Bruno Durand
email: bruno.durand@lif.univ-mrs.fr

Andrei Romashchenko
email: andrei.romashchenko@lif.univ-mrs.fr

Alexander Shen
email: alexander.shen@lif.univ-mrs.fr

D EFFECTIVELY CLOSED SUBSHIFTS AND 2D TILINGS

Michael Hochman showed that every 1D effectively closed subshift can be simulated by a 3D subshift of finite type and asked whether the same can be done in 2D. It turned out that the answer is positive and necessary tools were already developed in tilings theory.

Simulation

Let A be a finite alphabet and let F be an enumerable set of A-strings. Consider all biinfinite A-sequences (i.e., mappings of type Z → A) that do not contain substrings from F . The set of these sequences is effectively closed (its complement is a union of an enumerable set of intervals in Cantor topology) and invariant under (left and right) shifts. Sets constructed in this way are called effectively closed 1D subshifts.

Effectively closed 2D subshifts are defined in a similar way; instead of biinfinite sequences we have configurations, i.e., mappings of type Z 2 → A, and instead of forbidden strings we have forbidden patterns (rectangles filled with A-letters). Given the set F of forbidden patterns, we consider the set of all configurations where no elements of F appear. This set of configurations is closed under vertical and horizontal shifts. If F is enumerable, we get effectively closed 2D subshifts; if F is finite, we get 2D subshifts of finite type.

2D subshifts of finite type are closely related to tilings. A tile is a square with colored sides (colors are taken from some finite set C). A tile set is a set of tiles, i.e., a subset of C 4 , since each tile is determined by four colors (upper, lower, left, and right). For a tile set τ , we consider all τ -tilings, i.e., the tilings of the entire plane by translated copies of τ -tiles with matching colors.

Key words and phrases: effectively closed subshifts, subshifts of finite type, tilings. The paper was supported part by NAFIT ANR-08-EMER-008-01 grant, part by EMC ANR-09-BLAN-0164-01.

Tilings can be considered as a special case of 2D subshifts of finite type. Indeed, subshift is governed by local rules (forbidden pattern say what is not allowed according to these rules). In tilings the rules are extremely local: they say that the neighbor tiles should have matching colors, i.e., 1 × 2 rectangles where the colors do not match, are forbidden.

So every tile set determines a subshift of finite type (the alphabet is a tile set). The reverse statement is also true if we allow the extension of an alphabet. (It is natural since we have to simulate any local rule by a more restricted class of matching rules.) Formally, for every alphabet A and subshift S of finite type we can find:

• a set of colors C;

• a tile set τ ⊂ C 4 ;

• a mapping d : τ → A such that every τ -tiling after applying d to each tile becomes an element of S and every element of S can be obtained in this way from some τ -tiling. Such a correspondence between tilings and subshifts of finite type works in any dimension: k-dimensional tilings correspond to k-dimensional subshifts of finite type (modulo the alphabet extension). Now we want to compare subshifts in different dimensions. Let S be a 1D subshift. We can make a 2D subshift from it by copying each letter vertically. It is easy to see that an effectively closed 1D subshift becomes an effectively closed 2D subshift (we use rules that guarantee the vertical propagation, i.e., require that vertical neighbors should have the same letter, and the rules of the original 1D subshift in horizontal direction). This 2D shift, denoted by S, is not of finite type, if the original 1D shift was not of finite type. However, S is sofic, i.e., is a projection by a subshift of finite type in extended alphabet: Theorem 1.1. For every effectively closed 1D subshift S in alphabet A there exists an alphabet A ′ , a finite 2D subshift S ′ in alphabet A ′ , and a mapping d : A ′ → A such that the image of S ′ under d (applied in each place) is S. This theorem (with 3D instead of 2D, which makes it easier) was proved by Michael Hochman [START_REF] Hochman | On the dynamic and recursive properties of multidimensional symbolic systems[END_REF] who asked whether the same is true for 2D. His motivation came from ergodic theory.

It turned out that the tools needed to prove theorem 1.1 for 2D tilings were already developed in the framework of tilings theory when Hochman asked his question. Moreover, there are two different sets of tools that can be used; one was used by Nathalie Aubrun and Mathieu Sablik [START_REF] Aubrun | Simulation of recursively enumerable subshifts by two dimensional SFT and a generalization[END_REF] (and goes back to Leonid Levin [START_REF] Durand | Complex Tilings[END_REF]), the other one was used in [START_REF] Durand | Fixed-point tile sets and their applications[END_REF] (and goes back to Peter Gács [START_REF] Gács | Reliable Computation with Cellular Automata[END_REF]). In the sequel we discuss informally how these tools work, and what are the similarities and the differences.

Tools

Let us describe informally our problem. In 2D we have local rules that guarantee that each vertical line contains some letter. We need to add some other rules to guarantee that the emerging horizontal sequence of letters does not have substrings from some enumerable set F . We are allowed to superimpose additional structure to the configuration (by extending the alphabet: we let A ′ be a product of A and some other finite set). Rules for this extended configuration should guarantee that its base belongs to S. So we need to run a computation that generates F and some process that compares generated elements with substrings in the horizontal sequence. It is well known (since the first papers of Wang [START_REF] Wang | Proving theorems by pattern recognition, II[END_REF][START_REF] Wang | Dominoes and the ∀∃∀ case of the decision problem[END_REF] where the notion of a tile set was introduced) that tile sets can simulate computation easily: indeed, a time-space diagram of a Turing machine (or a cellular automaton) obeys local rules that guarantee that computation is performed correctly when started. The problem is to initiate the computation: there is no special point in the plane where the computation can be started, so we need to "break the translational symmetry" somehow.

This problem was solved by Berger [START_REF] Berger | The undecidability of the domino problem[END_REF] who proved that there exists an aperiodic tile set, i.e., a tile set τ such that τ -tilings exist but all are aperiodic. (A tiling is periodic if there is a non-zero translation that does not change it. One can show that if a tile set has a periodic tiling then it has a 2-periodic tiling where some finite block is repeated horizontally and vertically.) Berger used a complicated multi-level construction that was later simplified in different ways by Robinson [13] and others. The simplification made clear that Berger's construction is essentially based on selfsimilarity: any tiling can be divided into blocks that behave like individual tiles. (In the original construction this similarity was obscured by some irregularities; the cleaned versions could be found in [START_REF] Ollinger | Two-by-two Substitution Systems and the Undecidability of the Domino Problem, Computability in Europe[END_REF] or [START_REF] Durand | Local rules and global order, or aperiodic tilings[END_REF].)

