Bruno Frka-Petesic

Jérôme Fresnais

Jean-François Berret

Vincent Dupuis

Régine Perzynski
email: regine.perzynski@upmc.fr.

O Sandre

Stabilization and controlled association of superparamagnetic nanoparticles using block copolymers

Keywords: Colloids, Nano-composites, Block copolymers, Magnetic liquids, Self-assembly

Mixing in aqueous solutions polyelectrolyte-neutral block copolymers with oppositely charged species, spontaneously forms stable core-shell complexes, which are electrostatically driven. We report here on the structural and orientational properties of such mixed magnetic nanoclusters made of magnetic iron oxide nanoparticles (MNP) and polyelectrolyte-neutral block copolymers. Small Angle Neutron Scattering and Transmission Electron Microscopy experiments allows to probe the innercore nanoparticle organization, leading to an average inter-particle distance and confirming the hierarchical internal structure of the clusters. Thanks to the MNP optical anisotropy, we also probe the under-magnetic field orientational properties of the core-shell clusters and their dynamical rotational relaxation.

Introduction

Building up hybrid organic/inorganic nano-and meso-clusters of controlled size and shape [1][2] is an important issue for many applications. In [3][4] Boal et al have suggested to use inorganic nanoparticles as building blocks combined together to synthesize colloidal or supracolloidal assemblies in a polymer-mediated "Bricks and Mortar" approach, to take benefit of the nanoparticle properties and to control as well the stability of complexes. Indeed, inorganic nanoparticles are emerging as central constituents for future nanotechnological developments spreading over a wide variety of material science applications, from catalysis and opto-electronics to biosciences. Some core-shell structures have already been investigated using polyelectrolyte-neutral copolymer as a mortar and either oppositely charged surfactants [5] or nanoparticules as "bricks" [6][7][8]. The core is Journal of Magnetism and Magnetic Materials 321 (2009) 667-670 then described as a complex coacervation microphase made of "bricks" connected with the polyelectrolyte blocks, while the shell is made of neutral chains organized in a dense brush corona that insures the stability of the whole aggregate. Employed as such fundamental bricks, magnetic nanoparticles (MNPs) made of iron oxides drive particular interest in material science and biomedicine as MRI contrast agents [6], useful for magnetic cell labelling [9] or hyperthermia [10]. We study here the structural and orientational properties of such MNP hybrid clusters .

Synthesis of the co-assembly

The MNP's used in this work to built up the particle/polymer clusters are negatively charged γ-Fe 2 O 3 maghemite nanoparticles, chemically synthesized in LI2C following the Massart's protocol for ionic ferrofluid (FF) synthesis. It consists in a polycondensation of iron salts in alkaline aqueous medium [11]. The obtained nanoparticles can be stabilized in strong acidic conditions (pH<2) thanks to their positive structural surface charges. To obtain a stable colloidal FF dispersion at neutral pH, the MNPs are then coated with citrate ions, thanks to an adsorption equilibrium. Zetametric measurements show that the MNP's bear a strong and negative surface charge with a Zeta potential of -30mV. The typical charge density of these MNP's is 1.7 electronic charges per nm 2 at a concentration of free citrate species [Na 3 Cit]=10 mM and pH=7 [12]. The nanoparticles are magnetic monodomains with a saturation magnetization m S of the order of 2.5 10 5 A/m. Their FF dispersions are superparamagnetic. Probed by vibrating sample magnetometry they allow determining the size distribution of the roughly spherical MNP's, which is obtained by an adjustment of the FF dispersion magnetization to a 1 st Langevin law weighted by a log-normal distribution of diameter. The distribution is characterized by a mean diameter d 0 = exp<ln(d)> and a standard deviation σ of ln(d) [13]. Two MNP's are used here (see table 1).

The polyelectrolyte-neutral di-block copolymer is a poly(trimethylammonium ethylacrylate methyl sulfate) -b-poly(acrylamide) or PTEA-b-PAM. It is synthesized by controlled radical polymerization in solution and is provided by Rhodia [14]. Fig. 1 gives the chemical formula of the two monomers. In solution, the polyelectrolyte PTEA portion is positively charged whereas the PAM portion is neutral. We thus expect that positively charged PTEA blocks would adsorb on the surface of the negatively charged nanoparticles, and then make the nanoparticles aggregate into a dense core, while the neutral PAM blocks provide a steric repulsion limiting the cluster to a finite size and stabilizing it in the solution. Indeed the PAM supplies the clusters with a neutral corona of much lower density. The synthesis is performed by one-shot mixing at low concentration (1% wt max.) of stock nanoparticle and co-polymer solutions using PTEA-PAM di-blocks of various molecular weights to produce clusters of various sizes and shapes. The relative amount of each species is monitored by the charge ratio, which is fixed to unity [6]. Three different molecular weights for PTEA-b-PAM block combinations are probed to synthesize MNP-copolymer nanoclusters: (A) 2kD-60kD, (B) 5kD-30kD and (C) 11kD-30kD.

Structural properties

Small Angle Neutron Scattering, performed on PACE and PAXY spectrometers in LLB-CE Saclay-France, is first used to investigate the structural properties of the synthesized nanoclusters as a function of the scattering vector q . As the clusters are dispersed in H 2 0, the predominant contribution to the overall neutron scattering cross-section is expected to come from the nuclear scattering of the maghemite nanoparticles, that are constituting the cluster cores. Figure 2 presents the typical qdependent intensity scattered by a cluster dispersion Fig. 2 -q dependence of the neutron intensity, scattered by a dispersion of MNP1/B-copolymer clusters ( here based on nanoparticles MNP1 and B-type copolymers). The spectra present three general scattering features : i) a strong forward scattering at low q's (related to the overall cluster shape), ii) a shoulder in the intermediate q-range (marked by an arrow at q 0 in fig. 2) and iii) a q -4 decrease of the intensity at high q's (to be associated to the MNP form factor in the Porod regime). Interpreting the wave vector q 0 as the first order of the particleparticle structure factor leads to a mean interparticle distance inside the cluster of the order of d int = 2π/q 0 = 78 Å, meaning a volume fraction Φ int of magnetic particles inside the cluster core of the order of 27%.

Some complementary transmission electron cryomicroscopy, performed in Centre de Recherches Aubervilliers-Rhodia-France, on the same dispersion as that of Figure 2 shows clusters with a dense internal organization of MNP (see Figure 3).

Here also the main contrast comes from MNPs, which appear distributed in separate clusters, well apart from each other, the invisible PAM corona preventing their agglomeration. If in Figure 3, the clusters appear polydisperse in size and in shape, they all present a similar internal structure. The densely series A, B and C lead to the cluster structure factors S(q) of Figure 4. S(q) is here obtained by dividing the intensity scattered by the cluster dispersion by that of the initial ferrofluid based on MNP2 nanoparticles alone (and renormalized by the respective volume fraction of NMP's in the two dispersions). The low q behavior of the three S(q) strongly depends on the copolymer used to build up the coacervates. With copolymer A, the clusters are small and contains only few nanoparticles (dimers on average with a gyration radius of the order of 80 Å, roughly twice that of the initial ferrofluid). With copolymers B and C, the cluster mean size grows too much to be precisely evaluated by such a scattering technique, the socalled Guinier regime being not reached at low q's.

Moreover as MNP2 nanoparticles are more polydisperse than MNP1, no bump at the interparticle distance inside the clusters is here visible. However, as it is also observed for NP1/B-copolymer clusters, the structure factors of NP2/B-and NP2/Ccopolymer clusters present a marked correlation hole that can be associated to the hierarchical structure of the clusters. Indeed the coacervates have mean sizes larger than 100 nm and contain densely packed nanoparticles of much smaller size. The correlation hole in the cluster structure factor S(q) results from the competition between the cluster form factor (dominant at low q's) and the internal structure factor of the MNP inside the cluster (dominant at large q's).

To illustrate this point, Figure 5 compares the structure factor of a 30% attractive ferroglass based on nanoparticles of characteristics very close to these of MNP2 [15] with S(q) of NP2/C-copolymer clusters. If at low q's the cluster form factor rules up the cluster structure factor, at more local spatial scales (i.e. at large q's) the S(q) of the clusters clearly joins that of the ferroglass, comforting us with our previous Φ int determination of the order of 27%.

Orientational properties

The maghemite nanoparticles are optically anisotropic, their optical axis being linked to their magnetic moment through an anisotropy energy of the order of a few 10 -21 J . Under an applied field H, dilute FF dispersions present an H-dependent magneto-optical birefringence Δn, which is directly linked to the statistical orientation of the particles by the field [16]. Δn is proportional to the volume fraction of nanoparticles and is well described by a 2 nd Langevin function, weighted by the particle size distribution. Dispersions of NP2/A-and NP2/B-copolymer clusters are optically probed under an applied magnetic field up to H = 800 kA/m. They remain stable under field (contrarily to dispersions of NP2/Ccopolymer clusters which destabilize). Figure 6 compares the field dependence of their magnetooptical birefringence to that of to the mother FF solution, in a reduce representation Δn/Φ (Φ being the total MNP volume fraction in the dispersions). If the birefringence of the dispersion of NP2/Acopolymer clusters remains very close to that of the pure ferrofluid FF, that of NP2/B-copolymer clusters largely differs. For this latter sample, an extra contribution from the shape orientation of the cluster structure is observed that should be addressed to a cluster shape anisotropy similar to that of the cryo-TEM picture of Figure3.

