Gianluca Artini
email: gianluca.artini@cea.fr

Daniel Broc
email: daniel.broc@cea.fr

AN HOMOGENISATION METHOD FOR A FSI PROBLEM APPLICATION TO A TUBE BUNDLE ROW

A research program is ongoing at CEA in the framework of ASTRID project, for developing and validating tools for the analysis of the mechanical dynamic behaviour of Fast Breeder Reactor (FBR) cores. In this context, Fluid Structure Interaction is a recurring problem, especially in case of tube bundles in nuclear reactor cores or steam generators. The large number of elements makes modelling difficult to perform and any direct numerical simulation at whole core scale unachievable. In order to overcome this limitation, we resort to homogenisation techniques which allow to describe the global dynamic behaviour of the whole fluid-structure system. The purpose of this study is to better understand the effects of the spatial variation of the tube displacement field on the whole bundle dynamics. Thanks to multiple scales, a complete development of the problem is conducted. An analysis of the effects of spatial variations of the long scale displacement field on the tube bundle's dynamics is led. As a first step we consider the homogenisation of Euler linear equations that gives us the possibility to make a thorough assessment of the problem with the extension to follow to Navier-Stokes equations.

INTRODUCTION

A research program is ongoing at the French Commissariat à l' Énergie Atomique et aux Énergies Alternatives (CEA) for the * Address all correspondence to this author. development of the new prototypical Sodium Fast Breeder Reactor (Na-FBR) ASTRID.

The ASTRID reactor core presents a large amount of fuel elements (∼ 300) ranged in a very narrow gap configuration (∼ 3 mm) due to neutronic reasons. They are surrounded by about 300 reflectors elements and 500 PNL (Lateral Neutronic Protection) shielding elements.

In FBR configuration, the simulation of whole core results to be necessary for a complete safety assessment of the power plant. For example, phenomena occurring during seismic events or during core flowering involve a strong coupling in the whole core scale. Then, for a complete design and safety analysis understanding the fluid structure interaction phenomena occurring at whole core scale is of crucial relevance. It is well know that the presence of a fluid strongly changes the dynamic behaviour of the whole reactor core. The fluid presence implicates a strong coupling between reactor core elements in the whole core scale. In addition, the presence of a large number of tubes and the narrow gap configuration make the dynamic analysis very difficult to assess. In this perspective, the homogenisation techniques have a crucial role in the dynamic analysis of a large number degrees of freedom systems, as the FBR cores. This is one of the main reason through we resort to homogenisation techniques [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structure[END_REF] or [START_REF] Bergman | Les méthodes de lèhomogénéisation: théorie et applications en physique[END_REF].

In this paper, the authors obtain an homogenised equation system at long scale for pressure and displacement field in an enclosed tube bundle. Starting from the Euler linearised equations, thanks to the introduction of two scales development for pressure and displacement, a series of elementary problems describing the system at the short scale level is obtained. They report numerical simulations for these elementary problems and they discuss the main dynamic characteristics of the tube bundle displacement field obtained by the proposed homogenised model.

HOMOGENISATION METHODS: STATE OF THE ART

The literature on fluid structure interaction for a single tube immersed in an inviscid or viscid fluid is largely widespread, as in, for example, [START_REF] Paidoussis | Fluid-Structure Interactions : Slender Structures and Axial Flow[END_REF] and [START_REF] Axisa | Modelling of Mechanical Systems: Fluid-Structure Interaction[END_REF].

It is well known, the presence of fluid involves on oscillating structure two different effects: inertial and dissipative effects.

The fluid flow around an oscillating body may be described by Navier-Stokes equations or under some simplifying hypothesis by Euler or linear Euler equations. For example, Fritz, in [START_REF] Fritz | The Effect of Liquids on the Dynamic Motions of Immersed Solids[END_REF] analyses an inviscid fluid initially at rest enclosed in two oscillating cylinders. Thanks to Lagrange formulation, the author obtains an analytical expression of the fluid reaction force on the solid body in the assumption of small displacement amplitude. A proportionality between solid body acceleration and fluid force is shown. The proportional coefficient is the so called added mass coefficient.

On the other hand, in [START_REF] Chen | Added Mass and Damping of a Vibrating Rod in Confined Viscous Fluids[END_REF], Chen et al. analyse the problem of a cylindrical tube immersed in a viscid fluid. In the case of imposed displacement of inner tube's wall, the authors obtain an analytical expression of the fluid force, where two different terms are shown. The first one, representing the inertial effects due to the fluid presence, is in phase with the solid acceleration; the second one, in phase with the solid body velocity, is due to the fluid internal friction and it is representative of dissipative effects.

In a large number of industrial facilities, periodic structures immersed in a fluid are often encountered (nuclear reactors, steam generators, etc.). Especially when the tubes number largely increases, the coupling provoked by the fluid represents the main issue. Moreover, the large amount of degree of freedom makes any direct numerical simulations unachievable. For this reason, homogenisation method are developed.

A variety of methods for homogenisation are found in literature. The main goals of homogenisation is to obtain an equivalent model maintaining the local characteristics of the medium [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structure[END_REF] and [START_REF] Bergman | Les méthodes de lèhomogénéisation: théorie et applications en physique[END_REF]. Hereafter we recall the main homogenisation methods for FSI problems.

In [START_REF] Planchard | Homogénéisation des matrices de masse ajoutée dans le cas de grands faisceaux tubulaires[END_REF], Planchard proposes a method for homogenising the added mass matrix as the bundle is composed of a large number of tubes, identical and regularly spaced. This techniques consists in homogenising the added mass matrix only, and maintaining unchanged the tubes dynamic equation.

Shinohara and Shimogo in [START_REF] Shinohara | Vibrations of Square and Hexagonal Cylinders in a Liquid[END_REF] and Jacquelin and al. in [START_REF] Jacquelin | Homogenization of a Non-Linear Array of Confined Beams[END_REF], establish a bi-dimensional analytical homogenised model for square [START_REF] Shinohara | Vibrations of Square and Hexagonal Cylinders in a Liquid[END_REF][START_REF] Jacquelin | Homogenization of a Non-Linear Array of Confined Beams[END_REF] or hexagonal [START_REF] Shinohara | Vibrations of Square and Hexagonal Cylinders in a Liquid[END_REF] section tubes. The model proposed is limited to elements with plane faces; in fact, it is based on the hypothesis that the translation movement of elements leads to fluid velocity fluctuations in the direction parallel to solid faces. This hypothesis is valuable only in case of high confinement configurations (BWR or FBR cores). Expressions for frequencies of the bundle are obtained as functions of the main physical characteristics and parameters.

Hammami in [START_REF] Hammami | Contribution a l' Étude de l'Interaction Fluide Structure Dans Les Faisceaux de Tubes Par Une Technique d'Homogénéisation[END_REF] and Conca et al. after, in [START_REF] Conca | Problèmes Mathématiques En Couplage Fluide-Structure Applications Aux Faisceaux Tubulaires[END_REF], utilise the method of multiple scales based on the asymptotic development of the main variables, pressure p and structure displacement s, in order to obtain an homogenised two phase model. The model proposed by the authors is based on the periodicity of tube bundle geometry. The system obtained is representative of the long scale variation of the main quantities maintaining, at the same time, the effects of short scale phenomena.

In [START_REF] Broc | Fluid Structure Interaction for Tube Bundles, Physical Meaning of a Rayleigh Damping[END_REF] and in [START_REF] Broc | Fluid Structure Interaction Modelling for the Vibration of Tube Bundles: Part II-Homogenization Method Based on the Navier Stokes Equations[END_REF], the authors utilise a physical approach to the homogenisation. Thanks to the analysis of an elementary cell corresponding to one tube sourrounded by fluid, the authors are able to describe the global behaviour in terms of mean fluid structure velocity (the Darcy velocity) and homogenised global pressure. In [START_REF] Broc | Fluid Structure Interaction for Tube Bundles, Physical Meaning of a Rayleigh Damping[END_REF], Broc introduces the Rayleigh damping in homogenised equation system, in order to take into account the friction effects within the fluid using an Euler equation based model. Moreover, in [START_REF] Broc | Fluid Structure Interaction Modelling for the Vibration of Tube Bundles: Part II-Homogenization Method Based on the Navier Stokes Equations[END_REF], Broc and Desbonnets propose an homogenised model based on the Navier -Stokes equations, where the small amplitude assumption is not made. Via volume average of main quantities, equations for long scale variations are obtained in terms of Darcy velocity, long scale homogenised pressure and tube displacement.

This last two approaches, the mathematical one of Conca and Hammami and the physical one of Broc and Desbonnets, are completely equivalent. In fact, it may be demonstrated that the final equation systems obtained with this two techniques are equivalent to each others.

All the previous homogenised models are able to describe the global fuel bundle movement and, to the author knowledge, no homogenised model can describe the effects of the tube differential movements on the dynamic of the bundle itself. In this paper, the authors propose a homogenised model where it is possible to take into account these latter effects.

PROBLEM POSITION

In this paper, we analyse the dynamic behaviour of a tube bundle row, immersed in an inviscid fluid. The tubes are enclosed in a large vessel with a characteristic length L and there is no fluid flow within the bundle itself. The tubes are disposed in a periodic array with a pitch length equal to ε (Fig. 1).

From a mechanical point of view, the tubes are identical to each other and they are represented as rigid bodies and their dynamics is described by the Fundamental Principle of the Dynamic, where the forces acting on the tubes are the recalling elas- F F,i ). This means that each tube of the bundle is assimilated to an oscillator of one degree of freedom with a mass m s and a stiffness k s representative of elastic recalling and constrain forces whit no structure damping

m s sc,i + k s s c,i = F F,i (1)
where s c the tube's axis displacement. In the hypothesis of inviscid fluid, the fluid reaction force is given only in terms of pressure corresponding to the inertial effects due to the fluid presence [START_REF] Fritz | The Effect of Liquids on the Dynamic Motions of Immersed Solids[END_REF].