This self-similarity creates some kind of a skeleton that can be used to initiate computations. However, the problem is that we necessarily initiate them in many different places, and these "geometrically parallel" computation should be organized to achieve some goal. Berger used them to prove the undecidability of the domino problem (to determine whether a given tile set has at least one tiling); for that purpose it is enough to initiate multiple copies of the same computation: all are limited in time and space, but among them there are computations of arbitrary length. For that we split the plane into different zones used for different computations. It is possible to find such an arrangement; in each zone the standard local rules for a computation are used but zones are not contiguous. So we need additional efforts to transmit the information from one zone to another one. This all can be done (with limited overlap, so the total density of information in a given cell remains finite).

Then Hanf [START_REF] Hanf | Nonrecursive tilings of the plane, i[END_REF] and Myers [START_REF] Myers | Nonrecursive tilings of the plane, ii[END_REF] proved that there are tile sets that admit only non-recursive tilings (a much stronger statement than the existence of an aperiodic tile set). This was done by embedding a separation problem for two inseparable enumerable sets, and for this we need that all the parallel computations not only share the same (finite) program, but also share the same (infinite) input. Therefore, some additional machinery is needed to synchronize the inputs of all the computations (each computation gets a finite part of the infinite input sequence, but these finite parts are consistent pieces of an infinite input).

When simulating 1D effectively closed subshift, we need more: the input is given to us externally (the contents of the vertical lines that carry A-letters) and we need to check this input against all possible forbidden substrings. This means that we are very limited in space (and cannot distribute pieces of input sparsely over the entire plane as before).

Robinson-type solution

The way to do this was developed in [START_REF] Durand | Complex Tilings[END_REF]. At each level of self-similarity we have computation squares that are arranged in computational stripes. Such a stripe is infinite in vertical direction and carries an infinite computation of a finite-space cellular automaton. (One can wonder whether it makes sense to have an infinite computation in a finite space. Indeed, it is not really infinite; it runs for some time (exponential in the width of the stripe) and then is restarted. The repeated computations are not necessarily identical, since they interact with the other computations which could be different.) Each stripe performs some checks for the part of the horizontal sequence that is near it. When the level (and the size of the stripe) increases, the checked zone and the time allowed for the computation increase. Working together, the stripes can check the horizontal sequence against all forbidded substrings.

In [START_REF] Durand | Complex Tilings[END_REF] this technique was used for one specific 1D effectively closed subshift with a binary alphabet (for some fixed α < 1 we forbid all sufficiently long strings whose Kolmogorov complexity is less than α times length). However, this technique is quite general and can be used for any 1D effectively close subshift modulo some technical problem.

This technical problem is that the underlying self-similar structure may be "degenerate" in the sense that the plane is divided in two parts that have no common ancestors. In this case we need some additional tricks (extending the zone of responsibility of each stripe) that were not needed for the specific subshift of [START_REF] Durand | Complex Tilings[END_REF]. The reason why they were not needed: if a string of low complexity (compared to length) is split into two parts, one of then has low complexity, too.

So the technique of [START_REF] Durand | Complex Tilings[END_REF] is not enough. The final construction was discovered (in fact, independently from [START_REF] Durand | Complex Tilings[END_REF]) by N. Aubrun and M. Sablik.

Fixed-point solution

There is a different way to organize the computations that uses fixed-point selfsimilar tiling. The idea of a self-similar fixed-point tile set can be explained as follows. We already know (since Wang papers) that tiling can be used to simulate computations. This computation, in its turn, can be used to guarantee the desired behavior of bigger blocks, called macro-tiles. So for a desired behavior of macrotiles we can construct tiling rules (i.e., tile set) that guarantees this behavior. If, by chance, these tiling rules coincide with the rules for macro-tiles, we get self-similarity as a consequence.

But there is a classical tool to get this coincidence intentionally, not by chance: the Kleene fixed-point construction. It was used by Kleene in the recursion theory and later by von Neumann to construct self-reproducing automata. Usually it is illustrated as follows: for every program p (in fact, for every string p) there exists a program p ′ that prints the text of p. Kleene's theorem guarantees that one can find p such that p ′ is equivalent to p, i.e., the program p prints its own text. The same trick (though not just the statement of Kleene's recursion theorem) can be used for 2D computations. This was done first by Gács [START_REF] Gács | Reliable Computation with Cellular Automata[END_REF] in a complicated setting (error-correction in 2D computations); we use the same idea in a much simpler environment. For each tile set τ one can construct a set τ ′ of tiles that force macrotiles to behave like τ -tiles; Kleene's trick can then be used to make τ isomorphic to τ ′ . This construction is explained in [START_REF] Durand | Fixed Point and Aperiodic Tilings[END_REF].

Then some additional structure can be superimposed with this self-similar skeleton (by adding some other computations); Kleene's trick can still be used to achieve self-similarity (in some extended sense). This construction is rather flexible and can be applied to different problems, see [START_REF] Durand | Fixed-point tile sets and their applications[END_REF]. The differences and similiarities between two constructions are summarized in the following comparison table.

Comparison table

Problem

This work is licensed under the Creative Commons Attribution-NoDerivs License. To view a copy of this license, visit http://creativecommons.org/ licenses/by-nd/3.0/.


Adeline Crépieux

Fedor Simkovic

Benjamin Cambon

Fabienne Michelini

Fedor Šimkovic

Enhanced thermopower under time-dependent gate voltage

Keywords:

come

Since their discoveries in 1821 by Seebeck [1] and in 1834 by Peltier [2], thermoelectric effects have been exploited for many applications, such as heat voltage converters, thermocouples or refrigerators. The Seebeck coefficient, or thermopower S, measures the voltage induced by a temperature gradient through an open circuit, whereas the Peltier coefficient Π measures the heat flow induced by an applied current for no temperature gradient. In the linear response regime, the Onsager relation gives Π = -ST , where T is the average temperature of the sample.

