D Vescovi

D Berzi

P Richard

N Brodu

Plane shear flows of frictionless spheres: Kinetic theory and 3D Soft-Sphere Discrete Element Method simulations

We use existing 3D Discrete Element simulations of simple shear flows of spheres to evaluate the radial distribution function at contact that enables kinetic theory to correctly predict the pressure and the shear stress, for different values of the collisional coefficient of restitution. Then, we perform 3D Discrete Element simulations of plane flows of frictionless, inelastic spheres, sheared between walls made bumpy by gluing particles in a regular array, at fixed average volume fraction and distance between the walls. The results of the numerical simulations are used to derive boundary conditions appropriated in the cases of large and small bumpiness. Those boundary conditions are, then, employed to numerically integrate the differential equations of Extended Kinetic Theory, where the breaking of the molecular chaos assumption at volume fraction larger than 0.49 is taken into account in the expression of the dissipation rate. We show that the Extended Kinetic Theory is in very good agreement with the numerical simulations, even for coefficients of restitution as low as 0.50. When the bumpiness is increased, we observe that some of the flowing particles are stuck in the gaps between the wall spheres. As a consequence, the walls are more dissipative than expected, and the flows resemble simple shear flows, i.e., flows of rather constant volume fraction and granular temperature. 1

I. INTRODUCTION

Granular materials are collections of discrete particles characterized by loss of energy whenever the particles interact. Their mechanical behavior is very complex even in the case of simple flow conditions (i.e., elementary geometries, stationary motions) or when the granular matter is particularly treatable (i.e., dry, no complex shapes of the grains and no polydispersity, etc). Due to their microscopic, discrete nature and their macroscopic behavior, granular materials are treated in both the frameworks of discontinuum (Discrete Element simulations) and continuum mechanics. Among the latter, kinetic theories of granular gases represent the most fundamental approach. Classic kinetic theories have been derived [START_REF] Jenkins | A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles[END_REF][START_REF] Lun | Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres[END_REF][START_REF] Garzó | Dense fluid transport for inelastic hard spheres[END_REF][START_REF] Goldhirsch | Rapid granular flows[END_REF] assuming that the energy of the system is dissipated through binary, instantaneous collisions between smooth spheres, and have been proved to succeed at low to moderate solid volume fractions. When the granular material becomes denser, the assumption of chaotic, binary, instantaneous collisions fails; 5-7 also, force chains can develop within the medium. [START_REF] Majmudar | Contact force measurements and stress-induced anisotropy in granular materials[END_REF][START_REF] Da Cruz | Rheophysics of dense granular materials: Discrete simulation of plane shear flows[END_REF] Several modifications to the classic kinetic theories have been recently proposed, in order to take into account the role of velocity correlation [START_REF] Jenkins | Dense shearing flows of inelastic disks[END_REF][START_REF] Jenkins | Dense inclined flows of inelastic spheres[END_REF] and the development of force chains. [START_REF] Johnson | Frictional-collisional constitutive relations for granular materials, with application to plane shearing[END_REF][START_REF] Johnson | Frictional-collisional equations of motion for particulate flows and their application to chutes[END_REF][START_REF] Louge | Model for dense granular flows down bumpy inclines[END_REF][START_REF] Berzi | Constitutive relations for steady, dense granular flows[END_REF][START_REF] Vescovi | From solid to granular gases: the steady state for granular materials[END_REF] The steady plane shear flow of granular materials, in absence of gravity and pressure gradient, serves as test case for the theories. Numerical simulations of simple shear flows (i.e., characterized by homogeneous shearing obtained by imposing the Lees-Edwards 17 periodic boundary conditions in the shearing direction) of disks or spheres have been performed using Event-Driven molecular dynamics (ED) and the Soft-Sphere Discrete Element Method (SS-DEM). [START_REF] Mitarai | Velocity correlations in the dense granular shear flows: Effects on energy dissipation and normal stress[END_REF][START_REF] Da Cruz | Rheophysics of dense granular materials: Discrete simulation of plane shear flows[END_REF][START_REF] Ji | Characteristics of temporalspatial parameters in quasisolid-fluid phase transition of granular materials[END_REF][START_REF] Chialvo | A modified kinetic theory for frictional granular flows in dense and dilute regimes[END_REF] Inhomogeneous shearing can be obtained in numerical simulations [START_REF] Shojaaee | Shear flow of dense granular materials near smooth walls. I. Shear localization and constitutive laws in the boundary region[END_REF][START_REF] Karion | Wall stresses in granular Couette flows of mono-sized particles and binary mixtures[END_REF][START_REF] Liu | General features of granular Couette flow and intruder dynamics[END_REF][START_REF] Xu | Solutions of the kinetic theory for bounded collisional granular flows[END_REF] and physical experiments [START_REF] Orlando | Effect of particle size and boundary conditions on the shear stress in an annular shear cell[END_REF][START_REF] Miller | Eddy viscosity in dense granular flows[END_REF] when the granular material is sheared between two solid parallel planes, one at rest and the other moving at constant velocity. In this paper, we numerically solve the Extended Kinetic Theory (EKT), in the form proposed by Berzi [START_REF] Berzi | Extended kinetic theory applied to dense, granular, simple shear flows[END_REF] , for wall-bounded, plane shear granular flows of identical, frictionless spheres, and compare the field variables profiles with those obtained by performing 3D SS-DEM simulations. The boundaries are made bumpy by gluing spheres, identical to the moving particles, at the walls; the inter-particle collisions are characterized by the coefficient of restitution, e, the ratio of the relative velocity between two impending particles after and before a collision. We first use existing numerical results on simple shear flows to slightly modify the constitutive relations of EKT, and then, we analyze both the influence of the coefficient of restitution and the bumpiness. Recently, also Chialvo and Sundaresan [START_REF] Chialvo | A modified kinetic theory for frictional granular flows in dense and dilute regimes[END_REF] have proposed corrections to the kinetic theory of Garzó and Dufty 3 on the basis of 3D SS-DEM simulations of simple shear flows of frictionless and frictional spheres. The main differences between our work and theirs are: (i) we focus on flows where the influence of the boundaries cannot be neglected and propose corrections to the boundary conditions originally developed for nearly elastic spheres glued at the walls [START_REF] Richman | Boundary conditions based on a modified maxwellian velocity distribution function for flows of identical, smooth, nearly elastic spheres[END_REF] for two extreme values of the bumpiness;

(ii) we propose a different radial distribution function obtained from a combination of the classic Carnahan and Starling [START_REF] Carnahan | Equation of state for nonattracting rigid spheres[END_REF] 's and Torquato [START_REF] Torquato | Nearest-neighbor statistics for packing of hard spheres and disks[END_REF] 's expressions, which fits also the numerical data of Chialvo and Sundaresan 19 ; (iii) we use an expression for the correlation length recently obtained from the analysis of ED simulations of simple shear flows [START_REF] Berzi | Extended kinetic theory applied to dense, granular, simple shear flows[END_REF] which does not require additional parameters besides the coefficient of restitution; (iv) unlike Chialvo and Sundaresan [START_REF] Chialvo | A modified kinetic theory for frictional granular flows in dense and dilute regimes[END_REF] , we show that there is no need for correcting the constitutive relation of the shear stress, provided that the coefficient of restitution is lower than 0.95. The paper is organized as follows. In Sec. II we introduce the EKT and the boundary conditions. Sec. III is devoted to describe the simulation method. In Sec. IV we derive the definition of a new radial distribution function and compare the results of the SS-DEM simulations with those obtained from the numerical integration of the equations of EKT. Finally, concluding remarks are summarized in Sec. V.

II. GOVERNING EQUATIONS

We focus on the steady motion of a mixture of identical, frictionless spheres sheared between two parallel planes, one at rest and the other moving at constant velocity V (Fig. 1).

V y l

x 0 H Figure 1. Sketch of the constant-volume wall-bounded plane shear flow configuration. A granular material confined between two horizontal solid planes is sheared by moving one plane at constant velocity V (x are y are respectively the flow and shear directions). The two planes are made bumpy by gluing grains at their surface in a regular hexagonal array, where l is the distance between the edges of two adjacent spheres.

We take x and y to be the flow and the shearing directions, respectively, and ignore variations along the transversal direction z. In what follows, all the quantities are made dimensionless using the particle diameter d and density ρ p and the wall velocity V . Spheres having the same properties of the moving particles are glued at the walls in a regular hexagonal array, where l is the distance between the edges of two adjacent spheres. The bumpiness of the wall is measured by ψ, with sin ψ = (1 + l)/2. [START_REF] Richman | Boundary conditions based on a modified maxwellian velocity distribution function for flows of identical, smooth, nearly elastic spheres[END_REF] We take y = 0 to be at the top of the particles glued at the resting wall, and y = H to be at the bottom of the particles glued at the moving wall. The hydrodynamic mean fields are the solid volume fraction ν, the velocity along the x direction u, the pressure p and the shear stress s. For frictionless particles, momentum is exchanged only through collisions, [START_REF] Vescovi | From solid to granular gases: the steady state for granular materials[END_REF] and a description based on kinetic theory 1,2,4 is suitable. We adopt a constant coefficient of restitution e. The continuous velocity field is first coarsed-grained at the scale of a grain diameter. The mean square of the velocity fluctuations against this field, averaged at the same scale, then defines a "granular temperature" T field, which measures the degree of agitation of the system. In the absence of external forces, and in steady conditions, the momentum balance trivially asserts that the pressure and the shear stress are constant along y. The balance of the fluctuating energy reads

su ′ = Q ′ + Γ. ( 1
)
where Q is the fluctuating energy flux and Γ is the rate of dissipation associated to collisions. Here and in what follows, a prime indicates the derivative with respect to the y direction. In order to close the problem, we need constitutive relations for p, s, Q and Γ. Kinetic theory 3 gives

p = f 1 T, (2)
s = f 2 T 1/2 u ′ , (3)
Γ = f 3 L T 3/2 , (4)
and

Q = -f 4 T 1/2 T ′ -f 5 T 3/2 ν ′ , (5)
where f 1 , f 2 , f 3 , f 4 and f 5 are explicit functions of the volume fraction and the coefficient of restitution and are listed in Tab. I. There, g 0 is the radial distribution function, whose expression is given in section IV on the basis of numerical results. In Eq. ( 4), L is the correlation length, which accounts for the decrease in the rate of collisional dissipation due to the correlated motion of particles that occurs at large volume fraction. [START_REF] Mitarai | Velocity correlations in the dense granular shear flows: Effects on energy dissipation and normal stress[END_REF][START_REF] Kumaran | Dynamic of dense sheared granular flows. Part 1. Structure and diffusion[END_REF] Taking into account this effect, i.e.,