Some hypothesis on the geometry and on the mechanical and dynamical behaviour are assumed:

The dimension of the pitch ε is small compared to the main characteristic length of the tube bundle L,

ε < L (2)
The small amplitude hypothesis is assumed for the tube displacement, i.e.:

|s c | ε 1 (3)
Moreover, we assumed the displacement field of the tube bundle only in the horizontal direction x 1 ; thus we can write for each tube:

s c,i = s c,1 = s c (4)
The fluid, in the inviscid and incompressible hypothesis, is governed by the Euler equations. In our case, the hypothesis of

         ∂ w i ∂ x i = 0 in Ω F ∂ w i ∂t = - 1 ρ ∂ p ∂ x i in Ω F (5a) (5b)
where w i is the ith scalar component of the fluid vector velocity, ρ is the fluid density and p is the pressure fluid field. Moreover, the small displacement assumption (Eq. 3) allows to declare the slip boundary conditions in the non-displaced geometrical configuration:

w i n i = ṡc,i n i at each γ c ( 6
)
where n i is the scalar component of the normal vector n n n at each surface γ C . Coupling together mass balance equation (Eq. 5a) and momentum equation (Eq. 5b), we obtain the equation governing the fluid flow in term of pressure field only with appropriate boundary conditions:

             ∆ p = 0 in Ω F 1 ρ ∂ p ∂ n = -sc,i n i at each γ c ∂ p ∂ n = 0 at ∂ Ω F (7a) (7b) (7c)
where ∂ p ∂ n is the normal derivative of the pressure field to the surface γ c of each tubes.

HOMOGENISATION PROCESS

The equations governing the problem are written for the whole system, where, due to the geometrical configuration of the bundle, the main unknowns (pressure field p and tube displacement s) strongly vary in the scale of an elementary cell. Therefore, the variations of the unknowns may be considered depending on two different scales: (x, y). The "long scale" x corresponds to a variation in the whole bundle, while the "short scale" y is representative of variations in a single elementary cell Y (Fig. 5).

The dimensionless short scale variable y is defined by the following formula:

y 1 = x 1 ε , y 2 = x 2 ε ( 8
)
The introduction of two scales for the variation of main quantities leads to the "chain derivation rule" (Fig. 2) as seen in [START_REF] Conca | Problèmes Mathématiques En Couplage Fluide-Structure Applications Aux Faisceaux Tubulaires[END_REF] or in [START_REF] Hammami | Contribution a l' Étude de l'Interaction Fluide Structure Dans Les Faisceaux de Tubes Par Une Technique d'Homogénéisation[END_REF]:

d dx i = ∂ ∂ x i + 1 ε ∂ ∂ y i (9)
The "chain derivation rule" leads to a new formulation for the Laplace operator and directional derivatives, that are now written in the following form:

∆ = ∂ 2 ∂ x 2 1 + 2 ε ∂ 2 ∂ x 1 ∂ y 1 + 1 ε 2 ∂ 2 ∂ y 2 1 + ∂ 2 ∂ y 2 2 d dn = n 1 ∂ ∂ x 1 + 1 ε n 1 ∂ ∂ y 1 + n 2 ∂ ∂ y 2 (10)

Displacement Field

The solution for the displacement field s c , in the perspective of homogenisation, is searched in term of a continuous function s(x 1 ) in the long scale variable x 1 only. The function must satisfy the following relationship with the discrete vector field s c (Fig. 3):

s c = s(x c ) ∀ x c ( 11
)
However, we need a piece-wise function for the successive short scale analysis. We can achieve this purpose thanks to the following elementary Taylor development in the two scale x, y for the displacement field:

s ε (x 1 , y 1 ) = s(x 1 ) -εy 1 ∂ s(x 1 ) ∂ x 1 + ε 2 y 2 1 2! ∂ 2 s(x 1 ) ∂ x 2 1 -ε 3 y 3 1 3! ∂ 3 s(x 1 ) ∂ x 3 1 + O ε 4 (12) ε s(x 1 ) s ε (x, y)
x 1

FIGURE 3. SCHEMATIC REPRESENTATION OF THE DIS-PLACEMENT FIELD DEVELOPMENT. A MULTIPLICITY OF SO-LUTIONS SATISFIES THE CONDITIONS ASSUMED ON THE FUNCTION s(x 1 ) (SOLID AND DASHED LINES ARE TWO POS-SIBLE SOLUTION FOR THE DISPLACEMENT FIELD)

where the function s ε (x 1 , y 1 ) is a piece-wise constant function in each cell (solid black line in Fig. 3). The development proposed is made of by a function in long scale variable x 1 s(x 1 ) continuous on the whole tube bundle domain and and its successive derivatives. The chosen development represents the link between the discrete vector displacement field s c of each tubes and the continuous function s(x 1 ). In fact, independently from the form of s(x 1 ), the development represents a piece-wise functions in each elementary cell. We can check it thanks to the "chain derivation rule":

ds ε dx 1 (x 1 , y 1 ) = ∂ s ε ∂ x 1 + 1 ε ∂ s ε ∂ y 1 = 0.
and so the function s ε (x 1 , y 1 ) is constant in each elementary cell Y . Nevertheless, the condition imposed by Eq. 11 to s(x 1 ) leads to an infinity of possible solutions for the long scale function that may satisfy the proposed development of Eq. 12. An analysis of the uniqueness of the solution for s(x 1 ) will be discussed hereinafter.

Pressure Field

As made for the displacement field, we search for the pressure field a solution in the form of a development in long and short scales x, y. We choose the development of Eq. 13 that, due to the physics of the problem, describes a pressure field as a linear combination between shape functions and long scale tube acceleration and its successive derivatives. The development represents a pressure field where its variations are provoked only by

p ε (x 1 , y) = b 0 ρ x 1 0 s(x) dx + ερ s(x 1 ) b 1 + f 1 (y 1 , y 2 ) + ε 2 ρ ∂ s(x 1 ) ∂ x 1 b 2 + f 2 (y 1 , y 2 ) + ε 3 ρ ∂ s(x 1 ) ∂ x 1 b 3 + f 3 (y 1 , y 2 ) + O ε 4 (13)
The proposed development for pressure field p is a sum of variations in the two scales: the long scale behaviour is given by b i constant coefficients while the fast variations at the level of an elementary cell are described via the functions f i (y 1 , y 2 ) defined in the unitary elementary cell fluid region Y * (see Fig. 5).

In order to ensure the continuity of the pressure field between adjacent cells, condition of periodicity are imposed on the functions f i . Moreover, the functions f i have got a zero average value at the boundary ∂Y of the unitary elementary cell Y for ensure the uniqueness of the function itself.

A specific change of variable is introduced for the functions f i : This specific change of variables find its justification in the following; in fact, thanks to it, we are able to declare an easier problems series for short scale analysis in terms of function χ i (y) rather than in terms of f i (y).

f 1 (y) = χ 1 (y) -b 0 y 1 f 2 (y) = χ 2 (y) + b 0 y 2 1 2 -b 1 y 1 + b 0 8 -y 1 χ 1 f 3 (y) = χ 3 (y) -b 0 y 3 1 6 + b 1 y 2 1 2 -b 2 y 1 + b 1 8 -y 1 χ 2 + y 2 1 2 χ 1 - b 0 8 y 1 . . . (14
Moreover, the change of variables of Eq. 14 with Eq. 12 leads to a new development for pressure field p determined up to a constant value:

p ε (x c , y) = ερ s(x c )χ 1 (y 1 , y 2 ) + ε 2 ρ ∂ s(x c ) ∂ x 1 χ 2 (y 1 , y 2 )+ ε 3 ρ ∂ 2 s(x c ) ∂ x 2 1 χ 3 (y 1 , y 2 ) + O ε 3 (15)
The latter development of pressure field p, as the previous one, represents a pressure field where tubes accelerations and its successive derivatives are the cause of pressure fluctuations and there is no global fluid flow across the bundle.

The pressure field, due to the physics of the problem, is assumed as a linear combination of structure acceleration s(x 1 ) and shape function χ i (y). However, in this development, the solid acceleration and its successive derivatives are evaluated at the tube's axis ( s(x 1 ) = s(x c )). The χ i functions are functions of the short scale variable y only and they are defined in Y * , fluid region of the unitary elementary cell Y (Fig. 5).

Construction of the short scale problem series

We now proceed to the homogeneisation of the fluid structure interaction problem. Firstly the fluid problem is analysed from which we can obtain the description of the fluid force acting on the structures in terms of long scale variables only.

The introduction of Eq. 13 in the Laplace operator for p of Eq. 7a with the application of the "chain derivation rule" (Eq. 9) gives rise to a series of problems in short scale variable y in the fluid region Y * of the unitary elementary cell Y .

