Piermarco Cannarsa
email: cannarsa@mat.uniroma2.it

Francesco Marino

Peter Wolenski
email: wolenski@math.lsu.edu

and Impulsive Systems

Keywords: Differential Inclusion, Semiconcavity, Optimality, Minimum Time Function. AMS

In this paper we consider the Minimum Time Problem with dynamics given by a differential inclusion. We prove that the minimum time function is semiconcave under suitable hypotheses on the multifunction F.

Introduction

The minimum time problem consists of minimizing the time T over all trajectories of a controlled dynamical system that originate from an initial point x ∈ R n and terminate on a compact target set K ⊆ R n . The main goal of this paper is to develop and extend aspects of the considerable theory of this subject to situations where the controlled dynamics take the form of a differential inclusion. Specifically, we shall study the problem min T,

where the minimization is over all absolutely continuous arcs y(•) that satisfy the differential inclusion

ẏ(t) ∈ F y(t) a.e. t ∈ [0, T ] (2)
and the initial and terminal conditions y(t) = x and y(T ) ∈ K.

Here and throughout the paper, F : R n ⇒ R n is a given multifunction. As a function of the initial parameter x, the minimum time function T (x) equals the optimal in (1) and is our main object of study.

There is an extensive literature on the minimum time problem, perhaps originating with Hermes and Lasalle [START_REF] Hermes | Functional analysis and time optimal control[END_REF], but having since grown beyond our capacity here to mention everyone who has contributed to the topic. The relationship between the regularity of the minimum time function and the local controllability property is well known in classical control theory. The first definition of Petrov Condition is due to [14,[START_REF] Petrov | The Bellman problem for a time-optimality problem[END_REF], in which the author investigated the Lipschitz property of T when the target set is a singleton. More general results have been obtained over the last two decades. In [START_REF] Bardi | An approximation scheme for the minimum time function[END_REF], under the regularity assumption for the target set, T (•) is shown to be Lipschitz continuous if and only if controllability conditions are satisfied. The Hölder continuity of the minimum time function when the target is a lowerdimensional manifold is studied in [17]. Further regularity properties of the minimum time function for a nonlinear control system can be found in [START_REF] Cannarsa | Convexity properties of the minimum time function[END_REF][START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF] in which under a Petrov controllability assumption, T (•) is shown to be semiconcave if the distance function from a target set is semiconcave. Moreover a condition of Petrov type for a general target is shown to be necessary and sufficient for the Lipschitz continuity of T . A Necessary and sufficient condition for local Lipschitz continuity of the value function is provided in [18] when the target is an arbitrary closed set and the dynamics is given by a differential inclusion with a measurability assumption on t. In [19] a Petrovtype modulus condition is also considered to prove the local continuity of T near the target set. The semiconcavity of a value function for a general exit time problem is given in [16], where suitable smoothness assumptions on the dynamics of the system are required but the target set can be completely general. In [5] structural properties of a nonlinear control system ensuring that the attainable set satisfies a uniform interior sphere condition are detected. As a consequence of this result, a semiconcavity property for the value function with a general target is obtained. See [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF] and [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF] for further references on time optimal control problems and semiconcave functions.

Our main goal here is to extend the semconcavity result from [6, 7] to a class of differential inclusions that were recently introduced in [START_REF] Cannarsa | Semiconcavity of the value function for a class of differential inclusions[END_REF]. In a companion paper to this one, we shall address the other main results from [6,[START_REF] Cannarsa | Convexity properties of the minimum time function[END_REF], where the dynamic framework contains a control parametrization with C 1+ -dependence; that is, the gradient in the state variable exists and is locally Lipschitz. However, the conclusions in these results have a nature that depends only on the admissible velocity sets F (x) and not on a given parametrization. Thus it is natural to seek conditions only on the multifunction F that ensure similar conclusions. A full discussion of this issue as it relates to the Mayer problem is presented in [START_REF] Cannarsa | Semiconcavity of the value function for a class of differential inclusions[END_REF], and those considerations are pertinent for the minimum time problem that is being addressed in the current paper.

