other directions in order to obtain smoother metamodel surfaces in this direction. The exponent ν controls the covariance function and more particularly its gradients. When ν = 1 / 2 the covariance can be approximated by a negative exponential, exp (-αx ) and when ν goes to infinity it behaves as exp (-αx 2 ) . In the present study, the best (i.e. smoother) results are obtained for ν equal to 1.9. The above parameters have been chosen in order to avoid unphysical or unrealistic behavior of the apparent permeability such as, for instance, negative values or steep, spurious local maxima/minima. The method above is implemented in OpenTURNS [START_REF] Baudin | OpenTURNS: An Industrial Software for Uncertainty Quantification in Simulation[END_REF].

B3. Robustness of the metamodel

A procedure called k -fold, belonging to the class of crossvalidation methods, has been used in order to prove the robustness of the metamodel. The k -fold method starts with the full database S n = ( x i , y ( x i )) , for i = 1 , . . . , n, split into two complementary sets of size n 1 and n 2 , such that S n = S n 1 ∪ S n 2 . Then, a new metamodel is built using only the points present in the set S n 1 . For the sake of clarity, the metamodel built with only the subset S n 1 will be called from now on ˆ f n 1 , and the metamodel built with all the database will be indicated as ˆ f n . The idea now is to use the points in the set S n 2 as test, since they are essentially "new" for the metamodel ˆ f n 1 . The division of the subset is performed picking points in a random way, and is repeated k times in order to rule out any possible "lucky" combination.

Thus, the metric used for the error computation is the following:

ξ cv = 1 k n 2 k i =1 n 2 j=1 ˆ f n i (x j ) -ˆ f n 2 i (x j ) 2 ,
quantifying the quadratic error between the original metamodel and the one built each time with a different set that belongs different folds.

The metric is also averaged over all the test points n 2 present in all the k folds. The relative mean error can be computed as:

E cv % = 100 ξ cv mean (| ˆ f n i | ) .
In our case the number of points used to test the model n 2 is equal to √ N ≈ 12 as recommended for kriging metamodels by [START_REF] Wang | Review of metamodeling techniques in support of engineering design optimization[END_REF] . The number of folds has been varied from 5 to 25 and in all the cases tested the E cv % has been found to decrease below 6% when we use at least 16 folds (which means leaving out 7 to 8 points from the metamodel construction), which is more than acceptable to prove that our kriging method is a robust approximation ( Fig. B.20 ).

Fig. 1 .

 1 Fig. 1. Illustration of the REV concept.

Fig. 2 .

 2 Fig. 2. REV for the fiber geometry investigated.

Fig. 3 .

 3 Fig. 3. Permeability versus porosity for a square arrangement of cylinders. The scaling of the permeability is 2 and is explicitely indicated in the vertical axis.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Permeability versus porosity for a staggered arrangement of cylinders. Note that the permeability component is here scaled with d 2 .

Fig. 6 .

 6 Fig. 6. REV configurations. Left: 2 × 2 × 1 arrangement; centre: 1 × 1 × 1 arrangement (reference); right 1 × 1 × 3 arrangement.

Fig. 7 (

 7 top row) the local x 1 velocity component is drawn for the two-dimensional flow when ε = 0 . 6 , for three Reynolds numbers, to cover the transition from the Stokes to the inertial regime. In all plots, the velocities are rendered non-dimensional by the corresponding value ofH 11 ν β || f ||, where H 11 is the first component of the tensor H . When inertia is absent, the flow has a central symmetry; by increasing the Reynolds number, only the symmetry with respect to the x 1 axis is maintained ( x 1 is the direction of the forcing pressure gradient), with the wake's length which increases with Re d . When Re d is of order 100 the wake spreads to the downstream boundary of the REV, re-entering, because of periodicity, at the upstream side. This Re d represents the upper limit of validity for the cubic unit cell of side ; larger values of Re d could only be investigated with longer/larger/thicker REV's.The non-dimensional microscopic M 11 fields for the same parameters are displayed in Fig.7(mid row). All values in the figures arise from scaling M with 2 . Visually, these local fields are strongly correlated to the local streamwise velocity component in the whole Re d range. This is not unexpected since the local velocity drives the convective term of system (10) . The central symmetry of all components of M in the Stokes regime is coupled to the rotational invariance of the apparent permeability tensor in twodimensional flows.The effect of varying the porosity is shown in Fig.7(bottom row) where ε is taken equal to 0.4. Even at such a low porosity the stretching of the wake can be noticed, and it increases with Re d . Interestingly, this effect is milder when the forcing is inclined by an angle φ, since the tighter packing of the inclusions causes

Fig. 7 .

 7 Fig. 7. Top row: plane view of the dimensionless x 1 component of the local velocity field v β for the case θ = 0 , φ = 0 , ε = 0 . 6 and for three Reynolds numbers Re d = 0, 10, 50, from left to right. Mid row: microscopic M 11 fields corresponding to the images in the top row. Bottom row: M 11 fields for the same Euler angles and Reynolds number as in the top two rows, and smaller porosity ( ε = 0 . 4 ).

Fig. 8 .

 8 Fig. 8. Right: Non-dimensional M 21 field for θ = 0 , φ = 0 , Re d = 10 , ε = 0 . 8 , left: Non-dimensional M 12 field for θ = 22 . 5 • , φ = 45 • , Re d = 50 , ε = 0 . 4 .Table 2 Directions of the forcing tested and property of the solutions. Index θ φ

Fig. 9 .

 9 Fig. 9. Non-dimensional M components fields for the case θ = 22 . 5 • , φ = 45 • , Re d = 50 , ε = 0 . 6 .

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Diagonal elements of the apparent permeability H as function of the Reynolds number for porosity ε = 0 . 8 . The forcing direction is represented through the couple of Euler angles ( θ, φ) (cf.Table 2 for the case index). Left column: low-Re d regime; right column: inertial regime.

Fig. 12 .

 12 Fig. 12. Same as Fig. 10 with porosity ε = 0 . 4 .

Fig. 13 .

 13 Fig. 13. Response surfaces of H 11 with φ = 0 • for porosity ε = 0 . 4 , 0 . 6 , 0 . 8 , from left to right.

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Response surfaces of H 11 with θ = 0 • for porosity ε = 0 . 4 , 0 . 6 , 0 . 8 , from left to right.

Fig. 16 .

 16 Fig. 16. Response surface of H 11 ; in the left frame φ = θ = 0 • , in the centre frame φ = 90 • , θ = 0 • and on the right φ = 45 • , θ = 22 . 5 • .