The expression of the problem series for the f i functions results to be complex where each problem recalls the previous ones in the series making its resolution difficult to achieve. However, the change of variable between the f i and χ i functions (Eq. 14), makes the problem series largely simpler. In fact, as s(x 1 ) act as a parameter, we obtain the following expression for each χ i (y) problems, after regrouping them in terms of ε powers:

ε 0 ∆ y χ 1 (y 1 , y 2 ) = 0 ε 1 ∆ y χ 2 (y 1 , y 2 ) = 0 ε 2 ∆ y χ 3 (y 1 , y 2 ) = 0 . . . (16)
We evaluate the boundary conditions at the surface γ (tube's surface γ c scaled by a factor ε) in the unitary elementary cell Y (Fig. 5). For this reason, we introduce the development of Eq. 13 in boundary conditions of Eq. 7c and thanks to the relationship of Eq. 10 we obtain a series of boundary conditions for each function f i . These boundary conditions are expressed, as for the definition of the functions f i , in a complex form. Once again, thanks to the change of variable of Eq. 14, we are able to obtain a series of simple boundary conditions for the functions χ i at the tube surface γ in the elementary cell Y :

ε 0 ∂ χ 1 ∂ n y (y 1 , y 2 ) = n 1 (y 1 , y 2 ) ε 1 ∂ χ 2 ∂ n y (y 1 , y 2 ) = 0 ε 2 ∂ χ 3 ∂ n y (y 1 , y 2 ) = 0 . . . ( 17
)
where n 1 is the horizontal component of the normal vector at the boundary γ and ∂ χ i ∂ n y are the normal derivatives with respect to y. To properly close the problem, repetitive boundary conditions are obtained for left and right boundaries of the elementary cell ∂Y . We declare the boundary conditions in order to assure the continuity of pressure field p and fluid flow exiting and entering the unitary cell for the χ i functions. For the functions χ i we obtain the following relationship:

ε 0 χ R 1 = χ L 1 ε 1 χ R 2 -Ŷ χ R 1 = χ L 2 + Ŷ χ L 1 ε 2 χ R 3 -Ŷ χ R 2 + Ŷ 2 2! χ R 1 = χ L 3 + Ŷ χ L 2 + Ŷ 2 2! χ L 1 . . . (18)
and for the derivative with respect to y 1 we obtain:

ε 0 ∂ χ R 1 ∂ y 1 = ∂ χ L 1 ∂ y 1 ε 1 ∂ χ R 2 ∂ y 1 - Ŷ ∂ χ R 1 ∂ y 1 = ∂ χ L 2 ∂ y 1 + Ŷ ∂ χ L 1 ∂ y 1 ε 2 ∂ χ R 3 ∂ y 1 - Ŷ ∂ χ R 2 ∂ y 1 + Ŷ 2 2! ∂ χ R 1 ∂ y 1 = ∂ χ L 3 ∂ y 1 + Ŷ ∂ χ L 2 ∂ y 1 + Ŷ 2 2! ∂ χ L 1 ∂ y 1 . . . (19)
where χ L i and χ R i are the values of χ i functions at left and right boundaries of the unitary elementary cell Y , i.e. :

χ L i = χ i (-Ŷ , y 2 ) χ R i = χ i (+ Ŷ , y 2 )
where Ŷ = 0.5 (20)

Numerical Simulations of Short Scale Problems

We are going to show the numerical simulations for the elementary cell problem series of Eq. 16 with the appropriate boundary conditions. These simulations are obtained thanks to the Finite Elements code Cast3M [START_REF]Cast3M Code de Calcul Aux Éléments Finis[END_REF] developed at the CEA of Saclay.

In Fig. 6, we show results for the first three problems of the series and in Fig. 7 we present fluid accelerations fields relative to each elementary problem. The numerical simulations show that the pressure field of the first problem only is due to the tube's movement whereas the others elementary pressure field are due to pressure (odd problems) or to fluxes (even problems) at the boundary of the elementary cell ∂Y (see boundary conditions imposed by Eq 17 Eq. 19 and Eq. 18).

Figure 7 shows that the structure movement in the first elementary problem χ 1 generates in the fluid domain Y * a mean acceleration proportional to the fluid volume displaced by the tube itself |γ| (the surface of the tube in the elementary unitary cell Y ). Instead, for the successive elementary problems, where the structure it is assumed fixed, the fluid presents a zero mean acceleration field: it maintains a stagnant global behaviour. The boundary conditions imposed for χ i even problems lead to an elementary pressure field provoked by opposite fluid fluxes imposed at the boundary ∂Y . The obtained pressure field generates a zero resultant force on the tube wall. On the other hand, the boundary conditions for the odd χ i problems impose to the fluid a pressure drop across the elementary cell. In this case, the fluid reaction force is proportional to this pressure drop.

The convergence of the χ i problems is also illustrated by the values of the coefficients a i (see below) corresponding to the fluid reaction force of the pressure field for the odd problems (Fig. 8).

From Fig. 8 we can see that the coefficient's series presents a trend that can be approximated by the convergent series:

a 2i ≈ ∞ ∑ i 1 (2i)! (

Fluid Force

We introduce the development of the pressure field of Eq. 13 in the fluid force expression:

F F,i = ε γ ερ s(x c )χ 1 (y) + ε 2 ρ ∂ s(x c ) ∂ x 1 χ 2 (y)+ ε 3 ρ ∂ 2 s(x c ) ∂ x 2 1 χ 3 (y) + O ε 4 n i (y) dγ (22)
where we use the relationship between the integral at the surface γ c and the structure surface γ in the unitary elementary cell Y ( γ c (.) dγ c = ε γ (.) dγ).

As the structure acceleration field appears here as a function of the long scale variable x 1 only, the integral in the short scale y of the forces can be decomposed as follow:

γ χ i n j dγ = Y * ∂ χ i ∂ y j dy - ∂Y χ i n j d∂Y (23)
thanks to the Green theorem, where i = 1, 2, 3, . . . and j = 1, 2.

The previous decomposition allows us to express the fluid force acting on the tube's wall by constant values deriving from short scale analysis in the elementary cell Y .

For the first elementary problem χ 1 (y) we obtain:

γ χ 1 n 1 dγ = | γ| -a 0 ( 24
)
where | γ| is the structure's surface in the elementary unitary cell and a 0 is a constant value (see below).

The structure motion generates in the fluid an acceleration that puts on movement the fluid itself. This fluid acceleration is proportional to the structure surface (| γ| in the recursive elementary element Fig. 5). The other terms of Eq. 24 (a 0 ) represents the mean value of the pressure drop across the elementary cell.

For the successive problems (i = 2, 3, . . . ), the fluid reaction force is expressed proportionally to the pressure drop in the elementary cell Y only:

γ χ i n 1 dγ = -a i-1 for i = 2, 3, . . . (25)
where the a i-1 are constant coefficients.

In conclusion, the component of the fluid force in the horizontal direction is expressed by the following formulation:

F F,1 (x 1 ) = ε 2 ρ s(x 1 ) (| γ| -a 0 ) -ε 4 ρ ∂ 2 s(x 1 ) ∂ x 2 1 a 2 -ε 6 ρ ∂ 4 s(x 1 ) ∂ x 4 1 a 4 + O ε 8 (26)
where | γ| is the surface on the elementary unitary cell, the a i are constant values obtained by the numerical simulations and s(x 1 ) is the continuous acceleration field. Moreover, in Eq. 26 we resort to the result that the mean pressure drop is zero for each even problems: in fact the odd a i are equal to zero. As expected, the vertical component of the fluid force is a zero resultant force (F F,2 = 0).

Dynamic Equation for the Tube Bundle

Thanks to the analysis at the level of a single elementary cell, we are able to express the fluid force in term of the long scale variable x 1 only. The hydrodynamic force of Eq. 26 is composed by a first term proportional to the tube acceleration field s(x 1 ) representing the inertial effects caused on the oscillating structure by the presence of a dense fluid [START_REF] Fritz | The Effect of Liquids on the Dynamic Motions of Immersed Solids[END_REF]. In addition, the expression of the fluid force presents a series of successive terms involving the spatial derivatives of the continuous acceleration field s(x 1 ).

We introduce the following quantities,

m s = m 0 ε 2 k s = k 0 ε 2 (27)
valuable due to the proportionality of the tube mass and stiffness to the tube surface and the expression of the hydrodynamic force (Eq. 26) in the dynamic equation (Eq. 1). Thus, we obtain the following expression for the dynamic equation of the whole tube bundle:

m 0 s(x 1 ) + k 0 s(x 1 ) = -s(x 1 )M add -ε 2 ρ ∂ 2 s(x 1 ) ∂ x 2 1 a 2 -ε 4 ρa 4 ∂ 4 s(x 1 ) ∂ x 4 1 + O ε 6 Fluid Reaction Force (28)
where:

M add = -ρ (| γ| -a 0 ) (29)
is a constant positive value proportional to the solid acceleration hence the name of Added Mass coefficient.

DISCUSSION: MODAL ANALYSIS OF THE DYNAMIC EQUATION

As already said, the function s(x 1 ) is not unequivocally defined: in fact, there are an infinity of solutions that can satisfy Eq. 28 and Eq. 11 (see section "Displacement Field").

We illustrate the effects of spatial variations of the continuous displacement field s(x 1 ) on bundle's water frequency. In order to achieve this purpose, a modal analysis of Eq. 28 is performed by replacing time and spatial derivatives with the formula:

∂ ∂t = iω, ∂ ∂ x 1 = i λ ε ( 30
)
where ω is the tube angular frequency, ε/λ the spatial period of s(x 1 ) and i is the imaginary unit. The assumptions made on the spatial and time derivatives correspond in searching a solution for s(x 1 ) in the form of:

s(x 1 ,t) = sin λ ε x 1 e iωt ( 31
)
The introduction of the formula for space and time derivatives in Eq. 28 leads to:

-ω 2 m 0 s(x 1 ) + k 0 s(x 1 ) = ω 2 M add s(x 1 ) + ω 2 ρ ∞ ∑ n (-1) n λ 2n a 2n s(x 1 ) (32)
In order to have a solution of Eq. 32, whatever values are assumed by the function s(x 1 ), the following statement must be -

ω 2 m 0 + M add + ρ ∞ ∑ n (-1) n λ 2n a 2n + k 0 = 0 (33)
Two different occurrences are possible corresponding to two distinct conditions on the parameter λ ; in fact, we may have λ = 0 or λ = 0 . When λ = 0, the bundle is oscillating as a whole and there are no spatial variations of the displacement field; instead when λ = 0 spatial variations of displacement field are important and not neglected.

In the first case, where λ is equal to zero, the dynamic equation is reduced to a simple formulation and the resulting water frequency is given by:

f H 2 O = 1 2π k 0 m 0 + M add (34)
Thanks to this formula, the name of Added Mass for M add is well explained: the presence of the fluid acts on the structure as an added mass that makes the structure heavier leading to a decrease of bundle's water frequency (inertial effects [START_REF] Fritz | The Effect of Liquids on the Dynamic Motions of Immersed Solids[END_REF]). This condition is represented by the gray dashed line of Fig. 9.