A well-developed theory for differential inclusions exists (see e.g. [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF][START_REF] Clarke | Nonsmooth analysis and control theory[END_REF])

under the following collection of Standing Hypotheses:

(SH)      1) F (x)
is nonempty, convex, and compact for each x ∈ R n , 2) F is locally Lipschitz with respect to the Hausdorff metric, 3) there exists r > 0 so that max{|v| :

v ∈ F (x)} ≤ r(1 + |x|).
Recall the Hamiltonian H : R n × R n → R associated to F is defined by

H(x, p) = sup v∈F (x) v, p , (4)
and that there is a one-to-one correspondence between Hamiltonian functions that are convex and positively homogeneous in p and multifunctions F with closed and convex values. The relationship is

v ∈ F (x) ⇐⇒ v, p ≤ H(x, p) ∀p ∈ R n .
Assumptions on H are therefore intrinsic to F and do not depend on a particular parameterization. The following additional hypotheses will be invoked, and are stated in terms of Hamiltonian H: For each compact convex subset K ⊆ R n and p = 0 we have

(H)         
1) there exists a constant c ≥ 0 (depending on K) so that x → H(x, p) is semiconvex on K with constant c|p|, 2) the gradient ∇ p H(x, p) exists and is Lipschitz in x on K, and uniformly so over p in compact subsets of R n \ {0}.

The definition of semiconvexity and other preliminaries will be recalled in the next section. A method to generate examples of multifunctions satisfying (SH) and (H) is given in [START_REF] Cannarsa | Semiconcavity of the value function for a class of differential inclusions[END_REF]. The main proof technique behind the semiconcavity results in [4, 6, 8] and other earlier work relied on a priori estimates of solutions to smoothly parameterized differential equations. Our proof technique here relies heavily on the maximal principle. The main contribution of this paper and its companion is to show how the broad theory of minimum time optimal control can be carried out when smooth parameterization assumptions are replaced by the assumption (H).

We provide background material and preliminary results in Section 2, and our main semiconcavity result is given and proved in Section 3.

Preliminaries

We review the basic concepts from nonsmooth analysis used in the sequel. Standard references are [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF][START_REF] Clarke | Nonsmooth analysis and control theory[END_REF].

General considerations and semiconcavity

We denote by B the unit ball of R n centered at 0, and by B r (x) := x + rB the ball centered at x of radius r. The closed balls are denoted similarly but replace B by B. Hereafter, S and int(S) stand for the closure and interior, respectively, of any given set S ⊂ R n .

Let K ⊆ R n be a closed set and x ∈ K. A vector ν ∈ R n is a proximal normal to K at x provided there exists σ > 0 so that ν, y-x ≤ σ y-x 2 for all y ∈ K. The proximal normal cone to K at x consists of all the proximal normals to K at x and is denoted by N P K (x). One has ν ∈ N P K (x), ν = 1 if and only if there exists r > 0 so that B r (x + rν) ∩ K = {x}. The Clarke normal cone N K (x) is defined as

co ζ : ∃x i → x, ζ i → ζ, ζ i ∈ N P K (x i ) ,
where co denotes taking the closed convex hull. Now suppose f : R n → (-∞, +∞] is lower semicontinuous and

epi f = {(x, α) : α ≥ f (x)}
is its (closed) epigraph. For x ∈ dom f := {x : f (x) < ∞}, the set of proximal subgradients of f at x is denoted by ∂ P f (x) and equals

ν ∈ R n : (ν, -1) ∈ N P epi f (x, f (x)) .
The limiting subgradient ∂ L f (x) equals

ν : ∃x i → x, ν i → ν, ν i ∈ ∂ P f (x i ) ,
and the Clarke subgradient ∂f (x) is defined as ∂f (x) = co ∂ L f (x). Recall f is locally Lipschitz provided for all compact sets K ⊂ R n , there exists a constant k (called the rank of f on K) so that x, y ∈ K implies |f (x)-f (y)| < k|x-y|).

For a closed set K ⊆ R n , the distance function d K (•) : R n → R is defined by d K (x) := inf{ y -x : y ∈ K}, and is globally Lipschitz of rank one and satisfies ∂d K (x) = N K (x) ∩ B for all x ∈ bdry K.

The property of semiconcavity has several characterizations for both sets and functions, and the following proposition lists some of the functional properties that will be used in the sequel.