  K ν (.) is a modified Bessel function and (.) is the gamma function. The parameters that can be used to tune the metamodel are the amplitude parameter σ , the exponent ν and the scale vector λ. The kriging metamodel outputs can show different behaviours for different selections of the above three parameters and their setting is thus crucial. The amplitude parameter σ is chosen to be equal to 1; larger value lead to steeper gradients and undesirable local extrema around the data points. The vector λ = (λ θ , λ φ , λ Re d , λ ε ) is a scaling parameter for the distance | x ix j | . In this study, through systematic variations of the

Fig. B. 20 .

 20 Fig. B.20.Relative mean error computed using the k -fold approach presented against the number of folds k used to divide the dataset.

Table 1

 1 Relatives errors in the effective permeability validation.

		Regular REV		Staggered REV
	ε	e r K 11	e r K 33	e r K 11
	0.4	2.34%	3.22%	4.13%
	0.6	1.58%	0.38%	2.03%
	0.8	4.10%	2.03%	2.17%

Table B

 B It is then possible to compute angle between the forcing term, ∇p , and the average velocity vector, represented in Fig.B.19 for the two-dimensional case, φ = 0 . This is achieved by taking the ratio between the first two components of Darcy's equation, calling γ If the ratio between the two permeability components is equal to one, the angle γ vanishes. The correlation between H 11 and H 22 controls the deviation of the flow in the ( x 1 , x 2 ) plane, and the argument can easily be extended to H 11 / H 33 and H 22 / H 33 for deviation angles in three-dimensions.Using a linear correlation such as that shown in Table B.5 and Fig. B.18 , it is observed that in the low porosity case (ε = 0 . 4) the ratio can become very large indicating a strong deviation of the flow from the forcing direction, because of the strong constraint provided by the inclusions. As the porosity increases, the ratio does not differ much from unity, which means that the deviation remains limited. It is simple to see that the deviation angle, for ex-°for any θ . It should however be kept in mind that trends based on these ratios are valid only as long as Darcy's law and linear correlations are acceptable. Cases exists for which such trends are violated; for example, a flow with θ = 45 • and φ = 0 • has deviation angle γ equal to zero, for whatever porosity. In this case H 11 / H 22 is equal to one and such a point is an outlier in the regression plots of Fig. B.18 .

			.4	Table B.5
	Sampling parameters. Permeability components ratio for three
	parameter values of the porosity. The permeability values ratios here are given by the angular co-
	θ			0 °22.5 °45 °φ efficients of the linear correlations dis-
				0 °22.5 °45 °67.5 °90 °Re played in Fig. B.18 .
	ε	d		ε	0 0.4	10 0.6 H 11 / H 22	50 0.8 H 11 / H 33	100 H 22 / H 33
				0.4		1.57	11.06	96.03
				0.6		1.50	1.62	0.99
	plot, and the permeability points are represented with their linear 0.8 1.20 0.82 0.66
	regression on top. This kind of plot is common in statistical anal-
	ysis to determine if correlations in the data are present. The per-ample in the ( x 1 , x 2 ) plane, satisfies the approximate relation meability components display a positive correlation, with the data points which lie reasonably close to a straight line. This result has a physical implication. Remembering the diago-nal dominance of the permeability tensor, we have in the low Re d limit: tan γ = 1 -H 11 tan θ H 22 H 22 + tan 2 θ H 11 ,	Fig. B.19. Explanatory sketch for the relation between mean pressure gradient and mean velocity field.
	u β so that for β , v β β , w β H 11 H 22 ways below 12	β ∼ H 11 equal to, say, 1.5, the largest deviation remains al-∂ p ∂x 1 , H 22 ∂ p ∂x 2 , H 33 ∂ p . (B.1) ∂x 3
	the angular flow deviation with respect to the mean forcing. We
	thus have:
	tan (θ + γ ) =	H 22 H 11	tan θ .			(B.2)

Fig. B.18. Scatter matrix plot for the collected numerical data of the apparent permeability tensor.

Acknowledgment

The authors acknowledge the IDEX Foundation of the University of Toulouse for the financial support granted to the last author under the project Attractivity Chairs. The computations have been conducted at the CALMIP center, Grant no. P1540 . The complete database is available at the link: https://github.com/appanacca/ porous _ solvers _ OF.git .

Appendix A. Mesh convergence analysis

The mesh has been computed using the internal OpenFOAM mesher named snappyHexMesh . The final grid is mainly composed by hexahedral cells with a refined regular grid in the boundary layer regions next to the solid surfaces. Three different mesh sizes, with 0.65 × 10 6 , 10 6 and 1.5 × 10 6 elements, have been tested in order to demonstrate spatial convergence. This has been assessed using the Grid Convergence Index ( GCI ) introduced by [START_REF] Roache | Verification and Validation in Computational Science and Engineering[END_REF] .

Images of the coarsest mesh used are shown in Fig. A.17 . On the right frame a close up of the grid in the neighbourhood of the fiber's boundary is displayed: twenty points are used in the structured portion of the mesh along the wall-normal direction.

The GCI method is based upon a grid refinement error estimator derived from the theory of generalized Richardson extrapolation. It measures the ratio between the computed value of a quantity over the asymptotic numerical value, thus indicating how far the solution is from the asymptotic ("exact") value. The procedure is simple and provides a method to estimate the order of the spatial convergence, based on two or three different grid sizes. First of all, the grids must be generated with the same algorithm and they must have the same final quality. In each simulation the intrinsic average velocity in the porous medium is sampled. The method follows the following four steps:

1. Estimate the order of convergence of the procedure, defined as

where r is the grid refinement ratio each grid (it is computed as the ratio between the number of elements of two consecutive grids; the approach imposes that r should remain constant between any couple of consecutive grids and be larger than 1.1), and f i represents the amplitude of the intrinsic average velocity in each grid (1 = coarse, 2 = medium and 3 = fine).

2. Compute the relative error between grid i and j :

r p -1 , with F s a safety factor equal to 1.25 if the grids are three, and equal to 3 if the grids are only two [START_REF] Roache | Verification and Validation in Computational Science and Engineering[END_REF]. 4. Check whether each grid level yields a solution that is in the asymptotic range of convergence; this means that the quotient

1 r p should be as close as possible to one. The results are summarized in Table A.3 . From the table it can be seen that the intrinsic velocity difference is very small from one grid to the next and the coarse grid provides results close to the expected asymptotic value. This is taken as a sufficiently convincing argument to carry out all the computations in the paper with a grid density equal to that of grid 1.