The recent achievement of nanoscale systems has invigorated research activities in this field and renewed the quest of the great thermopower (see Ref. 3 for a recent review). On the one hand, stationary Seebeck coefficients have been measured in different nanoscale systems: quantum dot [4], atomic-size contacts [5], spin valves [6], nanowires [7], and carbon nanotube [8]. The Landauer-Büttiker formalism used for the electrical conductance was extended to model thermal transport in microstructures with many terminals [9,10], including inelastic effects [11]. Validity of the conventional thermodynamics linear equations was deeply questioned in mesoscopic systems: the Onsager relations between heat and charge transport coefficients [9,12], the Wiedemann-Franz law which links electrical and thermal conductances [13], and the Fourier law [14]. On the other hand, time-dependent electric transport also benefits from active research works. Single-electron time-control has been demonstrated experimentally [15,16], with a fair agreement with earlier theoretical developments [17,18]. More recently, deeper issues as memory effects [19] or interplay between multiple time-modulations [20] have been addressed theoretically. Besides, calculation of time-dependent heat current in a linear phonon chain has recently been achieved [21]. However, thermopower dynamics lacks both experimental as well as theoretical investigations.

This letter gives a first insight into time-dependent nonequilibrium thermoelectric transport. As a major result, illustrated in Fig. 1, we show that the thermopower can be strongly enhanced during the transient regime in a metal/dot/metal device, presented in Fig. 2. Time evolution of the thermopower exhibits promising features. Indeed, it can be significantly modified and controlled by changing the dot energy from ε0 to ε0 + γ0 at time t 0 = 0: starting from the stationary value at t < t 0 , it increases during a finite time, and then it converges toward its new stationary value at t → ∞. Time-dependent heat current through left (L) or right (R) reservoir in equilibrium reads

I h L,R (t) = I E L,R (t) - µ L,R (t) e I e L,R (t) , (1)
where I E L,R (t) is the energy current, I e L,R (t) the electric current, and µ L,R the chemical potential. Timedependent Seebeck coefficient can be obtained from the ratio between voltage gradient ∆V and temperature gradient ∆T between the two reservoirs, when both left and right time-dependent electric currents cancel

S(t) = - ∆V ∆T I e L (t) = I e R (t) =0 , (2)
whereas time-dependent Peltier coefficient is defined as

Π(t) = I h L (t) -I h R (t) I e L (t) -I e R (t) ∆T =0
.

(3)

The general system we consider consists of N energy levels in an interacting central region connected to noninteracting left and right leads. The total hamiltonian reads

H = H L + H R + H c + H T with H L,R = k∈L,R ε k (t)c † k c k , (4)
H c = n ε n (t)d † n d n + H int , (5)
H T = p=L,R k∈p,n V kn (t)c † k d n + h.c. , (6)
where

c † k (d † n ) and c k (d n )
are the creation and annihilation operators for the leads (dot). H int is the interacting part of H c (Coulomb interactions, phonons coupling, etc...). The energy band ε k of the reservoirs can be time-dependent through a bias change (source-drain voltage). Energy levels of the central region ε n can be time-dependent through a modulation of gate voltage. For the sake of generality, hopping amplitudes V kn are also allowed to be time-dependent. An extension to multiterminal system with additional degrees of freedom, e.g. spin, is straightforward. In this calculation, we only consider the electron contribution to the energy current. The energy current operator is related to the time derivative of the hamiltonian describing the leads [22] by

t t (t) 0 γ 0 ε ε 0 + 0 ε 0 ε 0 (t) L µ Γ R I I R E R T R T L µ R
I E L,R = -ḢL,R . Calculating commutators [H L,R
, H], we end up with ( = 1)

I E L,R (t) = i k∈L,R,n ε k (t)V kn (t)c † k d n + h.c. - k∈L,R εk (t)c † k c k . (7)
Thus, the average energy current reads

I E L,R (t) = 2Re k∈L,R,n ε k (t)V kn (t)G < nk (t, t) -Im k∈L,R εk (t)G < kk (t, t) , (8)
where

G < kk ′ (t, t ′ ) = i c † k ′ (t ′ )c k (t)
is the lead Green function, and where we have introduced the mixed Green function t1) are the Green functions of the isolated leads. Expression of the energy current becomes

G < nk (t, t ′ ) = i c † k (t ′ )d n (t) which obeys the Dyson equation G < nk (t, t ′ ) = n ′ ∞ -∞ dt 1 V * kn ′ (t 1 ) G r nn ′ (t, t 1 )g < k (t 1 , t ′ ) + G < nn ′ (t, t 1 )g a k (t 1 , t ′ ) . (9)
G < nn ′ (t, t ′ ) = i d † n ′ (t ′ )d n (t) is the dot Green function. g < k (t, t ′ ) = if (ε k )e -i t t ′ dt1ε k (t1) and g a k (t, t ′ ) = iΘ(t ′ - t)e -i t t ′ dt1ε k (
I E L,R (t) = 2Re Tr ∞ -∞ dt 1 G r d (t, t 1 )Ξ < L,R (t 1 , t) +G < d (t, t 1 )Ξ a L,R (t 1 , t) -Im Tr ĖL,R (t)G < L,R (t, t) , (10)
where we have defined the self-energy associated to energy transfer as

Ξ a,< L,R (t, t ′ ) = k∈L,R V * k (t)g a,< k (t, t ′ )ε k (t ′ )V k (t ′ ).
In these expressions, ĖL,R and V k are vectors whereas G r,< d , G < L,R , Ξ a,< L,R are matrices. Last term in Eq. ( 10) is a pure reservoir contribution to the energy current.

We now consider a non-interacting metal/dot/metal junction with a single dot level ε 0 connected to reservoirs with constant energy bands. This model is suitable for experiments in which Coulomb interaction is weak in the dot with strong coupling to reservoirs [15,23]. In that case, the energy current takes the form

I E L,R (t) = 2Re ∞ -∞ dt 1 ∞ -∞ dε 2π iεe iε(t-t1) ×Γ L,R (ε, t 1 , t) G r d (t, t 1 )f L,R (ε) + G < d (t, t 1 )Θ(t -t 1 ) , ( 11
)
where Θ is the Heaviside function, f L,R the Fermi-Dirac distribution function, ρ L,R the density of states, and

Γ L,R (ε k , t, t ′ ) = 2πρ L,R (ε k )V * k (t)V k (t ′
) measures the strength of the coupling between the dot and each lead.