Table I. List of auxiliary coefficients in the constitutive relations of kinetic theory. the breaking of the molecular chaos assumption, is the peculiarity of EKT. [START_REF] Jenkins | Dense shearing flows of inelastic disks[END_REF][START_REF] Jenkins | Dense inclined flows of inelastic spheres[END_REF][START_REF] Berzi | Extended kinetic theory applied to dense, granular, simple shear flows[END_REF][START_REF] Jenkins | Dense inclined flows of inelastic spheres: tests of an extension of kinetic theory[END_REF][START_REF] Berzi | Surface flows of inelastic spheres[END_REF][START_REF] Jenkins | Kinetic theory applied to inclined flows[END_REF] The expression for L has been suggested by Jenkins 11 on the basis of a simple heuristic argument,

f1 = 4νGF f2 = 8J 5π 1/2 νG f3 = 12 π 1/2 1 -e 2 νG f4 = 4M νG π 1/2 f5 = 25π 1/2 N 128ν G = νg0 F = 1 + e 2 + 1 4G J = 1 + e 2 + π 32
[5 + 2(1 + e)(3e -1)G] [5 + 4(1 + e)G] 24 -6 (1 -e) 2 -5(1 -e 2 ) G 2 M = 1 + e 2 + 9π 144 (1 + e) G 2 5 + 3G (2e -1) (1 + e) 2 [
L = max 1, L * u ′ T 1/2 , (6)
where L * is a function of the volume fraction and the coefficient of restitution. When L is equal to one, the molecular chaos assumption is valid and EKT reduces to classic kinetic theory. Berzi 26 has suggested an expression for L * on the basis of previous results of ED simulations of simple shear flows:

L * = f 2 f 3 1/2 2 (1 -e) 15 (g 0 -g 0,f ) + 1 3/2 , (7)
where g 0,f is the value of g 0 at the freezing point, ν = 0.49, i.e., the lowest value of the volume fraction for which a transition to an ordered state is first possible. [START_REF] Torquato | Nearest-neighbor statistics for packing of hard spheres and disks[END_REF] From the constitutive relations for the shear stress (3) and the pressure (2), we obtain the differential equation governing the velocity,

u ′ = s p f 1 f 2 T 1/2 . ( 8
)
By deriving Eq. (2) and using Eq. ( 5), the differential equation for the volume fraction results

ν ′ = Q T 1/2 f 2 1 f 4 pf 1,ν 1 - f 5 f 1 f 4 f 1,ν -1 , (9)
where f 1,ν represents the derivative of f 1 with respect to the volume fraction. Finally, using Eqs. ( 1), ( 2), ( 4) and (8), the differential equation for the energy flux reads

Q ′ = pT 1/2 f 1 f 2 s p 2 - f 3 Lf 1 . (10)
We also introduce an additional differential equation for the partial mass hold-up, defined as m = y 0 νdz,

m ′ = ν. (11)
Then, the value of the average volume fraction ν along y can be implemented as a boundary condition for m. We numerically solve the set of the four differential equations Eqs. ( 8)-( 11) using the function 'bvp4c' implemented in MATLAB, and fixing the gap H. We treat the pressure and the shear stress as parameters, so that we need six boundary conditions to solve the problem. As already mentioned, we implement the fixed average volume fraction as a boundary condition for the partial mass hold-up, i.e., m H = νH, while, at the resting wall, m 0 = 0. Here and in what follows, the index represents the coordinate y at which the quantity is evaluated. We allow the particles to slip at the bumpy walls, so that, for symmetry, u 0 = u w and u H = 1 -u w , where Richman 27 obtained, in the case of rigid, nearly elastic semi-spheres attached to a flat wall,

u w = π 2 h s p T 1/2 0 , (12)
with

h = 2 3 1 + 5F 0 (1 + B) sin 2 ψ 2 √ 2J 0 2 (1 -cos ψ) sin 2 ψ + 5F 0 √ 2J 0 , (13)
where

B = π [1 + 5/ (8G 0 )] / 12 √ 2
, and J 0 , F 0 and G 0 are obtained from the corresponding expressions of Tab. I with ν = ν 0 . The bumpy walls act either as a sink or a source of fluctuating energy to the system. The two boundary conditions for the energy flux are

Q 0 = Q w and Q H = -Q w , where Richman 27 proposed Q w = su w - π 2 pT 1/2 0 (1 -e) 2 (1 -cos ψ) sin 2 ψ . ( 14
)
The results of the numerical integration will be compared with those obtained from SS-DEM simulations described in Sec. III.

III. SS-DEM SIMULATIONS

We have carried out 3D SS-DEM simulations using our own code [START_REF] Brodu | Shallow granular flows down flat frictional channels: Steady flows and longitudinal vortices[END_REF] to make comparisons with the results of EKT. Although this method is well known and can be found in many papers, [START_REF] Silbert | Granular flow down an inclined plane: Bagnold scaling and rheology[END_REF][START_REF] Taberlet | The effect of sidewall friction on dense granular flows[END_REF][START_REF] Taberlet | Diffusion of a granular pulse in a rotating drum[END_REF][START_REF] Rycroft | Physical test of a particle simulation model in a sheared granular system[END_REF][START_REF] Majmudar | Jamming transition in granular systems[END_REF][START_REF] Hirshfeld | Granular flow from a silo: Discrete-particle simulations in three dimensions[END_REF][START_REF] Richard | Relevance of numerical simulations to booming sand[END_REF][START_REF] Richard | Rheology of confined granular flows: Scale invariance, glass transition, and friction weakening[END_REF] we present it here to support our discussion on the comparison of numerical and theoretical results (Fig. 2). In this method, each grain i is a soft sphere of diameter d i , mass m i , moment of inertia I i , position r i , velocity v i and angular velocity ω i . For a pair of particles {i, j}, we define the relative distance vector r ij = r ir j , their separation r ij = |r ij |, the relative velocity

j n   n ij j r i r n k (b) (a) i
v ij = v i -v j ,
and the normal unit vector n ij = (r ir j )/r ij . These two particles are in contact if their normal overlap

δ n ij = max(0, d i /2 + d j /2 -r ij
) is strictly positive. In general, the force on particle i from the interaction with particle j is the sum of a normal and tangential contribution :

f ij = f n ij + f t ij .
However, the present work deals with frictionless particles for which the contact force is purely normal. Therefore, the grains are submitted to neither tangential forces nor torques. For the normal force, we use the standard spring-dashpot interaction model: [START_REF] Luding | Introduction to discrete element methods: Basics of contact force models and how to perform the micro-macro transition to continuum theory[END_REF]

f n ij = k n δ n ij n ij -γ n v n ij ,
where k n is the spring constant, γ n the damping coefficient and v n ij the normal relative velocity

v n ij = (v ij • n ij )n ij .

The damping is used to obtain an inelastic collision. For a purely normal collision, the collision time t

c is equal to π/ k n /m ij -γ 2 n / 4m 2 ij 1/2 , with the reduced mass m ij = m i m i /(m i + m j ).
The normal restitution coefficient is given by e = exp [-t c γ n /(2m ij )] . The total force on particle i is then a combination of contact forces with other particles and the boundaries and an eventual resulting external force F ext . The resulting force f i is given by

f i = F ext + N j=1,j =i f ij ,
where N is the total number of flowing spheres. Once the forces are calculated for all the particles, the Newton's equations of motion, m i d 2 r i /dt 2 = f i , for the translational degrees of freedom are integrated. We use a velocity Verlet integration scheme with a time step ∆t = t c /30. The grains have all the same size and density. As already mentioned, the numerical results are given in nondimensional units: distances, times, velocities, forces, elastic constants and viscoelastic constants are, respectively, measured in units of d, d/V , V , ρ p d 2 V 2 , ρ p dV 2 and ρ p d 2 V . All the simulations have been performed in a rectangular box of length L x = 20, width L z = 10 and height L y = 20so that H = L y -2 = 18 -with N = 3132. The bumpiness has been generated by gluing, in a regular hexagonal array, a total of 340 particles at the two walls in the case of ψ = π/5, and 154 in the case ψ = π/3. Hence, taking into account the extra-space accessible to any flow particle in between the wall-spheres, ν = 0.45 when ψ = π/5 and ν = 0.44 when ψ = π/3. The particle stiffness of the linear spring model has been set equal to 2 • 10 5 . The non-dimensional ratio of the particle stiffness over the particle pressure is greater than 10 5 in all the simulations. This ensures that the contact time during a collision is much less than the flight time in between two successive collisions, so that the latter can be considered instantaneous. [START_REF] Berzi | Constitutive relations for steady, dense granular flows[END_REF][START_REF] Otsuki | Behavior of pressure and viscosity at high densities for two-dimensional hard and soft granular materials[END_REF] The value of γ n is adjusted to obtain the chosen normal restitution coefficient. Periodic boundary conditions are employed in the x and z directions and the horizontal flat walls are located at y = -1 and y = H + 1, the latter moving at constant horizontal velocity V . Those walls are treated as spheres of infinite size and density and the grains glued on their surface to create the bumpiness are treated like spheres of diameter 1 and infinite density. We focus on the steady state of sheared granular flows, that we consider achieved when the space-averaged granular temperature T becomes approximately constant (fluctuations around the time-averaged value less than 10%). The space-averaged granular temperature is computed as

T = 1 3N   N i=1 v i 2 - N i=1 v i 2   ,
where • denotes the Euclidean norm of a vector. Simulations have been performed by changing the coefficient of restitution (e = 0.20, 0.50, 0.60, 0.70, 0.80, 0.92, 0.98) and the bumpiness (ψ = π/5 and π/3).