Proposition 1. For a convex set K ⊆ R n , a function f : K → R, and a constant c ≥ 0, the following properties are equivalent:

(1) for all x 0 , x 1 ∈ K and 0 ≤ λ ≤ 1, one has

(1 -λ)f (x 0 ) + λf (x 1 ) -f (x λ ) ≤ cλ(1 -λ)|x 1 -x 0 | 2 , where x λ = (1 -λ)x 0 + λx 1 ,
(2) f is continuous and, for all x ∈ K and z ∈ R n with x ± z ∈ K, one has

f (x + z) + f (x -z) -2f (x) ≤ 2c|z| 2 , (3) the map x → f (x) -c|x| 2 is concave.
Moreover, if K is open and f is locally Lipschitz, then any of the above properties holds true if and only if (4) for each x ∈ K and ν ∈ ∂f (x), one has

f (y) ≤ f (x) + ν, y -x + c|y -x| 2
for all y ∈ K.

See [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF] for further details and a thorough development. A Lipschitz function f satisfying property (1) of Proposition 1 is called (linearly) semiconcave on K with constant c. The above "concave" concepts have "convex" counterparts by reversing the inequalities and the signs in the quadratic terms. In other words, f is semiconvex on K if and only if -f is semiconcave on K.

We note that semiconcavity of f is the same as the property f being lower C 2 that is presented in [15, Definition 10.29], where f is given a representation as an infinum of a parameterized family of C 2 functions. Related to that definition is the following simple fact which can be found in [8, Proposition 2.1.5].

Proposition 2. Suppose {f α (•)} α∈A is a family of semiconcave functions defined on the convex set K with a constant c independent of α belonging to the compact index set A. Then f (x) = inf{f α (x) : α ∈ a} is semiconcave on K with constant c.

Note that the previous result implies that if F is parameterized as F (x) = {f (x, u) : u ∈ U } where f has C 1+ -dependence in x and continuous in u with U compact, then for each p ∈ R n , one has x → H(x, p) is semiconvex. That is, such an F satisfies our new hypothesis (H1).

The Hausdorff distance between two compact subsets S 1 and S 2 of R n is defined as

dist H (S 1 , S 2 ) = max dist + H (S 1 , S 2 ), dist + H (S 2 , S 1 )
,

where dist + H (S, S ) = inf ε : S ⊆ S + εB . A multifunction F : R m ⇒ R n with compact values is Lipschitz on K ⊆ R m provided there exists a constant k (called a Lipschitz rank of F ) so that dist H F (x), F (y) ≤ k|x -y| for all x, y ∈ K. This holds if and only if for each p ∈ R n , x → H(x, p) is a Lipschitz function on K with constant k|p|. The mid-point property for a multifunction F on a convex subset K is that dist + H 2F (x), F (x + z) + F (x -z) ≤ c|z| 2 (5)
holds whenever x, x ± z ∈ K. This is equivalent to x → H(x, p) being semiconvex on K for each p ∈ R n with constant c|p|, which is the new assumption (H1) we shall impose in proving our main result.

If the above function or multifunction is defined on all of R n , then all of the above function and multifunction concepts can be quantified as being local, by which is meant that the said property holds in every convex compact neighborhood K of each point in R n with the constant depending on K.

The first part of the following result comes from [10, Prop 3.2.4(e)], but we give a simpler proof based on the assumption (H1). The second part is a partial inclusion analogous to [10, Prop 3.2.4(d)], but with convexity replaced by semiconvexity. Proposition 3. Suppose F satisfies (SH) and H : R n × R n → R is as in (4) and satisfies (H1).

(1) For any x ∈ R n and for any p ∈ R n , we have for λ > 0 that H(x, λp) = λH(x, p) and

∂ x H(x, λp) = λ∂ x H(x, p).
(

) For (u, v) ∈ ∂H(x, p), one has u ∈ ∂ x H(x, p) and v ∈ ∂ p H(x, p). That is, ∂H(x, p) ⊆ ∂ x H(x, p) × ∂ p H(x, p) 2
Proof. The positive homogeneity property H(x, λp) = λH(x, p) for all λ > 0 holds as seen directly from the definition (4). Suppose u ∈ ∂ x H(x, λp) with λ > 0. By (H1), the following inequality holds for any y in a neighborhood of x:

H(y, λp) -H(x, λp) ≥ u, y -x -o(y -x)
This is equivalent by positive homogeneity to

H(y, p) -H(x, p) ≥ u λ , y -x -o(y -x), which says that u λ ∈ ∂ x H(x, p); that is u ∈ λ∂ x H(x, p), which proves (1). The proof of (2) is given in [9, Lemma 2.1]