We investigate the time-dependent thermoelectric response to a unique change ε 0 (t) = ε0 + γ 0 (t) with γ 0 (t) = γ0 Θ(tt 0 ). This models a dot energy switching from ε0 to ε0 + γ0 by applying a gate voltage at t 0 (see Fig. 2). The time-dependent heat current defined by Eq. ( 1) for p = L, R is now expressed in terms of the spectral function A(ε, t) as

I h p (t) = - 1 h Γ p 2 ∞ -∞ (ε -µ p )f p (ε)Im{A(ε, t)}dε + p ′ =L,R Γ p ′ ∞ -∞ (ε -µ p ′ )f p ′ (ε)|A(ε, t)| 2 dε , (12)
with [17] A(ε, t

) = ε -ε0 + iΓ/2 -γ0 e i(t-t0)(ε-ε0-γ0+iΓ/2) (ε -ε0 + iΓ/2)(ε -ε0 -γ0 + iΓ/2) , (13)
and Γ = Γ L + Γ R , for which we assume that Γ p does not depend on energy, and simply reduces to Γ p = 2πρ p |V p | 2 . The integration over energy in Eq. ( 12) has been performed numerically. In Fig. 3 is plotted the time evolution of electric and heat currents when the dot energy is modified abruptly at t 0 = 0. Starting from constant values at t < 0, left and right currents converge toward constant values at t → ∞. Between these two limits, currents show strong time-dependent variations. Currents through the zero-temperature, T R = 0, right lead (, see dashed lines in Fig. 3) exhibit time oscillations whose period is related to ε0 and γ0 , as Eq. ( 13) explicitly indicates. These oscillations of the electric current have been measured through a Ge dot [16]. Concerning the heat current, experimental results are still needed. In the left lead (see solid lines in Fig. 3), these oscillations disappear due to thermal effects given by T L = 0.

Using particle number conservation, the average dot occupation number is calculated from electric currents as N d (t) = e -1 I e dis (t) dt, where I e dis (t) = I e L (t) + I e R (t) is the displacement current [16]. In the top inset of Fig. 3, N d (t) globally follows an exponential decrease N d (∞) (1e -t/τr ) + N d (0) e -t/τr (see dotted line) characterized by the relaxation time τ r = /Γ: weaker is the coupling between the dot and the leads, longer is the relaxation time. Time evolution of N d (t) shows oscillations around this decrease, that have been already observed in experiments [16].

Eq. ( 1) comes from thermodynamic relations in leads at equilibrium: dH L,R = dQ L,R + µ L,R dN L,R , where N L,R is the lead occupation number and Q L,R is the lead heat. Similarly, for the dot out-of-equilibrium, we write dH c = dQ dµ L dN Lµ R dN R : the energy change in the dot reflects a balance between heat variation and charges leaving the dot times their energies. Thus, we define and numerically calculate an average heat in the dot as

Q d (t) = I h d (t) dt, where I h d (t) = I h L (t) + I h R (t) -ḢT (t)
. This definition perfectly agrees with energy conservation including contribution from tunneling. In the bottom inset of Fig. 3, the time

evolution of ∆ Q d (t) = Q d (t) -Q d (0)
shows a behavior similar to the dot occupation number (top inset of Fig. 3). However, dramatic differences occur in the stationary regimes, for which N d (t) is always constant: dot heat varies linearly in time, without any violation of conservation laws.

In the linear response limit, the time-dependent Seebeck coefficient, defined by Eq. ( 2), can be obtained from the approximate Fermi-Dirac distribution function [9]:

f L,R (ε) ≈ f 0 (ε) + f ′ 0 (ε)(µ L,R -(ε -ε F )T L,R /T ))
, where f 0 is the Fermi-Dirac distribution function for the leads when µ L = µ R . Taking left and right electric currents equal zero, we obtain the linear response for S(t) in the case of strong coupling to reservoir and small energy variation:

S(t) = - ∞ -∞ dεf ′ 0 (ε)(ε -ε F )T (ε, t) eT ∞ -∞ dεf ′ 0 (ε)T (ε, t) , (14)
where T (ε, t) = -4Γ L Γ R Im{A(ε, t)}/Γ is the timedependent transmission coefficient. This result is a generalization of the Seebeck coefficient expression obtained in the stationary case [3,9] including time-dependence of the transmission coefficient. For the steady-state, we have I e L = -I e R (constant N d (t) ), as can be seen in the top graph of Fig. 3. But in the time-dependent case, I e L (t) = 0 does not imply I e R (t) = 0 because of the dis-placement current. Since the Seebeck coefficient is measured in an open circuit, we must find the adequate ∆V and ∆T which simultaneously cancel both currents. It is important to emphasize that Eq. ( 14) is only valid under the following assumptions: i) linear response (small ∆V and ∆T in comparison to ε0 and T ); ii) high transmission through the barriers (large Γ L and Γ R in comparison to other energies); and iii) small gate voltage time-variation γ0 , in comparison to ε0 . Indeed, these assumptions allow to approximate |A(ε, t)| 2 ≈ -2Im[A(ε, t)]/Γ, and, hence, to cancel both I e L (t) and I e R (t) at any time. Similar expressions can be obtained for the timedependent Peltier coefficient defined by Eq. ( 3)

Π(t) = ∞ -∞ dε(ε -ε F )(f L (ε) -f R (ε))T (ε, t) e ∞ -∞ dε(f L (ε) -f R (ε))T (ε, t) ∆T =0 , (15)
following the same assumptions. In the linear response regime, Eqs. ( 14) and ( 15) verify the Onsager relation Π(t) = -T S(t) at any time. Fig. 1 is obtained using Eq. ( 14) of the linear response. It is shown an increase of the thermopower after a step-shaped gate-voltage pulse was applied. The reason is that in the transient regime, the system is much more sensitive to temperature or electrostatic variations. Furthermore, we measure the thermoelectric benefit of the transient regime calculating the percentage (S tran max -S stat ave )/S stat ave where S tran max is the maximum value of S and S stat ave = (S(t < t 0 ) + S(t → ∞))/2. In Fig. 4, we have plotted the percentage of thermopower increase as a function of γ0 /ε 0 . This ratio plays an important role since it does control the thermopower increase. In such a junction, the transient thermopower can be tuned by both ε0 and γ0 , which depend on the dot structural properties and of the applied gate voltage (see Fig. 2). Higher is the ratio, higher is the thermopower increase. Here, an increase up to 40% is obtained at small temperature.

We have proposed a first approach to heat dynamics in nanoscale junctions. General formula for the time-dependent heat and energy currents flowing through an interacting resonant-tunneling system have been obtained. We show that an enhanced thermopower can be generated during the transient regime in a metal/dot/metal junction, and that its maximum value can be tuned by both the dot energy and the gate voltage. With such numerical investigations, it will be possible to go beyond the linear response for the Seebeck and Peltier coefficients, and further determine non-linear thermodynamic laws. Moreover, we shall consider interacting systems in order to analyze phonon bath contribution [21], impact of electron-phonon interaction [11], and to study influence of charging effects in other experiments [24].