Appendix B. Building the metamodel

B1. DACE sampling

The first step to build a metamodel is the collection of relevant samples. The quality of the final metamodel strongly depends on the samples collected and their number and distribution is of primary importance.

The apparent permeability tensor, H , depends on four independent variables; the samples have been generated starting from the set of parameters given in Table B.4 .

One of the best options to generate the relevant database would be to use a full factorial design approach in which all the combinations of the four variables from Table B.4 are computed. Because of the large number of computations required, this approach has not been retained. We have resorted to the methodology known as DACE (Design and Analysis of Computer Experiments), a technique to fill in the best possible way the space of the parameters of the problem. The Dakota library [START_REF] Adams | Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 6.0 Theory Manual[END_REF] has been selected for the purpose and the Monte-Carlo incremental random sampling algorithm [START_REF] Giunta | Overview of modern design of experiments methods for computational simulations[END_REF] has been chosen, in order to make efficient use of the cases already computed. This incremental approach selects in a quasi-random way the new samples to generate, starting from the existing ones. In the end, the set of samples comprises 118 cases.

In the scatter plot of


Dugast-Darzacq Claire

Grange Thierry

Schreiber-Agus Nicole

Ph.D Claire Dugast-Darzacq

Differential effects of Mxi1-SRα and Mxi1-SRβ in Myc antagonism

Keywords: Mxi1, Myc, isoforms, REF assay, transcription

Mxi1 belongs to the Myc-Max-Mad transcription factor network. Two Mxi1 protein isoforms, Mxi1-SRα and Mxi1-SRβ, have been described recently to share many biological properties. Here we assign differential functions to these isoforms with respect to two distinct levels of Myc antagonism. Unlike Mxi1-SRβ, Mxi1-SRα is not a potent suppressor of the cellular transformation activity of Myc. Furthermore, while Mxi1-SRβ exhibits a repressive effect on the MYC promoter in transient expression assays, Mxi1-SRα activates this promoter. A specific domain of Mxi1-SRα contributes to these differences. Moreover GAPDH interacts with Mxi1-SRα and enhances its ability to activate the Myc promoter. Our findings suggest that mxi1 gains functional complexity by encoding isoforms with shared and distinct activities.

Introduction

Members of the Myc oncoprotein family function as transcription factors that control various aspects of cellular behavior including cell growth, proliferation, differentiation, apoptosis, genomic stability, and tumorigenesis (reviewed recently in [START_REF] Lee | Myc target transcriptomes[END_REF][START_REF] Cole | Transcriptional activation by the Myc oncoprotein[END_REF][START_REF] Kleine-Kohlbrecher | Mechanisms of transcriptional repression by Myc[END_REF][START_REF] Wade | c-Myc, genome instability, and tumorigenesis: the devil is in the details[END_REF]). Deregulation of Myc contributes to the pathogenesis of a large proportion of human cancers (reviewed in [START_REF] Pelengaris | The c-MYC oncoprotein as a treatment target in cancer and other disorders of cell growth Expert[END_REF][START_REF] Nesbit | MYC oncogenes and human neoplastic disease[END_REF]). This deregulation has been shown to occur at multiple levels including those that affect myc gene expression, Myc protein stability, and Myc biological activity. Normal regulation of Myc activity occurs by mechanisms that influence the Myc protein per se (for examples see [START_REF] Sears | The life cycle of C-myc: from synthesis to degradation[END_REF]), and also through the functions of related members of the extended Myc-Max-Mad protein network (reviewed recently in [START_REF] Rottmann | The mad side of the Max network: antagonizing the function of Myc and more[END_REF]; note that the Mad subfamily recently has been renamed the Mxd subfamily).

The Mxi1 (a.k.a. Mxd2) protein first was described as a member of the Myc/Mad/Max network by virtue of its having a basic helix-loop-helix leucine zipper (bHLH/LZ) region similar to that of Myc and of its interaction with the obligate Myc DNA binding partner, Max [START_REF] Zervos | Mxi1, a protein that specifically interacts with Max to bind Myc-Max recognition sites[END_REF]. In early models that defined Mxi1's function within this network, Mxi1 (as well as the related Mad family proteins) was proposed to be a Myc antagonist. This was based upon its ability to bind competitively with Myc both to the Max protein and, once complexed with Max, to shared DNA sequence motifs (E-boxes; CANNTG). Beyond this, it was realized that whereas Myc could transactivate gene expression at the E-box through the recruitment of various co-activators (see [START_REF] Cole | Transcriptional activation by the Myc oncoprotein[END_REF] and references therein), Mxi1 could repress gene expression there through its interaction with Sin3/HDAC (histone deacetylase) complexes ( [START_REF] Schreiber-Agus | An amino-terminal domain of Mxi1 mediates anti-Myc oncogenic activity and interacts with a homolog of the yeast transcriptional repressor[END_REF][START_REF] Alland | Role for N-CoR and histone deacetylase in Sin3-mediated transcriptional repression [see comments[END_REF] and reviewed in [START_REF] Rottmann | The mad side of the Max network: antagonizing the function of Myc and more[END_REF]). The antagonism by Mxi1 on the molecular level correlated well with its ability to be a potent suppressor of Myc transformation activity in the rat embryo fibroblast (REF) assay, a surrogate assay for neoplastic transformation [START_REF] Schreiber-Agus | An amino-terminal domain of Mxi1 mediates anti-Myc oncogenic activity and interacts with a homolog of the yeast transcriptional repressor[END_REF]. Interestingly, a

Ce manuscrit a été publié dans FEBS Journal 274: 4643-4653 (2007) naturally occurring mouse Mxi1 protein isoform lacking the Sin3 recruitment domain (a.k.a. the SID), called Mxi1-WR, was unable to potently suppress Myc cotransformation activity in the REF assay [START_REF] Schreiber-Agus | An amino-terminal domain of Mxi1 mediates anti-Myc oncogenic activity and interacts with a homolog of the yeast transcriptional repressor[END_REF]. This suggested that Myc antagonism and growth suppression was linked to the presence of a SID and its ability to recruit co-repressors.