We have checked that the steady state does not depend on the initial configuration, by preparing two different initial states, consisting of N spheres uniformly distributed in the volume. In the first case the spheres are initially at rest; in the second case, we assign a linear distribution (from 0 to 1) of the x-velocity of the spheres. This second configuration corresponds to a higher value of the initial energy, i.e., of the initial space-averaged granular temperature. In both cases, we have achieved the same steady state, i.e., with the same value of space-averaged granular temperature and the same distributions of the field variables. The time at which the steady state is reached increases when the coefficient of restitution decreases (e.g., see Fig. 3 for the case ψ = π/5). For sufficiently small coefficients of restitution (case e = 0.2 in Fig. 3), the mean granular temperature continues to decrease, without reaching a steady state. The slope of the curve approaches the value -2 that characterizes the Homogeneous Cooling State (HCS), [START_REF] Goldhirsch | Rapid granular flows[END_REF] where the rate of change of the granular temperature in the balance of fluctuating energy is only due to the collisional dissipation and the granular temperature obeys the Haff's law, [START_REF] Haff | Grain flow as a fluid-mechanical phenomenon[END_REF] T ∝ (1 + t) -2 . We will discuss in Sec. IV this finding. Once the steady state is reached, measurements are averaged in time, over at least 2000 time steps, and over the lengths of the domain along the x and z directions, using 20 horizontal slices. Given that the averaging is sensitive to the amplitude of the spatial discretization, [START_REF] Weinhart | Coarse-grained local and objective continuum description of 3D granular flows down an inclined surface[END_REF] we chose a number of slices that does not affect the results. Example of profiles of ν, u, T and u ′ are plotted in Fig. 4 for ψ = π/5 and e = 0.80, when 20 or 40 horizontal slices are employed. Also shown are the results of the numerical integration of the EKT described in Sec. IV. The velocity profile has a characteristic S-shape, in agreement with recent physical experiments performed on disks. [START_REF] Miller | Eddy viscosity in dense granular flows[END_REF] Also, the profile of the shear rate very much resembles the experimental findings. The volume fraction increases and the granular temperature decreases with distance from the walls. The core of the flow is dense, i.e., the volume fraction is larger than 0.49, and there the molecular chaos assumption breaks down. All those features are well captured by kinetic theory.

IV. RESULTS AND COMPARISONS

We first use the numerical results obtained on simple shear flows of frictionless spheres by Mitarai and Nakanishi 6 and Chialvo and Sundaresan [START_REF] Chialvo | A modified kinetic theory for frictional granular flows in dense and dilute regimes[END_REF] to derive the expression of the radial distribution function g 0 . Mitarai and Nakanishi 6 performed ED simulations of inelastic hard spheres, whereas Chialvo and Sundaresan 19 used a SS-DEM code with a linear spring-dashpot model. In both works, the Lees-Edwards 17 boundary conditions were implemented in the shearing direction, in order to allow for the system to remain homogeneous during the shearing. From the constitutive relation for the pressure (2) and the expressions of Tab. I,

g 0 = 1 2ν(1 + e) p νT -1 , (15)
so that the radial distribution function can be obtained from the numerical values of pressure, volume fraction and granular temperature. For small volume fractions, g 0 obeys the Carnahan and Starling's expression, 28

g 0,cs = 2 -ν 2 (1 -ν) 3 , (16)
whereas Torquato's [START_REF] Torquato | Nearest-neighbor statistics for packing of hard spheres and disks[END_REF] proposed, on the basis of numerical results on elastic particles,

g 0,t =    g 0,cs if ν < 0.49, (2 -0.49) 2 (1 -0.49) 3 (ν rcp -0.49) (ν rcp -ν) otherwise. ( 17
)
with ν rcp = 0.636 the value of volume fraction at random close packing. Fig. 5 shows the radial distribution function obtained from the numerical simulations of Mitarai and Nakanishi 6 and Chialvo and Sundaresan 19 on simple shear flows and the present SS-DEM simulations of bounded shear flows. Eq. ( 17) fits well the numerical results in the case of nearly elastic particles (Fig. 5a), while underestimates the data for dense flows of particles when e ≤ 0.95 (Fig. 5b).

In the latter case, we propose to use the following expression:

g 0 = f g 0,cs + (1 -f ) 2 ν rcp -ν , (18)
where f is a function of the volume fraction which makes g 0 equal to the Carnahan and Starling's expression when the volume fraction is less than a limit value, ν m ,

f =    1 if ν < ν m , ν 2 -2ν m ν + ν rcp (2ν m -ν rcp ) 2ν rcp ν m -ν 2 m -ν 2 rcp otherwise. ( 19
)
We take ν m = 0.4; the quadratic expression for f when ν ≥ ν m ensures that the first derivative of g 0 is continuous, facilitating the numerical integration of the equations. In simple shear flows, the divergence of the flux of fluctuating energy in Eq. ( 1) can be neglected, and the correlation length reduces to:

L = f 3 T 3/2 su ′ . (20)
In Fig. 6 we plot the quantity f 3 T 3/2 / (su ′ ) as a function of the volume fraction, where s and T are those measured by Chialvo and Sundaresan 19 in their SS-DEM simulations, while f 3 is evaluated from the expression of Tab. I, using Eq. ( 18) and the measured values of the volume fraction. There, the lines represent the theoretical expression of the correlation length, which, in simple shear flows, using Eqs. ( 3),( 6) and (20), is

L = max 1, f 1/3 2 f 1/3 3 L * 2/3 . ( 21
)
The agreement between the numerical data and the theoretical expression of L is remarkable. In Fig. 6(b) we also plot, for comparison, the correlation length obtained from the modification of the kinetic theory suggested by Chialvo and Sundaresan 19 when e = 0.7. In simple shear flows, an algebraic relation between the shear rate and the granular temperature exists,

T u ′ 2 = f 2 f 3 L. ( 22
)
Substituting Eq. ( 22) in Eq. ( 2) and Eq. ( 3) leads to the following expressions for the pressure,

p = f 1 f 2 f 3 Lu ′ 2 , (23)
the shear stress

s = f 3 2 f 3 L 1/2 u ′ 2 , (24)
and the stress ratio

s p = f 2 f 3 f 2 1 L 1/2 . ( 25
)
The quantities T /u ′ 2 , p/u ′ 2 and s/p, obtained from the numerical simulations of Mitarai and Nakanishi 22), ( 23) and ( 25) with the radial distribution function given by Eq. ( 18) and the correlation length given by Eq. ( 21). In Figs. 7(b), 8(b) and 9(b) we also plot, for the case e = 0.7, the predictions of the present theory if the breaking of the molecular chaos is not accounted for (i.e., L = 1) and the predictions from the theory of Chialvo and Sundaresan [START_REF] Chialvo | A modified kinetic theory for frictional granular flows in dense and dilute regimes[END_REF] . Except for large coefficients of restitution (e > 0.95), the granular temperature, the pressure and the stress ratio are well predicted by kinetic theory in the entire range of volume fraction, if the expressions (18) for g 0 and (21) for L are adopted. Replacing Eq. ( 18) with Eq. ( 17) would allow a good fitting also for the case of nearly elastic particles (e > 0.95). Finally, Figs. 10(a) and 10(b) depict, respectively, the quantity p/T and s/ T 1/2 u ′ as functions of the volume fraction, where p, T , s and u ′ are those measured by Mitarai and Nakanishi 6 and Chialvo and Sundaresan [START_REF] Chialvo | A modified kinetic theory for frictional granular flows in dense and dilute regimes[END_REF] , when e = 0.70, together with the theoretical expressions of f 1 and f 2 of Tab. I, with g 0 given by Eq. (18). Also the data obtained from the present SS-DEM simulations on bounded shear granular flows (with e = 0.70 and ψ = π/5) are shown. All the numerical data collapse, independently of the simulation method and the flow configuration, and are in very good agreement with the theoretical curves. Similar agreement is obtained for other values of the coefficient of restitution. In particular, Fig. 10(b) indicates that there is no need to modify the constitutive relation of the shear stress of kinetic theory, at least if the particles are sufficiently inelastic. [START_REF] Berzi | Extended kinetic theory applied to dense, granular, simple shear flows[END_REF] We now compare the results of the numerical integration of Eqs. ( 8)-( 11), with the SS-DEM simulations in terms of profiles of volume fraction, velocity and granular temperature, distinguishing between small and large bumpiness.