A probing of the rotational diffusion of the clusters can be performed by analyzing the relaxation of magnetic birefringence after the application of a pulse of magnetic field of low amplitude (5 kA/m) [17]. Such relaxations are presented in Figure 7 for of NP2/A-and NP2/B-copolymer clusters. As it could Fig7 -Relaxation of magneto-optical birefringence for dispersions of NP2/A-copolymer clusters and of NP2/B-copolymer ones.

be expected from cryo-TEM pictures, the magnetooptical relaxations of Figure 7 reveal very polydisperse distributions of rotational relaxation times that we analyse in terms of stretched exponentials (stretch exponent a given in Fig. 7). It leads to an hydrodynamic rotational diameter d R equal to 37 ± 4 nm for NP2/A-copolymer clusters (shown to bear only two MNPs on average) and to 200 ± 20 nm for the very anisotropic NP2/Bcopolymer clusters. These values are both coherent with SANS measurements and cryo-TEM pictures.

Conclusions

Coupled SANS, cryo-TEM and magnetic birefringence measurements give an insight on the internal structure of the magnetic hybrid coacervates successfully synthesized here by the electrostatic association of polyelectolyte-neutral copolymers and magnetic iron oxide nanoparticles. Inside the core, the magnetic nanoparticles are glued together by the polyelectolyte blocks with a nanoparticle density of the order of 27%. The neutral branches of the diblock copolymers constitute the shell of much lower density. This corona provides the coacervates with colloidal stability in aqueous media. It is here probed hydrodynamically by the relaxation of magnetooptical birefringence, induced by applying a magnetic field on the dispersions of magnetic clusters.

Fig. 1 -

 1 Fig. 1 -Chemical structure of the PTEA and PAM blocks entering in the composition of the co-polymers.

Fig. 3 -

 3 Fig. 3 -Cryo-TEM picture of a dispersion of MNP1/B-copolymer clusters packed MNP remain all well apart from each other inside the clusters, being as if glued together by the PTEA chains.Figure 3 fairly agrees with an internal volume fraction of MNP's Φ int of the order of 27% determined by SANS Another series of SANS experiments performed with MNP2 nanoparticles and the whole copolymer

Figure 3

 3 fairly agrees with an internal volume fraction of MNP's Φ int of the order of 27% determined by SANS Another series of SANS experiments performed with MNP2 nanoparticles and the whole copolymer

Fig 4 -

 4 Fig 4 -Nanoparticle structure factor of cluster dispersions based on MNP2 nanoparticles and A, B or C copolymer.

Fig. 5 -

 5 Fig. 5 -Comparison berween the structure factor of MNP2/Ccopolymer clusters (square symbols) and the structure factor of an attractive ferroglass at a nanoparticle volume fraction Φ = 30% from [BrJP].

Fig6-

  Fig6 -Static magneto-optical birefringence Δn normalized by the volume fraction Φ of nanoparticles in the dispersions. Lower curves : pure ferrofluid FF and dispersion of NP2/A-copolymer clusters; upper curve dispersion of NP2/B-copolymer clusters.

Table 1 -

 1 MNP size characteristics.

	MNP Sample	do (nm)	σ
	MNP1	6.3	0.23
	MNP2	8.5	0.29

Acknowledgements

We acknowledge support from Rhodia, from ANR "ITC-nanoprobe" and from ECO-NET program n°6274QL. The authors thank A. Vacher and M. Airiau (C. R. Aubervilliers, Rhodia, France) for the cryo-TEM experiments and F. Cousin (LLB-CE Saclay-France) for his help in the SANS experiment.


Nicolas Normand

Myriam Servières

Jeanpierre Guédon

How to obtain a lattice basis from a discrete projected space

Euclidean spaces of dimension n are characterized in discrete spaces by the choice of lattices. The goal of this paper is to provide a simple algorithm finding a lattice onto subspaces of lower dimensions onto which these discrete spaces are projected. This first obtained by depicting a tile in a space of dimension n -1 when starting from an hypercubic grid in dimension n. Iterating this process across dimensions gives the final result.

Introduction

Regular lattices constitute the cornerstone for the building of discrete geometry tools and also for bases of classical continuous space of functions. Of course, these regular lattices can be defined without any outside reference. However, the definition of the resulting lattice is not obvious when a problem is designed in a discrete space and when it is mandatory to go back and forth from this first space to an other discrete space using a given discrete transform. When the same problem is entirely defined in a discrete manner, the lattice identification problem can become even harder. This paradigm was used to construct a cryptographic/signature scheme using the NTRU lattice [START_REF] Hoffstein | NTRU A Ring-Based Public Key Cryptosystem[END_REF].In this case, the lattice identification leads to an NP-problem.

Lattice identifications have also been investigated by Conway and Sloane for sphere packing which results are mainly employed for vector quantization in the multimedia coding area [START_REF] Conway | Sphere Packings, Lattices and Groups[END_REF]. In Sect. 2, the starting point to review the literature will lie into the continuous/discrete correspondence firstly established by Shannon and generalized by Unser-Aldroubi. This work allows to start with a basis through a tensorial product, to sample the continuous space to give a lattice onto which a Riesz functional basis will be defined.

The aim of this paper is then to demonstrate how to obtain an unitary tile with a regular lattice on the projection hyperplane with discrete projection directions. This will be performed in a general manner in Sect. 3. The fact that we restrict the projection operator to discrete projection is based on the attempt to get a simple way to obtain a lattice onto the hyperplane. The difficulty is then to extract the lattice from regular projection grids, i.e. not to oversample the hyperplane grid with unused points nor to undersample a grid from which the lattice would not be obtained.

In other words, each point of the discrete projection plane must have a predecessor in the initial space and each point of the initial lattice is projected onto an existing point.

Related Work

Starting with the construction of orthonormal bases that give regular tiling leads to the Gram-Schmidt orthonormalisation procedure that can be found in almost any algebra textbook [START_REF] Harris | Handbook of Mathematics and Computational Science[END_REF]. Because of the normalization, the resulting basis can be easily discretised and replicated to give a regular tiling.

Following this path (starting from the continuous point of view and discretizing afterward) any n dimensional continuous space will give regular tiling from this unconstrained orthonormal continuous grid. As a matter of fact, Unser and Aldroubi have generalized the Shannon-Whittaker-Kotelnikov Sampling Theorem starting with this n-dimensional orthogonal basis then lattice [START_REF] Aldroubi | Sampling procedures in function spaces and asymptotic equivalence with Shannonś sampling theory[END_REF][START_REF] Unser | Polynomial Spline Signal Approximations: Filter Desing and Assymptotic Equivalence with Shannonś Sampling Theorem[END_REF]. The initial purpose of this theorem is to use other functional bases than the Riesz bases {(sinc (kx), k ∈ Z)}.

This theorem is described in Fig. The first step corresponds to the orthogonal projection of the L 2 function f (x) onto a closed subspace of functions generated by a Riesz basis {η(kx), k ∈ Z}. In other words, these functions already need to be defined from a tiling (the first versions of the theorem corresponds to cardinal functions as the sinc for Shannon or cardinal splines for Unser-Aldroubi).

The second step also uses the same tiling but explicitly since it just picks up the values onto the tiling and throw out the rest of the continuous functions.

The third step re-generates this previous continuous function from three different informations:

1. the dual functional basis η (defined onto the tile) 2. the sample f η (k) (defined onto the tile) 3. the tile which allows to perform the discrete convolution that lies onto the box 3.

There are two major points with this great theorem:

1. The tiling is the subsequent material that links continuous and discrete words. The strength of this theorem is to allow to work only into a discrete word f η (k) and go back into a continuous word only when mandatory. 2. The only tilling known to allow the conditions of the theorem are obtained by tensorial products over higher dimensions. In other words, the Riesz basis structure in one dimension {η(kx), k ∈ Z} can not be used in two dimensions as {η(

((k -l) 2 + (l -y) 2 )), (k, l) ∈ Z 2 } but the only known extension is {η(k -l).η(l -y), (k, l) ∈ Z 2 }.
In this latter extension, Fig. 1 where x is a n-dimensional vector still holds.

As a consequence using specific discrete operator (in our case a projector operator) between step 2 and step 3 must be done with a correspondence between grids not to loose information and the benefits of the theorem. This correspondance has been applied for the continuous and discrete Radon transforms defined into spline spaces leading to new filtered backprojection algorithms [START_REF] Guédon | Bandlimited and Haar Filtered Back-Projection Reconstuction[END_REF][START_REF] Guédon | Pixel Intensity Distribution Models for Filtered Back-Projection[END_REF].