Recently, our group and others have recognized that in addition to the Mxi1-WR isoform, other Mxi1 protein isoforms exist both in mouse and man [START_REF] Dugast-Darzacq | Mxi1-SRalpha: a novel Mxi1 isoform with enhanced transcriptional repression potential[END_REF][START_REF] Engstrom | Mxi1-0, an alternatively transcribed Mxi1 isoform, is overexpressed in glioblastomas[END_REF][START_REF] Kawamata | Mxi1 isoforms are expressed in hematological cell lines and normal bone marrow[END_REF]. Many of these isoforms appear to arise from alternative exon 1 (and promoter) usage within the mxi1 genomic locus. One new isoform that we have described, Mxi1-SRα, exhibits many of the biological properties attributed originally to Mxi1 and outlined above (we have renamed the original Mxi1 isoform Mxi1-SRβ) [START_REF] Dugast-Darzacq | Mxi1-SRalpha: a novel Mxi1 isoform with enhanced transcriptional repression potential[END_REF]. Specifically, Mxi1-SRα can also bind to Max and to Sin3, and can function as a transcriptional repressor upon various reporter plasmids including synthetic E-box reporters. With respect to expression profiles, Mxi1-SRα and Mxi1-SRβ transcripts can be found together in the majority of newborn and adult mouse tissues examined on the gross level. However, tissue-specific expression patterns were also observed, including that Mxi1-SRα appears to be the predominant transcript in the adult intestine and in the developing embryo, while Mxi1-SRβ transcripts predominate in the adult liver and kidney [START_REF] Dugast-Darzacq | Mxi1-SRalpha: a novel Mxi1 isoform with enhanced transcriptional repression potential[END_REF].

In our initial description of Mxi1-SRα and its comparison to Mxi1-SRβ [START_REF] Dugast-Darzacq | Mxi1-SRalpha: a novel Mxi1 isoform with enhanced transcriptional repression potential[END_REF], we speculated that despite their apparent functional overlap in the assays employed in that study, the possibility existed for distinct functions for these two isoforms. In the present study, we have compared further the Mxi1-SRα and Mxi1-SRβ isoforms at the levels of Myc antagonism in the REF assay, subcellular localization, and transcriptional activity. Some of these analyses have assigned differential functions to the two isoforms, and we show that the unique amino terminal extension on Mxi1-SRα contributes to these differences. A possible basis for these differences may lie in the ability of Mxi1-SRα (but not of Mxi1-SRβ) to Ce manuscrit a été publié dans FEBS Journal 274: 4643-4653 (2007) recruit specific protein partners such as the nuclear GAPDH protein.

Results

Mxi1-SRα lacks the strong suppressive activity of Mxi1-SRβ in the REF assay

In our earlier report describing Mxi1-SRα, this isoform appeared functionally homologous to Mxi1-SRβ in that both could bind to Max and Sin3 and repress both basal and Myc-activated transcription of various reporter plasmids [START_REF] Dugast-Darzacq | Mxi1-SRalpha: a novel Mxi1 isoform with enhanced transcriptional repression potential[END_REF]. Based on these properties, we predicted that Mxi1-SRα would act like its Mxi1-SRβ counterpart to suppress Myc+Ras cotransformation activity in the REF assay. Expression constructs were generated encoding Myc-tagged versions of these two isoforms, as well as of the Mxi1-WR isoform that lacks the SID, and shown to give rise to proteins of the expected size expressed at similar levels (Fig. 1A). Each of these constructs (or empty vector) was introduced along with Myc and Ras into primary REFs, and the extent of foci formation was assessed ~10 days post-transfection. As shown in Figure 1B, the addition of Mxi1-SRβ to Myc+Ras transfections resulted in the expected 5-fold reduction in foci number relative to that obtained in the Myc+Ras+empty vector point (compare Fig. 1B black "SRβ" bar to Fig. 1B open "empty" bar; see also [START_REF] Schreiber-Agus | An amino-terminal domain of Mxi1 mediates anti-Myc oncogenic activity and interacts with a homolog of the yeast transcriptional repressor[END_REF]).

Surprisingly, the addition of Mxi1-SRα to Myc+Ras transfections resulted in at best a 2-fold reduction in foci number relative to the Myc+Ras+empty vector point (compare Fig. 1B grey "SRα" bar to Fig. 1B open "empty" bar). Indeed, Mxi1-SRα behaved comparably to Mxi1-WR in these assays (compare Fig. 1B grey "SRα" bar to Fig. 1B dotted "WR" bar). This was unexpected given that (i) the inability of Mxi1-WR to potently suppress Myc cotransformation has been attributed to its lack of a SID [START_REF] Schreiber-Agus | An amino-terminal domain of Mxi1 mediates anti-Myc oncogenic activity and interacts with a homolog of the yeast transcriptional repressor[END_REF] and (ii) Mxi1-SRα harbors a Sin3interacting SID that is ~70% homologous to the SID of Mxi1-SRβ [START_REF] Dugast-Darzacq | Mxi1-SRalpha: a novel Mxi1 isoform with enhanced transcriptional repression potential[END_REF].

We considered the possibility that the difference in suppression potential between Ce manuscrit a été publié dans FEBS Journal 274: 4643-4653 (2007) introduced Mxi1-SRα and Mxi1-SRβ could relate to disparities in their expression levels at the onset of foci formation. As such, we generated several expression constructs for Mxi1-SRα containing different lengths of 5'UTR, and tested these in in vitro transcription/translation assays followed by Western blotting, and in the REF assay (data not shown). Many of these constructs produced levels of Mxi1-SRα protein comparable to or greater than those of Mxi1-SRβ, yet they could still not potently suppress in the REF assay.

To address this in another way, we tested whether the introduction of lower amounts of As another gauge of suppression potential, we examined the introduced Mxi1 isoforms in stable transformed cell lines established from foci that had emerged in the various transfection points of the REF assay (Fig. 1C). Transformed cell lines established from the Myc+Ras+Mxi1-SRα points consistently expressed detectable levels of introduced Mxi1-SRα as assessed by Western blotting analysis (Fig. 1C, arrowed ~52 kDa band in lanes α1 and α2 in comparison to corresponding area in lane E1 which is from cell lines established from the Myc+Ras+empty vector point). Once again, this finding of expressed exogenous Mxi1-SRα resembled what is seen with the SID-less Mxi1-WR isoform as we have reported previously [START_REF] Schreiber-Agus | An amino-terminal domain of Mxi1 mediates anti-Myc oncogenic activity and interacts with a homolog of the yeast transcriptional repressor[END_REF]. In contrast, transformed cell lines established from the Myc+Ras+Mxi1-SRβ points failed to express detectable levels of introduced Mxi1-SRβ (Fig. 1C, lanes β1 and β2; and see also [START_REF] Schreiber-Agus | An amino-terminal domain of Mxi1 mediates anti-Myc oncogenic activity and interacts with a homolog of the yeast transcriptional repressor[END_REF]). These results suggest that strong selective pressure against the expression of introduced Mxi1-SRβ, but not of Mxi1-SRα (or Mxi1-WR), exists during the Ce manuscrit a été publié dans FEBS Journal 274: 4643-4653 (2007) course of cellular transformation induced by Myc.