A. Small bumpiness

Figs. 11(a), 12(a) and 13(a) show that, at small bumpiness (ψ = π/5), and using the boundary conditions of Richman [START_REF] Richman | Boundary conditions based on a modified maxwellian velocity distribution function for flows of identical, smooth, nearly elastic spheres[END_REF] , EKT only qualitatively reproduces the SS-DEM results, when ν = 0.45 as in the simulations. Those boundary conditions were developed for nearly elastic particles. Actually, the slip velocity and the volume fraction are underestimated, and the granular temperature is strongly overestimated when the coefficient of restitution is far from unity. In general, the SS-DEM simulations show that the volume fraction increases with the distance from the wall (Fig. 11a), and the walls are always "hotter" than the interior (Fig. 13a), i.e., the boundaries are energetic (the fluctuating energy flux is directed towards the interior of the flow); for very inelastic particles, a dense core surrounded by two more dilute layers appear (Fig. 11a). Also, the slip velocity increases as the coefficient of restitution decreases: for e = 0.50 the granular material roughly moves as a plug (Fig. 12a). Fig. 14(a) depicts the value of the slip velocity u w as a function of the coefficient of restitution. For e = 0.5, the slip velocity approaches the value 0.5, for which there is a condition of perfect slip at the walls: in that case, the particles do not touch the walls, so that no exchange of energy with the boundaries is possible. This is the reason why, for e lower than 0.5, the energy initially put into the system is entirely dissipated in collisions and the evolution of the mean granular temperature obeys the Haff's law (Fig. 3).

Fig. 14(b) shows the ratio of the quantity u w p/ T 1/2 0 s obtained from the SS-DEM simulations to the coefficient h obtained from Eq. ( 13) using ψ = π/5 and the numerical values of the volume fraction at the walls. The boundary condition on the slip velocity of Richman 27 must be corrected in order to reproduce the measurements. On the basis of best fitting, we propose to use

u w T 1/2 0 s/p = h exp(7.3 -8.6e). (26)
which represents the solid line in Fig. 14(b). If we employ Eq. ( 26) instead of Eq. ( 12) as a boundary condition, when numerically integrating the equations of EKT, the agreement with the numerical simulations is remarkable even in the case of very inelastic particles (Figs. 11b, 12b and 13b). We expect the numerical coefficients in Eq. ( 26) to depend on the bumpiness and, perhaps, the particle stiffness. We postpone to future works a systematic investigation on the role of those quantities in determining the correction to the slip velocity.

B. Large bumpiness

The SS-DEM simulations indicate (Figs. 15, 16 and 17) that, at large bumpiness (ψ = π/3), the volume fraction and the granular temperature are rather uniform, and the velocity profile is linearly distributed with zero slip velocity (Fig. 16), as for simple shear flows. Predictions of EKT in the case ψ = π/3 when the boundary conditions Eqs. (12) and ( 14) are employed strongly disagree with the SS-DEM results (Figs. 15a, 16a and 17a). Visual observation of the particle motion suggests that for large enough bumpiness, some of the flowing particles get stuck in the gaps between the particles glued at the walls; those trapped particles contribute then to create a "disordered" bumpy wall, similar to that employed in the numerical simulations of Silbert et al. [START_REF] Silbert | Granular flow down an inclined plane: Bagnold scaling and rheology[END_REF] , which is far less energetic than the "ordered" bumpy wall of Richman [START_REF] Richman | Boundary conditions based on a modified maxwellian velocity distribution function for flows of identical, smooth, nearly elastic spheres[END_REF] . In the cases e = 0.70 and 0.92, the walls are even slightly colder than the interior (Fig. 17a), i.e., the boundaries are dissipative (the fluctuating energy flux is directed towards the walls). If we use the mean volume fraction obtained by averaging the SS-DEM profiles and u w = Q w = 0 instead of Eqs. ( 12) and ( 14), i.e., we assume that the boundaries are neutral (they do not furnish nor subtract fluctuating energy), as boundary conditions, the numerical integration of EKT, which coincides with the analytical solution of simple shear flows, provides a fairly good agreement with the SS-DEM simulations (Figs. 15b, 16b and 17b). To check our intuition about the particles being trapped at the walls, we have also performed SS-DEM simulations, with e = 0.7, when random conformations of particles are glued at the walls (the details for the generation of this kind of boundaries are given in Silbert et al. [START_REF] Silbert | Granular flow down an inclined plane: Bagnold scaling and rheology[END_REF] ). The distribution of the volume fraction (and of the other quantities, not shown here for sake of brevity) is very similar to the case ψ = π/3 (Fig. 18a). The mean volume fraction is different in the two cases, because the space accessible to the flowing particles, whose number is constant and equal to 3132, is different. Also, the fact that the mean volume fraction measured in the SS-DEM simulatons νDEM is, in general, less than the theoretical value 0.44, that would characaterize the ψ = π/3 case when N = 3132, is an indication of particle trapping. Indeed, a rough estimate of the thickness ∆ of this trapped particle layer is

∆ = N 2L x L z 1 - νDEM 0.44 . (27)
Fig. 18(b) shows that ∆ goes to zero as e approaches one. Also, the thickness ∆ saturates to a constant value for coefficients of restitution lower than 0.7. Once again, we postpone to future works the generalization of these findings to other values of the bumpiness and the particle stiffness. Finally, Fig. 19 shows the influence of the coefficient of restitution on the stress ratio, s/p. Contrary to results reported for 2D plane shear flows of frictional grains submitted to imposed pressure, 9 the coefficient of restitution strongly affects the stress ratio. In the range 0.50 ≤ e ≤ 0.98, the stress ratio obtained from the present SS-DEM simulations is a decreasing function of the coefficient of restitution for large bumpiness (ψ = π/3); while s/p has a maximum around e = 0.80 for small bumpiness (ψ = π/5). The predictions of EKT when Eqs. ( 26) and ( 14) are employed as boundary conditions for ψ = π/5, and u w = Q w = 0 for ψ = π/3 are, once again, in a fairly good agreement with the simulations. The drop in the stress ratio for small bumpiness and coefficients of restitution less than 0.8 is due to the already mentioned increasing of the slip velocity, with the corresponding approaching to the Homogeneous Cooling State, in which the shear stress, and consequently the stress ratio, vanishes. 8)-( 11) for e = 0.50 (dashed line), e = 0.70 (solid line) and e = 0.92 (dot-dashed line) when: (a) the boundary conditions are Eq. ( 12) and Eq. ( 14); (b) the boundary conditions are uw = Qw = 0. In both cases, the mean volume fraction is that measured in the simulations.

. Same as in Fig. 15, but for the profile of velocity.

V. CONCLUSIONS

In this paper, the Extended Kinetic Theory is numerically solved for the shear flows of identical, frictionless particles bounded between two parallel, bumpy planes, at constant volume (plane shear flow). The bumpiness is due to spheres identical to those of the flow, glued at the walls in a regularly spaced, hexagonal array. The numerical solutions are compared with 3D SS-DEM simulations, and the roles of the coefficient of restitution and the bumpiness of the boundaries are investigated. We propose an expression for the radial distribution function to be used when e ≤ 0.95 which coincides with the Carnahan and Starling's 28 at small volume fraction, and diverges as the volume fraction approaches the shear rigidity as the Torquato's, 29 but, unlike the latter, its derivative is continuous in the entire range of volume fraction. We have shown that the proposed expression fits well the results of ED and SS-DEM simulations of simple shear flows. Also, we adopt a recently suggested expression for the correlation length in the dissipation rate of fluctuating energy, which depends only on the coefficient of restitution. At small bumpiness, the SS-DEM simulations show that the volume fraction increases with the distance from the wall, for every value of the coefficient of restitution, and the wall is always "hotter" than the interior. The slip velocity at the boundaries decreases with the elasticity of the particles, and, for coefficients of restitution less than 0.5, the slip is perfect: the boundaries do not touch the flowing particles, so that the system evolves accordingly to the Homogeneous Cooling State (it is not possible to obtain a steady shear flow). Also, the measured stress ratio is a non monotonic function of the coefficient of restitution, and reaches a maximum for e = 0.80. The results of the numerical integration of EKT agree well with the simulations, if a correction to the expression of the slip velocity depending on the coefficient of restitution is introduced in the boundary conditions derived for nearly elastic particles by Richman [START_REF] Richman | Boundary conditions based on a modified maxwellian velocity distribution function for flows of identical, smooth, nearly elastic spheres[END_REF] . At large bumpiness, the SS-DEM simulations show nearly uniform profiles of volume fraction and granular temperature, and linear distributions of the velocity field, as for simple shear flows. This is due to the fact that, when the gaps between the spheres glued at the walls are large enough, some of the flowing particles get stuck, making the bumpy wall more "disordered", and, then, more dissipative than expected. Even in the case of large bumpiness, EKT is able to reproduce the simulation results, if both the slip velocity and the fluctuating energy flux at the walls are taken to be zero. Summarizing, we have shown that Extended Kinetic Theory has the capability of quantitatively reproducing the flow of frictionless spheres in the entire range of volume fraction for which the collisions can be considered nearly instantaneous and random (i.e., the entire fluid-like regime of granular flows). Tests of proposed extensions to EKT to deal with friction, non-instantaneous collisions and enduring contacts will be the subject of future works.

Figure 2 .

 2 Figure 2. Sketches of two particles at contact (a) and of the contact forces used (b)

Figure 3 .

 3 Figure 3. Time evolution of the mean granular temperature for different values of the coefficient of restitution when ν = 0.45 and ψ = π/5.

Figure 4 .

 4 Figure 4. Profiles of ν, u, T and u ′ obtained from SS-DEM simulation when H = 18, ν = 0.45, ψ = π/5 and e = 0.80, when the domain along the y-direction is divided into 20 (open circles) and 40 (crosses) slices to perform the averaging. The solid lines are the results of EKT when Eq. (18) and Eq. (6) are employed. The dashed line in (a) is the value of the volume fraction at the freezing point, ν = 0.49.