A.C. thanks I. Safi for discussion and E. Bernardo for her help in bibliographic researches.

FIG. 1 :

 1 FIG.1: Increase of the Seebeck coefficient in the transient regime of a metal/dot/metal junction shown in Fig.2: ε0 = 0.5 with γ0 = 0.05 (solid line), γ0 = 0.1 (dashed line), and γ0 = 0.15 (dotted line). We take symmetric barriers: ΓR = ΓL, t0 = 0, εF = 0, kBT = 0.1. The unit for energies is Γ.

FIG. 2 :

 2 FIG.2:(Left) schematic of metal/dot/metal junction with current directions for each lead; (right) time-dependence of the dot energy level. We define chemical potentials and temperatures of the left and right leads as: µL,R = εF ± e∆V /2 and TL,R = T ± ∆T /2. The Fermi energy is εF and T = (TR + TL)/2 is the average temperature.

FIG. 3 :

 3 FIG. 3: (Top graph) electric current and (bottom graph) heat current through left lead (solid lines) and right lead (dashed lines) as a function of time t, for ΓR = ΓL, t0 = 0, ε0 = 0.5, γ0 = 2.5, kBTL = 1, kBTR = 0, µL,R = ±0.5. Insets show (top inset) the dot occupation, and (bottom inset) the dot heat. Dashed lines indicate the stationary limits at t → ∞.

FIG. 4 :

 4 FIG.4: Percentage of increase of the Seebeck coefficient maximum in the transient regime S tran max as a function of γ0/ε0 for kBT = 0.05 (solid line), kBT = 0.15 (dashed line) and kBT = 0.5 (dotted line). We take ΓR = ΓL and εF = 0.


Christèle Dubourg
email: christele.dubourg@chu-rennes.fr

Damien Sanlaville

Martine Doco-Fenzy

Cédric Le Caignec

Chantal Missirian

Sylvie Jaillard

Caroline Schluth-Bolard

Emilie Landais

Odile Boute

Nicole Philip

Annick Toutain

Albert David

Patrick Edery

Anne Moncla

Dominique Martin-Coignard

Catherine Vincent-Delorme

Isabelle Mortemousque

Bénédicte Duban-Bedu

Sèverine Drunat

Mylène Beri

Jean Mosser

Sylvie Odent

Véronique David

Joris Andrieux

31 microdeletion syndrome in 14 French patients with mental retardation

Keywords: 17q21.31 microdeletion, MAPT, mental retardation

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Introduction

Array comparative genomic hybridization (aCGH) has revolutionized the diagnosis of mental retardation through the use of a "reverse phenotyping" approach making it possible to identify recurrent genomic rearrangements before their clinical description [START_REF] Slavotinek | Novel microdeletion syndromes detected by chromosome microarrays[END_REF]. Such techniques led to the recent characterization of novel microdeletion and microduplication syndromes [START_REF] De Vries | Diagnostic genome profiling in mental retardation[END_REF][START_REF] Stankiewicz | Use of array CGH in the evaluation of dysmorphology, malformations, developmental delay, and idiopathic mental retardation[END_REF][START_REF] Slavotinek | Novel microdeletion syndromes detected by chromosome microarrays[END_REF]. A recurrent 17q21.31 microdeletion [Mendelian Inheritance in Man (MIM) #610443], resulting in the loss of a 500 to 650 kb region between low-copy repeat (LCR) regions, was first described in 2006 in mentally retarded individuals with a clearly recognizable clinical phenotype of mental retardation, hypotonia and characteristic facial features [START_REF] Koolen | A new chromosome 17q21.31 microdeletion syndrome associated with a common inversion polymorphism[END_REF][START_REF] Shaw-Smith | Microdeletion encompassing MAPT at chromosome 17q21.3 is associated with developmental delay and learning disability[END_REF][START_REF] Sharp | Discovery of previously inserm-00541962, version 1 -1 Dec 2010 unidentified genomic disorders from the duplication architecture of the human genome[END_REF]. The complex genomic architecture of this microdeletion, with large clusters of LCRs at the breakpoints, suggested an underlying mechanism of non allelic homologous recombination (NAHR) [START_REF] Stankiewicz | Genome architecture, rearrangements and genomic disorders[END_REF]. A common 900 kb inversion polymorphism occurs in this region, and chromosomes with the inverted fragment in different orientations correspond to two different haplotypes, H1 and H2 [START_REF] Stefansson | A common inversion under selection in Europeans[END_REF]. The direct orientation of LCRs flanking the deleted fragment in the H2 haplotype predisposes the offspring to NAHR, favoring the generation of this microdeletion [START_REF] Stefansson | A common inversion under selection in Europeans[END_REF]. This mechanism is similar to that described for predisposing inversion polymorphisms in other microdeletion syndromes, such as Williams-Beuren syndrome (WBS), Angelman syndrome, Sotos syndrome [START_REF] Feuk | Structural variation in the human genome[END_REF], and 15q13.3 microdeletion syndrome [START_REF] Sharp | A recurrent 15q13.3 microdeletion syndrome associated with mental retardation and seizures[END_REF]. For several chromosomal loci subject to NAHR, reciprocal deletion and duplication syndromes have been identified, including WBS and 7q11.23 duplication [START_REF] Somerville | Severe expressive-language delay related to duplication of the Williams-Beuren locus[END_REF], Smith-Magenis syndrome and Potocki-Lupski syndrome [START_REF] Potocki | Molecular mechanism for duplication 17p11.2-the homologous recombination reciprocal of the Smith-Magenis microdeletion[END_REF], Prader-Willi syndrome/Angelman syndrome and 15q11q13 duplication [START_REF] Long | Triplication of 15q11-q13 with inv dup(15) in a female with developmental delay[END_REF], Velocardiofacial syndrome and 22q11.2 duplication [START_REF] Edelmann | A common molecular basis for rearrangement disorders on chromosome 22q11[END_REF]. However, deletions are generated at a higher rate than their reciprocal duplications [START_REF] Turner | The rates of de novo meiotic deletions and duplications causing several genomic disorders in the male germline[END_REF]. The first case of duplication of the 17q21.31 region was not described until 2007 [START_REF] Kirchhoff | A 17q21.31 microduplication, reciprocal to the newly described 17q21.31 microdeletion, in a girl with severe psychomotor developmental delay and dysmorphic craniofacial features[END_REF], in a patient with severe psychomotor developmental delay, facial dysmorphism, inserm-00541962, version 1 -1 Dec 2010 macrocephaly, abnormally broad fingers and toes and hirsutism. Four other patients with 17q21.31 microduplication and behavior problems and poor social interaction have also recently been reported [START_REF] Grisart | 17q21.31 microduplication patients are characterised by behavioural problems and poor social interaction[END_REF].

