Rishi Kumar

Sarshad Rommel

David Jauffrès

Pierre Lhuissier

Christophe L Martin

Effect of packing characteristics on the discrete element simulation of elasticity and buckling

Keywords: Discrete Element Method, buckling, elasticity

Discrete Element Method (DEM) simulations are used to model the elastic properties of a continuous material. The preparation route of the particle packing is shown to have a significant effect on the macroscopic properties. We propose simple relations, generalized from the mean field solution, that are able to fit DEM results. These relations introduce only basic microstructural features such as the coordination number and the packing density. When the tangential to normal stiffness ratio increases above unity, the material becomes potentially auxetic. Buckling is also explored with DEM, and results on cylindrical bars are compared to the classic Euler results for critical stress.

Introduction

Originally designed for granular materials [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF], the Discrete Element Method (DEM) has been employed in the recent years to model continua. DEM, * Corresponding author. Email: christophe.martin@grenoble-inp.fr

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when used to model continuous materials, bears some resemblance with lattice methods, which model a solid as a set of nodes (with no volume or mass) connected by beam elements through which both attractive and repulsive forces and torques can be transmitted [START_REF] Schlangen | Fracture Simulations of concrete Using Lattice Models: Computational Aspects[END_REF]. This type of method has proven very efficient for simulating fracture, although crack closure may be an issue since nodes carry no volume. A comparable route has been adopted by models to simulate the behavior of cohesive materials with bonded particles under a static or kinematic assumption [START_REF] Griffiths | Modelling of elastic continua using a grillage of structural elements based on discrete element concepts[END_REF][START_REF] Jefferson | The elastic response of a cohesive aggregate -a discrete element model with coupled particle interaction[END_REF][START_REF] Misra | Micromechanical model for cohesive materials based upon pseudo-granular structure[END_REF][START_REF] Kuzkin | Vector-based model of elastic bonds for simulation of granular solids[END_REF]. Peridynamics is also a similar method proposing a theory of continuum mechanics with a nonlocal model that can simulate fractures and discontinuities [START_REF] Silling | A meshfree method based on the peridynamic model of solid mechanics[END_REF]. However, the simulation of Poissons ratios other than 1/4 is not straightforward with Peridynamics [START_REF] Liu | Discretized peridynamics for linear elastic solids[END_REF].

In DEM , the solid is represented by bonded particles that rearrange under applied deformation according to Newton's second law. In general, these particles are spherical to ease distance calculation and to benefit from the existing literature on spherical contact forces [START_REF] Wang | A finite deformation method for discrete modeling: Particle rotation and parameter calibration[END_REF][START_REF] Tan | Discrete element method (DEM) modeling of fracture and damage in the machining process of polycrystalline SiC[END_REF][START_REF] André | Discrete element method to simulate continuous material by using the cohesive beam model[END_REF][START_REF] Jerier | Normal contact between rough surfaces by the Discrete Element Method[END_REF]. In specific applications, it is more interesting to use other shapes, like tubes for example [START_REF] Leblicq | Discrete element modelling of bendable tubes[END_REF]. Nevertheless, spheres have the advantage of ensuring isotropy of the contact network when packed randomly [START_REF] Tavarez | Discrete element method for modelling solid and particulate materials[END_REF][START_REF] André | Discrete element method to simulate continuous material by using the cohesive beam model[END_REF]. Also, they may be used to obtain a mesh for complex geometries, for example when starting from three-dimensional images of real objects [START_REF] Hedjazi | Experimental investigation and discrete simulation of fragmentation in expanded breakfast cereals[END_REF].

Most of the above-cited examples of DEM applications for continuum are actually interested in breaking the initial continuum material into discrete entities to model damage and fracture. This is because DEM has shown a great ability to model fracture in continuous material. Indeed, the inher-ent discrete nature of DEM provides an effective formulation with in-built natural discontinuities. DEM has the additional advantage, as compared to the lattice or to the finite element methods, of dealing naturally with post-fracture stages where crack closure may play a role.

In this paper, we aim to provide relations that link the bulk elastic properties of a solid simulated by a packing of bonded particles to the microscopic and macroscopic characteristics of the packing. These relations introduce macroscopic basic parameters that define the structure of the packing, such as the coordination number (the average number of bonds per particle) and the packing density. The microscopic properties of the bonds, i.e. their normal stiffness and their tangential to normal stiffness ratio, are also included.

These relations are generalization of the classical mean field solutions applied to a random packing of spherical particles. They should provide simple and useful guidelines for the generation of particle packings. We explore also the capability for DEM to reproduce auxetic (negative Poisson's ratio) materials. Finally, the possibility to use DEM to model buckling in quasistatic conditions is demonstrated. DEM-type simulations were carried out on half-spherical shell under dynamic conditions, but were not compared quantitatively to analytical results [START_REF] Kuzkin | Vector-based model of elastic bonds for simulation of granular solids[END_REF]. Here we attempt a direct comparison with Euler's formulation under quasi-static conditions on a cylindrical bar.

Model description

Spherical particles with radius R are connected to each other through solid bonds that transmit normal and tangential forces and resisting moments (Fig. 1). An equivalent radius at the bond between two spheres of radii and R q is defined as:

R p R p T N M T M N R q l pq
R * = R p R q R p + R q (1)
The normal and tangential forces acting on the bond are given by simple linear elastic laws:

N = -K N δ N n = -2Σ N R * δ N n (2)
T = -K T δ T t = -2Σ T R * δ T t (3)
where δ N and δ T are the normal and tangential relative displacements between the two particle centers and n and t are the unit vectors normal and parallel to the contact plane, respectively. The normal and tangential stiffnesses K N and K T are size dependent. Thus, we prefer using material parameters Σ N = K N 2R * and Σ T = K T 2R * , which have the unit of stress and allow macroscopic elastic properties to be independent of the sphere size. Noting:

α = Σ T Σ N (4)
we restrict our investigation to the range 0 < α ≤ 10. The standard case is given by 0 < α < 1, while we will show that α ≥ 1 potentially leads to auxetic properties (negative Poisson's ratio). Forces are taken positive in tension while the tangential force opposes the accumulated tangential displacement.

Damping forces are included for computational convenience. They oppose the normal relative displacement velocity at the bond:

N v = -η n dδ N dt n (5)
where η n is the damping coefficient, which is chosen as a fixed fraction ξ of the critical damping:

η n = 2ξ 2mΣ N R * (6)
A value of 0.05 for ξ ensures underdamped behavior together with negligible effect on macroscopic stresses in quasi-static conditions. Similarly, damping forces oppose the tangential relative displacement velocity with ξ = 0.025.

Bonds transmit resisting moments, M N and M T , in the normal and tangential directions with a formulation originally proposed by Potyondy and

Cundall [START_REF] Potyondy | A bonded-particle model for rock[END_REF]:

M N = -4Σ T (R * ) 3 θ N (7)
M T = -2Σ N (R * ) 3 θ T ( 8
)
where θ N and θ T are the accumulated relative rotations in the normal and tangential directions in the contact framework (Fig. 1). Note that in a packing with multimodal size particles, the stiffness and resisting moments from one bond to another are distributed due to the use of the equivalent radius R * in Eqs. [START_REF] André | Discrete element method to simulate continuous material by using the cohesive beam model[END_REF] to [START_REF] Gaspar | A granular material with a negative Poissons ratio[END_REF].

The macroscopic stress tensor in the packing is calculated from Love's formulation [START_REF] Weber | Recherches concernant les contraintes intergranulaires dans les milieux pulvérulents[END_REF][START_REF] Christoffersen | A micromechanical description of granular material behavior[END_REF]:

Σ ij = 1 V contacts F i l pq,j , (9)
where the summation is made on all contacts and where V is the sample volume, F i is the i th component of the total contact force (with normal and tangential terms), and l pq,j is the j th component of the l pq vector connecting the centers of two particles p and q (Fig. 1). The simulations described hereafter have been carried out on our in-house code dp3D.

Sample preparation

Three dimensional cubic samples were prepared by packing 5000 spherical particles within a box bounded by periodic conditions. Under periodic boundary conditions, when a particle protrudes outside the periodic cell through a given face, it interacts with the particles on the opposite face. The first preparation step consists of creating a gas of particles. This is attained by locating particles randomly into the volume with the only constraint that they do not contact each other. Using a packing of nearly monomodal particles ( R ± 5%) with an average radius R, a relative density of D 0 = 0.3 is initially obtained. Ten such initial samples were generated using different random seeds to compute standard deviations for each set. As described in the next section, the standard deviation on the elastic properties was small enough to consider that ten samples for each set of conditions are sufficient.

Following this first step, a relative density of 0.65 is sought for, using three different preparation routes. The lengths of the cubic simulation box, after densification to 0.65, are L x = L y = L z = 32 R. We found that, under periodic conditions, this size together with 5000 particles is enough to obtain statistically meaningful results.

The first route simply prescribes an affine displacement to all particles to impose a macroscopic isostatic densification until the preset relative density is attained (D = 0.65). This densification technique leads to significant relative interpenetration between particles. We define this packing preparation route as unjammed.

In the second method, starting from the initial relative density of 0.3, the initial gas of particles is isostatically densified by moving inwards the periodic bounds. No normal tensile force nor tangential friction nor resisting moment is transmitted (i.e. only normal repulsive forces are used) at this stage. Weak jamming is obtained by imposing a control pressure P c to the packing (using Eq. ( 9)) with a proportional controller that dictates the hydrostatic strain-rate to the periodic bounds as described elsewhere [START_REF] Martin | Influence of adhesion and friction on the geometry of packings of spherical particles[END_REF].

The control pressure is several orders of magnitude smaller than the stiffness

(P c /Σ N = 1.0E -07
). The stiffness of particles is thus large enough to ensure that relative interpenetration between particles is negligible at this stage.