On the other hand, when spatial variations of the displacement field are not neglected (λ = 0), the hydrodynamic force explicitly depends on the spatial period λ . In this case, the bundle water frequency is expressed as follows:

f H 2 O = 1 2π k 0 m 0 + M add + ρ ∑ ∞ n (-1) n λ 2n a 2n (35)
where the appearance of an extra contribution to the added mass term, depending on the parameter λ and the coefficients a 2n , is shown.

From the numerical results, the series of the coefficients a 2n appearing in the extra added mass term of Eq. 35, can be approximated by the series 1/2n! (Eq. 21). Consequently, the contribution ρ ∑ ∞ n (-1) n λ 2n a 2n to the added mass due to the spatial variations of the tube bundle presents a periodic appearance ass seen in Fig. 10. Consequently, periodicity of the extra added mass leads to periodicity in water frequency as shown in Fig. 9.

As Fig. 9 shows, when the parameter λ increases, corresponding to a decreasing of the spatial period of s(x 1 ), the frequency in water decreases until it reaches a minimum value. The minimum value is obtained for a value of λ equal to π. This is the case of the black solid line of Fig. 11. In this configuration, the spatial period is equal to two times the size of an elementary cell. Tubes are oscillating in opposition provoking the higher added mass leading to the higher water frequency drop. The same configuration may be obtained with a spatial period smaller than the size of an elementary cell as shown by the grey solid line of Fig. 11. This configuration corresponds to the case where the parameter λ assumes the value 3π. From Fig. 9, we see that, when the parameter λ is equal to 3π, the water frequency presents a minimum of its values. This minimum value, due to periodicity condition, is the same of λ = π. More solutions of Eq. 28 are possible for a donate condition on s(x 1 ), but the frequency obtained, due to the periodicity, is the same.

From Fig. 9, as the parameter λ increases over π, water frequency increases, until it reaches a maximum value when λ = 2π. This is the case of black solid line of Fig. 12. In fact, the spatial period is equal to the size of an elementary cell thus each tubes present the same acceleration value and they are moving as an unique body. This condition is comparable to a constant value of tube displacement field, where no spatial variations are taken into account. Indeed, the frequency obtained is equal to the frequency of no spatial variations (grey dashed line of Fig. 9).

Moreover, in this latter case it may be demonstrated that the

s(x c ) = 1 s(x c ) = -1 s(x c ) = 1 ε s(x 1 ) s(x 1 )
x 1

FIGURE 11. UNIQUENESS OF THE SOLUTION s(x 1 ). TWO DIF-FERENT SOLUTION GIVE THE SAME DYNAMIC EFFECTS ON THE TUBE BUNDLE FREQUENCY

problem describing the pressure field is the elementary problem χ 1 (y) only. In fact, for a sinusoidal movement whit a spatial period of ε/2π (see Fig. 12), we can introduce a new elementary problem ζ 1 , such that:

ζ 1 = χ 1 (y) + ∞ ∑ n (-1) n (2π) 2n χ 2n+1 (36)
with its proper boundary conditions:

ζ 1 (-0.5, y 2 ) = ζ 1 (0.5, y 2 ) at ∂Y ∂ ζ 1 ∂ n y = n i at γ (37)
The elementary problem defined by the function ζ 1 corresponds precisely to the χ 1 elementary problem. This results confirms what found with the previous analysis; in fact, as the bundle oscillates as a whole (for each tube the acceleration is the same) the pressure field correspond exactly to the elementary pressure field obtained by the function χ 1 only.

In conclusion, we can deduce some fundamental results from the modal analysis conducted. Firstly, the spatial variation of the tube displacement field leads to extra contribution on the hydrodynamic force, affecting directly the tube bundle frequency. The extra contribution gives rise to a decrease in tube bundle frequency corresponding to an extra added weight to the structure.

The contribution to the fluid force from spatial variations of s(x 1 ) presents a periodic behaviour. The maximum frequency

s(x c ) = 1 s(x c ) = 1 s(x c ) = 1 ε s(x 1 ) s(x 1 )
x 1

FIGURE 12. TUBE BUNDLE OVERALL MOVEMENT: A TUBE MOVEMENT s(x 1 ) DESCRIBED BY CONSTANT VALUE OR BY SPATIAL VARIATIONS GIVES RISE TO THE SAME EFFECTS IN TERM OF FLUID REACTION FORCE AND FREQUENCY

drop is obtained when the spatial period of s(x 1 ) is equal to two times the size of an elementary cell corresponding to tubes opposition movement. Instead, the minimum frequency drop is obtained when the bundle oscillates as a whole. However, due to periodicity of the bundle's frequency, the maximum and minimum values are obtained also when the displacement field spatial period is lower than the cell size. In this case, the water frequency assumes values already assumed in λ ∈ [0, π].

Therefore, for an infinity solutions of s(x 1 ) for a given configuration s c , we can find one value of water frequency in the interval [0 : π]: i.e. the dynamic behaviour of the bundle is not influenced by the infinity of solutions for the continuous function s(x 1 ).

CONCLUSION AND PERSPECTIVES

In the present paper, the authors analysed, thanks to an homogenisation method, a fluid-structure interaction problem. The simple geometrical configuration allows to investigate and better understand the effects of spatial variations of tube bundle displacement field on its dynamics.

The authors developed the displacement and pressure field, unknown of the FSI problem, according to two different scales: long and short scale. These latter describe different kinds of variations of the main unknowns due to the particular geometrical configuration of the tube bundle. Thanks to the study at the local scale of an elementary unitary cell, the authors are able to introduce a series of problems, by means of which they describe fluid pressure and tube bundle displacement at long scale.

An equation characterising the homogenised dynamic behaviour of a tube bundle immersed in a stagnant fluid, where there is no global flow across the bundle, is obtained. The au-thors point out the appearance in the expression of the fluid force of a term proportional to solid acceleration corresponding to the inertial effects. In addition, the expression obtained for the fluid reaction force presents a series of terms directly proportional to spatial variations of long scale tube bundle acceleration.

A modal analysis of the homogenised dynamic equation is conducted, where the lowering of the water frequency due to spatial variations is illustrate. Furthermore, thanks to the modal analysis, we highlighted the uniqueness of the solution for the dynamic equation in terms of frequency.

The analysis conduct allows the authors to assess the problem of fluid structure interaction via an homogenisation method and to identify the main dynamic characteristic described by an homogenised method. Moreover, the analysis conducted leads to a solid base for the further step in whole reactor core homogenisation: the application to Navier-Stokes equations in order to describe with an higher detail the fluid flow and introduce the dissipative effects naturally described with this later model.

ACKNOWLEDGMENT

The present work has been supported by the CEA of Saclay in the frame of the RNR-Na (Sodium Fast Cooled Reactor) research and development program. The authors wish to thank Hélène Dumontet and Arnault Monavon for the precious support in the advancement of these works.

FIGURE 1 .

 1 FIGURE 1. TUBE BUNDLE ROW: GEOMETRICAL CONFIGU-RATION AND MAIN PARAMETERS

χ 3 FIGURE 6 .

 36 FIGURE 6. ISOVALUES OF THE FUNCTIONS χ i FOR THE FIRST 3 ELEMENTARY PROBLEMS

(a) Fluid Accelerations due to χ 1 (b) Fluid Accelerations due to χ 2 (c) Fluid Accelerations due to χ 3 FIGURE 7 .

 1237 FIGURE 7. FLUID ACCELERATIONS FOR THE FIRST 3 ELE-MENTARY PROBLEMS. ONLY THE χ 1 PROBLEM PRESENTS A NON ZERO MEAN ACCELERATIONS FIELD

FIGURE 9 .

 9 FIGURE 9. WATER TUBE BUNDLE FREQUENCY VS λ FOR m 0 = 10.kg m -2 AND f air = 10.Hz IN THE APPROXIMATION FOR THE COEFFICIENTS a i OF Eq. 21

FIGURE 10 .

 10 FIGURE 10. EXTRA ADDED MASS VS λ IN THE APPROXIMA-TION OF THE COEFFICIENTS a i OF Eq. 21

  )

		y 2
	Y *		∂Y
	1	n n n	| γ|	y 1
			γ
		1
	FIGURE 5. ELEMENTARY UNITARY ELEMENT Y SCALED BY
	A FACTOR ε. THE ELEMENTARY CELL REPRESENTS A SINGLE
	TUBE SURROUNDED BY FLUID. IT IS THE BASIC REPETITIVE
	ELEMENT CONSTITUTING THE TUBE BUNDLE. THE FIGURE
	SHOWS THE MAIN GEOMETRICAL PARAMETERS OF THE UNI-
	TARY CELL


Noël Burais

François Buret

Nicolas Faure
email: nicolas.faure@ec-lyon.fr

Laurent Nicolas

Ronan Perrussel
email: ronan.perrussel@ec-lyon.fr

Comparison between finite difference and finite element methods: application to biological cell exposure

Abstract -The finite element and finite difference methods are compared for the modelling of a biological cell exposed to an electric pulse.

I.INTRODUCTION

In this work, the performances and the ease of implementation of the Finite Difference (FD) and the Finite Element (FE) methods are compared for the modelling of biological cell exposed to electric fields. The electroquasistatic formulation (1) is considered in both timeharmonic and time-transient versions. The model of the cell is the simplest: a conductive cytoplasm surrounded by an insulating membrane [START_REF] Kotnik | Theorical evaluation of the voltage inducement on internal membranes of biological cells exposed to electric fields[END_REF]. This model is 3D axi-symmetric. The comparison is performed in terms of the computed transmembrane potential (TMP), which is defined as the difference between the outside and inside faces of the membrane.