With the use of higher resolution arrays, taking frequent copy number variations in normal controls into account, the minimal critical region involved in this 17q21.31 microdeletion syndrome was recently refined to a 424 kb region encompassing at least six genes: C17orf69, the corticotropin-releasing hormone receptor 1 gene (CRHR1) (MIM #122561), the intramembrane protease 5 gene (IMP5) (MIM #608284), the microtubule-associated protein tau gene (MAPT) (MIM #157140), the saitohin gene (STH) (MIM #607067), and KIAA1267 [START_REF] Koolen | Clinical and molecular delineation of the 17q21.31 microdeletion syndrome[END_REF]. MAPT seems to be the most interesting of these genes for further study, because it is strongly expressed in the brain and has been implicated in several neurodegenerative diseases [START_REF] Rademakers | The role of tau (MAPT) in frontotemporal dementia and related tauopathies[END_REF].

The largest series of patients with 17q21.31 microdeletion syndromes reported to date was studied by Koolen et al. [START_REF] Koolen | Clinical and molecular delineation of the 17q21.31 microdeletion syndrome[END_REF], who identified a clinically recognizable phenotype, including developmental delay, childhood hypotonia and facial dysmorphism. This phenotype was recently expanded to include aortic root dilatation, recurrent joint dislocation, conductive hearing loss, dental abnormalities, the persistence of fetal fingertip pads [START_REF] Tan | Phenotypic expansion and further characterisation of the 17q21.31 microdeletion syndrome[END_REF], iris-choroid coloboma and partial situs inversus [START_REF] Sharkey | 17q21.31 microdeletion syndrome: further expanding the clinical phenotype[END_REF].

We present here a new series of 14 patients with the 17q21.31 microdeletion syndrome ascertained through the network of French array CGH platforms (http://www.renapa.univ-montp1.fr/) and the BACH database (https://www.genopole-lille.fr/bach/menu.php), confirming the described phenotype, refining the minimal critical region and providing further inserm-00541962, version 1 -1 Dec 2010 support for the role of MAPT in this syndrome. We also discuss the results of H1/H2 genotyping and parent-of-origin analysis, comparing our results with published findings.

Materials and methods

Patients

A collaborative study involving the French array CGH platforms analyzing patients with mental retardation (MR), dysmorphic features and multiple congenital abnormalities (MCA) was set up for the molecular and clinical characterization of all unpublished French patients carrying the 17q21.31 microdeletion. Molecular cytogenetics analyses of 2672 MR/MCA patients showed that 14 of these patients had a 17q21.31 microdeletion. Informed consent for genetic testing was obtained from all tested patients and their relatives.

Molecular cytogenetics

Two platforms were used for genomic copy number analyses, which were carried out according to the manufacturer's protocol (Agilent Technologies, Santa Clara, CA, USA): the Human Genome CGH Microarray 44K (for all patients except patient 7) and 244K (patient 7).

Data were processed with Feature Extraction (v. 9.1) software and results were interpreted with CGH analysis (v. 4.0) software (Agilent ® ). When a 17q21.31 deletion was identified by array CGH, the microrearrangement was confirmed either by FISH with various probes (RP11-597L17 or RP11-656O14 targeting CRHR1, RP11-413P22, RP11-669E4 or RP11-769P22 targeting MAPT) or by multiplex ligation-dependent probe amplification (MLPA) with the SALSA kit P245 from MRC-Holland (Amsterdam, the Netherlands) (www.mlpa.com).

Genotyping for H1 and H2, and parent-of-origin analysis inserm-00541962, version 1 -1 Dec 2010

When material from the parents was available, a family study was performed (12 patients). H1/H2 genotyping was carried out for individuals with the 17q21.31 deletion and their parents, by PCR amplification of a region encompassing a potential 238 bp deletion in the intron 9 of the MAPT gene associated with the H2 background [START_REF] Stefansson | A common inversion under selection in Europeans[END_REF][START_REF] Koolen | Clinical and molecular delineation of the 17q21.31 microdeletion syndrome[END_REF].

The parental origin of the chromosome with the deletion was determined with the DG17S142 dinucleotide marker and three to five additional variable number tandem repeats within the deleted region, together with two flanking short tandem repeats, D17S810 and D17S920 [START_REF] Koolen | Clinical and molecular delineation of the 17q21.31 microdeletion syndrome[END_REF].

Marker analyses were carried out according to standard procedures and data were interpreted with GeneMapper (v. 4.0) software (Applied Biosystems, Foster City, CA, USA).

Results

Array CGH results

All the patients studied had an overlapping heterozygous deletion of the 17q21.31 interval (Fig. 1). Oligonucleotide array CGH data identified a 493 kb region (chr17:41,073,486-41,566,599) (hg18-build36) as deleted in 11 affected individuals. In one patient (patient 7), the deletion was larger, being somewhere between 675 kb and 810 kb in size and extending to the distal part of the chromosome. However, the distal breakpoint region was a site of frequent copy number variation (CNV) in normal controls. The proximal breakpoint was also found to be located in a variable region. This may account for the subtle shifts in the breakpoints observed, as in patient 10 (chr17:41,212,860-41,566,599) (hg18-build36). By contrast, patient 3 had a significantly smaller deletion, of only 205 kb (chr17:41,310,123-41,515,622) (hg18-build36).

inserm-00541962, version 1 -1 Dec 2010

FISH studies confirmed the 17q21.31 deletion in patients 1, 2 and 4 to 13, with RP11-597L17 and/or RP11-656O14, RP11-413P22, RP11-669E4 or RP11-769P22 found to be deleted. The deletion identified in patient 3 was validated by MLPA, which confirmed the presence of a heterozygous deletion for the two probes targeting MAPT exons 8 and 11, whereas the signal obtained for the CRHR1 probe (exon 8) showed that this gene was not deleted (data not shown).