Force equilibrium is imposed in a quasi-static configuration by resolving explicitly the second law of Newton as classically done in the DEM and by ensuring that the total force applied on particles is smaller than 10

-3 N max ,
where N max is the maximum contact force on the particle. This scheme is imposed until a preset relative density of D 0 = 0.5 is attained. This relative density is much below the Random Close Pack relative density (0.63-0.64).

Such a low value is still interesting because it ensures a fast sample preparation since particles rearrange easily. This packing is further densified, using affine displacement, up to D = 0.65. We define this packing preparation as weakly jammed.

The third method aims at obtaining a fully jammed packing by continuing the densification scheme described above at the control pressure P c until the densification rate attains very low values [START_REF] Martin | Influence of adhesion and friction on the geometry of packings of spherical particles[END_REF]. At this point, we consider that further densification is prevented by the full jamming of the particle assembly. A few particles (rattlers) with less than 4 contacts are still observed (< 2%). All other particles cannot be displaced without generating an increase in contact forces with neighboring particles, thus qualifying this packing as jammed [START_REF] Torquato | Multiplicity of Generation, Selection, and Classification Procedures forJammed Hard-Particle Packings[END_REF]. The relative density of this jammed packing is D = 0.636 ± 0.0004 and the coordination number is Z = 6.03 ± 0.011. As with the two other methods, this packing is further slightly densified with affine displacement of particles to attain D = 0.65.

The final affine displacement imposed to particles to attain the final packing density 0.65 allows all three packings to have the exact same density. It modifies only very slightly the microstructure of the jammed packing. We have studied the fabric tensor [START_REF] Satake | Fabric tensor in granular materials[END_REF] of the packings generated through this methodology and found that they did not exhibit any sign of anisotropy.

We have thus three preparation methods: unjammed, weakly jammed, and fully jammed, which lead, after the last affine densification to three packings that exhibit the same number of particles and the same relative density 0.02, 6.56 ± 0.02, respectively. Fig. 2 shows the typical packings associated with each preparation route. It is difficult to discern differences between the three packings. However, the coordination number of each particle is a clear indication of the marked characteristics of the jammed microstructure as compared to the unjammed and weakly jammed.

The differences between the preparation routes is also clear when examining Fig. 3, which plots the distribution of interparticle distances between contacting particles. Fig. 3 shows that the unjammed packing exhibit a very large distribution, entirely resulting from the affine densification process imposed to the initial gas of particles. On the opposite, the jammed packing presents a very narrow range of interparticle distance and these distances are close to the diameter of particles (small interpenetration of particles).

The weakly jammed packing shares important qualitative features with the jammed packing. Although it exhibits a broader distribution and smaller interparticle distances as compared to the jammed packing, it still presents the same trends as the jammed packing. In short, the unjammed packing is characterized by a small number of contacts that are very broadly distributed in size, while the jammed packings have a large number of smaller contacts.

From these three initial packings, bonds are installed between two particles p and q with radii R p and R q when the following criterion is met:

κ (R p + R q ) ≥ |l p,q | (10)
where |l p,q | is the center to center distance between particles p and q (Fig. 1), and κ is the interaction range (κ ≥ 1.0). Four values for κ were tested: κ = 1.000, 1.025, 1.050, 1.075. κ = 1.000 leads to the classic granular description of the packing where only particles in contact are bonded. For κ > 1, an interaction range bonds particles that are not geometrically in contact but are neighbors (Fig. 4). This feature provides the possibility to adjust the degree of interlocking of the particles forming the numerical medium as proposed by [START_REF] Jerier | Normal contact between rough surfaces by the Discrete Element Method[END_REF] or [START_REF] Scholtès | A DEM model for soft and hard rocks: Role of grain interlocking on strength[END_REF]. The average numbers of bonds per particle, denoted as Z b , are listed in Table 1. Z b is strictly equal to Z for κ = 1.000 and increases with increasing interaction range κ.

Elastic properties

The macroscopic elastic properties of the cubic numerical samples, of which the preparation has been described in the preceding section, have been obtained by testing them uniaxially under fully periodic conditions. Eq.

(.1) in the appendix is used to determine the macroscopic stress response to the imposed strain and the two macroscopic elastic constants, Young's modulus E and Poisson's ratio ν. The normal to tangential stiffness ratio (α = Σ T /Σ N ) has been varied from 0.005 to 10 to test the range of elastic constants attainable with potentially auxetic materials (ν < 0 for α > 1).

The two parameters discussed in the preceding section (D and Z b ), which describe the average microstructure of the packing of particles, can be advan-tageously used to propose a simple normalization of the Young's modulus:

Ẽ = E Z b DΣ N (11)
The rationale for Eq.( 11) is detailed in the appendix. It derives from the mean field assumption, which has been used to model particulate packings [START_REF] Christoffersen | A micromechanical description of granular material behavior[END_REF][START_REF] Liao | Stress-strain relationship for granular materials based on the hypothesis of best fit[END_REF][START_REF] Martin | Elasticity, fracture and yielding of cold compacted metal powders[END_REF] The effect of the ratio between the tangential and normal stiffnesses (α = Σ T /Σ N ) may be captured for the normalized Young's modulus by the following simple equation:

Ẽ = κ n 2π a 1 + a 2 α 4 + a 3 α (12)
where n is a rational number, and a i are fitted parameters. Similarly, for the Poisson's ratio:

ν = 1 -b 1 α b 2 + b 3 α (13)
where b i are fitting parameters. The simple form of Eqs. ( 12) and [START_REF] Jerier | Normal contact between rough surfaces by the Discrete Element Method[END_REF] originate from a generalization of the equations that describe the elastic behaviour of an aggregate of bonded particles under the approximations of a mean field or of a static solution [START_REF] Liao | Stress-strain relationship for granular materials based on the hypothesis of best fit[END_REF], as detailed in the Appendix.

We first specialize this study to α values in between 0 and 1, which is the standard case. For this range, Fig. 5 shows the normalized Young's modulus multiplied by 2πκ -n and the Poisson's ratio ν for the four values of interaction ranges κ and with the three preparation methods. The value of n is fitted to obtain an approximate master curve for all simulation points. Error bars are calculated from the standard deviations on ten simulations. The standard deviations are less than 0.05 and 0.01 for the normalized Young's modulus and Poisson's ratio, respectively. They are thus barely discernible in Fig. 5, indicating small dispersion from one random sample to another. The fact that all curves fall on a master curve indicates that our simple normalization of the Youngs modulus by the coordination number and by the relative density (Eq. ( 11)) is effective.

For all packings, the normalized Young's modulus increases and the Poisson's ratio decreases as the tangential stiffness increases as predicted qualitatively by the mean field model (see appendix) and by the static hypothesis model from Liao et al. [START_REF] Liao | Stress-strain relationship for granular materials based on the hypothesis of best fit[END_REF]. Still, the normalized Young's moduli of unjammed and weakly jammed prepared packings are significantly smaller than those predicted by the mean field and static models. The fully jammed packing is much closer to the mean field model. In particular, the slope is very similar in a large range of α values. This is because the mean field model has been derived under the assumption of point contacts between particles that are distributed with uniform probability over their surface [START_REF] Walton | The effective elastic moduli of a random packing of spheres[END_REF]. The unjammed and weakly jammed packings hardly satisfy these two assumptions at all. Discrete element simulations consistently lead to a softer response as compared to the mean field model. This is expected since quasi-static DEM ensures force equilibrium, in contrast with the mean field method, which does not. It is expected that the mean field method, because of its imposed assumption on uniform strain, leads to a stiffer response than the true behavior.

Table 2 lists the values of the fitting parameters a i and b i . These equations allow the macroscopic elastic properties of a packing of particles to be predicted with simple forms using important microstructural parameters (bond coordination number, interaction range of these bonds, and density).

Of particular interest is the value of the a 1 parameter, which dictates the behavior of the packing when the bond tangential stiffness vanishes. As discussed by Liao et al. [START_REF] Liao | Stress-strain relationship for granular materials based on the hypothesis of best fit[END_REF], if a 1 tends to zero (as in the static hypothesis or for unjammed packings), a packing with smooth particles (zero bond tangential stiffness) has no shear modulus but retains bulk modulus if its Poisson's ratio tends to 0.5. In that case, the packing resists isotropic bulk stresses but no deviatoric stress. It thus behaves as a fluid. Reversely, if a 1 is non-zero, packings do retain shear modulus and behave solid-like.

The same methodology is used to obtain the behaviour of packings in the range 1 ≤ α ≤ 10. These packings potentially lead to auxetic materials (negative Poisson's ratio) as shown previously in 2 dimensions from analytical considerations similar to those developed in the appendix [START_REF] Bathurst | Note on a random isotropic granular material with negative poisson's ratio[END_REF] or by discrete element simulations [START_REF] Gaspar | A granular material with a negative Poissons ratio[END_REF]. The same tendencies are observed in this range of α values than in the standard 0 < α ≤ 1 range (Fig. 6). The normalized Young's modulus is in relatively good accordance with the mean field model for the fully jammed packing, while it deviates markedly from this model and from the static hypothesis model for the two other preparations. Also, the Poisson's ratio is further decreasing with increasing stiffness ratio α values down to negative values. Poisson's ratio of the order of -0.5 are obtained for α = 10. Introducing the interaction range parameter κ into the fitting Eq. ( 12) allows a master curve Ẽκ -n to be obtained. However, this was not possible with the Poisson's ratio, and we prefer to present directly the Poisson's ratio values as a function of α without proposing a complex correlation between Poisson's ratio and α.