3 Obtaining a Tile in a m-Dimensional Space from a n-Dimensional Space

From n-Dimension to (n -1)-Dimension

The initial space L is a lattice in a Euclidean space. This n-dimensional discrete space can be seen as the regular sampling of a continuous n-dimensional space structured by a hypercubic grid {i 1 , . . . , i n }. Each point (b 1 , . . . , b n ) ∈ Z n in the discrete space corresponds to the point b 1 ×i 1 +. . .+b n ×i n in the continuous space (each lattice point is described by a n-dimensional vector b = {b 1 , . . . , b n } relative to the lattice basis {i 1 , . . . , i n }). The continuous space is translation invariant according to any vector i m or any integer combination of vectors i 1 , . . . , i n . By projecting the initial n-dimensional space along a line direction, we create a (n -1)-dimensional hyperplane. It can be easily seen (Fig. 2) that if the line direction is discrete (an integer combination of i 1 , . . . , i n ), then the hyperplane has a regular discrete structure: it is also a lattice. It is translation-invariant along any integer combination of i 1 , . . . , i n , where each i m is the projection of i m on the hyperplane. However, the set {i 1 , . . . , i n } is not a base (its dimension is n -1) and a n -1-vector subset does not generally define a tile.

The purpose is to extract a n -1-vector basis from {i 1 , . . . , i n } that defines a lattice basis for the projected hyperplane. Equivalently, it will lead to a discrete (n -1) × n projection matrix.

The proposed method will conceptually use the set of vectors {i 1 , . . . , i n }, obtained by projecting {i 1 , . . . , i n } along the line direction (v 1 , . . . , v n ) onto the hyperplane. The relationship that links these vectors together is given by the projection direction:

v 1 × i 1 + . . . + v n × i n = 0 . ( 1
)
In the following, we will assume that the subset {i 1 , . . . , i n-1 } is a vector basis (i.e. that i n is a linear combination of these vectors). Hence, i 1 , . . . , i n-1 Fig. 2. Some examples of 3D projected grids with projection directions (0, 1, 1), (1, 1, 2) and (1, 2, 2)

generates the continuous projected hyperplane. This is always true except when the last component of the projection direction, v n , is zero. In this particular case, (1) creates a linear dependence between the vectors i 1 , . . . , i n-1 . But since the projection direction is not the null vector, there is at least one non-zero v m component. The previous assumption can be ensured by permuting the vectors twice: before applying the method in order to put i m at the end and after applying the method to put the vectors back in the initial order.

The method iteratively creates a set of lattice basis (j 1 , . . . , j k ) with increasing dimension k. Each intermediate basis (j 1 , . . . , j k ) generates the part of L contained in the space spanned by (i 1 , . . . , i k ). At step k, the k -1 previously found vectors build a tile i.e. a parallelepiped which vertices correspond to projected discrete points and which interior does not contain any discrete points.

First Basis Vector. The first vector j 1 is chosen following the i 1 direction from the origin point O. The end point of j 1 is the visible point in i 1 direction (the closest point from O in this direction). Since the dimension of the projection matrix is n -1 for a line projection, there are exactly two linearly independant ways to follow this direction, either with i 1 or with a linear combination of i 2 , . . . , i n according to (1):

v 1 × i 1 = -v 2 × i 2 -. . . -v n × i n .
(

The second term of this equation can be written as an integral linear combination with a division by the greatest common divisor of its coefficients:

v 1 gcd(v 2 , . . . , v n ) × i 1 = - v 2 × i 2 + . . . + v n × i n gcd(v 2 , . . . , v n ) . ( 3
)
All the linear combinations of i 1 and v 1 i 1 / gcd(v 2 , . . . , v n ) are obviously collinear to i 1 . The visible point from the origin O in this direction is given by the minimal linear combination with integer coefficients. Let's remark that gcd(v2,...,vn) gcd(v1,...,vn) and v1 gcd(v1,...,vn) are integers and relatively prime. Following the Bézout's theorem, there exist α 1 , β 1 ∈ Z such that:

α 1 × gcd(v 2 , . . . , v n ) gcd(v 1 , . . . , v n ) + β 1 × v 1 gcd(v 1 , . . . , v n ) = 1 α 1 + β 1 v 1 gcd(v 2 , . . . , v n ) = gcd(v 1 , . . . , v n ) gcd(v 2 , . . . , v n ) . ( 4
)
The minimal integral combination of i 1 and v1i 1 gcd(v1,...,vn) is then chosen as the first vector of the projected lattice basis:

j 1 = gcd(v 1 , . . . , v n ) gcd(v 2 , . . . , v n ) i 1 .
From α 1 and β 1 (obtained with the extended Euclidean algorithm) we can find two points that project onto j 1 :

A 1 = α 1 , -β 1 v 2 gcd(v 2 , . . . , v n ) , . . . , -β 1 v n gcd(v 2 , . . . , v n ) , B 1 =   α 1 + β 1 v 1 gcd(v 2 , . . . , v n ) , 0, . . . , 0 n-1   . ( 5
)
Conversely to B 1 , A 1 always has integer components and thus belongs to L.

The k th Basis Vector. Let assume that vectors j 1 to j k-1 have already been found. The lattice spanned by (j 1 , . . . , j k-1 ) is the subset of L restricted to the continuous subspace generated by (i 1 , . . . , i k-1 ). The vector i k introduces a new dimension because i 1 , . . . , i k are linearly independent. j k must have the smallest non null component in this new direction. There are two independent ways to move along i k , following i k itself or a linear combination of i k+1 , . . . , i n according to (1):

v 1 i 1 + . . . + v k i k gcd(v k+1 , . . . , v n ) = - v k+1 i k+1 + . . . + v n i n gcd(v k+1 , . . . , v n ) . ( 6
)
The closeness of the hyperplanes to the origin is measured by the projection onto i k and gives respectively 1 and vn) . The minimum integral combination is given by α k and β k :

v k gcd(v k+1 ,...,
α k + β k v k gcd(v k+1 , . . . , v n ) = gcd(v k , . . . , v n ) gcd(v k+1 , . . . , v n ) . ( 7
)
The new basis vector j k is directly derived:

j k = α k i k + β k v 1 i 1 + . . . + v k i k gcd(v k+1 , . . . , v n ) . ( 8
)
Two antecedents of j k can be obtained by:

A k =   0, . . . , 0 k-1 , α k , -β k v k+1 gcd(v k+1 ..v n ) , . . . , -β k v n gcd(v k+1 ..v n )    , B k =    β k v 1 gcd(v k+1 ..v n ) , . . . , β k v k-1 gcd(v k+1 ..v n ) , a k + β k v k gcd(v k+1 ..v n ) , 0, . . . , 0 n-k    .(9)
Projection Matrix. Any point M (a 1 , . . . , a n ) in L is projected on the hyperplane to a point p(M ) = M with integer coordinates (b 1 , . . . , b n-1 ) in the (j 1 , . . . , j n-1 ) basis. The preimage of M is the set of points in L that are aligned with M relative to the projection direction V . These points can be reached from the known point b

1 A 1 + . . . + b n-1 A n-1 : p -1 (M ) = {m|p(m) = M } = b 1 A 1 + . . . + b n-1 A n + k V gcd(v 1 ..v n ) , k ∈ Z . (10) (A 1 , . . . , A n-1 , V / gcd(v 1 ..v n )
) is a basis of the initial lattice L. An extra row corresponding to the direction of the projection is added:

A = A 1 | . . . |A n-1 | V gcd(v 1 ..v n ) =           α 1 0 . .
          . (11) P = [Id n-1 |0].A -1 . ( 12
)
Figure 3 pictures the results of our algorithm for a simple 3D to 2D projection of angle direction [START_REF] Guédon | Separable and radial bases for medical image processing[END_REF]10,15). The projection matrix is given by:

P = 5 6 -6 0 3 -2 ,
and

A =   -1 0 6 -2 1 10 -3 1 15   .
Fig. 3. The result of the projection of a 3D lattice with angle direction (6, 10, 15). The three dashed lines vectors represent the projection of the initial 3D basis vectors whereas the two plain lines vectors are the result of the computed lattice

From (n -1)-Dimension to m-Dimension

To go from a n-dimensional space to a m-dimensional space (m ≥ 1), there are (nm) projection directions V :

V =    v 1,1 . . . v 1,n . . . . . . v n-m,1 . . . v n-m,n    . ( 13
)
The described method is followed along the first direction:

V 1 = [v 1,1 . . . v 1,n ] , (14)
and gives the transformation of the basis (i 1 , . . . , i n ) to (j 1 , . . . , j n-1 ). Each V i , i ∈ {2, . . . , (nm)} is projected onto (j 1 , . . . , j n-1 ). They give the projection directions in the (n -1)-dimensional space with the (j 1 , . . . , j n-1 ) basis. The process is iterated to reach a m-dimensional space.

The Projection Matrix

We will define the projection matrix from a n-dimensional space onto a mdimensional space on the regular grid defined before.

To obtain the final projection matrix, the projection directions can be followed in different order. One projection direction is chosen, the other projection directions are projected following this first direction and the projection matrix following this direction is derivated. In the projected direction, an other direction is chosen and the same process is iterated. Finally the projection matrix from the n-dimensional space to the m-dimensional space is obtained by putting together the intermediate matrix.

To go from a n-dimensional space to an m-dimensional space (m ≥ 1), there are (nm) projection directions V k .

To obtain the projection matrix from a n-dimensional to an m-dimensional space we will calculate the projection matrix step by step by lowering the dimension. The final nD to mD projection matrix is composed by products of all the projection matrices describing hyperplane lattices. This matrix being the product of integer matrices has only integer coefficients.