Figure 5 .

 5 Figure 5. Numerical (symbols) radial distribution function (after Mitarai and Nakanishi 6 , Chialvo and Sundaresan 19 and present SS-DEM simulations) as a function of the volume fraction for: (a) e = 0.98 and 0.99; (b) 0.5 ≤ e ≤ 0.95. Also shown are Eq. (18) (solid line) and the expressions of Carnahan and Starling (dot-dashed line) and Torquato (dotted line).

6

 6 and Chialvo and Sundaresan 19 , are shown in Figs. 7(a), 8(a) and 9(a), respectively, for different values of the coefficient of restitution. The lines represent Eqs. (

Figure 6 .Figure 7 .Figure 8 .Figure 9 .Figure 10 .

 678910 Figure 6. (a) Numerical (symbols, after Mitarai and Nakanishi 6 and Chialvo and Sundaresan 19 ) and theoretical (lines, Eq. (20)) correlation length as a function of the volume fraction, for different values of the coefficient of restitution: e = 0.70 (circles and solid line); e = 0.80 (squares and dashed line); e = 0.90 (stars and dotted line); e = 0.95 (diamonds and dot-dashed line). (b) Same as in Fig. 6(a) for the case e = 0.7. The dashed line represents the theory of Chialvo and Sundaresan 19 .

Figure 11 .

 11 Figure 11. Profile of volume fraction obtained from the present SS-DEM simulations (symbols) for ψ = π/5, ν = 0.45 and various coefficients of restitution. The data are compared with the numerical integration of Eqs. (8)-(11) for e = 0.50 (dashed line), e = 0.70 (solid line) and e = 0.92 (dot-dashed line) when: (a) the boundary condition on the slip velocity is Eq. (12); (b) the boundary condition on the slip velocity is Eq. (26).

Figure 12 .

 12 Figure 12. Same as in Fig.11, but for the profile of velocity.

Figure 13 .Figure 14 .

 1314 Figure 13. Same as in Fig.11, but for the profile of granular temperature.

Figure 15 .

 15 Figure15. Profile of volume fraction obtained from the present SS-DEM simulations (symbols) for ψ = π/3 and various coefficients of restitution. The data are compared with the numerical integration of Eqs. (8)-(11) for e = 0.50 (dashed line), e = 0.70 (solid line) and e = 0.92 (dot-dashed line) when: (a) the boundary conditions are Eq. (12) and Eq. (14); (b) the boundary conditions are uw = Qw = 0. In both cases, the mean volume fraction is that measured in the simulations.

Figure 17 .Figure 18 .Figure 19 .

 171819 Figure 17. Same as in Fig.15, but for the profile of granular temperature.


A Theron
email: theron@univ-perp.fr

A Rognon

B Gourbal

G Mitta

Multi-parasite host susceptibility and multi-host parasite infectivity: a new approach of the Biomphalaria glabrata / Schistosoma mansoni compatibility polymorphism

Keywords:

In this study, we analyze the degree of susceptibility/un-susceptibility of five strains of Biomphalaria glabrata from different geographical origins successively challenged with a panel of 4 Schistosoma mansoni strains. A total of 20 homopatric and heteropatric hostparasite combinations were tested with exposure doses of 1, 10, 20, 30 and 50 miracidia per individual host. By doing this, we characterized each B. glabrata strain by its "multi-parasite susceptibility phenotype" that reflects better the efficiency of their defense mechanism against not only one, but a diversity of schistosome stocks. In the same time, all the S. mansoni strains used were characterized, by their "multi-host infectivity phenotype" that reflects the level of infectivity they display when confronted to diverse snail populations. Based on these results it is possible to select different homogenous stocks of snails with different spectrum of susceptibility/un-susceptibility for several parasite strains. This will be a useful tool for future functional studies conducted to understand the genetics and molecular basis of the compatibility polymorphism in this host/parasite model.

Introduction

Schistosomiasis remains an important health problem that affects over 200 million people worldwide (WHO, 2010). Schistosoma mansoni causes intestinal schistosomiasis and as all trematode species, needs mollusks as first intermediate hosts to accomplish part of its life cycle before infecting humans. For this species, only fresh water snails of the genus Biomphalaria are potentially susceptible to an infection by the larval stages of the parasite.

However, within a same snail species, for example B. glabrata, the main intermediate host of S. mansoni in South America, compatibility varies between populations, strains or individuals [START_REF] Basch | Intermediate host specificity in Schistosoma mansoni[END_REF][START_REF] Theron | Schistosoma mansoni: distribution patterns of miracidia among Biomphalaria glabrata snail as related to host susceptibility and sporocyst regulatory processes[END_REF]. Snail/schistosome compatibility reflects both snail susceptibility and schistosome infectivity and the observed polymorphism of compatibility results from the complex phenotype-by-phenotype interactions occurring between each individual host and each individual parasite [START_REF] Basch | Intermediate host specificity in Schistosoma mansoni[END_REF][START_REF] Theron | Are Biomphalaria snails resistant to Schistosoma mansoni ?[END_REF][START_REF] Theron | Effects of laboratory culture on compatibility between snails and schistosomes[END_REF]. The immunologic mechanisms underlying compatibility have remained largely unknown (see [START_REF] Bayne | Successful parasitism of of vector snail Biomphalaria glabrata by the human blood fluke (trematode) Schistosoma mansoni: A 2009 assessment[END_REF][START_REF] Yoshino | Immunobiology of Biomphalaria glabrata-trematode interactions[END_REF]. However recent studies have succeed in identifying host immune recognition receptors and parasite antigens displaying a high level of variability and which were described to be in interaction (Roger et al., 2008;[START_REF] Roger | Molecular determinants of compatibility polymorphism in the Biomphalaria glabrata/Schistosoma mansoni model: New candidates identified by a global comparative proteomics approach[END_REF][START_REF] Moné | A Large Repertoire of Parasite Epitopes Matched by a Large Repertoire of Host Immune Receptors in an Invertebrate Host/Parasite Model[END_REF]. It was hypothesised that these molecules could be involved in the expression of the B. glabrata / S. mansoni compatibility polymorphism [START_REF] Mitta | Compatibility polymorphism in snail/schistosome interactions: from field to theory to molecular mechanisms[END_REF][START_REF] Hanington | A somatically diversified defense factor, FREP3, is a determinant of snail resistance to schistosome infection[END_REF]Roger et al., 2008a,b,c;[START_REF] Moné | A Large Repertoire of Parasite Epitopes Matched by a Large Repertoire of Host Immune Receptors in an Invertebrate Host/Parasite Model[END_REF]2011). Different B. glabrata laboratory strains (or isolates) show various degrees of susceptibility to S. mansoni infection [START_REF] Basch | Intermediate host specificity in Schistosoma mansoni[END_REF][START_REF] Theron | Schistosoma mansoni: distribution patterns of miracidia among Biomphalaria glabrata snail as related to host susceptibility and sporocyst regulatory processes[END_REF] and different strains of S. mansoni display different levels of infectivity towards a particular strain of snail host.

Compatibility is strain specific [START_REF] Theron | Are Biomphalaria snails resistant to Schistosoma mansoni ?[END_REF][START_REF] Webster | Coevolution and compatibility in the snail-schistosome system[END_REF], one snail strain that is highly susceptible to one particular strain of schistosome could be partially or totally unsusceptible to another schistosome strain (Richards, 1984;[START_REF] Lie | Specificity of natural resistance to trematode infections in Biomphalaria glabrata[END_REF]. On the other hand, one parasite strain that is highly infective to a snail host strain could be less or un-infective to another one. In spite of this overview about variation of compatibility, it appears that very few studies have investigated the multi-parasite strain susceptibility of B. glabrata stocks and conversely the multi-host strain infectivity of S. mansoni stocks.

The aim of the present study was to characterize and compare the degree of susceptibility/un-susceptibility of different strains of B. glabrata when challenged with a panel of S. mansoni strains. On the other hand, the infectious capacity of the different strains of S. mansoni used was also evaluated when confronted to a panel of B. glabrata stocks. By this way we will be able to discriminate particular strains of snails that are more or less efficient in their defense mechanisms against not only a single strain of parasite but against a diversity of schistosome strains and then, to go beyond the classic approach that traditionally characterize a strain either resistant or susceptible. This approach, concerning both the hosts and the parasite may lead to the selection of snail phenotypes representative of different spectrum of susceptibility/un-susceptibility and schistosome strains representative of different capacities of infectivity. Such discreet phenotypes of multi-parasite susceptibility provide an invaluable resource for future comparative genetics and functional studies to better understand the genetics and molecular basis of the compatibility polymorphism in this host/parasite model.

Materials and methods

Parasites and snails

Four strains of S. mansoni (Sm) from different geographical origins were used in this study. Each was maintained in the laboratory since more than 10-20 years on their sympatric strains of B. glabrata (Bg) as intermediate host and in mice (SWISS OF1) as definitive host.

These four homopatric host/parasite combinations are termed as follow: BgVEN / SmVEN (origin Venezuela, Guaraca); BgGUA / SmGUA (origin Guadeloupe, Dans Fond); BgBRE / SmBRE (origin Brazil, Recife) and BgBAR / SmLE (origin Brazil, Belo Horizonte). One supplementary B. glabrata strain for which we do not dispose of its corresponding parasite strain was also used: the BgBS-90 strain (origin Brazil, Salvador).