Application example: buckling of a cylindrical bar

Using the sample preparation method described in section 3, we have built cylindrical samples with length l and radius r. However, instead of periodic conditions, a rigid cylinder and two planes were used as boundary conditions. Once jammed, the cylinder and the planes were removed and bonds were installed. Fixed-fixed conditions are applied as boundary conditions by blocking translation and rotation of the particles in contact with the two planes.

We investigated l/r ratios in between 10 and 120 with the same number of particles per unit length. Thus, for l/r = 10, 10 000 particles were used while 100 000 particles were used for l/r = 100. Three samples were generated for a given l/r value to estimate dispersion. This was simply obtained by using different random seeds for the initial gas of particles. We have limited the sample number due to the CPU intensiveness of the simulations involving buckling. Simulations with 100 000 particles and large strains (to attain buckling), although heavily parallelized, are too long to allow for good statistics with a larger number of samples.

As in the preceding section, the same preparation routes were tested (unjammed, weakly jammed and jammed). A value α = 0.2 was chosen for the tangential to normal stiffness and the interaction range κ was set to unity. These values lead to a Poisson's ratio of the order of 0.2. No initial imperfection in the geometry, in the material properties or in the loading direction were introduced. This in contrast with the method classically used in Finite Element Method [START_REF] Chajes | Stability Behavior Illustrated by Simple Models[END_REF][START_REF] Schenk | Buckling analysis of cylindrical shells with random geometric imperfections[END_REF][START_REF] Meimand | Imperfection Sensitivity and Reliability Using Simple Bar-Spring Models for Stability[END_REF], where a small imperfection in the element nodal points or in the load application is imposed to ensure the initiation of buckling. However, it should be clear that the randomness of the packing provides some imperfection at the microscale.

Euler's critical stress is given by [START_REF] Euler | Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti. Additamentum I. De curvis elasticis. MarcumMichaelem Bousquet[END_REF]:

σ Euler = π 2 E 4 k l r 2 (14)
where k = 0.5 for the fixed-fixed conditions used here. Alternative solutions have been proposed. For example, Mazzilli [START_REF] Mazzilli | Buckling and post-buckling of extensible rods revisited: A multiple-scale solution[END_REF] has shown that the analytical solution:

σ M azzilli = 1 -1 - π 2 k l r 2 E 2 (15)
compares better to the critical load computed by numerical integration of the exact equation of equilibrium than Euler's formulation. However, the difference is only noticeable for bars with small slenderness. Fig. 7 shows typical stress-strain curves. Axial stresses and strains are normalized by σ Euler and ǫ Euler respectively (ǫ Euler = σ Euler /E). Fig. 7 indicates that the critical strain is larger than the critical strain given by Euler's formulation. Also, the post-buckling behaviour depends on the l/r ratio with a more abrupt decrease of stress as the l/r ratio increases. As

shown in Fig. 7, buckling is characterized by a clear maximum in the stressstrain curve. However, a non-linear region exists prior to this maximum.

Still, we compare the critical Euler buckling stress to this maximum stress. in some dispersion illustrated by error bars in Fig. 8. The three preparation routes lead to larger maximum stresses than the Euler's formulation for large l/r. However, our definition of the critical stress (see typical stressstrain curves Fig. 7) should lead in any case to an upper bound. Also, the overestimation of the critical stress would suggest that dynamics may be at play in our simulations which attempt to approach the quasi-static limit.

Indeed, dynamics lead to buckling stresses that exceed Euler critical stress [START_REF] Karagiozova | Dynamic Elastic-Plastic Buckling of Structural Elements: A Review[END_REF]. We have investigated this issue by calculating the normalized kinetic energy per particle:

Ẽkin = E kin n max (N R) (16)
where E kin is the kinetic energy of the system, n the number of particles, N and R the normal force (Eq. ( 2)) and the particle radius, respectively. Ẽkin should stay reasonably below 10 -08 to ensure quasi-static conditions [START_REF] Agnolin | Internal states of model isotropic granular packings. I. Assembling process, geometry, and contact networks[END_REF]. Fig. 9 indicates that this condition is satisfied before buckling for the two lowest values of l/r. For the largest slenderness, dynamics may explain partially the stress overestimation exhibited by DEM simulations.

Concluding remarks

The elastic properties of numerical samples made of particles linked by elastic bonds have been investigated. The preparation route used for the samples clearly has an effect on these properties. We have shown that when the ratio of tangential to normal stiffness (α) is smaller than unity, the Young's modulus and the Poisson's ratio are well described by simple equations (Eqs. ( 12) and ( 13)), which introduce microstructural properties of the packing of spheres that has served to generate the numerical sample. These microstructural properties are the average coordination number and the packing density.

The proposed equations are generalizations of the mean field and static solutions. They give approximate values of the elastic properties that should be expected for a given numerical sample, knowing its packing history.

Our results also indicate that, with the standard bond model used here, it is difficult to attain Poisson's ratio above 0.3. This is inherent to the fact that the Poisson effect is not introduced at the micro level in the bonds, since discrete element simulations are based only on pair interactions. Taking into account a more accurate strain field would require a more complex coupling between bonds. This has been proposed analytically [START_REF] Jefferson | The elastic response of a cohesive aggregate -a discrete element model with coupled particle interaction[END_REF] and developed numerically with an explicit calculation of bond interaction on the elastic field [START_REF] Jauffrès | Simulation of the elastic properties of porous ceramics with realistic microstructure[END_REF].

The three preparation routes investigated here involve very different effort in terms of CPU time to create the packing that meshes the numerical sample.

The unjammed route is a very rapid method since it only requires the random deposition of spheres into the volume. The fully jammed preparation route is much more CPU demanding if a density close to the Random Close Pack (RCP) is sought for. The weakly jammed route is an interesting compromise.

It allows for a wider span of attainable Poisson's ratio (also in the auxetic domain) than the unjammed route but is much less CPU demanding since densification rates in the low density domain are much faster than close to the RCP.

For bonds with tangential to normal stiffness greater than unity, we have shown that hypothetical auxetic materials built from discs [START_REF] Bathurst | Note on a random isotropic granular material with negative poisson's ratio[END_REF][START_REF] Gaspar | A granular material with a negative Poissons ratio[END_REF], may also be generated in 3 dimensions. The requirement that bonds be softer in the normal direction than in the tangential direction is difficult to satisfy with standard materials. However, some interesting designs have been proposed in the literature for granular-like materials [START_REF] Gaspar | A granular material with a negative Poissons ratio[END_REF] .

As an application example, we have shown that the simple bond model proposed here, with no built-in buckling mechanism, is capable of simulating buckling. A buckling application example of DEM-type simulations was proposed on half-spherical shell under dynamic conditions [START_REF] Kuzkin | Vector-based model of elastic bonds for simulation of granular solids[END_REF]. Here, we have attempted to ensure quasi-static conditions to enable quantitative comparison with Euler's formulation. It should be clear that the DEM is not the most appropriate method for simulating buckling. The finite element method, supported by decades of research on the subject, is a far more effective method. However, when post-buckling is accompanied by fracture, the DEM may be an interesting option. This is because, the topological modifications (branching, bifurcation, and new surface generation) that come with fracture are easier to capture with DEM than with the finite element method. while the normalized Young's modulus is given by Ẽ = 5 2π α 2 + 3α (.6)

We use generalized forms of Eqs. ((.2), (.3), (.5) and (.6)) in section 4 to fit the numerical results for the Poisson's ratio and the Young's modulus.

Figure 1 :

 1 Figure 1: Sketch of the bond between two spherical particles p and q transmitting normal and tangential forces and resisting moments.

(

  D = 0.65), but their microstructures differ markedly as demonstrated by the increasing coordination number with jamming extent: Z = 4.52±0.02, 5

Figure 2 :

 2 Figure 2: Typical packings for the three preparation routes investigated, starting from the same gas of particles. Packing density is 0.65 after affine densification. Colors indicate the number of contacts per particle (coordination number).

Figure 3 :Figure 4 :

 34 Figure 3: Distributions of interparticle distance l pq normalized by the average diameter of particles, for the three preparation routes investigated. Packing density is 0.65 after affine densification. Average coordination are indicated for each preparation route.

Table 1 :

 1 Average number of bonds per particle for the 3 preparation routes at D = 0.65 for the 4 values of the interaction range κ Z b preparation κ = 1.000 κ = 1.025 κ = 1.050 κ = 1.075 unjammed 4.52±0.02 5.00±0.01 5.49±0.02 5.99±0.01 weakly jammed to 0.5 5.39±0.02 5.81±0.01 6.23±0.01 6.56±0.01 jammed to 0.637 6.56±0.02 7.46±0.01 8.06±0.01 8.57±0.02

Figure 5 :

 5 Figure 5: Evolution of (a) the normalized Young's modulus Ẽ and (b) the Poisson' ratio for the three preparation routes and for various values of the interaction range κ, in the 0 ≤ α ≤ 1 range.

Figure 6 :

 6 Figure 6: Evolution of (a) the normalized Young's modulus Ẽ and (b) the Poisson' ratio for the three preparation routes and for various values of the interaction range κ, in the 1 ≤ α ≤ 10 range .

Figure 7 :

 7 Figure 7: Stress-strain curve examples for three values of the ratio l/r.

Fig. 8

 8 Fig. 8 summarizes the simulation results for the three preparation routes investigated here. The critical stress obtained from the simulation is normalized by σ Euler . The randomness of the granular structure inevitably results

Figure 8 :

 8 Figure 8: Critical buckling stresses normalized by Euler's stress versus the l/r ratio for the three preparation routes investigated. Mazzilli's solution (Eq. (15) (---) is plotted for completeness[START_REF] Mazzilli | Buckling and post-buckling of extensible rods revisited: A multiple-scale solution[END_REF].

Figure 9 :

 9 Figure 9: Normalized kinetic energy evolution versus normalized strain for the three values of the ratio l/r, as in Fig. 7.