Clinical description (Table 1)

The most typical clinical features of 17q21.31 syndrome include hypotonia, developmental delay, overly friendly/amiable behavior and facial dysmorphism with a long face, a tubular or pear-shaped nose and a bulbous nasal tip (Fig. 2). The principal clinical characteristics of patients 1 to 14 are described in Table 1. Additional features are described below.

Patient 1 is the first and only child of unrelated parents. The pregnancy was complicated by intrauterine growth retardation (IUGR), which was interpreted as originating from a vascular cause. This child, a boy, was born at 39 weeks by cesarean section. Early problems included an inability to suck effectively and feeding problems, with enterocolitis causing ulcer and necrosis. Molecular analysis of the GJB2 and GJB6 genes were performed because of deafness, but no alteration was detected. For patient 14, the neonatal period was marked by feeding difficulties and hypotonia.

Genotyping

The genotyping of 24 parents (in 12 families) showed that the deletion was inherited from the mother in seven families and from the father in four families. In the remaining family, the panel of repeat markers across the deleted region could not distinguish between the parental chromosomes (Table 2).

Fifteen of the 24 parents tested were heterozygous for the H2 inversion, whereas four were homozygous for this inversion. The parent from which the inversion originated carried the H2 inversion polymorphism in at least a heterozygous state.

inserm-00541962, version 1 -1 Dec 2010

Discussion

All the patients with 17q21.31 microdeletion syndrome reported here were identified by array CGH screening, by the network of French array CGH platforms, of large heterogeneous cohorts of patients with mental retardation. The calculated prevalence of 17q21.31 syndrome in our French cohort was 0.52% (14 of the 2672 patients tested), which is of the same order of magnitude as previous estimates (0.64% for Koolen et al. [START_REF] Koolen | Clinical and molecular delineation of the 17q21.31 microdeletion syndrome[END_REF], 0.5% for Sharkey et al. [START_REF] Sharkey | 17q21.31 microdeletion syndrome: further expanding the clinical phenotype[END_REF]).

Given the copy number polymorphisms located at either end of the deleted region, we investigated the size of the deletion in 11 of our patients conforming to the DECIPHER description of 17q21.31 syndrome as a standard deletion extending from 40.99 to 41.57 Mb.

However, the critical region of 17q21.31 was recently refined and narrowed to a 424 kb genomic interval (chr17:41,046,729-41,470,954, hg18) by analyses based on SNP arrays and ultra-high density custom oligonucleotide arrays targeting the 17q21.31 region and excluding frequent copy number variations in normal controls [START_REF] Koolen | Clinical and molecular delineation of the 17q21.31 microdeletion syndrome[END_REF]. In our patient 3 with classical clinical characteristics, including facial features (Fig. 2), a smaller, partially overlapping deletion was observed (chr17:41,310,123-41,515,622, hg18-build36). This deletion encompasses only MAPT, STH and KIAA1267, leaving the patient with two copies of the CRHR1 gene, but the phenotype of this patient was no milder than that of the other patients.

Comparison of this 205 kb deletion with the 424 kb critical region described by Koolen et al. [START_REF] Koolen | Clinical and molecular delineation of the 17q21.31 microdeletion syndrome[END_REF] defined a new minimal critical region of 160.8 kb (chr17:41,310,123-41,470,954, hg18), providing strong evidence for the involvement of of the MAPT (microtubule-associated protein Tau) gene (MIM #157140) in this syndrome. Gain-of-function mutations in MAPT have been identified in neurodegenerative diseases, such as autosomal dominant frontaltemporal dementia with parkinsonism [START_REF] Hutton | Association of missense and 5'-spice-site mutations in tau with the inherited dementia FTDP-17[END_REF][START_REF] Souza | Missense and silent tau gene mutations cause frontotemporal dementia with parkinsonism-chromosome 17 type, by affecting multiple alternative RNA splicing regulatory elements[END_REF], supranuclear bulbar palsy and corticobasal inserm-00541962, version 1 -1 Dec 2010 degeneration [START_REF] Pittman | Linkage disequilibrium fine mapping and haplotype association analysis of the tau gene in progressive supranuclear palsy and corticobasal degeneration[END_REF], or Alzheimer's disease [START_REF] Myers | The H1c haplotype at inserm-00541962, version 1 -1 Dec 2010 the MAPT locus is associated with Alzheimer's disease[END_REF], in which abnormal deposition of insoluble tau protein is observed, with a decrease in the concentration of the normal soluble tau molecule in patients' brains, affecting the assembly, dynamic behavior and spatial organization of microtubules in neurons and glial cells [START_REF] Hutton | Molecular genetics of chromosome 17 tauopathies[END_REF]. The microdeletion event observed in 17q21.31 syndrome results in MAPT haploinsufficiency. In mice, Mapt haploinsufficiency in tau -/animals affects microtubule density and stability in small-caliber axons [START_REF] Harada | Altered microtubule organization in small-calibre axons of mice lacking tau protein[END_REF], resulting in a murine phenotype including muscle weakness in a wire-hanging test, hyperactivity in a novel environment and impairment in contextual fear conditioning [START_REF] Ikegami | Muscle weakness, hyperactivity, and impairment in fear conditioning in tau-deficient mice[END_REF]. However, screening for mutations in MAPT in 122 individuals with a phenotype suggestive of 17q21.31 syndrome identified no disease-associated variants [START_REF] Koolen | Clinical and molecular delineation of the 17q21.31 microdeletion syndrome[END_REF].