Conclusion

Tiling a discrete projected space from higher dimensional Euclidean spaces was the subject of this paper. We first demonstrated how to obtain a tile in a (n -1)-dimensional space from the Euclidean hypercubic grid of dimension n. The generalization across dimensions is quite straightforward. The obtained results are directly useful for discrete Radon transforms.

1

 1

Fig. 1 .

 1 Fig. 1. Representation of the Unser-Aldroubi theorem

The original publication is available at www.springerlink.com


Unite ´de Ge E Quillet
email: edwige.quillet@diamant.jouy.inra.fr.

In rainbow trout (Oncorhynchus mykiss), the acknowledged sex-determining system is genetic sex determination (GSD) with female homogamety (&XX-(XY). Subsequently, mitotic gynogens are all expected to be females. Unexpected maleness was fortuitously observed in a mitotic gynogenetic family of rainbow trout (13 males out of 27). An equal ratio of males and females suggested the possible segregation of some Mendelian sex-influencing factor. In order to perform a comprehensive analysis of the inheritance and expression of the factor involved, the transmission of maleness was studied across the next three generations, using both conventional and/or meiotic and mitotic gynogenetic offspring. On the whole, males as well as intersexes were observed in crosses between two expected carrier parents, and in gynogenetic offspring of expected carrier females, but not in crosses between one expected carrier parent and one normal XX control. Sex ratios in the different crosses often fitted Mendelian proportions, but not always. Both excess and lack of maleness were observed. The simplest hypothesis consistent with most results is a one-locus model, assuming the existence of a mutation (termed mal) of a sex-determining gene, which is able to override the primary XX mechanism of sex determination and to induce the development of testicular tissue in the gonads of expected XX individuals. The one-locus model requires that the mal mutation usually, but not systematically, behave as a recessive mutation and have a limited penetrance, that is, heterozygous (mal/؉) may be sex reversed, homozygous (mal/ mal) may remain female, and carrier individuals may undergo partial masculinization alone (many intersexes were recorded). Inconsistency in sex ratios among offspring of parents expected to respond the same way was recorded, indicating that other modifier loci may also be involved. Finally, the occurrence of both males and females in clonal progenies showed that epigenetic factors also likely influence the expression of maleness. The effects of the mal mutation are compared to similar mutations recently described in other fish species. The nature and location of the mal gene (carried by heterochromosomes or an autosomal pair) is briefly discussed in view of the knowledge recently acquired on the subject.

The acknowledged system of sex determination in rainbow trout (Oncorhynchus mykiss) is a GSD one (genetic sex determination versus ESD, environmental sex determination) with female homogamety (&XX-(XY) as in most other salmonids. Still, sex chromosomes are hardly recognizable on conventional karyotypes [START_REF] Chourrout | Improved methods of direct chromosome preparation in rainbow trout, Salmo gairdneri[END_REF][START_REF] Thorgaard | Chromosomal differences among rainbow trout Salmo gairdneri populations[END_REF]. It has been possible to recognize the sex chromosome pair only recently using more sophisticated techniques such as fluorescence in situ hybridization FISH [START_REF] Iturra | Identification of sex chromosome molecular markers using RAPDs and fluorescent in situ hybridization in rainbow trout[END_REF][START_REF] Moran | Sex chromosome linkage of 5S rDNA in rainbow trout (Oncorhynchus mykiss)[END_REF]. To our knowledge, there is as yet no unequivocally sex-specific molecular marker or probe available in rainbow trout [START_REF] Allendorf | Sex-linkage of two enzyme loci in Oncorhynchus mykiss (rainbow trout)[END_REF][START_REF] Iturra | Identification of sex chromosome molecular markers using RAPDs and fluorescent in situ hybridization in rainbow trout[END_REF]Nakaya-ma et al. 1994;[START_REF] Sakamoto | A microsatellite linkage map of rainbow trout (Oncorhynchus mykiss) characterized by large sex-specific differences in recombination rates[END_REF]. The first evidence of female homogamety came from investigations carried out in progenies of hormonally sex-reversed individuals ( XX males, XY females, or induced hermaphrodites; see review by [START_REF] Chourrout | Revue sur le de ´terminisme ge ´ne ´tique du sexe des poissons te ´le ´oste ´ens[END_REF]. The analysis of sex ratios in uniparental families (gynogens or androgens) further supported the model. As a matter of fact, gynogenetic offspring are usually all female, and studies of androgens have confirmed the existence of YY males [START_REF] Parsons | Induced androgenesis in rainbow trout[END_REF][START_REF] Scheerer | Genetic analysis of androgenetic rainbow trout[END_REF]).

Yet scattered observations that deviate from the simple XX-XY model have been reported in rainbow trout. Males have occasionally been detected among meiotic gynogens (1 out of 50 in [START_REF] Chourrout | Induced gynogenesis in the rainbow trout: sex and survival of progenies. Production of all-triploid populations[END_REF], as well as in the offspring of assumed XX males [START_REF] Feist | The production of functional sex-reversed male rainbow trout with 17␣-methyltestosterone and 11␤-hydroxyandrostenedione[END_REF][START_REF] Okada | Functional masculinisation of genetic females in rainbow trout[END_REF]Tsumura et al. 1991). On the other hand, females have sometimes been recorded in the offspring of expected YY supermales. [START_REF] Chevassus | Production of YY rainbow trout males by selffertilization of induced hermaphrodites[END_REF] found 2 females out of 399 progenies of 4 supermales obtained by self-fertilization. [START_REF] Scheerer | Genetic analysis of androgenetic rainbow trout[END_REF] recorded 12 and 20 females out of 16 and 22 offspring, respectively, of two homozygous androgenetic males. Data from other salmonid species confirm that female homogamety is the usual rule in the family, and it is difficult to explain the observations made in trout from those observations alone.

Such deviations from the monofactorial XX-XY system (as well as from the ZZ-ZW system) have been described in other fish species. Some are well documented, and several explanations could be proposed. The existence of mutant heterochromosomes ( Y* or W* with reduced dominance) has been evidenced from progeny testing in the platyfish (Xiphophorus maculatus; [START_REF] Kallman | A new look at sex determination in poeciliid fishes[END_REF] and suggested in Leporinus elongatus from the use of molecular probes [START_REF] Baroiller | Sex determination studies in two species of teleost fish, Oreochromis niloticus and Leporinus elongatus[END_REF][START_REF] Nakayama | Sex chromosome polymorphism and heterogametic males revealed by two cloned DNA probes in the ZW/ZZ fish Leporinus elongatus[END_REF].

Another explanation proposed is the presence of Mendelian sex-determining ''autosomal'' factors. Though authors have often termed those factors ''autosomal,'' in our opinion, data do not always unequivocally demonstrate that they are carried by autosomes, and the model with multiple allelic heterochromosomes should not be formally excluded. The existence of such factors was first identified in the guppy (Poecilia reticulata) by [START_REF] Winge | Colour inheritance and sex determination in Lebistes[END_REF]: after a few generations of selection from spontaneous XX males, the authors obtained an XX strain with a balanced sex ratio. They explained their result as the accumulation of numerous autosomal male-determining genes overriding the sex genes of the X chromosome. The same explanation was proposed in more recent studies, especially in carp (Cyprinus carpio) and tilapias (Oreochromis sp.). Studying mitotic gynogens and their progenies and backcrosses in carp, Komen et al. (1992aKomen et al. ( ,b, 1995) ) described a mutation (that they termed mas-1) that induces maleness in XX females. In tilapias, a number of articles by different authors support the conclusion that in Oreochromis niloticus, there are one or more minor sex-influencing loci that cause female to male sex reversal in XX individuals [START_REF] Hussain | Estimating gene-centromere recombination frequencies in gynogenetic diploids of Oreochromis niloticus L., using allozymes, skin color and a putative sexdetermination locus (SDL-2)[END_REF]Mair et al. 1991a;[START_REF] Mu ¨ller-Belecke | Sex determination in tilapia (Oreochromis niloticus) sex ratios in homozygous gynogenetic progeny and their offspring[END_REF][START_REF] Sarder | Production and propagation of fully inbred clonal lines in the Nile tilapia (Oreochromis niloticus L.)[END_REF], as well as occasional females in the progeny of ''YY'' males [START_REF] Mair | Genetic manipulation of sex-ratio for the large-scale production on all-male tilapia, Oreochromis niloticus[END_REF]. Similarly, a mutation preventing ZZ individuals from becoming males was suspected in Oreochromis aureus (Mair et al. 1991b).

Finally, the role of environmental factors in sex determination in fish is evidenced in an increasing number of studies. Environmental factors, and especially temperature, may play a crucial role in sex differentiation in some species, though this influence is highly variable from one species to the next (see [START_REF] Baroiller | Endocrine and environmental aspects of sex differentiation in fish[END_REF] for a recent review). A few studies on clonal fish have clearly demonstrated the role of epigenetic factors. Three of the four clones of carp tested by Komen et al. (1992a) contained a small proportion of intersexes or males, despite the genetic identity among individuals. Similarly [START_REF] Sarder | Production and propagation of fully inbred clonal lines in the Nile tilapia (Oreochromis niloticus L.)[END_REF] recorded males in one of six clones of Nile tilapia. In salmonids, GSD seems to be the rule, and very few reports on ESD modification of sex are available [START_REF] Craig | Evidence for temperature-dependant sex determination in sockeye salmon (Oncorhynchus nerka)[END_REF][START_REF] Nagler | High incidence of male specific genetic marker in phenotypic female chinook salmon from the Columbia River[END_REF].