Mxi1-SRα appears to be localized to the nucleus like Mxi1-SRβ

The findings of the REF assay suggested that Mxi1-SRα and Mxi1-SRβ may encode differential functions with respect to their ability to antagonize Myc function. As a first attempt to uncover the molecular basis for this difference, we performed immunofluorescence assays on three different cell types after transfection with Myc or FLAG-tagged versions of Mxi1-SRα or Mxi1-SRβ (Fig. 2A). As shown in Figure 2B, Myc-tagged Mxi1-SRα and Myc-tagged Mxi1-SRβ each exhibit speckled nuclear staining in transfected U20S cells.

Similar results were obtained after transfection of U20S cells with the Flag-tagged isoforms, as well as after transfection of any of these constructs into COS7 or NIH3T3 cells (Fig. 2C and data not shown). It should be noted that this Mxi1-SRα subcellular localization is not in complete agreement with a previous report from another group [START_REF] Engstrom | Mxi1-0, an alternatively transcribed Mxi1 isoform, is overexpressed in glioblastomas[END_REF] which described a primarily cytoplasmic localization, with some nuclear staining as well. However, Mxi1-SRα is predicted to be nuclear by programs such as PSORT II (reliability 94.1% by the [START_REF] Reinhardt | Using neural networks for prediction of the subcellular location of proteins[END_REF]). Due to the lack of available Mxi1 isoform-specific antibodies, we cannot determine the localization of the endogenous forms by immunofluorescence at this time.

Based on the data presented in Figure 2, we believe that exogenous Mxi1-SRα, like Mxi1-SRβ, is a nuclear protein. Thus, difference in the subcellular localization cannot provide a basis for the differential effect of the two isoforms on Myc induced cellular transformation.

The evolutionarily conserved, extended proline-rich domain of Mxi1-SRα affects its suppression potential

We predicted that the basis for functional differences between Mxi1-SRα and Mxi1-Ce manuscrit a été publié dans FEBS Journal 274: 4643-4653 (2007) SRβ could relate to the unique amino terminal 61 amino acid (aa) extension on Mxi1-SRα (preceding its SID). As shown in Figure 3A, this extension (here named the PRD for prolinerich domain) is conserved from fish to man, and, at least in mammals, is proline and alanine rich. Hypothesizing that this PRD of Mxi1-SRα could be playing a regulatory role or encoding novel functions, we asked whether the presence of this domain is responsible for the differential effects of introduced Mxi1-SRα and Mxi1-SRβ in the REF assay. An expression construct was generated encoding a Myc-tagged version of Mxi1-SRα lacking its proline-rich domain (Fig. 3B); this construct was shown to give rise to a protein of the expected size expressed at similar levels to its full length counterpart (data not shown).

When introduced with Myc+Ras in the REF assay, this construct suppressed cotransformation activity at least as well as Mxi1-SRβ (Fig. 3B, compare striped "SRαΔPRD" bar to black "SRβ" bar). Said another way, deletion of the 61 amino acid PRD from Mxi1-SRα converts Mxi1-SRα into a potent suppressor of Myc+Ras transformation. However, the

PRD does not appear sufficient on its own to convert the Mxi1-SRβ protein into a protein with α-like properties in other assays (see below).

The extended proline-rich domain of Mxi1-SRα affects its activity on the MYC promoter

Having seen this effect of the proline-rich domain on Mxi1-SRα function at the cellular level, we next asked whether the presence or absence of this domain affects Mxi1-SRα activity on the promoters of downstream target genes. Earlier we had shown that Mxi1-SRα and Mxi1-SRβ exhibited similar effects on two synthetic reporter constructs in 293T cells [START_REF] Dugast-Darzacq | Mxi1-SRalpha: a novel Mxi1 isoform with enhanced transcriptional repression potential[END_REF]. Here we extended these analyses to the E-box containing promoter ornithine decarboxylase (ODC) promoter; notably this is one of the few promoters reported to be regulated by Mxi1 (and also by Myc) [START_REF] Wu | Overexpression of Mxi1 inhibits the induction of the human ornithine decarboxylase gene by the Myc/Max protein complex[END_REF]. As shown in Figure 4A, the addition of Mxi1-SRα or Mxi1-SRβ effectors to 293T cells also carrying ODC-driven luciferase resulted in a 2-3-fold reduction in luciferase activity, consistent with what has been shown previously for Mxi1-SRβ on this promoter [START_REF] Wu | Overexpression of Mxi1 inhibits the induction of the human ornithine decarboxylase gene by the Myc/Max protein complex[END_REF]. It is known that this region of the ODC promoter bears two E-box elements that are repressed by Mxi1 but activated by Myc (see [START_REF] Pena | Regulation of human ornithine decarboxylase expression by the c-Myc.Max protein complex[END_REF] for example). As such, the similar effects of the two Mxi1 isoforms on this promoter are in the line with their similar effects on the synthetic E box reporter [START_REF] Dugast-Darzacq | Mxi1-SRalpha: a novel Mxi1 isoform with enhanced transcriptional repression potential[END_REF].

A second promoter shown previously to be regulated by Mxi1-SRβ is the MYC promoter [START_REF] Lee | Mxi1 is a repressor of the c-Myc promoter and reverses activation by[END_REF][START_REF] Luo | Autorepression of c-myc requires both initiator and E2F-binding site elements and cooperation with the p107 gene Ce manuscrit a été publié dans[END_REF]. For this promoter, regulation by Mxi1 has been proposed to occur not through E-box sequences but through initiator (Inr) elements and possibly also E2F binding sites present in cis [START_REF] Lee | Mxi1 is a repressor of the c-Myc promoter and reverses activation by[END_REF][START_REF] Luo | Autorepression of c-myc requires both initiator and E2F-binding site elements and cooperation with the p107 gene Ce manuscrit a été publié dans[END_REF]. Whether the action of Mxi1-SRβ on the MYC promoter is direct or indirect remains to be elucidated. Consistent with what has been reported previously, Mxi1-SRβ exhibited a mild, but reproducible, repressive effect on the full length human c-MYC promoter (Fig. 4B) [START_REF] Lee | Mxi1 is a repressor of the c-Myc promoter and reverses activation by[END_REF][START_REF] Luo | Autorepression of c-myc requires both initiator and E2F-binding site elements and cooperation with the p107 gene Ce manuscrit a été publié dans[END_REF]. Surprisingly, Mxi1-SRα activated this reporter (Fig. 4B). Deletion of the 61 amino acid PRD from Mxi1-SRα converted Mxi1-SRα from an activator to a potent repressor of the MYC promoter (Fig. 4B). Again, whether the action of Mxi1-SRα is direct or indirect remains to be elucidate. Of note, the same trend of Mxi1-SRβ and Mxi1-SRαΔPRD repressing, but Mxi1-SRα activating, was observed on the minimal MYC-P1P2 promoter (data not shown). Taken together, our findings suggest that on certain downstream target gene promoters in transient transfection experiments, Mxi1-SRα and Mxi1-SRβ exhibit distinct transcriptional effects, and these are correlated with the presence of the proline-rich domain on Mxi1-SRα.