Differential inclusions

There are several books (cf. [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF][START_REF] Aubin | Differential inclusions[END_REF][START_REF] Clarke | Nonsmooth analysis and control theory[END_REF]) that contain the basics of differential inclusion theory. It is well-known that the assumptions of (SH) imply that if T (x) < +∞, then the minimum time problem (1)-( 3) has an optimal solution. We next recall the Necessary Conditions for an optimal solution. See [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF]Theorem 3.2.6]. Actually recent and more general necessary conditions can be found in the literature, but this one is enough for our aims because (H1) implies the subgradients of H split into its (x, p) components, which relays the effect of Proposition 3(2).

Theorem 1 (Necessary Conditions). Assume that (SH) and (H1) hold. Suppose x(•) is an optimal solution for the minimum time problem. Then there exists an absolute continuous arc p :

[0, T ] → R n such that for a.e. t ∈ [0, T ]    ẋ(t) ∈ ∂ p H(x(t), p(t)) -ṗ(t) ∈ ∂ x H(x(t), p(t)) -p(T ) ∈ N K (x(T )) (NC)
In addition, p(•) can be normalized to satisfy |p(T )| = 1.

The last normalized statement follows from Proposition 3(1).

Remark 1. It is noteworthy that the dual arc p never vanishes. Indeed let C|p| be the Lipschitz constant for H(•, p), then

| ṗ(t)| ≤ C|p(t)| t ∈ [0, T ]
Therefore, by Gronwall's lemma, easily we obtain that for any t ∈ [0, T (x)] either p(t) = 0 or p(t) ≡ 0.

We next turn to an auxiliary ODE that resembles a "feedback map" associated with a particular dual arc. The differentiability statement in (H2) is equivalent to the argmax set of v → v, p over v ∈ F (x) being a singleton v. We denote this unique element by F p (x) = v, which equals ∇ p H(x, p), and note that p → F p (x) is continuous. The Lipschitz statement in (H2) says that x → F p (x) is locally Lipschitz. The following result is the key ingredient that allows us to replace smooth parameterizations by (H). We quote it from [9, Proposition 3] and give some important immediate consequences.

Proposition 4. Assume that (SH) and (H) hold, and p(•) is an absolutely continuous arc defined on [0, T ] with p(t) = 0 for all t ∈ [0, T ]. Then for each x ∈ R n , consider the initial value problem

ẋ(t) = F p(t) x(t) a.e. t ∈ [0, T ] x(0) = x (IVP)
(1) (IVP) has a unique solution y(•; 0, x) that is a solution of ( 2).

(2) The arc y(•; 0, x) := x(•) satisfies

p(t), ẋ(t) = H x(t), p(t) a.e. t ∈ 0, T ].
(3) As a function of the initial condition, x → y(t; 0, x) is locally Lipschitz on R n with constant independent of t ∈ [0, T ]. That is, for each M > 0 there exists a constant k so that

|x i | ≤ M , i = 1, 2, implies y(t; 0, x 1 ) -y(t; 0, x 2 ) ≤ k|x 1 -x 2 |.
The following is a direct consequence of (NC).

Proposition 5. If x(•) is an optimal solution to (1)-(3) and p(•) is its associated adjoint arc, then x(•) solves (IVP).

Assumptions on the target K

In addition to assumptions on the dynamic data F , we shall impose two assumptions on the target K. The first is the well-known Petrov condition (see [8, Definition 8.2.2]).

(PC) ∃δ > 0 so that H(x, -ν) ≥ δ ν , ∀x ∈ bdry K, ν ∈ N P K (x). Assumption (PC) turns out to be equivalent to the minimal time function T (•) being Lipschitz in a neighborhood of K, but is also useful in obtaining a bound of T (•) in terms of the distance function d K (x) := inf{ x-y : y ∈ K}. The following can be found at [8, Theorem 8.2.3]. Proposition 6. Assume F satisfies (SH) and K ⊆ R n is compact. Then (PC) is equivalent to the existence of constants ρ, m > 0 so that

T K (x) ≤ md K (x) ∀x ∈ K ρ , where K ρ := {x : d K (x) ≤ ρ}.
The second assumption we shall require is known as the Interior Sphere Property (ISP). It has been invoked in similar contexts as we are using here, for example in [START_REF] Cannarsa | Convexity properties of the minimum time function[END_REF] and [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF], and is stated as follows.