The facial features described by other groups [START_REF] Koolen | Clinical and molecular delineation of the 17q21.31 microdeletion syndrome[END_REF][START_REF] Tan | Phenotypic expansion and further characterisation of the 17q21.31 microdeletion syndrome[END_REF] was found in our patients, but become increasingly obvious with age. Consistent with one patient described by Tan et al. [START_REF] Tan | Phenotypic expansion and further characterisation of the 17q21.31 microdeletion syndrome[END_REF], two of our patients presented scaphocephaly revealing metopic craniostenosis. All our patients were hypotonic, with global developmental delay predominant over language problems. Other defects (ophthalmologic, cardiac, urologic, central nervous system, joint hypermobility, hip dysplasia) were commonly noted. Prenatal ischemic infarction was diagnosed in patient 3, as also reported for one of the patients described by Koolen et al. [START_REF] Koolen | Clinical and molecular delineation of the 17q21.31 microdeletion syndrome[END_REF]. Tan et al. [START_REF] Tan | Phenotypic expansion and further characterisation of the 17q21.31 microdeletion syndrome[END_REF] reported hearing defects resulting from chronic otitis. One of our patients (patient 1) was deaf, but this problem was due to a neurosensorial defect rather than conductive problems, despite the occurrence of numerous ENT infections in early infancy. The 17q21.31 genomic region contains a 900 kb inversion polymorphism generating two highly divergent haplotypes, H1 and H2, that are transmitted en bloc [START_REF] Stefansson | A common inversion under selection in Europeans[END_REF]. The frequency of the H2 lineage has been reported to be about 20% in the European population, whereas this inserm-00541962, version 1 -1 Dec 2010 haplotype is rare in Africans and almost never detected in East Asians. We indirectly identified this H2 inversion in 23 of the 48 chromosomes from parents tested and, for each trio, the parent from which the chromosome 17 deletion was inherited carried the H2 inversion polymorphism in either the homozygous or heterozygous state, giving a significantly higher frequency. Seven of the deletions were of maternal origin and four were of paternal origin. Combining these data with those obtained by Koolen et al. [START_REF] Koolen | Clinical and molecular delineation of the 17q21.31 microdeletion syndrome[END_REF], we can assume that there is no significant bias in the parental origin of the chromosome bearing the deletion. Unfortunately, we were unable to obtain DNA from the parents of patient 3. The H2 haplotype confers a genomic architecture in which LCR subunits are directly oriented, mediating the NAHR event leading to deletion [START_REF] Lupski | Genome structural variation and sporadic disease traits[END_REF]. However, as the frequency of 17q21.31 deletion in the offspring of H2 inversion carriers is low, the H2 allele is necessary, but not sufficient to generate the 17q21.31 deletion [START_REF] Koolen | Clinical and molecular delineation of the 17q21.31 microdeletion syndrome[END_REF].

Considering our French series of 14 patients with the 17q21.31 microdeletion syndrome and previous reports of 22 [START_REF] Koolen | Clinical and molecular delineation of the 17q21.31 microdeletion syndrome[END_REF] and 11 patients [START_REF] Tan | Phenotypic expansion and further characterisation of the 17q21.31 microdeletion syndrome[END_REF], it is clear that this microdeletion syndrome is clinically significant. Our study also narrowed the definition of the minimal deleted region to a restricted interval of 160.8 kb, strengthening the implication of the MAPT gene in this syndrome.

Patient 2

 2 is the second child of Moroccan and Breton parents. Prenatal imaging showed right hydronephrosis. A ventricular septal defect (VSD) observed at birth regressed spontaneously. This patient presented bronchiolitis and pyelonephritis at the age of six months, feeding problems, fainting due to hypotonia at the age of nine months, and epileptic seizures at the age of three years. He has a short stature and low weight (-2SD) and is prone to inserm-00541962, version 1 -1 Dec 2010 frequent falls. He has discrete aortic dilatation, with measurements at the upper limit of the normal range (27.3 mm aortic sinus at 15 ½ years of age, 28.5 mm at 17 years). Patient 3 is the first child of healthy unrelated parents. The pregnancy was characterized by a hygroma colli during the first trimester and multicystic right renal dysplasia. An ischemic infarction of the superficial middle cerebral artery territory, on the right side, was diagnosed in the neonatal period and probably occurred before birth. It resulted in fronto-parietal ischemic sequelae, initially in the form of symptomatic ipsilateral temporooccipital hemorrhage, subsequently progressing to cortico-subcortical atrophy. Explorations carried out during and after the early postnatal period demonstrated significant asymmetry in the development of the two hemispheres of the brain. At the age of 16 months, this patient still presents axial hypotonia, with an unstable sitting posture and retropulsion episodes. Patient 4 is a quiet child. Skin lesions were reported for this patient, with numerous nevi, including a hairy nevus on the face, a depigmented patch of the base of the neck and a pigmented spot on the left nipple. Patient 5 was born after a pregnancy complicated by IUGR. This patient, born to patients from a gypsy background, had a number of skin lesions, including generalized hyperpigmentation, a pigmented spot on the left side of the front of the neck, a depigmented patch above the left nipple and numerous nevi, including one on the right sole and another on the left hand. Patient 6 presented initial hypotonia, followed by seizures at the age of 14 months. His testes were located in the inguinal canal. Patient 7 was delivered at 31 weeks of gestation, by cesarean section due to acute fetal distress. Scaphocephaly was recorded, leading to the detection of metopic craniostenosis at the age of two years. inserm-00541962, version 1 -1 Dec 2010 Patient 8 presented feeding difficulties in the neonatal period. Severe speech dyspraxia became apparent in infancy. Patient 9 is the second child of non consanguineous parents. She was hypotonic as a baby, with tongue protrusion. She also presented feeding difficulties and constipation. Patient 10 is the third child of unrelated parents. The pregnancy was uncomplicated. A bilateral cleft lip was discovered at birth, with no damage to the alveolar ridge or palate. This patient is very sociable and cheerful. Her skin is very dry. Patient 11 presented delayed development of axial tonus at four months. Seizures first appeared at the age of eight months, with temporo-occipital foci. Patient 12 presented hypotonia and feeding difficulties during the neonatal period. He also displayed cryptorchidism requiring surgical treatment. Patient 13 presented developmental delay, more serious for psychomotor development than for language.

Figure 1 .

 1 Figure 1. Schematic diagram of 17q21.31 rearrangements, indicating relative sizes and

Figure 2 .

 2 Figure 2. Facial photographs of ten patients with 17q21.31 deletions: from left to right,

Figure

  Figure(s)

Table I .

 I Clinical characteristics of patients with 17q21.31 deletions

Table II .

 II Genotyping for H1/H2 and parent-of-origin results inserm-00541962, version 1 -1 Dec 2010

	CNV			CNV
	40.99	DECIPHER (about 580kb)		41.57
	41.04	Koolen et al 2008 (424 kb)	41.47
	41.07	Patients 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 14 (493 kb)	41.57
	41.01	675 kb < Patient 7 < 810 kb		41.71	41.84
		41.31	Patient 3 (205 kb)	41.51
	41.21	Patient 10 (353 kb)	41.57
		New minimal critical region
		(160.8 kb)

Acknowledgments

We thank all the staff of the Biogenouest  -IBiSA (Rennes) transcriptomics platform. We also thank our colleagues for referring patients, and the patients and their families for participating in this study. This research was supported by grants from FEDER, DHOS and STIC 2004/EGMAR. inserm-00541962, version 1 -1 Dec 2010

Electronic databases

Network of the