Mxi1-SRα is able to interact with GAPDH, and these two proteins synergise to activate the myc promoter

We hypothesized that the basis for the differential effects of Mxi1-SRα and Mxi1-SRβ in several functional assays could relate to differences in their protein interaction profiles. To Ce manuscrit a été publié dans FEBS Journal 274: 4643-4653 (2007) address this, we established inducible HeLa cell lines expressing FLAG-Mxi1-SRα or FLAG-Mxi1-SRβ under the control of the TET ON promoter. We monitored induction as well as expression levels of the isoforms by immunoprecipitation with an anti-FLAG antibody followed by anti-FLAG Western blot analysis (Fig. 5A). We then performed a FLAG pull down analysis, followed by resolution on SDS-PAGE gel and silver staining (see Fig. 5B). Several bands appearing to be present in the Mxi1-SRα but not Mxi1-SRβ lanes were subjected to mass spectrometry analysis (see Material and Methods). One candidate Mxi1-SRα interacting protein identified was the 38kDa GAPDH protein which obtained a very high score with 31 matching peptides (Data not shown). This interaction was confirmed by Western blotting analysis using the anti-GAPDH antibody [START_REF] Zheng | S phase activation of the histone H2B promoter by OCA-S, a coactivator complex that contains GAPDH as a key component[END_REF] on the FLAG immunoprecipitates (Fig. 5C). Interestingly, this 38 kDa GAPDH protein has recently been characterized to be part of a transcriptional co-activator complex [START_REF] Zheng | S phase activation of the histone H2B promoter by OCA-S, a coactivator complex that contains GAPDH as a key component[END_REF]. Accordingly, we next tested whether Mxi1-SRα and GAPDH could synergize to activate the myc promoter.

Whereas GAPDH overexpression had no activating effect on the P1P2myc promoter without effector (Fig. 5D, compare lane 2 with lane 1) or even in the presence of Mxi1-SRβ (Fig. 5D, compare lane 5 and 6 with lane 1), the co-expression of GAPDH with Mxi1-SRα led to enhanced activation (Fig. 5D, compare lane 4 with lane 3). Very interestingly, GAPDH over expression did not affect the activity of the Mxi1-SRα protein when its PRD was deleted (Fig. 5D, compare lane 7 and 8 with lane 1). Thus, all the specific properties of the full length Mxi1-SRα protein observed here appears to depend on the presence of the PRD. Taken together, it appears that Mxi1-SRα may recruit a unique set of proteins, including GAPDH, to participate in transcriptional activation of target genes.

Ce manuscrit a été publié dans FEBS Journal 274: 4643-4653 (2007)

Discussion

In this study, we have further compared the Mxi1-SRα and Mxi1-SRβ protein isoforms that have previously been described by us to be highly similar at the levels of tissuetype expression patterns, protein interaction profiles, and transcriptional repression activity [START_REF] Dugast-Darzacq | Mxi1-SRalpha: a novel Mxi1 isoform with enhanced transcriptional repression potential[END_REF]. Here we extend the similarity between these isoforms by showing that, at least when exogenously introduced, both isoforms localize to the nucleus (Figure 2) and both repress the promoter of a known Mxi1 (and Myc) downstream gene target, ODC (Figure 4A). However, Mxi1-SRα and Mxi1-SRβ also appear to encode differential functions, and those revealed in this study relate to two distinct levels of Myc antagonism. First, contrary to Mxi1-SRβ, Mxi1-SRα is not a potent suppressor of the cellular transformation activity of Myc (Figure 1). Second, while Mxi1-SRβ has a mild, but reproducible repressive effect on the MYC promoter in transient expression assays, Mxi1-SRα instead activates this promoter (Figure 4B).

The finding of these differential functions is in line with the dogma that the proteome gains functional complexity by encoding multiple isoforms of a given protein, with these isoforms having shared and distinct features (reviewed in [START_REF] Landry | Complex controls: the role of alternative promoters in mammalian genomes[END_REF][START_REF] Stamm | Function of alternative splicing[END_REF]). With respect to Mxi1-SRα and Mxi1-SRβ, this functional complexity may allow for differential regulation of Mycdependent processes. This could occur via alterations in the balance between the two isoforms in specific cell types, developmental stages, or even during cancer pathogenesis.

Regarding the latter, it is interesting to note that the Mxi1-SRα isoform (also know as Mxi1-0) was cloned initially as a gene upregulated in a neuroblastoma cell line. Moreover, in that study, the ratio between Mxi1-SRα/Mxi1-0 and Mxi1-SRβ in primary glioblastomas was shown to be increased relative to their ratio in normal brain [START_REF] Engstrom | Mxi1-0, an alternatively transcribed Mxi1 isoform, is overexpressed in glioblastomas[END_REF]. Future studies using isoform-specific reagents could determine whether this also holds true for other cancer types, Ce manuscrit a été publié dans FEBS Journal 274: 4643-4653 (2007) and could ascertain whether altering the levels of Mxi1-SRα or Mxi1-SRβ can differentially impact upon cellular processes including proliferation, apoptosis, differentiation, etc.

Isoforms of numerous proteins have been studied and compared in this manner, including alternative isoforms of members of the p53/p63/p73 (reviewed in [START_REF] Murray-Zmijewski | isoforms: an orchestra of isoforms to harmonise cell differentiation and response to stress Cell Death Differ[END_REF]) and the Bcl2 families (reviewed in [START_REF]splicing of Bcl-2related genes: functional consequences and potential therapeutic applications[END_REF]).