Table 2 :

 2 Parameters a i , n and b i of Eqs. (12) and (13) fitted for the three preparation routes (0 < α ≤ 1).

	preparation	a 1	a 2	a 3 n	b 1	b 2	b 3
	unjammed	0.18 5.45 9.64 5	0.24 3.38 6.91
	weakly jammed to 0.5 0.42 7.45 8.68 2	0.55 3.02 5.75
	jammed to 0.637	1.09 6.72 5.12 1	0.80 3.07 4.07
	mean field	2	3	1	0	1	4	1
	static hypothesis	0	10	6	0	1	2	3

Table 3 :

 3 Parameters a i , n and b i of Eqs. (12) and (13) fitted for the three preparation routes (1 ≤ α ≤ 10, potentially auxetic materials).

	preparation	a 1	a 2	a 3 n	b 1	b 2	b 3
	unjammed	1.10 0.86 0.72 6	0.54 6.41 1.10
	weakly jammed to 0.5 1.62 1.33 0.72 3	0.75 6.00 1.07
	jammed to 0.637	2.06 2.07 0.78 1.5	0.91 5.09 0.99

Appendix. Elastic behaviour of an aggregate of bonded particles under the approximation of a mean field solution

The spherical particle kinematics may be simply treated under the approximation of a mean field solution (Voigt hypothesis) for a monomodal packing. In that case, the motion of each particle center relative to a reference particle is dictated by the macroscopic deformation gradient and force equilibrium is not enforced. We start from the formulation derived by [START_REF] Liao | Stress-strain relationship for granular materials based on the hypothesis of best fit[END_REF], which considers the case of bonds transmitting both normal and tangential forces (but no resisting moment). The model is similar to those proposed in [START_REF] Christoffersen | A micromechanical description of granular material behavior[END_REF][START_REF] Walton | The effective elastic moduli of a random packing of spheres[END_REF][START_REF] Storakers | The viscoplastic compaction of composite powders[END_REF][START_REF] Martin | Elasticity, fracture and yielding of cold compacted metal powders[END_REF]. Starting from Eq. ( 9) under the assumption of a random packing of particles of uniform radius R, it leads to the stress-strain relationship under the mean field assumption (Eq. ( 38) in [START_REF] Liao | Stress-strain relationship for granular materials based on the hypothesis of best fit[END_REF]):

The Poisson's ratio is then readily given by

while the normalized Young's modulus simplifies to

An alternative expression has been derived by [START_REF] Liao | Stress-strain relationship for granular materials based on the hypothesis of best fit[END_REF], based on the static hypothesis (or best-fit model). It leads to:

Thus, for the static hypothesis the Poisson's ratio is ν = 1 -α 2 + 3α (.5)


Q Q Zhou

Jean-Paul Renard

Gaelle Le Friec

Vincent V Brochard

Nathalie Beaujean

Yacine Cherifi

Alexandre Fraichard

Jean Cozzi

Generation of Fertile Cloned Rats by Regulating Oocyte Activation

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The rat is a reference animal model for physiological studies and for the analysis of multigenic human diseases such as hypertension, diabetes and neurological disorders (1). Genetic manipulation in the rat is hampered by the lack of suitable technologies such as embryonic stem cells (ES) which are routinely used to generate targeted mutations in the mouse. Cloning through somatic cell nuclear transfer (SCNT) is a potential alternative approach in species for which ES technologies are unavailable. However, all previous efforts to clone rats have been unsuccessful, with developmental arrest at implantation stage (2 and references therein).

The fine-tuned coordination between nuclear transfer and timing of oocyte activation is critical to the outcome of somatic cloning. This coordination is hampered in the rat because almost all the oocytes spontaneously, although abortively, activate within 60 min of their removal from oviducts (3). Such rapid but incomplete activation process is not encountered in other cloned species. To allow embryo reconstruction before the onset of oocyte activation, we initially developed a one-step SCNT procedure for the rapid substitution of the endogenous meiotic metaphase by an exogenous mitotic one. This latter was isolated from synchronized cultured fetal CD-SpragueDawley fibroblasts (12.5 dpc). Individual mitotic nuclei were injected into a recipient OFA-SpragueDawley oocyte, from which the meiotic metaphase was withdrawn while removing the micropipette from cytoplasm after injection. However, within 30 min after recovery 70% of oocytes showed clear morphological evidence of spontaneous release from MII (Fig. 1A). When activation of cloned embryos (Fig. 1B) was induced and maintained by exposure to a cdc2-specific kinase inhibitor (Butyrolactone;150 µM, 2 h) (4), 201 out of 221 reconstructed embryos expelled the polar body and subsequently divided into two-cell embryos. Their transfer into OFA-SpragueDawley foster mothers (11 recipients, 221 embryos) resulted in 9 implantation sites, but no fetal development.

Forty percent of oocytes selected for SCNT turned to be already activated as evidenced by disjoined sister chromatids moving to opposite poles (Fig. 1C). These observations strongly supported the view that despite rapid manipulation, most of the oocytes were not suitable for cloning. As activation is triggered by the inactivation of MPF activity through a proteasome-mediated cyclin degradation pathway, we used MG132, a protease inhibitor that reversibly blocks the first meiotic metaphase-anaphase transition in the rat [START_REF] Josefsberg | [END_REF]. We found that this drug also reversibly stabilized most oocyte MII metaphases for up to 3 hours (77%; Fig. 1D).

We then collected oocytes in the presence of MG132 (5µM) (4) as described above. SCNT was performed within 30 minutes of drug removal. 876 embryos were implanted into 12 pseudopregnant foster female rats. At 12.5 dpc, the females were sacrificed, and four females contained 16 fetuses. Thirteen of the fetuses, obtained from 3 females, were viable with beating hearts (Fig 1E). In the next series of experiments, we transplanted 129 cloned embryos into two foster mothers and allowed them to go to term. Only one foster mother contained viable fetuses and this animal delivered three live male pups of fibroblast origin as unambiguously demonstrated by microsatellite marker analysis (4). One normal sized pup (5.9 g) died a few hours after birth. The other two pups grew to sexual maturity (Fig. 1F) and generated normal progeny.

We have also obtained normal progenies (size, weight and development) from two additional cloned female rats demonstrating the potential of the technique for the development of fertile rat lines of both sexes (supporting online text).

Our data highlight the importance of adapting the SCNT procedure to oocyte physiology for successful cloning. Recently, random mutagenesis has been proposed to generate knock-out rats (6). Our results however pave the way for more extensive genetic modifications such as conditional knock-out and gene replacement which are required to produce relevant models of human diseases.

Published online 25 September 2003; 10.1126/science.1088313 Include this information when citing this paper.

Fig.1. (A)

A freshly recovered rat oocyte with a marked cytoplasmic protrusion (arrowhead) revealing an ongoing activation process. (B) A rat oocyte used for micromanipulation. (C) DNA status of oocytes after recovery in standard conditions reveals the presence of two separate sets of chromatids whereas after recovery in MG132 supplemented medium (D) reveals a stabilized metaphase plate. Normal cloned fetus (E) at day 14.5 dpc and two adult cloned male rats (F) obtained using MG132 supplemented medium. Scale bar 10 µm (A) to (D) and 1 mm (E).

Developmental Biology and Reproduction Unit, INRA, Jouy en Josas, France.

Institute of Zoology, CAS, Beijing, RP of China.

genOway, Lyon, France.*To whom correspondence should be addressed. E-mail: renard@jouy.inra.fr

Generation of Fertile Cloned Rats by Regulating Oocyte Activation

Qi Zhou, 1,2 Jean-Paul Renard, 1 * Gaëlle Le Friec, 3 Vincent Brochard, 1 Nathalie Beaujean, 1 Yacine Cherifi, 3 Alexandre Fraichard, 3 Jean Cozzi 3


Sergio Luiz

Almeida Camargo

Tiphaine Aguirre-Lavin

Pierre Adenot

; Thamiris

Dornelas Araujo

Vivian Rachel

Araujo Mendes

Iuri Drumond

Nathalie Beaujean

Eliza Diniz Souza
email: elizadds@gmail.com

Heat shock during in vitro maturation induces chromatin modifications in the bovine embryo

Keywords: heat stress, oocyte, heterochromatin, histone methylation

Heat stress compromises bovine oocyte developmental competence but the effects of high temperature during oocyte maturation on embryo chromatin organization is unknown. In this study bovine oocytes were exposed to heat shock (41°C) for 12h during in vitro maturation and then submitted to in vitro fertilization. The heat shock did not affect (P>0.05) the cleavage but reduced (P<0.01) the blastocyst rate on Day 7 and Day 8. No effect (P>0.05) on total cell number was found but the heat shock increased (P<0.05) the proportion of apoptotic cells in blastocysts at Day 8. Immunofluorescence analysis of H3K9me3 and HP1 was performed in embryos at 52 h post in vitro fertilization. An accumulation of H3K9me3 in the nuclei of embryos derived from heat-shocked oocytes at 4-cell and 8cell stages was found. Also, a non-expected higher proportion (P<0.05) of 4-cell stage embryos displaying nuclei with increased HP1 fluorescence was observed, suggesting an abnormal chromatin compaction in embryos from heat-shocked oocytes. Embryos at 8-cell stage derived from heat-shocked oocytes displayed lower (P<0.05) relative amount of HSP40 transcripts than control ones. In conclusion, heat shock before fertilization has an effect on embryo chromatin, influencing the accumulation of H3K9me3 and HP1 in early embryos as well as further development.