( )

div grad 0 σ ε V = t   ∂   - +     ∂     (1)

II.TIME-HARMONIC COMPARISON

In FE, an approximate transmission condition is implemented in order to avoid meshing the thin membrane (5 nm of thickness) [START_REF] Poignard | Asymptotic for steady state voltage potentials in a bidimensionnal highly contrated medium with thin layer[END_REF]. In FD, the mesh is non-conforming to the membrane; a uniform grid is used with a five-point stencil whose coefficients are evaluated by harmonic averaging [START_REF] Leveque | The immersed interface method for elliptic equations with discontinuous coefficients and singular sources[END_REF]. Fig. 2 shows the electric potential along one symmetry axis (from the center of the cell to the upper electrode). The potential "jump" characterizes the TMP. The relative difference between results obtained with both FE and DF methods is less than 1%. Compared to analytical values [START_REF] Schwan | Electrical properties of tissue and cell suspension[END_REF], the error is below 4%.

III.TIME-TRANSIENT COMPARISON

The time discretization is based on the Crank Nicholson scheme in the FE case and on the implicit Euler scheme in the FD case.

The top of Fig. 2 depicts the 300 nanosecond pulse applied to the cell. The bottom shows the TMP computed with both FD and FE methods. In FD, results are obtained with different grid steps (from 200 nanometers to 50 nm). When considering the finest FD grid, both methods are in good agreement: the maximal difference is below 1%. From Table I, it is shown that a fine mesh is required in our implementation of the FD method.

IV.CONCLUSION

If electro-quasistatic formulation is sufficient to model the phenomena, the FE method gives the best result in terms of the ratio between accuracy and numerical costs (Table I). In this application, the solution of the full Maxwell system is not relevant with the FD method. Because of the mesh size, the Courant-Friedrichs-Levy stability condition imposes a time step which is too small compared to the duration of the pulse.

Fig. 1

 1 Fig. 1 Potential along one symmetry axis of the cell at 5 MHz computed with both methods.

Fig. 2

 2 Fig. 2 Time-transient response of the cell (Max. source Efield: 2.10 5 V/m).


Cyril Goudet
email: cyril.goudet@igf.cnrs.fr

Xavier Rovira
email: xavier.rovira@uvic.cat

Amadeu Llebaria
email: amadeu.llebaria@iqac.csic.es

Shedding light on metabotropic glutamate receptors using optogenetics and photopharmacology

Keywords: Receptor, Neurotransmitter, Glutamate, Optical control, Optogenetic, Optopharmacology Highlights

Metabotropic glutamate receptors (mGluRs) are a family of G protein-coupled receptors activated by glutamate, the main excitatory neurotransmitter of the mammalian central nervous system. These receptors are considered as potential therapeutic targets in many neurological diseases but a better understanding of their complex molecular dynamics and of their role in the normal and pathological functioning of the brain is still required.

Manipulating mGluRs with high spatial and temporal precision holds great promise for deciphering their physiological and pathological functions. This article reviews several recently developed optogenetic and photopharmacological solutions for the optical control of mGluRs and their applications, from the study of the molecular dynamics of receptor activation to the study of their roles in vivo.

Introduction

Glutamate is the major excitatory neurotransmitter of the mammalian central nervous system (CNS). It activates two categories of receptors: ionotropic and metabotropic glutamate receptors (iGluRs and mGluRs). While iGluRs are ligand-gated ion channels that mediate fast responses to glutamate, mGluRs are class C G-protein coupled receptors (GPCRs) responsible for slower neuromodulatory actions of glutamate.

Eight genes encoding mGluRs have been identified in mammalian genomes. In addition, several mGluR splice variants have been described, most of them altering the intracellular Cterminal tails [START_REF] Pin | Evolution, structure, and activation mechanism of family 3/C G-protein-coupled receptors[END_REF]. Based on sequence homology, signal transduction and pharmacology, mGluRs are subdivided into 3 groups. Group-I mGluRs (mGlu1, 5) are mainly coupled to Gq and, as such, activate PLC and generate intracellular calcium signals. Both group-II (mGlu2, [START_REF] Lindsley | Practical Strategies and Concepts in GPCR Allosteric Modulator Discovery: Recent Advances with Metabotropic Glutamate Receptors[END_REF] and group-III (mGlu4, [START_REF] Pin | Organization and functions of mGlu and GABAB receptor complexes[END_REF][START_REF] Kubo | Structural basis for a Ca2+-sensing function of the metabotropic glutamate receptors[END_REF][START_REF] Jiang | Extracellular calcium modulates actions of orthosteric and allosteric ligands on metabotropic glutamate receptor 1alpha[END_REF] mGluRs are mainly coupled to Gi/o and can therefore inhibit adenylate cyclase and regulate the activity of various ion channels [START_REF] Flor | Orthosteric versus allosteric GPCR activation: the great challenge of group-III mGluRs[END_REF][START_REF] Lindsley | Practical Strategies and Concepts in GPCR Allosteric Modulator Discovery: Recent Advances with Metabotropic Glutamate Receptors[END_REF].

The different mGluRs are widely distributed throughout the peripheral and central nervous system. In neurons, group I mGluRs are mostly localized at the post-synapse where they upregulate neuronal excitability, whereas group II and III mGluRs are mostly presynaptic receptors that reduce synaptic transmission. mGluRs are not only present in glutamatergic synapses but they have also been observed in other types of synapses, such as dopaminergic and GABAergic synapses. mGluRs have also been detected in astrocytes, oligodendrocytes and microglia (see [START_REF] Nicoletti | Metabotropic glutamate receptors: from the workbench to the bedside[END_REF] for review). In addition, mGluRs are distributed outside the CNS, for example in the heart, adrenal glands or lymphocytes (see [START_REF] Flor | Exciting times beyond the brain: metabotropic glutamate receptors in peripheral and non-neural tissues[END_REF] for review). mGluRs are complex allosteric machines composed of a seven transmembrane domain (7TMD), common to all GPCRs, connected to a large bilobate extracellular venus-flytrap domain (VFT) through a short and rigid cysteine-rich domain (CRD). Moreover, mGluRs are constitutive dimers cross-linked by a disulfide bridge at the level of the VFT and this dimerization is a prerequisite for mGluRs function [START_REF] Pin | Organization and functions of mGlu and GABAB receptor complexes[END_REF]. Glutamate binds in the cleft between the two lobes of the VFT. The glutamate-binding pocket is well conserved among the different mGluRs, making the design of subtype selective orthosteric ligands challenging, but several successful molecules have been described [START_REF] Flor | Orthosteric versus allosteric GPCR activation: the great challenge of group-III mGluRs[END_REF]. Positive, negative and neutral allosteric ligands that bind in the 7TMD have also been identified. Since this binding pocket is more variable than that of glutamate among mGluR subtypes, it has allowed the discovery of subtype selective allosteric ligands of mGluRs [START_REF] Lindsley | Practical Strategies and Concepts in GPCR Allosteric Modulator Discovery: Recent Advances with Metabotropic Glutamate Receptors[END_REF]. Allosteric modulators targeting the VFT have also been described, such as cations and anions [START_REF] Kubo | Structural basis for a Ca2+-sensing function of the metabotropic glutamate receptors[END_REF][START_REF] Jiang | Extracellular calcium modulates actions of orthosteric and allosteric ligands on metabotropic glutamate receptor 1alpha[END_REF][START_REF] Kuang | Ion dependence of ligand binding to metabotropic glutamate receptors[END_REF][START_REF] Vafabakhsh | Conformational dynamics of a class C G-proteincoupled receptor[END_REF][START_REF] Diraddo | A real-time method for measuring cAMP production modulated by Galphai/o-coupled metabotropic glutamate receptors[END_REF][START_REF] Tora | Allosteric modulation of metabotropic glutamate receptors by chloride ions[END_REF]. Moreover, mGluRs that were long believed to exclusively form homodimers can also assemble into heterodimers, multiplying the receptor combinations and their subsequent signaling and modulatory mechanisms [START_REF] Lindsley | Practical Strategies and Concepts in GPCR Allosteric Modulator Discovery: Recent Advances with Metabotropic Glutamate Receptors[END_REF].

Due to their important role in the regulation of synaptic transmission and neuronal excitability, mGluRs are considered potential therapeutic targets for treating neurological disorders, including anxiety, depression, schizophrenia, chronic pain and Parkinson disease (see [START_REF] Nicoletti | Metabotropic glutamate receptors: from the workbench to the bedside[END_REF][START_REF] Niswender | Metabotropic Glutamate Receptors: Physiology, Pharmacology, and Disease[END_REF][START_REF] Acher | Therapeutic potential of group III metabotropic glutamate receptor ligands in pain[END_REF] for reviews). However, despite the efforts of pharmaceutical companies, there are currently no marketed drugs targeting mGluRs. Drug development and clinical research would benefit from better knowledge of the complex molecular machinery of these receptors and their role in the normal and pathological functioning of the brain, including the defined anatomical location of receptors functionally involved in specific physiological responses.

Therefore, the precise control over mGluR activity is crucial for a better understanding of their dynamics and function either in cells or in living organisms. Over the past few years, different optical methods have been developed to manipulate mGluRs with accurate spatial and temporal resolution via the use of light and photoactive molecules as external triggers for receptor activation or inactivation. A first approach, optogenetic pharmacology, is based on covalently attached light-operated ligands while the second one, photopharmacology (also known as optopharmacology), is based on freely diffusible light-operated ligands. The present article will review the recent development of these two optical methods and their in vitro and in vivo applications to study mGluRs.

Optogenetic pharmacology

Optogenetic pharmacology is based on tethered photocontrolable ligands that bind covalently to genetically modified proteins [START_REF] Kramer | Optogenetic pharmacology for control of native neuronal signaling proteins[END_REF][START_REF] Leippe | Specificity and Speed: Tethered Photopharmacology[END_REF]. Once attached, these ligands enable rapid, reversible and reproducible photoactivation or photoinactivation in specific cells expressing their target (Fig. 1). Before being applied to mGluRs, this approach has been used to photocontrol various voltage-gated and ligand-gated ion channels [START_REF] Kramer | Optogenetic pharmacology for control of native neuronal signaling proteins[END_REF].