In the course of a research program on mitotic gynogenesis in rainbow trout, we fortuitously recorded an unexpected maleness in one mitotic family (13 phenotypic males out of 27 mitogens). Though the number of fish was quite small, the 1:1 sex ratio might suggest the Mendelian segregation of some sex-modifying genetic factor in that family. In order to analyze the inheritance and expression of the factor involved, we analyzed the sex ratios during the next three generations, using both conventional and/or meiotic and mitotic gynogenetic offspring as experimental groups.

Materials and Methods

Experimental Fish

Rainbow trout (O. mykiss) breeders belonged to the SY strain reared at the Institut National de la Recherche Agronomique ( INRA) experimental farm (Gournay-sur-Aronde, Oise, France). The strain is the result of the initial introduction of several domestic populations from the United States and France that were pooled and then maintained as a single population by random mating over several generations. Unless otherwise specified, the different experiments were carried out at this farm.

The first experiment ( Exp.A0) was performed in November 1991 as part of a program of production of homozygous mitot-ic gynogens. Twenty-two individual SY females were used to produce both mitotic gynogenetic lines and control families (fertilization of ova of each SY female with a mixture of milt from 7-12 males). When fish were 1 year old, mitotic lines ( EM) were mixed after fin clipping, while control families were gathered anonymously into three subgroups. Sixteen mitotic lines were studied for sex ratio and other traits. Fish from one particular mitotic line with a distinctive sex ratio ( line EM1, descended from female SY1, 13 phenotypic males out of 27 two-year-old individuals) were used as breeders for the present study. Experimental crosses were then performed across the next three generations.

Experiment A1 ( Exp.A1) was designed to study the sex ratio in offspring of EM1 males when mated with different types of females. The crosses were performed in November 1993 and 1994. In Exp.A1a, a single EM1 male was mated with a pool of 15 unrelated control females. In Exp.A1b, two EM1 males were mated with two of their mitotic sisters. As EM1 fish were homozygous mitotic gynogens, the resulting families were four clones reared all together as of the eyed stage. Sex ratio was then recorded within the whole mixed group which was called EM1 G2 . Each sexed fish was assigned to its parents from genotypes at the microsatellite loci. In Exp.A1c, 7 EM1 males were mated with 18 individual females from the SY control subgroup that contained their control sisters. These females were termed D, as daughters of SY mothers, that is, control daughters of SY1 anonymously mixed with the daughters of five other SY females. Several factorial crosses involving individual EM1 males and D females were performed. Because of experimental limitations, offspring were sent at the eyed stage to another experimental farm (SEMII, Finiste `re, France) where they were partially mixed (47 families distributed into 10 identical tanks) after adjustment for an equal number of eggs. Rearing conditions were controlled and standardized among tanks (water flow, density, feeding rate). Sex ratios were recorded for juveniles within each tank, and the parents that the fish originated from were assessed from the analyses of microsatellite markers in SY females, EM1 males, D females, and offspring.

Experiment A2 ( Exp.A2) was designed more precisely to study the offspring of D females that had given birth to males in Exp.A1c. Female D12 was the only one still available at the time of the experiment ( December 1996). It was mated with nor- &,(: both gonads are all female or male, respectively; I, intersexes. a Sex was determined by observing gonads under a microscope. b Sex was determined from the phenotypic appearance of mature fish. Only fish with a clear male phenotype were sacrificed to observe gonads (except in line EM1).

mal control males and XX unrelated neomales ( XX sex-reversed females), and reproduced by both meiotic and mitotic gynogenesis. Sex ratios were checked in the four types of progenies. Experiment A3 ( Exp.A3) was carried out to analyze the progenies of EM1 G2 fish. In December 1995, seven EM1 G2 males were individually mated with a pool of three control females. One year later, five EM1 G2 females were also tested. They were crossed with one unrelated XX sex-reversed male and one of the EM1 G2 males tested previously. They were also reproduced by both meiotic and mitotic gynogenesis. Yet, survival was quite low in most of the crosses (very likely because of the high consanguinity of the females themselves and of the resulting progenies), and only two EM1 G2 females could be used in the end.

Lastly, experiment A4 ( Exp.A4) was performed in November 1999. The goal was to use progeny testing to select females that carried the masculinizing factor in the meiotic gynogens of female D12 in order to maintain a carrying line. Eight gynogenetic females were used for a new round of mitotic gynogenesis, but only two produced mitotic offspring large enough to be kept until sexing ( F59 and F62).

Mitotic and Meiotic Gynogenesis

Meiotic gynogenesis was induced using the methods described in [START_REF] Chourrout | Induced gynogenesis in the rainbow trout: sex and survival of progenies. Production of all-triploid populations[END_REF]: sperm was ultraviolet ( UV) light inactivated and early heat shock (26.5ЊC for 20 min) was applied to prevent polar body extrusion. Mitotic gynogenesis was induced using late heat shock (31ЊC for 5 min) as described in [START_REF] Diter | Suppression of first egg mitosis induced by heat shocks in the rainbow trout[END_REF]. In each case, donor males used for UV irradiation were homozygous for the ''golden'' dominant gene of depigmentation, and haploid control batches (ova fertilized with irradiated sperm and no thermal shock) were produced for every single female.

Genetic Controls and Parental Assignment

In Exp.A0, the mitotic gynogenetic origin of EM fish was checked with six polymorphic isozymes: esterase ( EST-1), isocitrate dehydrogenase (sIDHP-1 and sIDHP-2), malate dehydrogenase (sMDH-B1,2), phosphoglucomutase (PGM2), and superoxide dismutase (sSOD1). Three microsatellites were analyzed as an additional control of the origin of individual fish that had a male phenotype: Str15INRA, Str60INRA, and Str73INRA (GenBank accession numbers AB001058, AB001057, and AB001056, respectively). In the EM1 line, every adult fish was checked for five microsatellite loci, whatever its sexual phenotype: OmyFGT2Tuf (previously FGT2; Sakamoto et al. 1994a), Str2INRA [START_REF] Estoup | Parentage assignment using microsatellites in turbot (Scophtalmus maximus) and rainbow trout (Oncorhynchus mykiss) hatchery populations[END_REF]), Omy18INRA (Gharbi K, unpublished), and OmyPuPuPyDU and Omy77DU [START_REF] Morris | Microsatellites from rainbow trout (Oncorhynchus mykiss) and their use for genetic study of salmonids[END_REF].

Microsatellites were used to assign each individual progeny to its parent couple when families had been mixed. Two microsatellites in Exp.A1b (OmyFGT2Tuf and Omy77DU) and three in Exp.A1c [FGT1 (Sakamoto et al. 1994b), OmyFGT2Tuf, and Omy77DU] were enough to trace the genetic origin of the fish in the required groups.

The electrophoretic procedures for horizontal starch-gel electrophoresis were as described in [START_REF] Guyomard | Electrophoretic variations in six populations of brown trout (Salmo trutta L.)[END_REF] and [START_REF] Krieg | Population genetics of French brown trout (Salmo trutta L.): large geographical differentiation of wild populations and high similarity of domesticated stocks[END_REF]. The genetic determinism of the electrophoretic variation in isozyme loci was demonstrated in [START_REF] Diter | Gene segregation in induced tetraploid rainbow trout: genetic evidence of preferential pairing of homologous chromosomes[END_REF] and Guyomard (1984).

Microsatellite polymorphism was analyzed using the usual techniques: DNA extraction (from blood or fin tissue), polymerase chain reaction (PCR) amplification, electrophoresis on standard DNA sequencing gel, and autoradiography. Primers, sequences, and specific conditions of PCRs have been published in the above-mentioned publications and in [START_REF] Estoup | CT ) n and (GT ) n microsatellites: a new class of genetic markers for Salmo trutta L ( brown trout)[END_REF].

Determination of Sex Ratios and Statistical Analyses

The sex ratios were most usually determined for juveniles (6-11 months old). Fish were sacrificed and both gonads were dissected, slightly squashed, and directly examined over their entire length under a microscope. The presence of ovarian lamellae was the criterion used to attribute female sex, whether oocytes were present or not. Fish were scored as intersexes when at least one of the gonads was an ovotestis or when one ovary and one testis were observed simultaneously. In some experiments, sex ratio was determined at adulthood: in these cases, sex determination relied on phenotype only (secondary sexual traits and/or spontaneous production of gametes by abdominal pressure).

Comparisons of observed sex ratios to expected proportions were performed using the chi-squared test corrected for continuity [START_REF] Sokal | Biometry: the principles and practice of statistics in biological research[END_REF]. Comparisons among sex ratios were performed using the two-way tables test of independence.