Introduction

Heat stress is one of the main causes of bovine low reproductive performance in the tropics as well as in hot seasons in subtropical countries. Its effects on reproduction is supported by in vivo and in vitro studies which show that bovine oocytes are sensitive to heat stress at various stages of follicular development. It can disturb the physiology of the follicle-enclosed oocyte during the lengthy period of follicular development, leading to an oocyte with low competence for fertilization and subsequent embryo development [START_REF] Roth | Physiology and endocrinology symposium: Cellular and molecular mechanisms of heat stress related to bovine ovarian function[END_REF].

Exposure of cultured cumulus-oocyte complexes (COCs) to elevated temperatures (41°C) during the first 12 h of in vitro maturation decreases the cleavage rate [START_REF] Roth | Involvement of apoptosis in disruption of developmental competence of bovine oocytes by heat shock during maturation[END_REF][START_REF] Roth | Disruption of nuclear maturation and rearrangement of cytoskeletal elements in bovine oocytes exposed to heat shock during maturation[END_REF] and the proportion of oocytes that develop toward blastocyst stage [START_REF] Edwards | Differential responses of bovine oocytes and preimplantation embryos to heat shock[END_REF][START_REF] Roth | Involvement of apoptosis in disruption of developmental competence of bovine oocytes by heat shock during maturation[END_REF][START_REF] Ju | Heat shock reduces developmental competence and alters spindle configuration of bovine oocytes[END_REF]. The heat shock impairs both nuclear and cytoplasmic maturation events, such as translocation of cortical granules to the oolemma [START_REF] Payton | Susceptibility of bovine germinal vesicle-stage oocytes from antral follicles to direct effects of heat stress in vitro[END_REF], cytoskeleton rearrangement [START_REF] Roth | Disruption of nuclear maturation and rearrangement of cytoskeletal elements in bovine oocytes exposed to heat shock during maturation[END_REF], and spindle formation [START_REF] Roth | Disruption of nuclear maturation and rearrangement of cytoskeletal elements in bovine oocytes exposed to heat shock during maturation[END_REF][START_REF] Ju | Heat shock reduces developmental competence and alters spindle configuration of bovine oocytes[END_REF]. Those alterations may potentially lead to incomplete nuclear maturation, fertilization failure, and/or abnormal zygote formation [START_REF] Payton | Susceptibility of bovine germinal vesicle-stage oocytes from antral follicles to direct effects of heat stress in vitro[END_REF][START_REF] Roth | Disruption of nuclear maturation and rearrangement of cytoskeletal elements in bovine oocytes exposed to heat shock during maturation[END_REF]. Moreover, heat shock-induced apoptosis has been documented for bovine oocytes exposed to elevated temperatures during maturation [START_REF] Roth | Involvement of apoptosis in disruption of developmental competence of bovine oocytes by heat shock during maturation[END_REF].

One of the little-studied effects of heat stress on oocytes is epigenetic modification. It is clear that the epigenome can be modulated by nutritional [START_REF] Murdoch | Nutritional influence on epigenetic marks and effect on livestock production[END_REF][START_REF] Longo | Of mice and humans through the looking glass: "reflections" on epigenetics of lipid metabolism[END_REF]) and environmental conditions [START_REF] Horowitz | Epigenetics and cytoprotection with heat acclimation[END_REF]. Indeed, the epigenome provides an interface between genes and the environment and may be considered as a potential mechanism for environmental adaptation [START_REF] Rissman | Minireview: transgenerational epigenetic inheritance: focus on endocrine disrupting compounds[END_REF]. For instance, heat acclimation can cause transcriptional changes in genes associated to chromatin remodeling, heat shock proteins and apoptosis in cardiac cells of rats, and those changes on chromatin structure may take part in an acclimation memory phenomenon [START_REF] Tetievsky | Physiological and molecular evidence of heat acclimation memory: a lesson from thermal responses and ischemic cross-tolerance in the heart[END_REF]. Maternal heat stress increases embryo mortality in hens but supplementation of zinc in the maternal diets can reduce such effect, which was suggested to be due to increased antioxidant ability via epigenetic activation [START_REF] Zhu | Maternal dietary zinc supplementation enhances the epigenetic-activated antioxidant ability of chick embryos from maternal normal and high temperatures[END_REF]. Heat shock can alter the expression of some target genes in bovine immature oocytes [START_REF] Gendelman | In vivo vs. in vitro models for studying the effects of elevated temperature on the GV-stage oocyte, subsequent developmental competence and gene expression[END_REF] and in vitro-fertilized embryos [START_REF] Camargo | Absence of sperm factors as in the parthenogenesis does not interfere on bovine embryo sensitiveness to heat shock at pre-implantation stage[END_REF], suggesting an epigenetic modification during oocyte growth and embryo development caused by the elevated temperature.

Epigenetic modifications related to chromatin structure include changes on DNA methylation and post-translational modifications on histones. Highly condensed chromatin (heterochromatin) is typically transcriptionally repressive whereas non-condensed chromatin (euchromatin) is permissive to transcription [START_REF] Cheung | Epigenetic regulation by histone methylation and histone variants[END_REF]. During early development the embryo undergoes genome activation that requires epigenetic modifications such as changes on histone methylation and chromatin organization [START_REF] Sepulveda-Rincon | Early epigenetic reprogramming in fertilized, cloned, and parthenogenetic embryos[END_REF]. In bovine, the major embryonic genome activation (EGA) occurs between 8-16 cell stages [START_REF] Frei | Qualitative and quantitative changes in protein synthesis occur at the 8-16-cell stage of embryogenesis in the cow[END_REF][START_REF] Graf | Genome activation in bovine embryos: review of the literature and new insights from RNA sequencing experiments[END_REF]) and is associated with chromatin remodeling (Pinchugin et al. 2010). Those modifications are essential to reprogram the parental genome to embryonic genome and then generate healthy embryos [START_REF] Beaujean | Epigenetics, embryo quality and developmental potential[END_REF]. In cloned bovine embryos derived by nuclear transfer, abnormal reprogramming of heterochromatin from the somatic donor cells has been observed (Pinchugin et al. 2010) and is believed to affect embryonic development (Pinchugin et al. 2010, Le Bourhis et al. 2010). We postulate that one of consequences of heat shock during oocyte in vitro maturation is modifications on chromatin organization of early stages embryos. Those changes could disturb the EGA, impairing further development toward the blastocyst stage.

In order to test whether heat shock during oocyte maturation could affect chromatin remodeling in the early stages embryo we evaluated the distribution of histone H3 lysine 9 trimethylation (H3K9me3) and heterochromatin protein 1 (HP1) by immunostaining of embryos at 4-cell and at 8-cell stages obtained by in vitro fertilization of heat-shocked oocytes. Histone H3 lysine 9 (H3K9) methylation is a transgenerational epigenetic modification conserved from yeast to humans [START_REF] Rissman | Minireview: transgenerational epigenetic inheritance: focus on endocrine disrupting compounds[END_REF]. H3K9 methylation has been shown to correlate with transcriptional repression and serve as a specific binding site for heterochromatin protein 1 (HP1), which recognizes H3K9-methylated chromatin, oligomerizes and forms a versatile platform that participates in diverse nuclear functions, ranging from gene silencing to chromosome segregation [START_REF] Eissenberg | The HP1 protein family: getting a grip on chromatin[END_REF][START_REF] Lachner | Methylation of histone H3 lysine 9 creates a binding site for HP1 proteins[END_REF][START_REF] Grewal | Heterochromatin revisited[END_REF]. During early development there is an asymmetry on H3K9 methylation between maternal and paternal chromatin in mice [START_REF] Liu | Regulation of histone H3 lysine 9 methylation in oocytes and early preimplantation embryos[END_REF]) and cattle [START_REF] Pichugin | Dynamics of constitutive heterochromatin: two contrasted kinetics of genome restructuring in early cloned bovine embryos[END_REF]. In bovine in vitro-fertilized embryos, H3K9 methylation is reduced between the 2-and 4-cell stages and increases from the 8-cell stage onward [START_REF] Santos | Epigenetic marking correlates with developmental potential in cloned bovine preimplantation embryos[END_REF][START_REF] Wu | Multiple histone site epigenetic modifications in nuclear transfer and in vitro fertilized bovine embryos[END_REF]. Despite the presence of methylated H3K9, there is no clear co-localization between HP1 and H3K9me3 in bovine embryos at the 2-and 4cell stages. In fact, co-localization between H3K9me3 and HP1 is only evident at the 8-cell stage [START_REF] Pichugin | Dynamics of constitutive heterochromatin: two contrasted kinetics of genome restructuring in early cloned bovine embryos[END_REF], concurrently with EGA. Changes in HP1 and/or H3K9 methylation levels could potentially disturb the EGA and impair embryonic development.

Thus, this study aimed to evaluate the effect of heat shock during oocyte in vitro maturation on development, gene expression and chromatin organization of in vitro-fertilized embryo.

Materials and Methods

Chemicals were purchased from the Sigma Chemical Co. (Saint Louis, MO, USA) unless otherwise indicated. All experimental procedures followed ethical guidelines for animal experimentation and were approved by local Committee (CEUA EGL 02/2015 protocol).

Experimental design

Three experiments were performed in order to evaluate the developmental competence of heatshocked oocyte after in vitro fertilization (n=569), the chromatin organization of 4-and 8-cell stage embryos derived from heat-shocked oocytes and the gene expression of 8-cell stage embryos derived from heat-shocked oocytes. Control groups were comprised of in vitro-fertilized embryos derived from oocytes not exposed to heat shock.