Light-controlled mGluRs have been designed thanks to the joint efforts of Isacoff's and Trauner's labs. They developed a first generation of "LimGluRs" based on photoswitchable tethered ligands (PTLs) that contain a maleimide for cysteine attachment at one end, and connected via a photoisomerizable azobenzene linker to a glutamate derivative at the other end. These ligands, called MAGs, covalently bind to mutated mGluRs where a geometrically appropriate cysteine-attachment point has been incorporated. Azobenzene possesses a stable trans isomer and a metastable cis isomer [START_REF] Beharry | Azobenzene photoswitches for biomolecules[END_REF]. Trans-to-cis isomerization of azobenzene can be induced by UV light illumination, while cis-to-trans isomerization can be induced by visible light or heat. Under illumination at appropriate wavelengths, azobenzene photoisomerization induces a change in the overall conformation of the MAG, enabling (or not) the interaction of the glutamate moiety with its binding pocket. Using this approach, several LimGluRs have been generated: light-agonized LimGluR2, LimGluR3 and LimGluR6 and a light-antagonized LimGluR2 [START_REF] Levitz | Optical control of metabotropic glutamate receptors[END_REF]. LimGluR2 has been expressed in heterologous cells, in rodent brain slices and in vivo in zebrafish where it controlled the function associated with mGlu2 receptors by light [START_REF] Levitz | Optical control of metabotropic glutamate receptors[END_REF]. To enable a more precise spatial control of LimGluRs, some MAGs have been modified to enable a more accurate spatial activation by 2-Photon techniques [START_REF] Carroll | Two-photon brightness of azobenzene photoswitches designed for glutamate receptor optogenetics[END_REF].

More recently, a second generation of light-controlled mGluRs has been developed using the SNAP-tag or CLIP-tag technology and photoswitchable orthogonal remotely tethered ligands (PORTL). In this method, the receptors are genetically modified to incorporate a SNAP-tag or a CLIP-tag at their N terminus. SNAP-tag [START_REF] Keppler | A general method for the covalent labeling of fusion proteins with small molecules in vivo[END_REF] is a monomeric protein of 182 residues (20 kDa), derived from the enzyme AGT, that recognizes and irreversibly binds to O 6 -benzylguanine (BG) derivatives. A SNAP derivative, called CLIP-tag, has been developed to specifically react with O 4 -benzylcytosine (BC) derivatives [START_REF] Gautier | An engineered protein tag for multiprotein labeling in living cells[END_REF]. Since SNAP-tag and CLIP-tag possess orthogonal substrate specificities, SNAP-and CLIP-tagged proteins can be labeled simultaneously and specifically with different molecular probes in living cells. This has proven to be a highly useful and potent approach in life sciences for multiple applications ranging from protein detection in microscopy to the single molecule study of protein-protein interaction (see [START_REF] Reymond | Visualizing biochemical activities in living cells through chemistry[END_REF] for review). In the field of mGluRs, this technology has notably been used to study homodimerization [START_REF] Maurel | Cell-surface protein-protein interaction analysis with time-resolved FRET and snap-tag technologies: application to GPCR oligomerization[END_REF], to discover specific heterodimeric mGluRs [START_REF] Doumazane | A new approach to analyze cell surface protein complexes reveals specific heterodimeric metabotropic glutamate receptors[END_REF], to develop mGluRs biosensors [START_REF] Doumazane | Illuminating the activation mechanisms and allosteric properties of metabotropic glutamate receptors[END_REF][START_REF] Scholler | HTS-compatible FRETbased conformational sensors clarify membrane receptor activation[END_REF] and to probe receptor activation dynamics at a single molecule level [START_REF] Olofsson | Fine tuning of sub-millisecond conformational dynamics controls metabotropic glutamate receptors agonist efficacy[END_REF]. For optogenetic pharmacology, PORTLs have been designed to covalently bind to the SNAP tag or CLIP tag incorporated into the targeted receptor. They are composed of a glutamate moiety, followed by a long flexible linker containing an azobenzene and terminated with a BG or BC that specifically react with the SNAP tag or CLIP tag domains. This strategy has first been applied to mGlu2 permitting the photoactivation of mGlu2 and the control of excitability in heterologous cells or transfected neurons [START_REF] Broichhagen | Orthogonal Optical Control of a G Protein-Coupled Receptor with a SNAP-Tethered Photochromic Ligand[END_REF]. Then, combining SNAP and CLIP tagged receptors and specific spectral variants of PORTLs, Levitz and colleagues have created a family of light-gated group II/III mGluRs [29], allowing multiplexed orthogonal optical control within receptor homo or heterodimers.

Photopharmacology

Photopharmacology (also called optopharmacology) is based on freely diffusible, druglike, light-operated ligands [30,31]. It allows the control of the function of the ligand on its target by light. Contrary to optogenetic pharmacology, no genetic modification of the targeted receptor and its exogenous expression are required, enabling the photocontrol of endogenous receptors. Two types of photo-regulated drugs have been developed for photopharmacology: photoactivable ligands and photoswitchable ligands (Fig. 1).

Photoactivable ligands

Photoactivable ligands are inactive 'caged' compounds that can be photoactivated following brief pulses of light. They are constituted of a ligand linked to a photo-labile protecting group that will be removed following illumination, enabling the uncaged ligand to bind to its receptor (Fig. 1). Therefore, these photoactivable ligands enable the precise control of the onset of drug activity at a specific location. This method was developed during the seventies and first applied to nucleotides [32,33].

Subsequently, the caged strategy was adapted to different neurotransmitters, including glutamate and GABA [34,[START_REF] Wieboldt | Synthesis and photochemistry of photolabile derivatives of gamma-aminobutyric acid for chemical kinetic investigations of the gamma-aminobutyric acid receptor in the millisecond time region[END_REF]. Photoactivation of caged glutamate greatly improved the spatial and temporal resolution of synaptic connectivity mapping in neuronal networks [START_REF] Callaway | Photostimulation using caged glutamate reveals functional circuitry in living brain slices[END_REF]. It has also been used to study the function of mGluRs [START_REF] Crawford | Activation of Ca2+-dependent currents in dorsal root ganglion neurons by metabotropic glutamate receptors and cyclic ADPribose precursors[END_REF]. However, the use of caged glutamate is somehow limited due to the lack of subtype selectivity. This led to the development of iGluR and mGluR selective compounds.

Recently, the first photoactivable ligand selective for an mGluR subtype was designed [START_REF] Font | Optical control of pain in vivo with a photoactive mGlu5 receptor negative allosteric modulator[END_REF]. This caged compound called JF-NP-26 is an inactive photo-caged derivative of raseglurant (ADX-10059), a negative allosteric modulator (NAM) of mGlu5 receptors.

Illumination of JF-NP-26 with violet light induces a photochemical reaction prompting the release of the active ligand, raseglurant, and the coumarin DEACM, and thus the blockade of mGlu5 receptors activity was observed in vitro in cell-based assays or in vivo in animal models of pain [START_REF] Font | Optical control of pain in vivo with a photoactive mGlu5 receptor negative allosteric modulator[END_REF].

Photoswitchable ligands

Photoswitchable mGluRs ligands are azobenzene-containing molecules that can rapidly and reversibly photo-isomerize under specific light, passing from an 'active' to an 'inactive' configuration, or vice versa (Fig. 1). Indeed, azobenzene N=N double bond geometry changes during photo-isomerization [START_REF] Beharry | Azobenzene photoswitches for biomolecules[END_REF], inducing a change in the configuration of the ligand which drastically affects the affinity for its target. In the dark or under white light, the azobenzene moiety is in a trans configuration while it reaches the cis configuration upon illumination at the adequate wavelength. Relaxation to the generally thermodynamically more stable transisomer can be induced by irradiation or by spontaneous thermal relaxation.

In the last few years, photoswitchable ligands have been developed for various ion channels and receptors, such as GABAA [START_REF] Stein | Azo-propofols: photochromic potentiators of GABA(A) receptors[END_REF] and NMDA receptors [START_REF] Laprell | Optical control of NMDA receptors with a diffusible photoswitch[END_REF], KATP [START_REF] Broichhagen | Optical control of insulin release using a photoswitchable sulfonylurea[END_REF], TRPA1 [START_REF] Kokel | Photochemical activation of TRPA1 channels in neurons and animals[END_REF] or GIRK channels [START_REF] Trads | Optical control of GIRK channels using visible light[END_REF], and GPCRs such as adenosine [START_REF] Bahamonde | Photomodulation of G proteincoupled adenosine receptors by a novel light-switchable ligand[END_REF], µ-opioid [START_REF] Schonberger | A photochromic agonist for mu-opioid receptors[END_REF] or cannabinoid receptors [START_REF] Westphal | Synthesis of Photoswitchable Delta9-Tetrahydrocannabinol Derivatives Enables Optical Control of Cannabinoid Receptor 1 Signaling[END_REF] among others.

Several freely diffusible photoswitchable ligands targeting mGluRs have been developed.

The first of them was Alloswitch-1, an mGlu5 receptor NAM [47], which was the first example of an allosteric photoswitchable ligand targeting a GPCR. Alloswitch-1 was developed based in the structure of VU0415374, an allosteric ligand of mGlu4 receptors [48] in which an azobenzene was formally inserted replacing an amido group. Illumination by green or violet light stabilizes either the trans-or the cis-configuration of the ligand which correspondsto high and low pharmacological activity on mGlu5 receptors, respectively, which has been observed both in vitro or in vivo. Similarly, Optogluram is a photoswitchable mGlu4 positive allosteric modulator (PAM) [49] which allows for selective, reversible and repeated optical manipulation of mGlu4 receptors activity with light both in vitro and in vivo. This ligand is also based in VU0415374 structure but with the azobenzene formally replacing a different amido group to that used in Alloswitch-1. The enhancement of mGlu4 receptors activity observed by optogluram in the trans-form was reduced upon isomerization to the cisconfiguration after irradiation with violet light. The potentiation of mGlu4 receptors is recovered after green light Optogluram photoisomerization from cis to trans.