On the molecular level, we show that the unique proline rich domain (PRD) on Mxi1-SRα contributes to the differential functions of Mxi1-SRα and Mxi1-SRβ in Myc antagonism Deletion of this domain converts Mxi1-SRα into a potent suppressor of Myc/Ras cotransformation activity (Figure 3) and also changes Mxi1-SRα's activity on the MYC promoter from activation to repression (Figure 4B). Relevant to this, we were intrigued by the proline-rich composition of the PRD of Mxi1-SRα, given that this is a recurring feature of transactivation domains. However, when we tested the PRD in the Gal4 heterologous reporter assay system, we did not observe it to have inherent transactivation potential ( [START_REF] Dugast-Darzacq | Mxi1-SRalpha: a novel Mxi1 isoform with enhanced transcriptional repression potential[END_REF] and data not shown). Thus the PRD is necessary but likely not sufficient for the transactivation activity of Mxi1-SRα. Consistent with this, a chimeric protein that we generated to contain the PRD hooked up to the Mxi1-SRβ isoform (PRD-Mxi1-SRβ) is not able to activate a myc promoter in transient transfection experiment (data not shown). Thus, the activation function of Mxi1-

SRα that depends on the integrity of the PRD appears to depend also on other features of the Mxi1-SRα proteins as it cannot be simply transferred to another protein, even as closely related as Mxi1-SRβ

We speculate that the PRD may be involved in regulating other functional domains of Mxi1 (e.g., the SID or the bHLH/LZ) and/or in the recruitment of other activities. However, in some assays, the presence of this domain does not appear to distinguish Mxi1-SRα from Mxi1-SRβ (this study and [START_REF] Dugast-Darzacq | Mxi1-SRalpha: a novel Mxi1 isoform with enhanced transcriptional repression potential[END_REF]). This suggests that the effects of the proline-rich domain are context sensitive, and could depend on variables including cellular milieu and promoter Ce manuscrit a été publié dans FEBS Journal 274: 4643-4653 (2007) environment. It is of interest in this regard that, in our hands, Mxi1-SRα and Mxi1-SRβ behaved similarly on E-box containing promoters which are thought to be repressed by Mxi1 (and related Mad family members) in a basic region-, Max-, and Sin3-dependent manner (for example see [START_REF] Wu | Overexpression of Mxi1 inhibits the induction of the human ornithine decarboxylase gene by the Myc/Max protein complex[END_REF]; also see [START_REF] Gunes | Expression of the hTERT gene is regulated at the level of transcriptional initiation and repressed by Mad1[END_REF]). In contrast, on the MYC promoter which is repressed by Mxi1-SRβ in an E-box-independent manner ( [START_REF] Lee | Mxi1 is a repressor of the c-Myc promoter and reverses activation by[END_REF][START_REF] Luo | Autorepression of c-myc requires both initiator and E2F-binding site elements and cooperation with the p107 gene Ce manuscrit a été publié dans[END_REF]), Mxi1-SRα exerts distinct effects. It is possible that this differential regulation of target genes contributes including the effect on transformation we have observed in the REF assay (Figure 1). A very analogous scenario has been recently for isoforms of the Wilms' tumor gene WT1. A newly identified WT1 isoform (WT1s) has been shown to arise from alternative promoter/leader exon utilization, similar to how Mxi1-SRα and Mxi1-SRβ arise.

While the full length WT1 protein encodes both transcriptional repression and activation domains, WT1s lacks the repression domain and consequently has different effects on downstream targets and in growth/cancer-related assays [START_REF] Hossain | N-terminally truncated WT1 protein with oncogenic properties overexpressed in leukemia[END_REF].

Our Flag pull down analysis showed that Mxi1-SRα, but not Mxi1-SRβ, is able to recruit nuclear GAPDH (Figure 5B and5C). Moreover, GAPDH seems to enhance the activating potential of Mxi1-SRα on the myc promoter, but has no effect on the repression effect of Mxi1-SRβ or a Mxi1-SRα protein deleted from its PRD (Figure 5D).

Interestingly, nuclear GAPDH/p38 has been shown previously to be recruited by Oct-1 in a co-activator complex implicated in the S phase transcription of histone H2B promoter [START_REF] Zheng | S phase activation of the histone H2B promoter by OCA-S, a coactivator complex that contains GAPDH as a key component[END_REF].

Thus, it is tempting to speculate that Mxi1-SRα may be able to recruit co-activator complexes containing GAPDH to specific target genes resulting in activation. More and more reports suggest that GAPDH is a multifunctional protein displaying diverse activities distinct from its conventional glycolytic activity. For example, it has been shown to be able to regulate cyclin B-cdk1 activity via its interaction with the protein SET [START_REF] Carujo | Glyceraldehyde 3-phosphate dehydrogenase is a SET-binding protein and regulates cyclin B-cdk1 activity[END_REF], to induce the pro-apoptotic mitochondrial membrane permeabilization which is essential for apoptosis [START_REF] Tarze | GAPDH, a novel regulator of the pro-apoptotic mitochondrial membrane permeabilization[END_REF] Ce manuscrit a été publié dans FEBS Journal 274: 4643-4653 (2007) or to prevent down-regulation of CSF-1 protein by binding to CSF-1 AU-rich element and thus increasing metastatic properties in ovarian cancer [START_REF] Bonafe | Glyceraldehyde-3-phosphate dehydrogenase binds to the AU-Rich 3' untranslated region of colony-stimulating factor-1 (CSF-1) messenger RNA in human ovarian cancer cells: possible role in CSF-1 posttranscriptional regulation and tumor phenotype[END_REF]. GAPDH is an abundant protein but its participation in many different complexes in different cellular compartment could make it limiting for some of its roles. Thus, some specific functions of GAPDH could be more sensitive than other to variation in its intracellular level or availability. Our observation that overexpression of GAPDH enhance the activity of overexpressed Mxi1-SRα indicates that it is amounts for Mxi1-SRα function. In this respect, we tested if the GAPDH protein could be the limiting factor preventing the PRD alone from having a transcriptional activity by itself in the gal4 assay (data not shown) and we found that even in the presence of overexpressed GAPDH, the PRD was not sufficient to activate transcription, emphasizing the contribution of other features of the Mxi1-SRα protein.

In the future, it would be important to uncover the full spectrum of differentially interacting proteins as well as the spectrum of downstream target genes regulated by Mxi1-SRα and/or Mxi1-SRβ, and assess the transcriptional effects of these isoforms on these targets. A better molecular grasp on these Mxi1 isoforms is necessary for understanding the precise role(s) of Mxi1 within the extended Myc network and in the context of development and cancer.