For the first experiment immature COCs were cultured in maturation medium at 38.5°C for 24 h (control) in 5.0% CO2 and humidified air or at 41.5°C for the first 12 h of a 24 h culture period (heat shock) in 7.0% CO2 and humidified air. After in vitro maturation (IVM), COCs were co-cultured with spermatozoa for 20 h for in vitro fertilization (IVF). Putative zygotes (n=470) were denuded of cumulus cells and then cultured in CR2aa at 5% CO2, 5% O2, and 90% N2 at 38.5°C in humidified air. Cleavage on day three (72 h post insemination) and blastocyst rates on day seven and day eight post-insemination was recorded. Hatching rates were recorded on day eight. Six replicates were performed. Total cell number and apoptosis index was determined in blastocysts at day eight. In the second experiment the heat shock, IVM, IVF and in vitro culture (IVC) were performed as described in the first experiment. Embryos at 4-and 8-cell stage at 52h post fertilization (n=58) were fixed in 4% (w/v) paraformaldehyde and chromatin staining was performed with primary and secondary fluorescent antibody to detect H3K9me3 and HP1. Embryos from four replicates were used in this experiment. For the third experiment, IVM, IVF and IVC were also performed as described in the first experiment. Embryos at 8-cell stage at 52h post fertilization were washed in PBS plus 0.1% polyvinyl alcohol and then rapidly frozen in liquid nitrogen and stored until total RNA extraction. Three pools of 10 embryos per group were used.

Oocytes recovery and in vitro maturation (IVM)

Oocytes recovery and IVM were performed as previously described by [START_REF] Camargo | Absence of sperm factors as in the parthenogenesis does not interfere on bovine embryo sensitiveness to heat shock at pre-implantation stage[END_REF], with a few modifications. Ovaries from predominantly Bos indicus crossbred cows were obtained at a local commercial slaughterhouse (Fripai, Juiz de Fora, MG, Brazil) and transported to the laboratory in saline solution at 34-36°C. Follicles with 3-8 mm diameter were aspirated and COCs with homogeneous cytoplasm and compact layers of cumulus cells were selected for IVM and randomly allocated into control (38.5°C) or heat shock (41.5°C for 12h) groups. In vitro maturation was performed for 24 h in tissue culture medium (TCM-199; Gibco Life Technologies, Inc., Grand Island, NY, USA) supplemented with 10% estrus cow serum and 20 μg/mL follicle stimulating hormone (FSH; Pluset, Calier, Barcelona, Spain) in 5% CO2 and humidified air. Heat shock was performed for 12h at 41.5°C under 7% CO2 followed by 12 h at 38.5°C under the same conditions of control group.

In vitro fertilization (IVF) of oocytes

Frozen/thawed motile spermatozoa from a single Holstein bull were obtained after centrifugation at 3.600×g for 7 min in Percoll (Nutricell Nutrientes Celulares, Campinas, SP, Brazil) discontinuous density gradient (45 and 90%). The pellet was then centrifuged again at 520×g for 3 min in Fert-TALP medium. For in vitro fertilization, in vitro-matured oocytes were co-cultured with 2 x 10 6 /mL spermatozoa for 20 h in Fert-TALP in a humidified atmosphere of 5% CO2 and 38.8°C in air.

In vitro embryo culture (IVC)

Presumptive zygotes were denuded in a PBS solution with 0.1% hyaluronidase and then cultured for eight days in modified CR2aa medium supplemented with 2.5% fetal calf serum (Nutricell Nutrientes Celulares, Campinas, SP, Brazil) in incubator at 38.5ºC under 5% CO2, 5% O2, 90% N2 and saturated humidity. Cleavage rate was evaluated on day three post-fertilization and blastocyst rate was evaluated on day seven and day eight as well as hatching rate.

TUNEL Labeling

Blastocysts on day eight from different replicates were submitted to terminal deoxynucleotidyltransferase-mediated dUTP-digoxigenin nick end-labelling (TUNEL) staining using a commercially available kit (Dead End Fluorimetric TUNEL System; Promega, Madison, WI, USA) according to manufacturer's instructions. Briefly, embryos were fixed in 4% paraformaldehyde at 4°C and then permeabilized with 0.2% Triton X-100 (Promega), both in PBS (Nutricell). Positive control embryos were previously treated with 8 units/mL DNase (Promega). After permeabilization, samples were incubated in 100 μL drops with reagent mix containing enzyme solution and 90% staining solution for 1 h at 37°C in a dark humid chamber. Negative control embryos were incubated only in the staining solution without enzyme solution. After that, embryos were stained with Vectashield (Vector Laboratories Inc., Burlingame, CA, USA) plus 406-diamidino-2-phenylindole (DAPI) and mounted on slides for evaluation by fluorescence microscopy. Total cell number and apoptotic cell number per embryo were counted, and apoptotic cell index was calculated as the proportion of apoptotic cell/ total cell number.

Immunofluorescence, confocal microscopy and image analysis

Immunofluorescence analysis of H3K9me3 and HP1 was performed in embryos with 4-and 8cells at 52 h post in vitro fertilization. Embryos were fixed in 4% paraformaldehyde and permeabilized with 1% Triton X-100 in PBS. Afterwards all samples were blocked with 2% BSA in PBS and then incubated with primary antibody. H3K9m3 labeling was performed with rabbit anti-H3K9m3 primary antibody (Millipore/Merk; 1:500 in PBS-BSA) and Cy3 anti-rabbit secondary antibody (1:200 in PBS+BSA). HP1 labeling was performed with mouse anti-HP1beta primary antibody (Euromedex, 1:200 in PBS-BSA) and FITC anti-mouse secondary antibody (1:200 in PBS+BSA).

Embryos were mounted on slide in Vectashield (Vector Laboratories, Burlingame, CA, USA) under conditions of preservation of their three-dimensional (3D) nuclear structure. They were observed with an inverted Zeiss LSM 700 confocal microscope (MIMA2 platform, INRA). The Z-stacks were acquired using a frame size of 1024x1024 with a pixel depth of 8 bits and 0.370 µm z-steps and with sequential multitrack scanning using the 405-, 488-and 555-nm wavelengths of the lasers. Images were processed using an in-house pipeline analysis software, developed with Python script language and ITK library (www.itk.org).

Spatial 3D datasets were imported into the pipeline using the Bio-Formats library (http://www.loci.wisc.edu), and multi-channels images were split to individual DNA, HP1 and H3K9me3 Z-stacks. Segmentation of nuclei, H3K9me3 and HP1 spots was done using three different processes. To outline correctly the nucleus, the DNA channel and the HP1 protein were combined (AddImageFilter operation) to improve the detection of the nuclear shape, because in cattle embryos, DAPI staining alone did not allow a perfect detection of the nucleus. The binary mask of each nucleus was obtained by selecting (threshold filter) the 10% of brightest pixels in the combined image and suppressing the smallest binary areas (opening filter). Then binary structures with empty areas were filled (fill holes filter) and nuclei were discriminating from artifacts by fixing a minimal size of objects (725 µm 3 ). Finally, a color label to identify each nucleus separately was assigned. This nuclear mask was the region of interest (ROI) in the following image processing of proteins channels. Since H3K9me3 was accumulated in bright spots in the nuclei, these ones were segmented using a maximum entropy threshold filter in the ROI. The H3K9me3 were labeled individually. For the HP1 signal, accumulation of proteins resulted in spots of very different sizes and of different brightness according to the embryo developmental stages. A 2D maximum entropy thresholding was performed to detect spots, but a lot of punctuate structures were detected when the HP1 signal was more homogeneous. To remove unspecific spots, a mask, which detected approximately the regions enriched in HP1, was used. It was obtained by applying a Gaussian filter (sigma = 0.3 µm) on the HP1 Z-stack, and a threshold filter keeping 5% of the brightest pixels. Then the HP1 spots were labeled individually. Measurements were done from the segmented nuclei, and spots image datasets.

In addition, detailed image analysis of H3K9me3 and HP1 fluorescent staining was performed in order to identify the proportion of embryos displaying different patterns of chromatin staining (Figure 1). Figure 1-A

RNA extraction, reverse transcription and relative quantification by polymerase chain reaction (RT-PCR)

Embryos at 8-cell stage were washed in PBS and then rapidly frozen in liquid nitrogen, and stored at -80°C. Three pools of ten embryos per group where used for total RNA extraction with RNeasy Micro Kit (Quiagen, Valencia, CA, USA) according to the manufacturer's instructions and treated with DNase to prevent DNA contamination. Elution was performed with 14µL of RNAase-free water. RNA samples (6 µL/reverse transcription reaction, equivalent to 4.3 embryos) were reverse transcribed using the SuperScript III First-Strand Synthesis Supermix (Invitrogen, Carlsbad, CA, USA) according to the manufacturer's instructions using oligo(dT)20 primers in a final volume of 20 µl (equivalent to 0.21 embryos/µL). Samples were first incubated at 65ºC for 5 min and then for 50ºC for 50 min. The reaction was terminated at 85ºC for 5 min and then chilled on ice. After that, RNase H was added to the samples and incubated at 37ºC for 20 min.