As the coefficient of light absorption by tissue depends on wavelengths and ultraviolet light which could potentially be damaging to irradiated tissues, it is interesting to design ligands that can photo-isomerize at red-shifted wavelengths. Recently, OptoGluNAM4.1, a blue light-sensitive mGlu4 receptors photoswitchable NAM has been described [50].

Interestingly, after irradiation with blue light, this compound relaxes very fast from its cis to trans configuration, recovering the initial configuration by simply switching-off the illumination thus allowing the use of a single light source. This compound is active both in vitro and in vivo, being able to block the action of mGlu4 receptor agonist in a light-dependent manner.

More recently, a series of photoswitchable mGlu5 receptor NAMs based on the phenylazopyridine scaffold present in Alloswitch-1 has been generated [51]. Most of the trans-isomers of these ligands are active both in vitro in cell-based assays and in vivo in zebrafish behavioral experiments. Interestingly, depending on their structures, the ligands exhibit different photo-isomerization properties with, notably, optimal illumination wavelengths varying from 360 to 500 nm.

Applications In vitro applications

In vitro, optogenetic pharmacology and photopharmacology have great potential in allow us to understand how receptors function at the molecular level, how they interact with each other and how they modulate the activity of a given cell. They can be of particular interest to study all of the kinetic aspects of receptor activation and function, which are impossible to be address by other means [52].

Firstly, light-controlled mGluRs that can be rapidly and repetitively switched on and off by light can be useful optogenetic tools to understand the activation mechanisms of mGluRs and their associated signaling. Using LimGluRs, Levitz and colleagues demonstrated the ability of LimGluR2 to optically modulate native downstream targets of mGlu2 receptors both in heterologous cells and in cultured hippocampal neurons [53]. This study also provides proof of concept that this strategy can be applied to study the synaptic activity of mGluRs in neural circuits. Indeed, the authors demonstrated that light-activation of LimGluR2 reduces excitability and inhibits neurotransmitter release in mouse brain slices [53]. Furthermore, the design of specific PTL's compatible with 2-photon microscopy will enable high-precision activation of LimGluRs, making them valuable tools to map the modulation of synaptic activity by these receptors in neuronal networks [START_REF] Carroll | Two-photon brightness of azobenzene photoswitches designed for glutamate receptor optogenetics[END_REF]. mGluRs are complex allosteric machines composed of two subunits each containing different modules which interact with each other during the activation process. Large conformational rearrangements occur upon activation to transmit the information from the extracellular domains binding glutamate to the transmembrane domains coupling G-proteins [START_REF] Pin | Organization and functions of mGlu and GABAB receptor complexes[END_REF]. Photoswitch has been used to probe the receptor dynamics during activation, demonstrating the cooperativity between the subunits of an mGlu2 homodimer or an mGlu2/3 heterodimer [53]. The system has been improved to enable dual optical control of multiple receptors using bioorthogonal tethering of PORTL either to SNAP or CLIP tags [START_REF] Broichhagen | Orthogonal Optical Control of a G Protein-Coupled Receptor with a SNAP-Tethered Photochromic Ligand[END_REF]29]. Levitz and colleagues recently achieved the independent photoswitch of a single or the two subunits composing an mGluR homodimer or heterodimers, demonstrating that these optogenetic tools can provide subunit, spatial and temporal precisions to the study of mGluRs dynamics and function.

Photoswitchable and photoactivable ligands enable the optical control of endogenous receptors in their native environment. Photolytic release of raseglurant from its cagedderivative JF-NP-26 effectively induces the block of the activity of heterologously expressed mGlu5 receptors in HEK293 cells and native ones in striatal primary neurons [START_REF] Font | Optical control of pain in vivo with a photoactive mGlu5 receptor negative allosteric modulator[END_REF].

Photoswitchable allosteric modulators of mGlu5 and mGlu4 receptors reversibly photoregulate the activity of their target in transfected HEK293 cells or in primary culture of neurons or astrocytes [47,[49][50][51]. Contrary to optogenetic pharmacology, no genetic modification of the targeted receptor and its exogenous expression are required, enabling the photocontrol of endogenous receptors. Photoactivable ligands have been used for precise stimulation of receptors at defined locations in brain slices, allowing mapping of the distribution and function of neurotransmitter receptors in neuronal networks. Such an application could be feasible in principle using the photoactivable mGlu5 receptors NAM JF-NP-26 and the photoswitchable allosteric modulators of mGlu4 and mGlu5 receptors.

In vivo applications

Combining optogenetic pharmacology and photopharmacology with the high spatial and temporal resolution of optical manipulation offers the possibility to investigate the physiopathological function of receptors at precise locations in living animals. However, in vivo optogenetic pharmacology and photopharmacology face several challenges. The first one concerns the genetic manipulation required by optogenetics to express the modified receptor to allow anchoring of the remotely tethered photoswitchable ligands in the targeted organisms. However, this point can be adressed by the use of viral vectors [54]. The second challenge resides in local ligh delivery in living organisms. Indeed, although relatively easy with small animal models, such as tadpoles or zebrafish, or in superficial tissues or retina, light delivery is more challenging in deep tissues of bigger organisms, such as rodents.

However, thanks to the advent of optogenetics, the development of technical solutions for optical control is in constant progress and overcomes this obstacle.

The small size and transparency of Zebrafish and Xenopus tropicalis tadpoles allow easy and non-invasive optical manipulations in the presence of a photoresponsive molecule.

Alloswitch-1, a photoswitchable NAM of mGlu5 receptors, allows photocontrol of the swimming behavior of the Xenopus tadpole [47]. Zebrafish, in particular, is a genetic model of choice since many years, enabling specific targeting of optogenetic tools in populations of cells and their subsequent monitoring by light [55]. The light-controlled mGluRs can be expressed in zebrafish. Photo-activation of LimGluR2 reversibly and repeatedly modulates the escape behavior in zebrafish [START_REF] Levitz | Optical control of metabotropic glutamate receptors[END_REF]. Concerning photopharmacological applications, zebrafish also presents the interest of possessing mGluRs with high sequence homology with the ones of mammals [START_REF] Haug | Phylogeny and expression divergence of metabotropic glutamate receptor genes in the brain of zebrafish (Danio rerio)[END_REF], reducing the chances of having inactive ligands due to species selectivity. Notably, the allosteric binding pocket of mGlu4 receptor is well conserved between rodents and zebrafish [50]. Photoswitchable allosteric modulators of mGlu4 and mGlu5 receptors enable the photocontrol of the motility behavior of zebrafish larvae [50,51].

One of the striking attributes of the zebrafish model is the robustness of the translation of the photoswitchable properties of the ligands determined in cell-based assays with the ones observed in vivo, as illustrated in the study by Gomez-Santacana and colleagues [51]. In this structure-activity relationship study, the photopharmacological properties of a series of photoswitchable NAMs of mGlu5 receptors have been determined both in vitro in heterologous cells and in vivo in zebrafish. The optimal wavelengths of illumination range from 360 to 500 nm. The "photoisomerization scores" (PIS) which rank the effectiveness of the azobenzene photoisomerization (from poor to high) and the "photoinduced potency shifts" (PPS) which quantify the effectiveness of the compound photoswitching (corresponding to the ratio of the IC50's under illumination and in the dark) of most of the ligands are comparable both in vitro and in vivo [51].

Using photoswitchable [49] or photoactivable mGluR drugs [START_REF] Font | Optical control of pain in vivo with a photoactive mGlu5 receptor negative allosteric modulator[END_REF], two pioneering studies have recently demonstrated the feasibility of controlling endogenous neuromodulatory mechanisms in the brain of freely moving mice. Zussy and colleagues have injected optogluram, a photoswitchable PAM of mGlu4 receptors, and photocontrolled the activity of native mGlu4 receptors in a specific area of the brain of freely behaving mice. This study revealed that it is possible to regulate persistent pain-related symptoms in a temporally and spatially restricted manner taking control of amygdala mGlu4 receptors by light with a photoswitchable PAM molecule and controlling the effects at a peripheral site. To our knowledge, this was the first study that established that photopharmacology with a small diffusible drug-like photoswitchable ligand can be used in vivo to regulate the behavior in a disease model. Another study has demonstrated the possibility to use photoactivatable mGluRs ligands in vivo [START_REF] Font | Optical control of pain in vivo with a photoactive mGlu5 receptor negative allosteric modulator[END_REF]. Font and colleagues use the caged mGlu5 receptors NAM JF-NP-26 to control inflammatory or neuropathic pain in mice by uncaging this compound either at the periphery, in the paw, or centrally in the thalamus. Strikingly, this study shows that photoreleasing the active drug at specific locations in the brain or in the periphery results in analgesia, whereas targeting other mGlu5 receptors-containing tissues is devoid of any effect on pain sensitivity, thus linking anatomically restricted mGlu5 receptor antagonism to analgesic effects in vivo.