(ISP)

∃r > 0 so that ∀x ∈ K, ∃y ∈ K such that x ∈ B r (y) ⊆ K.

The constant r > 0 in this definition is the size of realizability of (ISP), and when this dependence is relevant (as it is in the next result) we write (ISP) r for this assumption. There is an intrinsic relationship between (ISP) and the regularity properties of the distance function restricted to R n /K (see [8, Proposition 2.2.2] for the proof).

Proposition 7. If (ISP) r holds, then the distance function d K (•) is locally semiconcave on R n \ K with constant 1 r . Actually we will use a slightly stronger property than the previous result. Recall that the signed distance from the boundary of a nonempty set S ⊂ R n is given by dS

(x) = d S (x) -d R n \S (x) ∀x ∈ R n . (6)
Obviously for a ball B r (x 0 ), the signed distance turns out to be

dBr(x0) (x) = x -x 0 -r ∀x ∈ R n , (7)
and it can readily checked that it is semiconcave on each convex subset of B R (0) ∩ R n \ {B r 2 (x 0 )} with a constant C depending only on r and R > r. The interesting consequence of the (ISP ) r property that will be useful in the proof of our main result is the following. Proposition 8. Suppose K ⊂ R n is compact and satisfying (ISP ) r for some r > 0, and let R > r. Then there exists a constant C = C(r, R) > 0 such that [START_REF] Aubin | Differential inclusions[END_REF] dK

(x + z) + dK (x -z) -2 dK (x) ≤ C|z| 2 (8)
holds for all x, z ∈ B R (0) with x ∈ R n \ int(K).

(2) Let x ∈ bdry K and denote by B := B r (x B ) ⊂ K the ball touching the boundary of K at x. Then

dK (y) ≤ ∇ dB (x), y -x + C|y -x| 2 (9)
for all y ∈ B R (0).

Proof. Fix x, z ∈ B R (0) such that x ∈ R n \ int(K), and let x ∈ K satisfy d K (x) = |x -x|. By (ISP ) r , there exists x 0 ∈ K so that B := B r (x 0 ) satisfies x ∈ B ⊂ K. Suppose |z| < r 2 . Note that dK (x + z) + dK (x -z) -2 dK (x) ≤ dB (x + z) + dB (x -z) -2 dB (x) ≤ C|z| 2 , where C = C(r, R) is the semiconcave constant associated with dK (•) for each convex subset of B R (0) ∩ R n \ {B r 2 (x 0 )} .
The semiconcave inequality is valid in the last estimate due to the fact that x ± z -x0 ≥ r 2 . On the other hand, whenever |z| ≥ r 2 , we have that

dK (x + z) + dK (x -z) -2 dK (x) ≤ 4R 2r 2r 2 ≤ 16R r 2 |z| 2
since dK (x ± z) ≤ 2R, and the proof of ( 8) is thus complete. In order to prove (9), first we note that (see figure 1)

dK (y) ≤ dB (y) -dB (x) (10)
Since x → dB (x) is a smooth function (in particular C 1,1 ) away from the point x B , we obtain that dB (y) -dB (x) ≤ ∇ dB (x), y -x + C|y -x| 2 which in turns proves [START_REF] Cannarsa | Semiconcavity of the value function for a class of differential inclusions[END_REF].

It is easy to see that the above result implies Proposition 7. In particular if we consider the case in which x + z and x -z belong in R n \ int K then (8) holds with the signed distance function replaced by d K which in turns implies the semiconcavity. The Proposition 8 allows us to consider cases in which points along a line segment [x -z, x + z] belong to K. Remark 2. Another consequence of (ISP ) r is that the transversality condition on the adjoint arc p(•) can be further specified. If x(•) is optimal, then by the (ISP ) r assumption, there exists a ball B := B r (x B ) contained in K that touches the boundary of K at the point x T (where T = T (x)), and hence x(•) is also optimal if the target K is replaced by B. Applying (NC) to this new target says that we can find an adjoint arc with p(T ) = -∇ dB x(T ) .

The main result

We prove in this section that, under suitable assumptions, the minimum time function is locally semiconcave. First we prove the semiconcavity result in a neighborhood of the target set and then we will obtain the local semiconcavity in the whole controllable set.