Relative quantification was performed in triplicate using real-time polymerase chain reaction (ABI Prism 7300 Sequence Detection Systems, Foster City, CA, USA). Reactions were prepared using a mixture of SYBR Green PCR Master Mix (Applied Biosystems), 0.1µM primers, nuclease-free water and reverse transcribed cDNA. The volume of reverse transcribed cDNA for every primer was previously calculated and it is shown in microliter and as embryo equivalent. For YWHAZ (Tyrosine 3monooxygenase/tryptophan 5-monooxygenase activation protein zeta) and GAPDH (Glyceraldehyde-3-phosphate dehydrogenase) were added 0.32µL of reverse transcribed cDNA per PCR reaction (equivalent to 0.07 embryo/reaction); for ACTB (beta actin), HSP40 (Heat shock 40kDa protein) and OCT4 (Octamer-binding transcription factor 4) were added 0.64µL/PCR reaction (0.14 embryo/reaction); for HSF1 (Heat Shock Factor 1) was added 1.92µL/PCR reaction (0.42 embryo/reaction) and for HSP90A1 (Heat Shock Protein 90 KDa class A member 1) was added 0.96µL/PCR reaction (0.21 embryo/reaction). YWHAZ, ACTB and GAPDH genes were used as endogenous references. The cDNA template was denatured at 95°C for 15 min, followed by 45 cycles of 94°C for 15 sec, annealing at specific temperature for each primer for 15 sec (Table 1) and extension at 60°C for 30 sec. Negative controls, comprising the PCR reaction mixture without nucleic acids, were also run with each group of samples. After each PCR run, a melting curve analysis was performed to confirm that a single specific product was generated. Primer efficiency was calculated using LinRegPCR software [START_REF] Ramakers | Assumption-free analysis of quantitative real-time polymerase chain reaction (PCR) data[END_REF]

Statistical Analysis

The rates of cleavage were evaluated on day three and blastocysts on day seven and eight after fertilization. Data of cleavage, blastocyst production and spots number and size were compared by analysis of variance. Apoptosis index was analyzed by logistic regression and total cell and apoptotic cell number by linear mixed model. Proportion of embryos displaying nuclei with different staining patterns was compared by Fischer Exact test. Relative transcript quantification was performed by Comparative Ct method and the data analyzed by mixed model using the Wise Fixed Reallocation Randomization TEST in REST software [START_REF] Pfaffl | Relative expression software tool (REST) for group-wise comparison and statistical analysis of relative expression results in real-time PCR[END_REF]. Values are shown as mean ± standard error mean (S.E.M.) and the significance level was P<0.05.

Results

Embryonic development and blastocyst total cell number and apoptosis

The heat shock during in vitro maturation did not affect the cleavage rate, ranging from 64.5% to 74.2% between treatments. However, a significant reduction in the proportion of putative zygotes that reached the blastocyst stage was observed in the heat shock group on day seven (P<0.01) and day eight (P<0.05) after fertilization (Table 2). In addition, heat shock reduced (P<0.05) the proportion of hatched embryos at day 8 when compared to control, suggesting a reduction in the embryo quality (Table 2).

Blastocyst total cell number was not affected (P>0.05) by the heat shock during oocyte in vitro maturation but the apoptotic cells index increased when compared to control (Table 3). Figure 2 shows representative images illustrating the total cell number and apoptotic nuclei in blastocysts from control (A and A`) and heat shock (B and B`) groups.

Chromatin structure analysis of embryos at early stages

For the quantitative analysis, number and size of spots (representing accumulation of H3K9me3 or HP1) in each nucleus were counted and measured. The number of H3K9m3 spots in the nuclei of 4cell stage embryos was greater (P<0.05) in heat shock group than in control one, whereas the spots mean size was not different. At 8-cell stage the number of spots was significantly lower (P<0.01) but the mean size was higher (P<0.02) in the heat shock than in the control group (Figure 3). That increase of spots size with a decrease of spot number at 8-cell stage in the heat shock group is likely due to increase of H3K9 trimethylation that results in the fusion of several spots forming bigger but fewer spots than in the control group. The mean size of H3K9me3 spots increased between 4-and 8-cell stages for both control (385±65 and 509±44 voxels, respectively) and heat shock (347±32 and 692±42 voxels, respectively) groups.

The number and mean size of HP1 spots in the nuclei were not statistically different (P>0.05) between heat shock and control group (Figure 3). However, in both cases, the total number of HP1 spots slightly decreased suggesting that the same mechanisms of fusion occur for HP1 and H3K9me3. It is also noteworthy that while HP1 spots increased in size between 4-cell and 8-cell stages (from 250±39 to 398±47 voxels, respectively) in the control group, it decreased in the heat shock group (404±88 and 280±27 voxels for 4-cell and 8-cell stages, respectively). The analysis of fluorescence enabled to distinguish embryos displaying different patterns of chromatin staining between control and heat shock groups. For H3K9me3 immunofluorescence there was an increased (P<0.01) proportion of embryos at 4-cell stage from heat shock group displaying nuclei with globular pattern (43.7%) when compared to control group (5.5%), whereas no difference (P>0.05) was found for the fibrous and spread patterns (Figure 4-A). At 8-cell stage, distribution of H3K9m3 was similar (P>0.05) between control and heat shock groups for globular and fibrous patterns (Figure 4-B) and no spread distribution was found in both groups. For HP1, the heat shock group presented higher (P<0.05) proportion of 4-cell stage embryos with fibrous pattern and lower (P<0.05) proportion with spread pattern than the control group, and no globular pattern was found (Figure 4-C). At 8-cell stage there was no statistical difference between the groups except that no spread pattern was found in nuclei of embryos in the heat shock group (Figure 4-D). Those results show that the heat shock during in vitro maturation increases the proportion of embryos displaying nuclei with an elevated accumulation of H3K9me3 (globular pattern) and HP1 (fibrous pattern) at the 4-cell stage.

Overall, the higher number of immunofluorescence spots indicates that an increased accumulation of H3K9me3 in the nuclei of embryos derived from heat-shocked oocytes can be found at 4-cell stage, one cell cycle earlier than nuclei of embryos derived from non-heat-shocked oocytes. This resulted in larger spots but in a small amount at 8-cell stage probably due to the union of the several spots found at 4-cell stage. That precocious accumulation can also be perceived by the high proportion of 4-cell embryos displaying nuclei with strong immunofluorescence staining for H3K9me3 when they are derived from heat-shocked oocytes. Similarly, the higher proportion of embryos displaying nuclei with HP1 fibrous patterns in the heat shock group at the 4-cell stage correlates with the slight increase observed in the size of HP1 spots between embryos from heat shock and control groups at that stage (250±39 vs. 404±88 voxels, respectively).

Effect of heat shock on transcript levels in embryos

The relative expression of four target genes (HSF1, HSP90A1, HSP40 and OCT4) was investigated in order to associate the heat shock during in vitro maturation with gene expression in embryos at 8-cell stage. The amount of HSP40 transcripts was lower (P<0.05) in embryos derived from heat-shocked oocytes while no difference (P>0.05) between groups was found for HSF1, HSP90 and OCT4 genes (Figure 5). HSP90A1,HSP40 and OCT4 in [START_REF] Pfaffl | Relative expression software tool (REST) for group-wise comparison and statistical analysis of relative expression results in real-time PCR[END_REF].

Discussion

Heat stress has been a constant concern during hot seasons in both tropical and subtropical regions in cattle production systems because of its effects on reproductive and productive performance. The deleterious effect on reproduction is associated to oocytes with poor developmental competence. As previously reported [START_REF] Edwards | Differential responses of bovine oocytes and preimplantation embryos to heat shock[END_REF][START_REF] Hansen | Adverse impact of heat stress on embryo production: causes and strategies for mitigation[END_REF][START_REF] Roth | Involvement of apoptosis in disruption of developmental competence of bovine oocytes by heat shock during maturation[END_REF][START_REF] Roth | Disruption of nuclear maturation and rearrangement of cytoskeletal elements in bovine oocytes exposed to heat shock during maturation[END_REF] we found that the heat shock during in vitro maturation decreases oocyte ability to develop toward blastocysts stage after in vitro fertilization and compromises embryo quality by increasing the apoptosis index. Such low developmental competence has been associated to several cytoplasmic, molecular and nuclear alterations [START_REF] Payton | Susceptibility of bovine germinal vesicle-stage oocytes from antral follicles to direct effects of heat stress in vitro[END_REF][START_REF] Roth | Disruption of nuclear maturation and rearrangement of cytoskeletal elements in bovine oocytes exposed to heat shock during maturation[END_REF][START_REF] Roth | Physiology and endocrinology symposium: Cellular and molecular mechanisms of heat stress related to bovine ovarian function[END_REF]; however, little is known about the effect of heat shock during oocyte in vitro maturation on chromatin remodeling post-fertilization. In this study we showed that alterations on chromatin organization can be found in early embryos derived from oocytes exposed to heat shock during in vitro maturation. The immunofluorescence analysis revealed that the nuclei of 4-and 8-cell embryos had higher accumulation of H3K9me3. Also, a higher proportion of 4-cell embryos presented nuclei with strong fluorescence for H3K9me3 and HP1 when compared to embryos derived from control oocytes.

In bovine, H3K9 methylation is normally low in embryos at 4-cell stage and increases from 8cell stage onwards [START_REF] Santos | Epigenetic marking correlates with developmental potential in cloned bovine preimplantation embryos[END_REF][START_REF] Wu | Multiple histone site epigenetic modifications in nuclear transfer and in vitro fertilized bovine embryos[END_REF]. In our study we found that the size of H3K9me3 spots increased from 4-cell to 8-cell stage in embryos from control and heat shock groups, confirming those previous studies. Interestingly, the mean size of HP1 spots increased from 4-cell to 8-cell in the control group only. As H3K9me3 is a binding site for HP1 [START_REF] Lachner | Methylation of histone H3 lysine 9 creates a binding site for HP1 proteins[END_REF], the increase of heterochromatinization at the 8-cell stage was expected as in the control group. However, it was not observed in the heat shock group. The size of HP1 spots reduced between the 4-and 8-cell stages in the heat shock group (404±88 vs 280±27 voxels, respectively), despite the notable increase of H3K9me3 spots size. The large HP1 spots size as well as the proportion of embryos with the fibrous pattern nuclei suggest that heterochromatinization is stronger in 4-cell stage embryos from heat-shocked oocytes when compared to control group or even to the 8-cell stage. Those findings indicate that the heat shock during oocyte maturation may hasten H3K9 trimethylation and heterochromatin formation in 4-cell stage embryos.