Compatibility trials

As snail infection rates vary with the parasite dose [START_REF] Theron | Schistosoma mansoni: distribution patterns of miracidia among Biomphalaria glabrata snail as related to host susceptibility and sporocyst regulatory processes[END_REF]2008), compatibility tests were conducted by dose-response curves obtained by challenging individual snails (35-40 snails of 6-8 mm in diameter per treatments) with doses of 1, 10, 20, 30 and 50 freshly hatched miracidia (Mi) in 5 ml of water at 24-25° C for 8 h. Following exposure to miracidia, snails were replaced in their original containers until their infection status was assessed. For all experiments, infection of snails was evaluated by the presence and count of mother sporocysts at 15 days post-exposure as previously described [START_REF] Theron | Development of accessory sexual organs in Biomphalaria glabrata as related to infection timing by Schistosoma mansoni: Consequences on the energy utilisation patterns by the parasite[END_REF][START_REF] Allienne | Recovery of primary sporocysts in vivo in the Schistosoma mansoni / Biomphalaria glabrata model using a simple fixation method suitable for extraction of genomic DNA and RNA[END_REF]. Briefly summarized, snails were relaxed in pond water containing excess of crystaline menthol for 12 hr. The snail body was removed and fixed in modified Raillet-Henry solution. Using this technique, mother sporocysts were readily observable as translucent white bodies within an opaque brown tissue background. The number of developped mother sporocysts (containing young daughter sporocysts) present within each snail was determined following exhaustive dissection of the snail including deeper tissues. A previous histological study on BgGUA snails exposed to SmGUA strain and fixed 48 hr later, showed that among the miracidia that have penetrated, 22% appeared well developped without host reaction, 40% were encapsulated by the host hemocytes and 38% degenerated without host reaction [START_REF] Theron | Schistosoma mansoni: distribution patterns of miracidia among Biomphalaria glabrata snail as related to host susceptibility and sporocyst regulatory processes[END_REF]. One hour after exposure, 26% of the miracidia had not penetrated the snail. The level of compatibility was quantified by the proportion of infected snails and the intensity of infection by the number of developped mother sporocysts (MSp) per snails. All the host-parasite combinations identified as compatible by the presence of developing mother sporocysts at 15 days post-infection, were able to produce cercariae. This has been verified during the laboratory cycling of the various strains of S. mansoni on the different strains of snails and additionaly, during the experimental selection of compatible phenotypes of snail hosts currently in progress.

For each of the 5 snail strains, compatibility was tested separately with their homopatric schistosome strain and the 3 other heteropatric parasite strains respectively (for the BS-90 snail strain, the 4 parasite strains used were heteropatric). In total, the following 20 host-parasite combinations were tested with exposure doses of 1, 10, 20, 30 and 50 miracidia per host : (BgBAR / SmLE, BgBAR / SmVEN, BgBAR / SmBRE, BgBAR / SmGUA); (BgVEN / SmVEN, BgVEN / SmLE, BgVEN / SmBRE, BgVEN

Results

Variations of susceptibility between B. glabrata strains (Fig. 1)

For all the host/parasite combinations tested, the dose-response curves generated showed the same pattern: a rapid increase in infection rates between the doses 1 and 10-20 miracidia per snail then a plateau that leveled at different rates (Fig. 1).

Each snail strain exhibits a particular scope of susceptibility when confronted to the 4 strains of schistosomes (Fig. 1). The BgBRE strain was 100% susceptible with 3 (SmLE, SmVEN and SmBRE) of the 4 schistosome strains tested but very poorly susceptible to the SmGUA (around 5%). At the opposite, BgBAR strain, 100% susceptible with its sympatric schistosome strain (SmLE), was only 50% susceptible with SmVEN, 10% with SmBRE and totally unsusceptible to SmGUA. BgVEN and BgGUA strains showed a similar pattern of susceptibility (100% with SmLE and SmVEN, 80% with SmBRE and 40% with SmGUA).

The BgBS-90 stock was only partially susceptible to the SmLE strain of S. mansoni (53% of infection with 20Mi) and totally unsusceptible to the 3 other strains of S. mansoni.

Excepted for the homopatric couple BgGUA/SmGUA that showed the lowest compatibility level compared to the corresponding heteropatric combinations, the 3 other homopatric couples (BgBRE/SmBRE, BgVEN/SmVEN and BgBAR/SmLE) were equally or more compatible than their corresponding heteropatric couples.

Variations of infectivity between S. mansoni strains (Fig. 2)

By the side of the parasite, clearly the SmLE strain appeared as having the highest capacity of infection whatever the snail strain used. Infection rates of 100% were obtained with the dose of 10-20 Mi per snail with 4 snail strains BgBRE, BgGUA, BgVEN and BgBar.

SmLE was the only schistosome strain tested capable of infecting the BgBS-90 strain (53% of infection with 20Mi). At the opposite, the SmGUA strain appeared as the less infective even towards its sympatric host snails (40-50% of BgGUA infected at the plateau), equivalent infection rates were obtained only with the BgBAR stock. SmGUA was very few infective for BgBRE (around 5%) and un-infective towards the BgBAR and BgBS90 strains. SmVEN and SmBRE strains showed high level of infectivity with the BgBRE (100%), BgGUA (80%) and BgVEN (80%) snail stocks, low with the BgBAR strain (15%) and un-infective towards the BgBS-90 strain.

Variations of parasite intensities (Fig. 3, A, B)

Variations of parasite intensities (number of developed MSp per snails), are presented in figure 3 only for the BgBRE host strain challenged by the 4 parasite strains and for the SmBRE parasite strain infecting the 5 host strains at the different doses of miracidia. These two examples were representative of the other combinations. For compatible combinations, the number of MSp increases with the miracidial doses at exposure. For the homopatric couple SmBRE/BgBRE (Fig. 3 A) the mean number of MSp /snail for 10 Mi was of 3.58 ± 0.19 and raised to 16.18 ± 0.86 MSp for the dose of 50 Mi/snail. For the less compatible combination, SmBRE/BgBAR (Fig. 3 B) the number of MSp remained stable around 2.5 whatever the miracidial dose was.

For the same parasite strain and at equivalent dose of Mi, variations exist as a function of the host strain challenged and its level of susceptibility. Highest was the compatibility level, highest was the mean parasite intensity. As an example (Fig. 3 B), the relationship between mean parasite intensities and compatibility level for the SmBRE strain at the dose of 50 Mi/snail infecting the 4 host strains was as follow: 2.5 MSp and 16% of infection for SmBRE/BgBAR, 8.0 MSp and 75% for SmBRE/BgVEN, 10.0 MSp and 90% for SmBRE/BgGUA, 16.0 MSp and 100% for SmBRE/BgBRE. We note that only a low proportion of miracidia presented to the snail host succeed in developing to mother sporocysts and this proportion varies between parasite strains, host strains and miracidial doses at exposure (Fig. 4). Including data from all the host strains and all the miracidial doses, SmLE parasite strain showed the highest Mi/MSp transformation rate with an average of 35%, SmGUA the lower (20%). Including data from all the parasite strains and the miracidial doses, it is among the snails of the BgBRE strain that we observed the highest rate of developing miracidia (35%) and among the snails of the BgBAR strain the lowest (16%).

Discussion

Investigations on snail/schistosome compatibility between Biomphalaria species and S. mansoni started several decades ago with the pioneer works of [START_REF] Files | A study on the comparative susceptibility of snails vectors to strains of Schistosoma mansoni[END_REF], [START_REF] Abdel-Malek | Susceptibility of the snail Biomphalaria boissyi to infection with certain strains of Schistosoma mansoni[END_REF], then [START_REF] Richards | Genetics of a molluscan vector of schistosomiasis[END_REF][START_REF] Richards | Susceptibility of adult Biomphalaria glabrata to Schistosoma mansoni infection[END_REF], and [START_REF] Basch | Intermediate host specificity in Schistosoma mansoni[END_REF]. Later, numerous studies confirmed strong variations in host susceptibility or parasite infectivity, depending of the geographic origin of the host and parasite strains used [START_REF] Lie | Specificity of natural resistance to trematode infections in Biomphalaria glabrata[END_REF]Richard and Shade, 1987;[START_REF] Morand | Parasite-host coevolution and geographic patterns of parasite infectivity and host susceptibility[END_REF][START_REF] Webster | Coevolution and compatibility in the snail-schistosome system[END_REF]. Our study addressed another component of this snail/schistosome compatibility polymorphism to the extent that the susceptibility / unsusceptibility level of different strains of B. glabrata was determined when confronted to the same panel of different S. mansoni strains. By doing this, we were able to characterize each B. glabrata strain by what we defined as its "multi-parasite susceptibility phenotype" and we may better compare the efficiency of their defense mechanism against not only one, but a diversity of schistosome stocks. In the same time, we characterized each of the S. mansoni strain used by their "multi-host infectivity phenotype" that reflects the level of infectivity they display when confronted to a diversity of snail stocks. Clearly the results obtained, showed marked differences between both host and parasite strains in their capacity to resist or infect the diversity of parasites or hosts proposed, respectively. This offers the possibility to select experimentally new homogeneous susceptible/unsuceptible phenotypes of snails for further studies.

A matching phenotype-phenotype interaction

As previously demonstrated for one of the host/parasite combination tested in this study (i.e. SmGUA/BgGUA, see [START_REF] Theron | Effects of laboratory culture on compatibility between snails and schistosomes[END_REF], we note that all the dose-response curves obtained for each of the 20 host/parasite couples analyzed showed the same shape.