Theorem 2. Suppose F satisfies (SH), (H), and that K is compact. Assume further that the Petrov condition (PC) and the Interior Sphere Property (ISP) r both hold. Then there exists ρ > 0 so that T (•) is semiconcave on each convex set K ⊆ K ρ \ K with a constant independent of K.

Proof. The constant ρ > 0 is chosen as in Proposition 6. It suffices to show that there exists a constant c > 0 so that for every x and ε > 0 with x + εB ∈ K ρ \ K, we have

T (x + z) + T (x -z) -2T (x) ≤ c|z| 2 (11)
for all z < ε. So fix x and ε > 0 so that x + εB ⊆ K ρ \ K, and fix z with z < ε. We choose the constant C so that the conclusion of Proposition 8 holds for all vectors that we analyze.

There exists an optimal trajectory x(•) starting from x and terminating at x(T ) ∈ K, where T = T (x). By (NC) and Proposition 5, there exists an adjoint arc p(•) (normalized so that p(T ) = 1) for which the following differential equation holds.

ẋ(t) = F p(t) (x(t)) a.e. t ∈ [0, T ] x(0) = x. ( 12
)
With this same p(•), let x ± (•) be the solutions of (IVP) with initial conditions

x ± (0) = x ± z. That is, ẋ± (t) = F p(t) (x ± (t)) a.e. t ∈ [0, T ] x(0) = x ± z (13)
Recall Proposition 4(2), where a consequence of the above ODEs is that

H x(t), p(t) = p(t), ẋ(t) a.e. t ∈ [0, T ], (14)
where x(•) is any one among the arcs x(•), x -(•), and x + (•). Another feature of the ODEs is dependence on initial data, and Proposition 4(3) says that

|x(t) -x ± (t)| ≤ k |z| and |x + (t) -x -(t)| ≤ 2 k |z| ∀t ∈ [0.T ]. ( 15
)
As a first case, let us assume x ± (t) / ∈ K for t ∈ [0, T ]. To be consistent with notation of the second case below, it is convenient to denote the time when one of the arcs has hit the target by t * , and so in our case here, we have t * = T = T (x). We shall frequently have occasion to refer to arc representing teh midpoint between the arcs x ± (•), and so is given by

x(t) := 1 2 x + (t) + x -(t) for t ∈ [0, t * ]. ( 16
)
Two different subcases emerge depending on whether x(t * ) ∈ K or not.

In the first subcase where x(t * ) ∈ K, there exists t < t * such that x( t) ∈ bdry K. By the dynamic programming principle, we have

T (x + z) + T (x -z) -2T (x) ≤ 2 t + T x + ( t) + T x -( t) -2t * ≤ T x + ( t) + T x -( t) (17)
By Proposition 6, Proposition 8, and (15), we obtain

T x + ( t) + T x -( t) ≤ m d K x + ( t) + d K x -( t) = m d K x( t) + x + ( t) -x -( t) 2 + d K x( t) - x + ( t) -x -( t) 2 -2d K (x( t)) ≤ 1 4 m C x + ( t) -x -( t) 2 ≤ c|z| 2 with c = m C k 2 .
Combining this estimate with (17) yields [START_REF] Clarke | Nonsmooth analysis and control theory[END_REF].

The second subcase is where x(t * ) / ∈ K, and by the dynamic programming principle, we have

T (x) = t * ; T (x + z) ≤ t * + T (x + (t * )); and T (x -z) ≤ t * + T (x -(t * )).
Combining these gives

T (x + z) + T (x -z) -2T (x) ≤ T x + (t * ) + T x -(t * ) . (18)
Now since d K (•) bounds T (•) (Proposition 6), we have

T (x + (t * )) + T (x -(t * )) ≤ m d K x + (t * ) + d K x -(t * ) , (19)
and from Proposition 8 that

d K (x + (t * )) + d K (x -(t * )) = d K (x + (t * )) + d K (x -(t * )) -2d K (x(t * )) + 2d K (x(t * )) ≤ C |x + (t * ) -x -(t * )| 2 + 2d K (x(t * )) ≤ 4 k 2 C |z| 2 + 2d K (x(t * )) (20)
where the last inequality again follows from (15). We need to obtain the appropriate estimate on d K (x(t * )). Since x(t * ) / ∈ K, the signed distance function dK x(t * ) is equal to d K x(t * ) , and applying remark 2 and proposition 8(2) with y = x(t * ) and x = x(t * ), we obtain

d K (x(t * )) ≤ -p(t * ), x(t * ) -x(t * ) + C |x(t * ) -x(t * )| 2 ≤ -p(t * ), x(t * ) -x(t * ) + C k 2 |z| 2 (21)
where the second inequality follows from (15).