Methylation of H3K9 associated with heterochromatization has a repressive role on gene expression (Eissemberg & Elgin 2000[START_REF] Lachner | Methylation of histone H3 lysine 9 creates a binding site for HP1 proteins[END_REF][START_REF] Grewal | Heterochromatin revisited[END_REF] and this silencing is critical to ensure the appropriated gene activation during the EGA [START_REF] Nothias | Regulation of gene expression at the beginning of mammalian development[END_REF][START_REF] Mason | Chromatin and epigenetic modifications during early mammalian development[END_REF]. Previous study showed that there is a co-localization between H3K9me3 and HP1 at 8-cell stage in bovine embryos fertilized in vitro [START_REF] Pichugin | Dynamics of constitutive heterochromatin: two contrasted kinetics of genome restructuring in early cloned bovine embryos[END_REF]), which coincides with major EGA. The unexpected accumulation of H3K9me3 and HP1 in the nuclei of 4-cell embryos and the abnormal formation of numerous H3K9me3 condensed spots in 8-cell embryos derived from heat-shocked oocytes indicate alterations of chromatin remodeling that can perturb the EGA and, consequently, gene expression.

It was shown previously that expression of some genes, as POU5F1 and GAPDH, can be altered in 8-cell embryos derived from oocytes collected in the hot season [START_REF] Gendelman | In vivo vs. in vitro models for studying the effects of elevated temperature on the GV-stage oocyte, subsequent developmental competence and gene expression[END_REF]). In the present study we also found that the relative expression of specific gene can be altered in 8-cell embryos derived from oocytes exposed to heat shock during in vitro maturation. We analyzed the expression of genes that encode proteins associated to heat-shock (HSF1, HSP90 and HSP40) and pluripotency (OCT4). Despite there was no difference on relative abundance of HSF1, HSP90 and OCT4 transcripts between treatments, the expression of HSP40 was downregulated in those embryos derived from heatshocked oocytes, suggesting that the heat shock during oocyte maturation may perturb the expression of some genes in embryos at 8-cell stage. HSP40 is a co-chaperone with important role on cell protection under stress conditions [START_REF] Gotoh | hsp70-DnaJ chaperone pair prevents nitric oxide-and CHOP-induced apoptosis by inhibiting translocation of Bax to mitochondria[END_REF][START_REF] Lanneau | Apoptosis versus cell differentiation: Role of heat shock proteins HSP90, HSP70 and HSP27[END_REF]) and alterations on its expression may impair further embryonic development under in vitro suboptimal environment. However, alterations on gene expression caused by the heat shock during in vitro maturation may be not gene specific presuming that such alterations are due to a random alteration on chromatin remodeling. This may be one of the reasons for some early stage embryos develop toward blastocyst stage whereas others do not.

How the heat shock during IVM can affect embryo chromatin is not clear but it may be related to disturb on redox balance. Accumulation of reactive oxygen species (ROS) is known to cause DNA damage and affects histones and DNA methylation [START_REF] Niu | Oxidative stress alters global histone modification and DNA methylation[END_REF][START_REF] Mikhed | Redox regulation of genome stability by effects on gene expression, epigenetic pathways and DNA damage/repair[END_REF]. Recently it was shown that the paternal pronucleus retains the DNA methylation when the oocyte is fertilized with spermatozoa submitted to oxidative stress [START_REF] Wyck | Oxidative stress in sperm affects the epigenetic reprogramming in early embryonic development[END_REF]. As the heat shock increases the ROS production in bovine oocytes matured in vitro [START_REF] Ascari | Addition of insulin-like growth factor I to the maturation medium of bovine oocytes subjected to heat shock: effects on the production of reactive oxygen species, mitochondrial activity and oocyte competence[END_REF], the alterations found in our study could be in part associated to an increased level of ROS interfering in the chromatin organization in the first cell cycles of embryos derived from heat-shocked oocytes. This is the first study showing that the heat shock during oocyte in vitro maturation can alter the chromatin organization in early in vitro-fertilized embryos. We suggest that alterations on accumulation of H3K9me3 and HP1 caused by heat shock during in vitro maturation can negatively influence the chromatin remodeling during the EGA, affecting the expression of some genes, and contribute to impair the embryo development and quality, represented by lower blastocysts rate and higher apoptotic index.

In conclusion, the heat shock during in vitro maturation can induce alterations of H3K9me3 and HP1 accumulation in early bovine embryos, affecting embryo development and quality. Our data indicates that heat shock before fertilization can have an epigenetic effect on embryos and highlights the need of studies to understand how such alterations can influence the minor and major genome activation and how they can be modulated, in an effort to contribute to mitigate the heat stress effects on reproduction.

Disclosure statement

No potential conflict of interest was reported by the authors.

  illustrates representative merged images of immunofluorescence in 4-cell and 8-cell stages embryos. Figure1-B illustrates the patterns of chromatin compaction observed (as previously established in[START_REF] Pichugin | Dynamics of constitutive heterochromatin: two contrasted kinetics of genome restructuring in early cloned bovine embryos[END_REF]: spread (uniform staining), fibrous (small spots displaying various levels of staining) or globular (evident and big spots displaying strong staining).

Figure 1 .

 1 Figure 1. Representative images of the different chromatin compaction patterns established thanks to HP1 and H3K9me3 immunostaining in the nuclei of 4-cell (left) and 8-cell bovine embryos (right) showing that within each embryo the distribution pattern was similar within blastomeres (A, merged images). Scale bar: 15 µm. Detailed images analysis with computing tools was performed on individual nuclei (B) in which three different patterns were distinguished: (a) Spread: uniform staining, (b) fibrous: small spots displaying various levels of staining, and (c) globular: evident and big spots displaying strong staining. Those analysis were performed in 3D on confocal z-scan that were merged on these representative examples for better visualization with HP1 in green (a, b and c) and H3K9me3 in red (a`, b` and c`).

Figure 3 :

 3 Figure 3: Analysis of trimethylation of histone3 lysine 9 (H3K9m3) and heterochromatin protein 1 (HP1) in nuclei of four-cell and eight-cell stage bovine embryos derived from control oocytes or oocytes exposed to 12h of heat shock during in vitro maturation. Accumulation of HP1 or H3K9m3 is given by spot size and number per nucleus. Asterisks indicate differences between control and heat shock groups within the same patterns of staining (*P<0.05; **P<0.01). Spots size (volume) = voxels. Data of spots number and size were compared by analysis of variance.

Figure 4 :

 4 Figure 4: Percentage of four-cell and eight-cell stage bovine embryos displaying different patterns of histone 3 lysine 9 trimethylation (H3K9m3) and heterochromatin protein 1 (HP1) staining in the nucleus. Spread: uniform staining, fibrous: small spots displaying various levels of staining, and globular: evident and globular spots displaying strong staining. Control: embryos derived from nonheat-shocked oocytes; Heat shock: embryos derived from oocytes exposed to 12h of heat shock during in vitro maturation. N = number of embryos. Asterisks (*) indicate differences between control and heat shock groups within the same patterns of staining (P<0.01 for H3K9me3 and P<0.05 for HP1). Data of embryos displaying nuclei with different staining patterns was compared by Fischer Exact test.

  Figure 5: Relative abundance transcripts evaluation of HSF1, HSP90A1, HSP40and OCT4 in 8-cell stage bovine embryos derived from oocytes exposed to 12h of heat shock during in vitro maturation. Data of Heat Shock were compared to Control group (calibrator). Asterisk (*) indicates difference (P>0.05) between control and heat shock groups. Data from three replicates. Data of relative transcript quantification was performed by Comparative Ct method and analyzed by mixed model using the Wise Fixed Reallocation Randomization TEST in REST software[START_REF] Pfaffl | Relative expression software tool (REST) for group-wise comparison and statistical analysis of relative expression results in real-time PCR[END_REF].

Funding

  E.D. Souza was supported by scholarship from FAPES (Fundacao de Amparo a Pesquisa e Inovacao do Espírito Santo). This work was supported by National Council for Scientific and Technological Development (CNPq), Fapemig (APQ 543-14 and CVZ-PPM 107-17) and the Laboratoire d'Excellence Revive Labex (Investissement d'Avenir, ANR-10-LABX-73).

Table 1 .

 1 Primers used for real-time polymerase chain reaction

	for each reaction. The primer efficiency average was 1.92;

Table 2 .

 2 Cleavage rate (D3) and blastocysts rate at Day 7 (BLD7) and Day 8 (BLD8) and hatched blastocysts at Day 8 (HBLD8) for groups Control and Heat Shock during in vitro maturation. Values with different superscript letters in the same column differ (P<0.05). Values are shown as mean ± s.e.m.

	Treatments n	Cleavage (%) BLD7 (%)	BLD8 (%)	HBLD8 (%)
	Control	240	74.2±4.1 a	31.8±2.5 a	34.1±2.9 a	35.8±3.9 a
	Heat Shock 230	64.5±5.4 a	19.0±3.5 b	21.2±3.8 b	13.8±5.1 b

Table 3 .

 3 Means of total cells number, apoptotic cells number and apoptotic index of blastocysts to Control and Heat Shocked oocytes groups from oocytes in vitro maturated. Values with different superscript letters in the same column differ (P<0.05). Values are shown as mean ± s.e.m. N= embryos number.

	Treatments	n	Total Cells	ApoptoticCells	Apoptotic Index
	Control	20	113.15±6.62 a	10.05±0.99 a		8.75±0.69 b
	Heat Shock	21	96.29±6.59 a	12.48±1.16 a	12.34±0.80 a
				Figure 2. Representative images of
				DAPI	and	TUNEL-labeling
				embryos illustrating the total cell
				number and apoptotic nuclei
				(TUNEL positive) in bovine
				blastocysts derived from control
				oocytes or oocytes exposed to 12h of
				heat shock during in vitro
				maturation. Scale bar: 50 µm. DAPI
				labeling: A and B, TUNEL labeling:
				A` and B`. Control group: A and A`,
				heat shock group: B and B`.