Results

Experiment A0

Sex ratios recorded in the different EM lines are presented in Table 1. Considering the genetic determination of sex in rainbow trout ( XX-XY), only females were expected in gynogenetic progenies. Yet at 8 months, six of the nine sampled lines contained fish with completely or partially male gonads. Some of the females (about 4%) exhibited reduced fertility (few or no visible oocytes). At adulthood, fish of male appearance were again observed in several lines: 5 of the 13 lines contained males or intersexes. Only fish with a clear male b Some males produced milt by abdominal pressure (sp) while others did not (nsp). c Sex ratio based on observation of gonads of 8-month-old juveniles under a microscope. d Sex ratio based on observing phenotypic appearance during reproduction at 2 years. Fish classified as males may also have been intersexes, with no external evidence of female maturation.

phenotype were sacrificed (except in line EM1, where they were all kept for further crosses) and their gonads dissected. It is therefore possible that intersexes of ambiguous phenotype or limited male appearance were not identified. On the whole, males or intersexes were found in 7 of 16 EM lines. The mean proportion was about 10% in juveniles and 5% at adulthood (which may be an underestimation, as some intersexes may not have been detected by phenotype).

Residual paternal contribution would be the most obvious explanation for the presence of males in gynogenetic offspring. Yet all of the 99 fry that hatched in the 16 haploid control batches (16,000 ova overall) were normally pigmented, and no golden fry were observed in the mitotic batches themselves, which supports high-efficiency sperm irradiation. The normal fry (0.6%) very likely came from spontaneously unreduced diploid eggs. In the same way, controls performed at the isozymes and microsatellite markers in juveniles and 2-year-old adults showed that males and intersexes were all homozygous and carried only alleles present in the mother. These controls do not definitely rule out the risk of some paternal contribution, as genetic markers were not analyzed in golden donor males that may have shared common alleles with females. Yet EM fish were all homozygous: in that case, paternal contribution is not detected in each given offspring, only when it supplies the same allele as that supplied by the mother. For a given locus, the probability decreases when the probability that parents are heterozygous increases, which is the case with microsatellites which are far more variable than isozymes. For a given fish, the probability decreases as the number of markers increases. Considering the number of markers that were used, it seems unlikely that a substantial paternal contribution alone explains the results observed.

We then focused on line EM1 which had quite a large number of male progenies: 43% of the total sample (versus 5% as a mean value in other lines), which was not significantly different from a 1:1 sex ratio ( 2 1df ϭ 0.73; P ϭ .39). Males from other lines were not studied further.

Experiment A1

Experiment A1 was designed to check the assumption that SY1 was heterozygous for some sex-determining locus and, consequently, that EM1 males were homozygous at that locus. Mating of EM1 males with control females ( Exp.A1a) or with their EM1 mitotic sisters ( Exp.A1b) was expected to produce offspring of the same genotype as the SY1 mother, that is, all females. The sex ratios observed in the two experiments are presented in Table 2. In Exp.A1a, only females were recorded, though some were partially (greatly reduced number of oocytes) or completely sterile (1 and 2 of 91, respectively). In contrast, in Exp.A1b, many fish (16 of 24) displayed a male phenotype. Yet 8 of the 16 phenotypic males did not produce milt by abdominal pressure, and may have been intersexes. As survival in this experiment was very low (2% from the eyed stage to maturation at 2 years), we carried out microsatellite analyses to check the status of survivors and to assess the actual contribution of the four initial homozygous EM1 parents. Twenty-two of the 24 remaining fish had the same EM1 mother (&6), while the 2 EM1 males ((3 and (6) were equally represented. The two cloned progenies from &6 contained both females and males. The two remaining fish belonged to the other two clones (&4(3 and &4(6) and were phenotypic males.

The purpose of Exp.A1c was to check whether SY1 had actually transmitted some masculinizing gene to its control daughters. Such daughters would be heterozygous for the gene, and when mated with EM1 spontaneous males should give birth to 50% heterozygotes (remaining female, as did their mother and SY1 grandmother) and 50% homozygous offspring that would express maleness. As the experiment was not initially designed for this purpose, daughters of SY1 were anonymously mixed with daughters of five other SY females. There were 18 D females available in the control group: thus the expected number of SY1 daughters was three, and the number of heterozygous daughters for the gene of interest was half of that (i.e., one or two females in all probability), which was quite low. Consequently every D female was used in the experiment, and microsatellites were used for assignment to SY mothers. Sex ratios are reported in Tables 3 and4. Males or intersexes were observed in 4 of the 10 tanks. Clearly, maleness depended on the D female rather than on males. Besides the fact that tanks were managed under controlled conditions, the results in tank 1 strongly support the idea that expression of maleness is due to a female effect rather than a tank effect: D1 and D2 progenies were incubated in the same recirculating unit (10ЊC) and then reared in the same tank as the eyed stage, and only one contained males and intersexes (13 of 18 for D2 versus 0 of 20 for D1). Parental assignment was performed in tanks 1 and 6 only (detailed in Table 4). Three females had males and intersex offspring ( D2, D11, and D12). It is striking to note that all three were putative daughters of SY1. The frequency of sex-reversed fish was 72%, 64%, and 57% for D2, D11, and D12, respectively, which was not significantly different ( 2 1df Ͻ 0.50; P Ͼ .48). Similarly, no effect of EM1 males on sex ratio could be detected ( highest 2 1df ϭ 0.44; P ϭ .51). The overall mean frequency of males was not different from 0.5 (33 of 66). Yet the total frequency of sex-reversed fish (( ϩ I ϭ 64%) was different from 0.5 ( 2 1df ϭ 4.38; P ϭ .0364).

Families of tanks 3 and 9, containing only a few males or intersexes, were not studied further. Yet one should emphasize the fact that, in both tanks, female SY5 was involved as the putative mother and remember that line EM5 also contained males or intersexes (see Table 1). All these results show that SY1 (and SY5 very likely) has actually transmitted some masculinizing factor to its daughters through normal mating, and they rule out uncon- a A total of seven individual EM1 males were used for fertilization. At every date of fertilization, each sampled male was mated with each spawning D female. At the eyed stage, families were adjusted for an equal number of embryos and grouped according to experimental facilities. b Genotypes of SY and D females were examined at five loci. Each SY female was excluded as a putative mother when, at one or more of the five loci, the D female had no common allele with it. Except for D4, a single SY female compatible with the genotype of every D female could be identified. trolled parental contributions after gynogenesis as an explanation for the maleness observed in Exp.A0.

Experiment A2

Female D12 (assumed to be heterozygous for the masculinizing factor, just like SY1) was used for further analysis of the transmission of male phenotype in normal and gynogenetic progenies. The aim was to confirm the results obtained with SY1 using additional controls and to compare the frequency of maleness in the two types of gynogenetic progenies (results in Table 5).

The expected 1:1 sex ratio was observed in the cross with control males ( 2 1df ϭ 0.48; P ϭ .49), and only females were detected in the cross with XX males. Homozygous mitotic progenies contained 58% partially or completely male fish, a proportion that is not different from a 1:1 ratio ( 2 1df ϭ 1.07; P ϭ .30). Sex reversal (( ϩ I) was lower in meiotic gynogens (18%) than in mitotic gynogens ( 2 1df ϭ 18.8; P ϭ 2.10 Ϫ5 ).

Experiment A3

As maleness in the EM1 G2 group was unexpectedly high, additional crosses were performed to study transmission and expression in the next generation. Two EM1 G2 females (no. 6 from clone &6(3 and no. 7 from clone &6(6) and seven males (from the same clones plus clone &4(6) were used as breeders. In that particular experiment, gynogenesis was not completely successful, as two golden fry were observed in the haploid control (more than 500 ova). Yet all the gynogenetic groups were fertilized with the same pool of irradiated sperm, which made it possible to make between-group comparisons.

Crossing EM1 G2 males with control females produced only females (60 progenies per male; i.e., 420 fish sexed). Five had either no visible or few oocytes. The sex ratios of the offspring of the two EM1 G2 females are reported in Table 6. Only females were recorded in the crosses with the control XX neomale (110 females, of which 5 were partially or completely sterile). Males and intersexes were observed in all other groups. In female 6, sex ratios were not different from 1:3 in the cross with the EM1 G2 male and from 1:1 in mitotic progeny, respectively. On the other hand, sex ratios were female biased in progenies of female 7 ( Table 6), though the differences between the two females were not significant ( 2 1df ϭ 0.96 and 0.66; P ϭ .33 and .42 in the two types of crosses, respectively). The frequencies of (( ϩ I) in meiotic gynogenetic progenies were lower than in the mitotic groups in female 6 ( 2 1df ϭ 9.92; P ϭ .0017) and were the same in the two females ( 2 1df ϭ 2.73; P ϭ .098).

Experiment A4

Sex reversal was usually lower in meiogens than in mitogens of females assumed to be heterozygous for the masculinizing factor ( Exp.A2 and Exp.A3). One possible explanation is that meiotic recombination at the presumed locus produces heterozygotes (assumed to remain female), which limits the expression of the trait to the residual frequency of homozygotes. Under this assumption, females Control of sperm inactivation for gynogenesis: no golden fry were observed in meiotic or mitotic batches. Only three normally pigmented fry out of the 300 haploid control ova hatched. a Females that were hormonally sex reversed to functional XX males. in the meiogenetic progeny of D12 were expected to be either homozygous for the wild-type allele or heterozygous. Sex ratios observed in the mitogens from F59 and F62 are reported in Table 7. Offspring of F62 actually had an equal ratio of males and females (( ϩ I ϭ 58%; 2 1df ϭ 1.82; P ϭ .18), indicating a probable heterozygous status. Unexpectedly, offspring of F59 were almost sex reversed (( ϩ I ϭ 83%; 2 1df ϭ 34.26; P Ͻ 10 Ϫ6 ). Moreover, only 7 of the 14 females had ovaries containing oocytes. These results strongly suggest that female F59 was homozygous for the mutation despite its phenotypic sex (gonads were not examined). Similarly its mitotic offspring were homozygous, but few remained female nevertheless.