The following lemma will assist in getting the appropriate estimate on the inner product term in (21). Lemma 1. Let x(•) and p(•) be as above. Suppose 0 ≤ t 1 < t 2 ≤ T (x) and x(•) is a solution to (4) with initial condition x(t 1 ). Then

p(t 2 ), x(t 2 ) -x(t 2 ) -p(t 1 ), x(t 1 ) -x(t 1 ) ≤ c k 2 (t 2 -t 1 ) x(t 1 ) -x(t 1 ) 2 .
Proof. We have

p(t 2 ), x(t 2 ) -x(t 2 ) -p(t 1 ), x(t 1 ) -x(t 1 ) = t2 t1 d dt p(t), x(t) -x(t) dt = t2 t1 ṗ(t), x(t) -x(t) + p(t), ẋ(t) -ẋ(t) dt (22)
The second integrand in (22) satisfies

p(t), ẋ(t) -ẋ(t) = H x(t), p(t) -H x(t), p(t)
by ( 14). The necessary conditions (NC) have that -p(t) ∈ ∂ x H x(t), p(t) , and since x → H(x, p) is semiconvex (which is (H1)), we have by Proposition 1(4) that the previous quantity is

≤ -p(t), x(t) -x(t) + c |x(t) -x(t)| 2 . (23)
Now integrate (23) from t 1 to t 2 and insert this estimate into (22). The first integrand in (22) will cancel with the inner product term in (23), and we are left with

p(t 2 ), x(t 2 ) -x(t 2 ) -p(t 1 ), x(t 1 ) -x(t 1 ) ≤ c t2 t1 x(t) -x(t) 2 dt ≤ c k 2 (t 2 -t 1 ) x(t 1 ) -x(t 1 ) 2 ,
where the last inequality holds by Proposition 4(3).

Returning now to the inner product in (21), we have

-p(t * ), x(t * ) -x(t * ) = p(t * ), x(t * ) -x(t * ) = 1 2 p(t * ), x(t * ) -x + (t * ) + p(t * ), x(t * ) -x -(t * )
Recall [START_REF] Petrov | The Bellman problem for a time-optimality problem[END_REF], in which it is stated that both x ± (•) satisfy the conditions of Lemma 1. We apply Lemma 1 twice with t 1 = 0 and t 2 = t * , and noting

p(0), x(0) -x + (0) + p(0), x(0) -x -(0) = p(0), z -z = 0, deduce that p(t * ), x(t * ) -x(t * ) ≤ c k 2 t * z 2 (24)
Finally, combining the estimates in ( 21) and ( 24) we arrive at

d K x(t * ) ≤ k 2 (C + c t * ) |z| 2 .
In conjunction with (20), (18), and (19), we may conclude that (11) holds for c = 2m k 2 (3 C + c t * ). This finishes the proof of the first case where x ± (t) / ∈ K for all t ∈ [0, T ] We now turn to the case where one of x ± (t) hits K before x(•) does. For definiteness, suppose x + (•) reaches the target before the others, and so x + (t * ) ∈ K for some t * < T (x) and x(t) and x -(t) do not belong to K for all t ∈ [0, t * ]. The other case is similar where x -(•) hits first is similar. The case we are currently analyzing is portrayed in Figure 2. Therefore with x(•) defined as in (16), we have

T (x + z) + T (x -z) -2T (x) ≤ T x -(t * ) -2T x(t * ) = T x -(t * ) -2T x(t * ) + 2 T x(t * ) -T x(t * ) (25)
To prove [START_REF] Clarke | Nonsmooth analysis and control theory[END_REF], we need to show both of the bracketed terms in (25) are bounded above by c |z| 2 . The proof of each estimate is somewhat lengthy.