Materials and Methods

Plasmid Generation

The Myc-tagged Mxi1-SRα and Mxi1-SRβ constructs were described in [START_REF] Dugast-Darzacq | Mxi1-SRalpha: a novel Mxi1 isoform with enhanced transcriptional repression potential[END_REF], and the Myc and Ras expression constructs and the pvNic vector were described in [START_REF] Schreiber-Agus | An amino-terminal domain of Mxi1 mediates anti-Myc oncogenic activity and interacts with a homolog of the yeast transcriptional repressor[END_REF]. The Myc tagged WR expression construct was generated by introducing the WR cDNA containing the full 5'UTR in pcDNA3.1. The coding region of Mxi1-SRα and Mxi1-SRβ were subcloned by an amino terminal flag tag. The Mxi1-SRα ΔPRD expression construct corresponds to the full Mxi1-SRα deleted for its first 61amino acids. The ODC and MYC reporter constructs were kind gifts of Dr. John Cleveland and Dr. Linda Penn, respectively. The HA-GAPDH expression vector was obtained by amplifying GAPDH cDNA by RT-PCR on RNA from HeLa cells, followed by subcloning in an HA-tag-containing expression vector. The Tet responsive Mxi1-SRβ and -SRα expression constructs were generated by cloning the FLAG-Mxi1 fusion protein behind a tet responsive promoter.

Further details of plasmids constructions are available upon request.

REF assay and foci studies

Primary REFS were prepared and transfected using calcium phosphate precipitation as described [START_REF] Mukherjee | Myc family oncoproteins function through a common pathway to transform normal cells in culture: cross-interference by Max and trans-acting dominant mutants[END_REF]. For each construct listed, 2µg was used per plate except in the "SRβ low" point where only 0.4µg of plasmid DNA was used. The number of foci obtained for each plate was counted 10-15 days after transfection. Individual foci were picked, subcloned, and expanded as described [START_REF] Mukherjee | Myc family oncoproteins function through a common pathway to transform normal cells in culture: cross-interference by Max and trans-acting dominant mutants[END_REF].

Immunofluorescence

U2OS cells were transfected with 100 ng of DNA with Fugene, and immunofluorescence was performed as described in [START_REF] Shav-Tal | Dynamic sorting of nuclear components into distinct nucleolar caps during transcriptional inhibition[END_REF] using the anti-FLAG (M2, Sigma-Aldrich, St Louis, MO, USA) or the anti-myc (Upstate, Millipore, Bedford, MA, USA) primary antibodies and the anti-rabbit coupled to FITC (Jackson Immuno Research) or the anti-mouse coupled to FITC (Jackson Immuno Research, West Grove, PA, USA) secondary antibodies, respectively.

Images were acquired with an Olympus BX61 epifluorescence microscope (Olympus America, Melville, NY) and a Roper Scientific CoolSNAP HQ camera (Roper Scientific, Tucson, AZ).

Transcriptional reporter assays

293T cells were transfected by the calcium phosphate precipitation method, and luciferase activity was assessed 48 hours post transfection as described in [START_REF] Dugast-Darzacq | Mxi1-SRalpha: a novel Mxi1 isoform with enhanced transcriptional repression potential[END_REF]. The luciferase values were concentration as assessed by a Bradford assay.

Protein preparation and Western blotting analysis

Protein preparation and Western blotting analysis were performed as described in [START_REF] Dugast-Darzacq | Mxi1-SRalpha: a novel Mxi1 isoform with enhanced transcriptional repression potential[END_REF]. In vitro transcription/translation was performed using the TNT transcription / translation system (Promega, Madison, WI, USA).

Establishment of inducible cell lines

HeLa TET ON cells (Clontech, Takara, Mountain View, CA, USA) were transfected with inducible constructs expressing FLAG-Mxi1-SRα, FLAG-Mxi1-SRβ or empty vector using Fugene. Three days after transfection, cells were selected using puromycin (1µg/mL), and on day 15 post-transfection clones were picked and expanded. After induction with doxycycline (1µg/mL), the individual clones were tested for their expression of the protein of interest.

Flag pull down / Silver Staining / Mass spectrometry Analysis

For each stable cell line, ten 15-cm-diameter plates at 80% confluence were induced with doxycycline (1µg/mL) for 24h. Immunoprecipitation was performed using the FLAG agarose antibody (Sigma #A2220) in 50mM Tris pH 7.5, 150 mM NaCl, 0.5% NP40, 5 mM EDTA and 1 mM DTT. The immunoprecipitated protein was eluted from the beads according to the Manufacter's instruction and the supernatant was run on a 12% SDS-PAGE gel. Silver staining of the gel was performed as described in [32]. Mass spectrometry was performed by Ce manuscrit a été publié dans FEBS Journal 274: 4643-4653 (2007) the Rockefeller University Proteomics Resource Center (New York). Ce manuscrit a été publié dans FEBS Journal 274: 4643-4653 (2007)

Titles and legends to figures

Mxi1-

  SRβ would compromise its suppression potential. Introduction of one-fifth the usual amount of Mxi1-SRβ to Myc+Ras transfections resulted in the same 5-fold reduction in foci formation observed with the usual dose of Mxi1-SRβ (compare Fig. 1B black "SRβ low" bar to Fig. 1B black "SRβ" bar). Together, these findings suggested that the differential effects of Mxi1-SRα and Mxi1-SRβ in the REF assay likely relate to variables aside from expression levels.

Figure 1 :Figure 2 :

 12 Figure 1: Contrary to Mxi1-SRβ, Mxi1-SRα is not a potent suppressor of cellular

Figure 3 :

 3 Figure 3: An Mxi1-SRα protein deleted for its proline-rich domain is able to potently

Figure 4 :Figure 5 :

 45 Figure 4: Common and distinct transcriptional effects of Mxi1-SRα and Mxi1-SRβ on

Ce manuscrit a été publié dans FEBS Journal 274:4643-4653 (2007)

Acknowledgments

The authors thank members of the Schreiber-Agus laboratory, as well as Dr. Andras Fiser, for stimulating discussions and helpful advice on the study. We thank Dr. Paul Corn for critically reading the manuscript. We thank Laina Freyer and Dr. Rachele Arrigoni their research contributions to the project, Dr. Singer for use of his microscopes, and Dr.

Xavier Darzacq for his contributions with the immunofluorescence analysis. This work was supported by NCI grant 1 R01 CA92558 (NSA) and the Association pour la Recherche contre le Cancer (ARC) (TG). CDD is a recipient of postdoctoral awards from the International Agency for Research on Cancer, the National Cancer Center and from the Association pour la Recherche contre le Cancer (ARC). Support from the Albert Einstein Cancer Center is also acknowledged.