Discussion

A Model to Explain the Data Unexpected maleness was observed in several mitotic gynogenetic lines of rainbow trout. The transmission of maleness through normal mating and across generations meant it was possible to rule out paternal contribution to gynogenetic progeny as the reason behind the observations. Inbreeding and the associated reduction in developmental homeostasis [according to the theory of [START_REF] Lerner | Genetic homeostasis[END_REF]] could also be proposed as an explanation for unexpected sexual phenotypes, and could account for occasional male mitogens in some lines. Yet the inbreeding ef-fect per se (i.e., not caused by homozygosity at some unknown modifying loci) could not easily explain the maleness we observed steadily across generations.

The simplest hypothesis to explain the results overall is a one-locus model in which the SY1 female was heterozygous at some sex-determining locus and carried a recessive allele (that we shall call mal hereafter) able to override the usual XX mechanism of sex determination and induce the development of testicular tissue in the gonads of expected XX individuals. The results of Exp.A1a are consistent with the hypothesis that EM1 males were homozygous XX (mal/mal). When crossed with normal females (ϩ/ϩ), they produced only heterozygous progeny, that were indeed all female, as was the SY1 mother. Data from Exp.A1c also support the one-locus model: D2, D11, and D12 females were heterozygous (mal/ϩ) daughters of SY1, and female D14, the last putative daughter of SY1 whose progeny contained only females, should therefore have received the wild-type allele from its mother. When crossed with any of the homozygous (mal/mal) EM1 males, heterozygous D females actually produced progenies of which about 50% reversed to males. Finally, the 1:1 sex ratio in mitogens of several females expected to be heterozygous (mal/ϩ) from their parentage is consistent with this assumption ( D12 in Exp.A2, EM1 G2 no. 6 in Exp.A3, and F62 in Exp.A4).

Yet in a number of cases, the observed sex ratios did not fit the ones expected with the one-locus model alone. In some cases, an excess of males (intersexes included) was observed, indicating that heterozygotes (mal/ϩ) can also be sex reversed. The proportion of such reversals was estimated at 27% in progenies of D females ( Exp.A1c) and up to 67% in Exp.A1b, where EM1 males (mal/mal) were mated with their EM1 ''sisters'' expected to be (ϩ/ϩ). On the other hand, sex ratios recorded in Exp.A3 (female EM1 G2 no. 7) and Exp.A4 ( F59 and its progeny) suggest that the opposite phenomenon can also occur, that is, homozygotes (mal/mal) may sometimes keep the primarily all-female gonads.

Because of the pedigree of our fish (a single carrier ancestor), we monitored the transmission and expression of the same allelic form of the gene across the different generations and types of crosses. Therefore, under the one-locus model, the conclusions are that the mal mutation usually, but not systematically, behaves as a recessive mutation and has limited penetrance. At the population level, the consequence is that there is no unequivocal relationship between the genotype at the locus involved and gonadal sex. At the individual level, it results in some fish undergoing only partial sex reversal and remaining intersexes. A few females with apparently abnormal ovaries were also observed throughout the study. In the crosses with control parents, they may have been sterile triploids, resulting from spontaneous diploid eggs. In other groups, inbreeding resulting from gynogenesis and/or mating between relatives is likely to account for reduced fertility or a delay in gonad development (indeed, all tested fish have SY1 as a common ancestor). Yet one can also assume that in some cases the normal expression of female phenotype was restricted by the presence of the masculinizing factor in carrier females. Control of sperm inactivation for gynogenesis: mitotic gynogenesis was included in a larger experiment involving other females. Altogether, haploid controls fertilized with the same irradiated sperm as F59 and F62 contained more than 2300 ova and no survivors, whether golden embryos or fry, were observed.

Although a one-locus model accounts for most results, it is possible that one or more other modifier loci may also be involved in the expression of maleness. As a matter of fact, the results of Exp.A3 suggested a possible effect of parents on sex reversal in offspring expected to respond in the same way with regard to their assumed genotype at the mal locus. While sex ratios fitted the expected ratios in offspring of female 6 (1:3 in the cross with the heterozygous EM1 G2 male and 1:1 in mitotic gynogens), they were systematically skewed toward females in the same groups from female 7. Thus specific genetic background may contribute to the inconsistency of maleness expression between and within groups. Yet in the particular case of clonal progeny ( Exp.A1b), genetic factors should not account for the occurrence of both males and females. Such a result indicates the role of epigenetic factors in the expression of maleness. The possible role of environmental factors on sex reversal was not investigated in the present study, but may explain some inconsistencies with data of experiments that were not carried out at the same time or same place. In particular, environmental differences may have contributed to the occurrence of apparently opposing penetrances in different experiments (maleness in mal/ϩ individuals versus the occurrence of mal/mal females).

Comparison of Results with Literature

The mutation we identified here in trout appears quite similar in terms of nature and effects to those described in other species. Mutations have essentially been described as recessive. The primary argument is that the detection of mutations usually relies on the analysis of heterozygous individuals that remained female. The second argument is that maleness is generally reduced or absent when carrier individuals are crossed with nonrelated individuals. An additional argument is that the proportion of males is very often lower in meiogens than in mitogens from the same mother. This is interpreted as the result of recombination between the minor sex-determining locus and the centromere, which produces heterozygosity at the gene and decreases the occurrence of expected males. For instance, in carp, Komen et al. (1992a,b) found 6.6% males in meiogens versus 47.7% in mitogens of the same parents. In tilapia, published data are quite consistent: 7.5% versus 47.5% with one female [START_REF] Hussain | Estimating gene-centromere recombination frequencies in gynogenetic diploids of Oreochromis niloticus L., using allozymes, skin color and a putative sexdetermination locus (SDL-2)[END_REF], 0% versus 33.3% and 66.6% for two females (Mu ¨ller-Belecke and Ho ¨rstgen-Schwark 1995), 12% versus 20% [START_REF] Sarder | Production and propagation of fully inbred clonal lines in the Nile tilapia (Oreochromis niloticus L.)[END_REF] for the mean of four females. In the present study, frequencies were 14% versus 40% (mean of three females).

Yet mutations often show incomplete penetrance rather than strict recessiveness. Some heterozygous progeny can undergo sex reversal, as was confirmed by Komen et al. (1992aKomen et al. ( ,b, 1995) ) in carp and in the present study. On the other hand, trout homozygous for the mutation may keep female gonads. There is no indication of that in carp, but in tilapia, the ratio of males is significantly lower than 50% in some, but not all, mitotic lines [START_REF] Sarder | Production and propagation of fully inbred clonal lines in the Nile tilapia (Oreochromis niloticus L.)[END_REF]. Similarly in this study we observed a lack of males in mitotic gynogens of female 7 ( Exp.A3). In mitotic lines, selective mortality might also contribute to unbalanced sex ratios. Yet results obtained with female F59 support the hypothesis of incomplete penetrance in some cases fairly well.

Limited penetrance also results in the association between the mal mutation and the development of intersex individuals. Species we refer to in the present article (carp, tilapia, and trout) are all gonochoristic species, and hermaphrodites or intersexes have been exceptionally observed under natural or normal breeding conditions. In salmonids, very few individuals were reported: one in chum salmon (Oncorhynchus keta; [START_REF] Hikita | A hermaphroditic chum salmon, Oncorhynchus keta, from the Chitose River with an example of its self-fertilization[END_REF] and another, more recently in a wild Scottish population of Arctic char (Salvelinus alpinus; Fraser 1997). Very few inter-sexes were observed in normal progenies of O. niloticus (9 of 2999 fish belonging to 59 families; Mair et al. 1991a), but in the same study, one particular female gave a progeny containing a high proportion of intersexes (5.8-25.6%). Of interest is that the experimental history led the authors to assume that intersexes were genetically female. In tilapia (O. aureus), [START_REF] Penman | Sex ratio of gynogenetic and triploid tilapia[END_REF] also recorded intersexes in one out of six meiogenetic offspring. Yet in other studies dealing with genetic sex determination in tilapias, intersexes were not recorded. In the course of their work on the mas-1 mutation in carp, Komen et al. (1992a,b) observed a fairly high proportion of intersexes in meiogens (3%) and to a greater extent in mitogens (25%). These were usually predominantly male, and in all cases were classified as homozygous for the mutation. In trout, intersexes were usually not observed in crosses with controls, but in all other cases. More generally, we found intersexes associated with most cases of unexpected maleness, whatever the mitotic line affected ( Exp.A0).