They are characterized by a rapid increase of the infection rates between 1 and 10-20 miracidia per snail, then an asymptote that leveled off at different values as a function of the strain used. For combinations with compatibility level lower than 100%, the parasite intensities increased with increasing miracidial doses but not the proportion of infected hosts (at 20, 30 and 50 miracidia/snail). This may indicate that there is no threshold dose for which the parasite would be able to overwhelm the host defenses. Similar pattern of dose-response curves with the plateau that leveled below 100% were obtained with higher miracidial doses (80 and 100 Mi/snail, data not shown). Increasing the miracidial dose simply involves sampling a larger fraction of the phenotypic diversity in the parasite strain. In other words, a dose of 20 miracidia per snail contains, for all the S. mansoni strains, most of the possible phenotypes that potentially match with the host phenotype challenged. These results agree with the hypothesis that the success or failure of a challenge by one parasite mainly depends on the matched/mismatched status of the host and parasite phenotypes (Bash, 1975;[START_REF] Theron | Are Biomphalaria snails resistant to Schistosoma mansoni ?[END_REF][START_REF] Theron | Effects of laboratory culture on compatibility between snails and schistosomes[END_REF][START_REF] Mitta | Compatibility polymorphism in snail/schistosome interactions: from field to theory to molecular mechanisms[END_REF]. Taking into account the little water volume used (5 ml) for the exposition of individual snails to the miracidia, we assume that all miracidia had contact with the host. Despite this high host/parasite contact, only a small proportion of miracidia (between 20 and 40%) is not recognized by the innate defense system of the host and develops into Msp, the other larvae are encapsulated or degenerated [START_REF] Theron | Schistosoma mansoni: distribution patterns of miracidia among Biomphalaria glabrata snail as related to host susceptibility and sporocyst regulatory processes[END_REF]. This means that within the same snail host, a specific defense response occurs against some incompatible entered miracidia but without effect on neighboring compatible larvae. The reason of the observed different fates of the nondeveloping MSp (encapsulated vs degenerated) has until now received little attention. The absence of hemocytic reaction around the degenerated MSp strongly suggests unidentified humoral factors responsible for the nondevelopment of these larvae [START_REF] Sullivan | Schistosoma mansoni, NIH-SM-PR2 strain, in susceptible and nonsusceptible stocks of Biomphalaria glabrata: comparative histology[END_REF][START_REF] Vasquez | Further characterization of passively transferred resistance to Schistosoma mansoni in the snail intermediate host Biomphalaria glabrata[END_REF]. Such humoral reaction against MSp was always observed in case of immune priming of B. glabrata (Sire el al., 1998;[START_REF] Portela | Evidence for specific genotype-dependent immune priming in the Lophotrochozoan Biomphalaria glabrata snail[END_REF]. Indeed, MSp of the challenge infection of B. glabrata degenerate progressively within the host tissues without cellular encapsulation. Then, in case of "simultaneous" pluri-miracidial exposure (in our case 8 hours of exposition), not all miracidia penetrate the snail in the same time and we could hypothesize that incompatible miracidia penetrating first and recognized as non-self should be rapidly encapsulated while those penetrating later after this primary hemocyte-mediated defense should be stopped in their development by the presence of toxic plasma molecules and died undergoing gradual degeneration. Physiological unsuitability of the intermediate host was also proposed as a possible factor for the nondevelopment of some MSp [START_REF] Vasquez | Further characterization of passively transferred resistance to Schistosoma mansoni in the snail intermediate host Biomphalaria glabrata[END_REF]. As emphasized by [START_REF] Basch | Intermediate host specificity in Schistosoma mansoni[END_REF], compatibility is tested independently for each entering miracidium and each exposed snail. The phenotype (susceptible vs unsusceptible) of a host is expressed as a function of the parasite genotype it harbors and reciprocally, the phenotype (infective vs uninfective) of a parasite is expressed as a function of the genotype of the particular host that it enters.

Variations in host susceptibility and parasite infectivity

According to the pattern of the dose-response curves, the value of the infection rates at the plateau for 20 Mi/snail was retained to define and compare the level of susceptibility/unsusceptibility between the snail/schistosome combinations studied and a radar graph presentation will facilitate such comparisons (Fig. 5 and6). By the side of the host strains (Fig. 5), a gradient of increasing multi-parasite susceptibility could be established. Snails of the BS90 strain appeared as those having the best efficient internal defense system against the four strains of S. mansoni tested. However we note that within this highly un-susceptible strain, that was historically considered as "totally resistant" to both new and old world S. mansoni strains [START_REF] Ittiprasert | Reversing the resistance phenotype of the Biomphalaria glabrata snail host Schistosoma mansoni infection by temperature modulation[END_REF] half of the B. glabrata snails was unable to eliminate S. mansoni of the SmLE strain. The BgBAR strain, unsusceptible to SmGUA is poorly susceptible to SmBRE and SmVEN, but totally susceptible to SmLE. The BgBRE strain is 100% susceptible to three parasite strains but highly unsusceptible to SmGUA.

BgGUA and BgVEN strains show similar pattern and appear as the most susceptible to all parasite strains. None of the snail strains used was totally (100%) susceptible to the four S. mansoni strains respectively (Fig. 5).

Marked variations are also observed as related to the infectivity spectrum displayed by the parasite strains, ranging from the SmGUA poorly infective or un-infective towards the 5 host strains, to SmLE that is able to infect 100% of the four host strains and half of the BgBS90 snails (Fig. 6). Except for the BgGUA/SmGUA combination (37% of infection), all the other homopatric host/parasite couples showed 100% of compatibility. Among the homopatric combinations it is interesting to note that for the snail stock BgBAR that yields the highest multi-parasite un-susceptibility level, its corresponding parasite strain (SmLE) displays on the contrary the highest multi-hosts infectivity level. Such negative correlation between host susceptibility and parasite infectivity was however not so apparent for the other snail/schistosome homopatric couples.

We must emphazized that all the S.mansoni and B. glabrata strains used in this study were maintained in the laboratory during several years and their infectivity and susceptibility phenotypes are not representative of the corresponding natural populations [START_REF] Theron | Effects of laboratory culture on compatibility between snails and schistosomes[END_REF][START_REF] Bech | Follow-up of the genetic diversity and snail infectivity of a Schistosoma mansoni strain from field to laboratory[END_REF]. However data collected during many years of routine maintenance of S. mansoni strains in our laboratory do not show significant change in their compatibility phenotypes after their establishment (personal observation). More precisely, previous molecular analyses using 15 microsatellite markers carried out on the SmGUA/BgGUA combination, have revealed that a loss of genetic diversity occured gradually from the field S. mansoni isolate to the last laboratory generation analysed after three years of cycling [START_REF] Bech | Follow-up of the genetic diversity and snail infectivity of a Schistosoma mansoni strain from field to laboratory[END_REF]. What was interesting is the fact that, while the compatibility level decreases sharply between the field isolate and the first laboratory generation, later the degree of compatibility does not evolve and remains at the same level during all the successive laboratory generations despite repeated bottlenecks at each maintenance cycle. Then, a substantial level of compatibility polymorphism can be maintained independently of the decrease of neutral genetic diversity [START_REF] Bech | Follow-up of the genetic diversity and snail infectivity of a Schistosoma mansoni strain from field to laboratory[END_REF]. Such phenomenon could be linked to recently described mechanisms able to generate high diversification levels in polymorphic antigens of S. mansoni [START_REF] Mitta | Compatibility polymorphism in snail/schistosome interactions: from field to theory to molecular mechanisms[END_REF] through somatic recombination events and regulatory mechanisms occuring during their expression (see below) .

Possible underlying molecular mechanisms

The co-evolution dynamics between hosts and parasites involves huge reciprocal selective pressures on both protagonists. This co-evolutionary dynamics leads to an arms race expected to shape molecular determinants playing a key role at the core of the interaction.

Relatively few reports have evaluated the impact of these reciprocal antagonistic pressures on the molecules involved in host defense mechanisms and parasite infectivity. Recent molecular approaches of the mechanisms underlying this compatibility polymorphism have identified a group of polymorphic antigens (the SmPoMucs ) expressed by the miracidia of S. mansoni that interact with polymorphic and/or diversified immune recognition molecules (the FREPs) of B. glabrata (Roger et al., 2008a, b, c;[START_REF] Moné | An exemple of molecular co-evolution: Reactive oxygen species (ROS) and ROS scavenger levels in Schistosoma mansoni/Biomphalaria glabrata interactions[END_REF][START_REF] Hanington | A somatically diversified defense factor, FREP3, is a determinant of snail resistance to schistosome infection[END_REF][START_REF] Mitta | Compatibility polymorphism in snail/schistosome interactions: from field to theory to molecular mechanisms[END_REF]. Co-evolutionary dynamics could also play on effector pathways as we demonstrated a reciprocal adaptation between host reactive oxygen species (ROS) and parasite ROS scavenger traits [START_REF] Moné | An exemple of molecular co-evolution: Reactive oxygen species (ROS) and ROS scavenger levels in Schistosoma mansoni/Biomphalaria glabrata interactions[END_REF]. Indeed, comparison between two homopatric couples of B. glabrata/S. mansoni with different compatibility level showed that higher ROS-production capacity by the host was associated with a higher ROS-scavenging ability by the parasite (highly compatible couple) and in contrast, lower ROS production by the host with lower ROS-scavenging by the parasite (less compatible combination).

Then compatibility polymorphism in B. glabrata/S. mansoni interaction could be the result of multigenic and "multi-mechanism" processes in which the relative weight of recognition mechanisms of the host and antigenic diversity of the parasite could be in balance with the effector ability (ROS production) of the host and anti-effector mechanism (ROS scavengers) of the parasite. This hypothesis could be tested in the different strains used in the present study by analysing the polymorphism occurring on FREPS and SmPoMuc to evaluate the investment of each strain in these recognition/antigenic variation processes. We could also envisage more global transcriptomic or proteomic approaches to take into account the mechanism variability we can have during co-evolutionary processes between these different populations. This would be facilitated by the use of discreet host phenotypes selected for their spectrum of susceptibility/un-susceptibility towards several strains of S. mansoni.