Finally, yet importantly, there have been two recent in vivo applications of optogenetics in the field of mGluRs. First, using a strategy known as opto-XR [START_REF] Airan | Temporally precise in vivo control of intracellular signalling[END_REF], Van Wyk and colleagues designed a chimeric receptor consisting of the intracellular domains of the ON-bipolar cellspecific mGlu6 receptor and the light-sensing domains of melanopsin that they called Opto-mGluR6 [START_REF] Van Wyk | intracellular domains of the ON-bipolar cell-specific mGlu6 receptor and the light-sensing domains of melanopsin called Opto-mGluR6. These results raise hope for future therapeutic applications of mGluRbased optogenetic to restore vision[END_REF]. This chimeric receptor responds to daylight by activating the GIRK signaling pathway in cells. Transgenic expression of opto-mGluR6 specifically in ON-bipolar cells by a viral approach and the GRM6/sv40 enhancer promoter restores vision in the blind rd1 mouse [START_REF] Van Wyk | intracellular domains of the ON-bipolar cell-specific mGlu6 receptor and the light-sensing domains of melanopsin called Opto-mGluR6. These results raise hope for future therapeutic applications of mGluRbased optogenetic to restore vision[END_REF]. More recently, Berry and colleagues restored vision of blind rd1 mice by transgenically expressing the light-activated SNAG-mGluR2 [START_REF] Broichhagen | Orthogonal Optical Control of a G Protein-Coupled Receptor with a SNAP-Tethered Photochromic Ligand[END_REF] in retinal ganglion cells using an AAV vector containing the SNAP-mGlu2 receptors under control of the synapsin promoter [START_REF] Berry | Restoration of patterned vision with an engineered photoactivatable G protein-coupled receptor[END_REF].

These two strategies raise hope for future therapeutic applications to restore vision.

Conclusion

As compared to conventional pharmacological approaches, the use of light to trigger receptor responses is a potent and unique tool to study their mechanisms of action, and their physiological and biological roles, at cellular, tissue and living animal experimental levels. In particular, optogenetic pharmacology and photopharmacology can greatly improve spatial and temporal resolution of the control over their target protein. The fundamental research tools presented in this review are of particular interest to study the molecular and kinetic aspects of mGlu receptor activation. They also hold great promise for exploring the role and therapeutic potential of mGluRs and their associated signaling in physiology and disease. In the near future, the number of available photocontrolled tools will likely increase, expanding the range of the current applications and helping to understand the biological particularities of the mGluRs extraordinary molecular machines glutamate moiety, followed by a long flexible linker containing an azobenzene and terminated with a BG that specifically react with the SNAP tag. Photoactivation of mGlu2 allows the control of excitability in heterologous cells or transfected neurons. This strategy opens the way to optical control of multiple receptors in the same preparation using different protein tags, such as SNAP or CLIP tags, and PORTL spectral variants. Structural model of a dimeric mGluR composed of Venus Flytrap (VFT), Cysteine Rich (CRD) and a seven transmembrane domain (7TMD) domain. A first subunit is in dark grey while the second one is in light grey. The orthosteric binding site is located between the two lobes of the VFT (yellow shadow) while most allosteric modulators bind in a pocket located the 7TMD (blue shadow). For clarity, the orthosteric and allosteric pockets are only displayed in a single subunit. Optical control of mGluRs can be achevied via two main strategies. The optogenetic pharmacology strategy is based on photoswitchable tethered ligands binding to a genetically modified mGluR, harboring either a geometrically appropriate cysteineattachment point which will covalently bind PTL (1) or a SNAP or CLIP tag which will anchor PORTL [START_REF] Flor | Orthosteric versus allosteric GPCR activation: the great challenge of group-III mGluRs[END_REF]. PTL contain a glutamate derivative at one end (orange circle) connected via a photoisomerizable azobenzene linker (green or purple circle) to a maleimide for cysteine attachment at the other end. PORTL are composed of a glutamate moiety, followed by a long flexible linker containing an azobenzene and terminated with a BG that will react specifically with the SNAP tag (or BC in case of a CLIP tag). In this figure, a PORTL is binding to its SNAP tag fused to the N terminus of the light grey subunit. The photopharmacology strategy is based on freely diffusible photoactivable (3) or photoswitchable ligands (4) that target endogenous mGluRs. Photoactivable ligands, also called 'caged' ligands, are constituted of a ligand linked to a photo-labile protecting group that will be removed following illumination, enabling the uncaged ligand to bind to its receptor. Photoswitchable mGluRs ligands are azobenzene-containing molecules that can rapidly and reversibly photo-isomerize under illumination at specific wavelenghts, passing from an 'active' to an 'inactive' configuration, and vice versa.

  29•. Levitz J, Broichhagen J, Leippe P, Konrad D, Trauner D, Isacoff EY: Dual optical control and mechanistic insights into photoswitchable group II and III metabotropic glutamate receptors. Proc Natl Acad Sci U S A 2017, 114:E3546-E3554. This study describes the generation of a family of light-controlled group II/III mGluRs using the PORTL strategy in combination with the SNAP/CLIP tag technology. Several variants of light-controlled mGluR2, 3, 6, 7, and 8 are designed. The article provides an analysis of the mechanisms of photoswitch and discusses the advantages of the PORTL vs. PTL approaches. 30. Broichhagen J, Frank JA, Trauner D: A roadmap to success in photopharmacology. Acc Chem Res 2015, 48:1947-1960. 31. Velema WA, Szymanski W, Feringa BL: Photopharmacology: beyond proof of principle. J Am Chem Soc 2014, 136:2178-2191. 32. Engels J, Schlaeger EJ: Synthesis, Structure, and Reactivity of Adenosine Cyclic 3',5'-Phosphate Benzyl Triesters. Journal of Medicinal Chemistry 1977, 20:907-911. 33. Kaplan JH, Forbush B, 3rd, Hoffman JF: Rapid photolytic release of adenosine 5'triphosphate from a protected analogue: utilization by the Na:K pump of human red blood cell ghosts. Biochemistry 1978, 17:1929-1935. 47. Pittolo S, Gomez-Santacana X, Eckelt K, Rovira X, Dalton J, Goudet C, Pin JP, Llobet A, Giraldo J, Llebaria A, et al.: An allosteric modulator to control endogenous G protein-coupled receptors with light. Nat Chem Biol 2014, 10:813-815. 48. Engers DW, Field JR, Le U, Zhou Y, Bolinger JD, Zamorano R, Blobaum AL, Jones CK, Jadhav S, Weaver CD, et al.: Discovery, synthesis, and structure-activity relationship development of a series of N-(4-acetamido)phenylpicolinamides as positive allosteric modulators of metabotropic glutamate receptor 4 (mGlu(4)) with CNS exposure in rats. J Med Chem 2011, 54:1106-1110. 49••. Zussy C, Gomez-Santacana X, Rovira X, De Bundel D, Ferrazzo S, Bosch D, Asede D, Malhaire F, Acher F, Giraldo J, et al.: Dynamic modulation of inflammatory pain-related affective and sensory symptoms by optical control of amygdala metabotropic glutamate receptor 4. Mol Psychiatry 2016 in press. This article describes the first example of deep brain optical control of native receptors in freely behaving mice using photopharmacology. Optogluram, an mGlu4 PAM, allows rapid and reversible alleviation of chronic pain related behavioral symptoms. 50•. Rovira X, Trapero A, Pittolo S, Zussy C, Faucherre A, Jopling C, Giraldo J, Pin JP, Gorostiza P, Goudet C, et al.: OptoGluNAM4.1, a Photoswitchable Allosteric Antagonist for Real-Time Control of mGlu4 Receptor Activity. Cell Chem Biol 2016, 23:929-934. This article describes the in vitro and in vivo photopharmacological properties of OptoGluNAM4.1, a blue light sensitive and fast relaxing mGlu4 photoswitchable NAM. Interestingly, this ligand rapidly recovers its trans 'active' configuration by simply switchingoff the illumination thus allowing the use of a single light source. 51•. Gomez-Santacana X, Pittolo S, Rovira X, Lopez M, Zussy C, Dalton JA, Faucherre A, Jopling C, Pin JP, Ciruela F, et al.: Illuminating Phenylazopyridines To Photoswitch Metabotropic Glutamate Receptors: From the Flask to the Animals. ACS Cent Sci 2017, 3:81-91. This article presents a series of photoswitchable mGlu5 NAMs based on the phenylazopyridine scaffold of Alloswitch-1 with different photo-isomerization properties. Interestingly, the photoswitchable properties determined in vitro in cell-based assays are robustly translated in vivo in zebrafish behavioral experiments.52. Lohse MJ, Hofmann KP: Spatial and Temporal Aspects of Signaling by G-Protein-Coupled Receptors. Mol Pharmacol 2015, 88:572-578. 53••. Levitz J, Habrian C, Bharill S, Fu Z, Vafabakhsh R, Isacoff EY: Mechanism of Assembly and Cooperativity of Homomeric and Heteromeric Metabotropic Glutamate Receptors. Neuron 2016, 92:143-159.. This study demonstrates the ability of the light-controlled LimGluR2 to modulate downstream effectors of mGlu2 in heterologous cells and in cultured neurons and proves that this strategy can be applied to study the synaptic activity of mGluRs in neural circuits in mouse brain slices. 54. Lerner TN, Ye L, Deisseroth K: Communication in Neural Circuits: Tools, Opportunities, and Challenges. Cell 2016, 164:1136-1150. 55. Portugues R, Severi KE, Wyart C, Ahrens MB: Optogenetics in a transparent animal: circuit function in the larval zebrafish. Curr Opin Neurobiol 2013, 23:119-126.

Figure 1 .

 1 Figure 1. Different strategies to photocontrol mGluRs.

Acknowledgments 17

The authors thank Ebba L. Lagerqvist for critical reading of the manuscript. We acknowledge financial support from the Agence Nationale de la Recherche (ANR-12-NEUR-0003 and ANR-16-CE16-0010-01), the ERANET Neuron LIGHTPAIN project, the Fondation Recherche Médicale (FRM team DEQ20130326522), the Centre National de la Recherche Scientifique, the Spanish Ministry of Economy, Industry Competitiveness (SAF2015-74132-JIN) and MINECO (CTQ2014-57020-R and CTQ2017-89222-R), cofinanced by the European Regional Development Fund (FEDER) and the Catalan Government (2014SGR109).