The effects of environmental factors on sex determination in fish are well documented [START_REF] Baroiller | Endocrine and environmental aspects of sex differentiation in fish[END_REF]), but in salmonids we only found two reports dealing with this. Female-biased sex ratios were observed by [START_REF] Craig | Evidence for temperature-dependant sex determination in sockeye salmon (Oncorhynchus nerka)[END_REF] in two genetically distinct forms of sockeye salmon (Oncorhynchus nerka) and were parsimoniously attributed to the effect of temperature shifts during embryonic development. Recently [START_REF] Nagler | High incidence of male specific genetic marker in phenotypic female chinook salmon from the Columbia River[END_REF] reported a high incidence of a male-specific DNA marker in wild phenotypic females of chinook salmon (Oncorhynchus tshawytcha) from the Columbia River: a survey of the local environment led the authors to conclude that genetic males had been sex reversed by exposure to adverse conditions during incubation (exposure to estrogens or to fluctuating temperatures, which would be more relevant for the present purpose). To our knowledge, there are no indications of such effects in rainbow trout. Conversely, several studies demonstrate the considerable stability of sex ratios under various thermal conditions [START_REF] Baroiller | Endocrine and environmental aspects of sex differentiation in fish[END_REF][START_REF] Van Den | Temperature and steroid effects on gonadal sex differentiation in rainbow trout[END_REF]. Nevertheless, environmental impacts on sex ratios are likely to occur when the primary sex-determining factors are bypassed ( XX-mal system). For instance, [START_REF] Winge | Colour inheritance and sex determination in Lebistes[END_REF] mentioned that the sex balance in the line selected from spontaneous XX males of guppy was more susceptible to external factors than in the XY system. Ko-men et al. (1995) also alluded to the influence of temperature and feeding level on the expression of the mas-1 mutation in carp.

Frequency and Nature of the Gene

In our strain we found 1 female out of 16 that unequivocally carried the mutation. Yet because of incomplete penetrance, other EM lines with some spontaneous males may also have originated from females carrying the same mutation or similar ones. In carp, Komen et al. (1992a) identified two nonallelic masculinizing factors, and the existence of such multiple mutations (allelic or not) in trout cannot be excluded. Moreover, deviations from expected sex ratios were hardly noticeable in adulthood, while the systematic observation of juvenile gonads revealed that maleness (associated to intersexuality) was not exceptional at all. Thus the actual frequency of mutation(s) may be largely underestimated in rainbow trout. As a matter of fact, the observations of [START_REF] Scheerer | Genetic analysis of androgenetic rainbow trout[END_REF] match the XX-mal model we propose here quite well (the authors had themselves suggested that their data might reflect autosomal gene influences): the androgenetic male parents they tested, and that gave birth to almost all female offspring when mated with control females, behaved as would XX (mal/ mal) individuals (the few males would be heterozygous, having undergone sex reversal). It is therefore very likely that the mutation (or a similar one) was also present in their original strain, which would indicate that the phenomenon is not that exceptional in rainbow trout. Yet the occurrence of deviations from the strict monofactorial XX-XY model of sex determination seems far less frequent in trout than in tilapia. In O. niloticus, for instance, different authors found 4 females out of 8 (Mair et al. 1991a), 10 out of 14 (Mu ¨ller-Belecke and Ho ¨rstgen-Schwark 1995), and 4 out of 4 [START_REF] Sarder | Production and propagation of fully inbred clonal lines in the Nile tilapia (Oreochromis niloticus L.)[END_REF]) that gave male mitogens, but all worked with the same population originating from Lake Manzala ( Egypt) and kept at the Institute of Aquaculture of Stirling, which may limit the implications of the data. The frequency of mas-1 in carp was not estimated [the single female reported in the experiment of Komen et al. (1992b) was heterozygous].

Meiotic offspring were produced in the present work with the hope that their sex ratios would provide an estimation of the recombination rate of the mal locus, and possibly an indication of whether it is car-ried by the Y chromosome (a ''weak'' Y) or by an autosomal one. Yet because of incomplete penetrance, phenotypic data are unreliable and recombination rates may be biased high. A tentative correction can be made from data of F59 mitogens: using the observed value of penetrance (0.83), the frequency of actual homozygotes can be reestimated from the observed sex reversal in the three meiotic gynogenetic families of Exp.A2 and Exp.A3 (6-19%, intersexes included). Corrected estimations of recombination rates remain quite high (more than 50%), which may indicate that the mal locus is quite distant from its centromere, as least in female meiosis.

On the other hand, genetic analyses of male meioses mapped the primary sex-determining locus very close to the centromere [START_REF] Allendorf | Sex-linkage of two enzyme loci in Oncorhynchus mykiss (rainbow trout)[END_REF] with enzyme loci, [START_REF] Sakamoto | A microsatellite linkage map of rainbow trout (Oncorhynchus mykiss) characterized by large sex-specific differences in recombination rates[END_REF] with microsatellites], while a more distal position (about 20 cM) was observed by [START_REF] Young | A detailed map of rainbow trout produced using doubled haploids[END_REF] from amplified fragment length polymorphism (AFLP) data. Yet there are considerable sex-specific differences in recombination rates [START_REF] Sakamoto | A microsatellite linkage map of rainbow trout (Oncorhynchus mykiss) characterized by large sex-specific differences in recombination rates[END_REF], with map distances usually being higher in female meiosis. Therefore no reliable conclusion can be drawn concerning the relative location of the mal locus and the primary sex-determining factor. Genome scanning with autosomal and sex-linked microsatellites is now planned to address the issue.

᭧Mutation

  2002 The American Genetic Association 93:91-99

Table 1 . Sex ratios recorded in mitotic gynogenetic lines from 16 SY females

 1

		Juveniles a			Adulthood b
	Line	&	I	(	&	I	(
	EM1	14	1	7	14		13
	EM2	21			25
	EM3				15
	EM5	22	1	4	24
	EM6				20
	EM7	26			23
	EM8	33	4	3
	EM11	51			14
	EM12				21
	EM14				20
	EM19	40	1	5	36	1
	EM23				16		1
	EM27				11
	EM29	43		1
	EM31				41	1
	EM32	48	3	4
	Total observed	299	10	24	280	2	14

Table 2 . Sex ratios recorded in Exp.A1a and Exp.A1b

 2 Sterile female (no visible oocytes on ovarian lamella).

	Experi-					( b
	ment	Cross	&	& st a	I	sp/nsp	Total
	A1a c	Single EM1 male ϫ 15 control females	91	2	0	0	93
	A1b d	EM1 G2 , clone (&6(3)	6	-	-	2/3	11
		EM1 G2 , clone (&6(6)	1	-	-	5/4	10
		EM1 G2 , clone (&4(3)	0	-	-	0/1	1
		EM1 G2 , clone (&4(6)	0	-	-	1/0	1
		EM1 G2 , total	8	-	-	8/8	24
	NB: paternal origin of 1 female was not definitively determined (&6, (3, or (6).

a

Table 3 . Sex ratios recorded in 9-month-old juveniles of Exp.A1c

 3

		EM1 males a EM1 no. (1 ϩ 2 ϩ 3)	EM1 no. (4 ϩ 5)		EM1 no. 4			EM1 no. (3 ϩ 6 ϩ 7)	Only possi-
	Tank	Daughter	&	I	(	&	I	(	&	I	(	&	I	(	ble mother b
	1	D 1	20	-	-										SY7
		D 2	5	2	1 1										SY1
		D 3													SY2
	2		2 0
		D 4													?
		D 5													SY5
	3		1 9	1	-
		D 6													SY2
		D 7													SY7
	4		2 0	-	-
		D 8													SY2
		D 9													SY5
	5					3 0	-	-
		D 10													SY2
	6	D 11				9	2	1 4							SY1
		D 12				10	5	8							SY1
		D 13													SY6
	7	D 14							30	-	-				SY1
		D 15													SY5
	8	D 16										40	-		SY7
	9	D 17										43	6	1	SY5
	10	D 18										40	-		SY5

Table 4 . Sex ratios determined after parental assignment in tanks 1 and 6 (Exp.A1c)

 4

		EM1 males
		no. 1	no. 2	no. 3	no. 4	no. 5
	Female	&:I:(	& :I:(	& :I:(	& :I:(	& :I:(
	D1	10:0:0	6:0:0	4:0:0	-	-
	D2	3:1:8	2:1:2	0:0:1	-	-
	D11	-	-	-	5:1:6	4:1:8
	D12	-	-	-	9:4:7	1:1:1

Table 5 . Sex ratios recorded in Exp.A2 (examination of gonads of 6-month-old juveniles from female D12)

 5

							(I ϩ ()
	Progenies of female D12	&	& st	I	(	Total	(%)
	Cross with 10 pooled control males	47	0	-	55	102	54
	Cross with 12 pooled neomales a	99	1	-	-	100	0
	Meiotic gynogenesis	55	1	5	7	68	18
	Mitotic gynogenesis	17	2	6	21	46	58

Table 6 . Sex ratios recorded in Exp.A3 (examination of gonads of 7-month-old juveniles in progenies of EM1 G2 females)

 6

	Sex

a Expected sex ratios assuming that all homozygotes for the masculinizing allele, but they alone, are at least partially sex reversed. b Chi-squared comparison test with the expected sex ratios.

Table 7 . Sex ratios recorded in Exp.A4 by examination of gonads of 1-year -old juveniles in mitotic gynogens of females F59 and F62

 7

	Meiotic
	gynogenetic
	female	&	& st	I	(	Total	( I ϩ () (%)
	F59	7	7	10	58	82	83
	F62	32	1	22	24	79	58