Selection of multi-parasite (un)-susceptible host phenotypes

A snail strain is not homogenous but composed of multiple host phenotypes in respect of their susceptibility/un-susceptibility towards the diversity of parasite phenotypes, then, the distribution of these phenotype frequencies within a host strain may be modified by selection.

Most of the studies on variation of snail susceptibility were interested in experimental selection of host stocks yielding either 0% or approaching 100% infection frequencies when exposed to only one particular strain of schistosome [START_REF] Mascara | Schistosoma mansoni: continuous variation in susceptibility of the vector snail of schistosomiais, Biomphalaria tenagophila I. Self-fertilization-lineage[END_REF][START_REF] Richards | Genetics of Biomphalaria glabrata and its effect on the outcome of Schistosoma mansoni infection[END_REF][START_REF] Webster | Selection and strain specificity of compatibility between snail intermediate hosts and their parasitic schistosomes[END_REF]. As we know that a host strain un-susceptible to one parasite strain could be totally susceptible to another one, such an approach gives limited information about the real efficiency level of its defense system against the diversity of parasite phenotypes. Based on the results we obtained by exposing different snail strains to different strains of schistosomes, the new idea would be to select different stocks of snails showing different spectrum of susceptibility/un-susceptibility vis-à-vis of several parasite strains. The figure 7 shows how selection patterns could be operated on the different strains of B. glabrata used in order to produce homogenous snail stocks of different multi-strain (un)susceptibility types. Within the BgBS-90 strain, by successive elimination of the snails susceptible to SmLE, we can select a snail stock with a phenotype (A) characterized by its high defense efficiency since capable to eliminate four different strains of schistosome.

Eliminating snails susceptible to SmBRE and SmVEN within the BgBAR strain, we can select another host phenotype (B), un-susceptible to three parasite strains (SmVEN, SmGUA and SmBRE). Within both, BgGUA and BgVEN strains, elimination of snails susceptible to SmGUA and SmBRE allows creating two snail stocks of phenotype (C) both un-susceptible to two parasite strains. In the same way using the BgBRE strain and by successive elimination of the few snails susceptible to SmGUA, we can select another phenotype (D) un-susceptible to only one strain of schistosome. Finally with snails of the BgGUA and BgVEN, but in this case by increasing the frequence of snails susceptible to both SmGUA and SmVEN, we will select a phenotype (E) with the less efficient defense system since totally susceptible to the four different strains of schistosomes.

Conclusion

Our study through reciprocal cross-infections between five strains of B. glabrata and four strains of S. mansoni (i.e. 20 different homopatric and heteropatric host-parasite combinations) confirms the high degree of compatibility polymorphism that occurs in this host/parasite system. Differences in dose-response curves that characterize each snail stock (i.e. increase rate of infection at low miracidial doses then level of the asymptote) could result from the variations of the number and frequence of compatible combinations within each strain of B. glabrata when confronted to the different strains of schistosomes. By using the same panel of parasite strains against each snail stock, our results demonstrate that some B. glabrata strain have defense mechanisms more efficient than other strains through probably a better capacity to recognize a larger diversity of parasite phenotypes present within each schistosome strains and perhaps also a higher effector ability in ROS production. Thus depending on the genotypic diversity in the host and parasite isolates, the snail-schistosome compatibility would be determined by a combination of changes and a balance in recognition and effector mechanisms that determine the success or failure of the infection.

The compatibility polymorphism exists not only at this inter-strain level but also between individual hosts of a same strain. A snail population is not homogenous with regard to the susceptibility to schistosome miracidia. Within a strain, each individual host possesses its specific capacity of defense against all (unsusceptibility) or solely a part (susceptibility) of the phenotypic diversity of the miracidia from a parasite strain. The same diversity exists also for the parasite strains with some of then being more infective toward the panel of B. glabrata stock than other used in this study. Each parasite strain discriminates between the different individual host phenotypes of a snail strain. This variation of susceptibility between and within snail strains gives the opportunity, through experimental directional selection, to develop new and more homogenous snail stocks differing by their multi-parasite (un-) susceptibility phenotype (i.e., the number of schistosome strains with which there are not partially, but totally (un-)compatible). This will be a useful tool for future comparative functional studies conducted on different effectors mechanisms or recognition processes involving other immune receptors or antigens to better understand the genetics and molecular basis of the compatibility polymorphism in this host/parasite model.

To conclude and as previously mentioned, we must keep in mind that levels of compatibility characterizing laboratory strains are not representative of their corresponding field populations [START_REF] Theron | Effects of laboratory culture on compatibility between snails and schistosomes[END_REF][START_REF] Bech | Follow-up of the genetic diversity and snail infectivity of a Schistosoma mansoni strain from field to laboratory[END_REF]. Then we should proceed with caution to the possibility of extrapolating results of such studies in order to discuss epidemiological implications as for example, geographical compatibility variations and colonization risk of new areas or disease spreading that might occur under effects of climate and global changes (Morley, 2011). However, such laboratory studies on a diversity of parasite and host strains, provide detailed informations on molecular determinants playing a key role in the host-parasite compatibility interaction. This may contribute to the search for molecular markers which might inform on the potential efficiency of the defense system of snails from a wild population and on the infectivity capacity of a schistosome population towards snail hosts .

Legends to Figures

Fig. 1

-

  Fig. 1 -Infection rates (% ± 1 standart error) of individual snails of the five Biomphalaria glabrata strains (BgBRE, BgVEN, BgGUA, BgBAR and BgBS-90) exposed to increasing doses of miracidia (Mi) from the four strains of Schistosoma mansoni (SmBRE, SmGUA, SmVEN and SmLE).

Fig. 2 -

 2 Fig.2-Infectivity (% of infected snails ± 1 standart error) of four strains of Schistosoma mansoni (SmBRE, SmGUA, SmVEN and SmLE) towards the five Biomphalaria glabrata host strains(BgBRE, BgVEN, BgGUA, exposed to increasing doses of miracidia (Mi).

Fig. 3 -

 3 Fig. 3 -Evolution of parasite intensities (mean number of mother sporocysts (Msp) per infected snails ± 1 standart error): A) after exposure of BgBRE snails to increasing doses of miracidia (Mi) from the four strains of Schistosoma mansoni (SmBRE, SmGUA, SmVEN and SmLE); B) after exposure of Biomphalaria glabrata snails from the five strains (BgBRE, BgVEN, BgGUA, BgBAR and BgBS-90) with increasing doses of miracidia (Mi) from the SmBRE strains of Schistosoma mansoni. Percentages in brackets correspond to infection rates of the host-parasite combinaison for a dose of 50 Mi/snail.

Fig. 4 -

 4 Fig.4-Average transformation rates of miracidia into mother sporocysts for the different doses of miracidia (Mi) at exposure time: A) for each schistosome strains, all host strains included; B) for each host strains, all schistosome strains included. Dashed lines represent the total average for each strain including all miracidial doses (except for 1 Mi).

Fig. 5 -

 5 Fig. 5 -Radar graph representation of the multi-parasite susceptibility phenotypes characterizing each Biomphalaria glabrata strain (BgBS-90, BgBAR, BgBRE, BgGUA and BgVEN) vis-à-vis of the four strains of Schistosoma mansoni (SmLE, SmVEN, SmBRE and SmGUA). Infection rate (%) and parasite intensity (number of mother sporocysts (Msp) per snail) values are those corresponding to an exposure dose of 20 miracidia/snail.

Fig. 6 -

 6 Fig. 6 -Radar graph representation of the multi-host infectivity phenotypes characterizing each strain of Schistosoma mansoni (SmBRE, SmGUA, SmVEN and SmLE) vis-à-vis of the five Biomphalaria glabrata strains (BgBS-90, BgBAR, BgBRE, BgGUA and BgVEN).Infection rate (%) and parasite intensity (number of mother sporocysts (Msp) per snail) values are those corresponding to an exposure dose of 20 miracidia/snail.

Fig. 7 -

 7 Fig. 7 -Selection patterns on each Biomphalaria glabrata strain, to create new discreet host phenotypes: (A) from the BgBs90 strain, un-susceptible to 4 Schistosoma mansoni strains (SmBRE, SmGUA, SmVEN and SmLE); (B) from the BgBAR strain, un-susceptible to 3 parasite strains (SmBRE, SmGUA and SmVEN); (C) from the BgGUA and BgVEN strains, un-susceptible to 2 parasite strains (SmBRE and SmGUA); (D) from the BgBRE strain, unsusceptible to one parasite strain (SmGUA); (E) from the BgVEN and BgGUA strains, susceptible to 4 parasite strains ( see text for more details).

Fig. 8

 8 Fig. 8

Acknowledgements

The authors are indebted to E. S. Loker (Center for Evolutionary and Theoretical Immunology, University of New Mexico, Albuquerque, USA) for providing the BS-90 stock of snails, to Diana Ballen (Instituto Venezolano de Investigaciones Científicas (IVIC), Caracas, Venezuela) for the snails and schistosomes from Venezuela and G. Oliviera (Instituto Nacional de Ciência e Tecnologia em Doenças Tropicais, Fundacao Oswaldo Cruz, Brazil) for the BgBAR and SmLE strains. This work was funded by the ANR (grant # 25402 BiomGenIm; ANR-07-BLAN-0214-03), the CNRS, and the UPVD. The funders had no role in the study design, data collection, data analysis, decision to publish, or preparation of the manuscript.