We consider the second term first. That is, we will show

T x(t * ) -T x(t * ) ≤ c |z| 2 . ( 26
)
Let x(•) be the solution of

ẋ(t) = F p(t) x(t) a.e. t ∈ [t * , T (x)]
x(t * ) = x(t * ), which (unlike x(•)) is a trajectory of (2). If x(•) reaches K before x(•) does, then T (x(t * )) < T (x(t * )) and ( 26) is trivial. So suppose x(t) / ∈ K for t ∈ [t * , T (x)], and set t := T (x) -t * . By the dynamic programming principle, we have

T (x(t * )) = t T x(t * ) ≤ t + T x( t) ,
and therefore

T x(t * ) -T x(t * ) ≤ T x( t) ≤ md K x( t) (27)
by Proposition 6. Applying remark 2 and proposition 8(2) with y = x( t) and x = x( t), the following estimate holds

d K x( t) ≤ -p( t), x( t) -x( t) + C x( t) -x( t) 2 . ( 28
)
Similar to the inequality in (15), we have by Proposition 4(3), and then using (15) twice, that

x(t) -x(t) ≤ k x(t * ) -x(t * ) = k 2 x + (t * ) + x -(t * ) -2x(t * ) ≤ k 2 |z| (29)
for all t ∈ [t * , T (x)]. Substituting (28)and ( 29) into (27) yields

T (x(t * )) -T (x(t * )) ≤ m p( t), x( t) -x( t) + m k 4 C |z| 2 (30)
To estimate the first term in the right hand side of (30), we invoke Lemma 1 with t 1 = t * and t 2 = t, and deduce

p( t), x( t) -x( t) ≤ p(t * ), x(t * ) -x(t * ) + c k 2 x(t * ) -x(t * ) 2 = 1 2 p(t * ), x(t * ) -x + (t * ) + p(t * ), x(t * ) -x -(t * ) + c k 2 x(t * ) -x(t * ) 2
The two terms in the brackets can also be estimated using Lemma 1 with t 1 = 0 and t 2 = t * . The initial values make no contribution this time since x(0) = 1 2 x + (0) + x -(0) . Therefore the bracketed term is bounded by c k 2 t * |z| 2 , and the last term is bounded by c k 4 |z| 2 . Substituting all of this into (30) yields [START_REF] Cannarsa | Optimality conditions and synthesis for the minimum time problem[END_REF]

with c := m k 2 c t * + c k 2 + C k 2 .
The last consideration to finish the proof of Theorem 2 is to appropriately bound the first term in (25). We must show

T (x -(t * )) -2T (x(t * )) ≤ c |z| 2 . ( 31
)
It is convenient at this point to introduce new notation because the estimates do not directly involve x and z, but rather depend on the values of the trajectories x ± (•) at time t * . We define y and w by setting

y := x + (t * ) and w := x(t * ) -x + (t * ) = x -(t * ) -x + (t * ) 2 ,
and note that y ∈ K, y + w = x(t * ), and y + 2 w = x -(t * ). Since |w| ≤ k|z| by (15), in order to prove (31), it suffices to show

T (y + 2w) -2T (y + w) ≤ c |w| 2 . (32)
Toward this end, let ȳ(•) be optimal solution to the minimal time problem starting from y + w, and now denote by p(•) its associated adjoint arc. If θ := T (y + w) satisfies T (y + 2w) ≤ 2θ , then (32) is trivial, so suppose this is not the case, and let y(•) be the solution of

ẏ(t) = F p( t 2 ) (y(t)) a.e. t ∈ [0, 2θ ] y(0) = y + 2w.
The dynamic programming principle says T (y + 2w) ≤ 2θ * + T y(2θ ) , and therefore (32) follows upon showing that T y(2θ ) is bounded by c|w| 2 . We have by Proposition 6 that

T y(2θ * ) ≤ m d K y(2θ * ) ≤ m y(2θ * ) -2 ȳ(θ * ) + y + d K 2ȳ(θ * ) -y = m (I) + (II) . (33)
To bound (I), note the fundamental theorem of calculus, a change of variables,

and assumption (H2) imply

(I) = 2θ * 0 ẏ(t)dt -2 θ * 0 ẏ(t)dt = 2 θ * 0 [F p(t) (y(2t)) -F p(t) (ȳ(t))] dt ≤ 2 θ * 0 |y(2t) -ȳ(t)|dt (34)
where is the Lipschitz constant contained in the assumption (H2). By standard ODE theory, there exists a constant k so that for all 0 ≤ t 1 , t 2 ≤ θ * so that

|y(2 t 1 ) -ȳ(t 2 )| ≤ k |w|. (35)
We also record the fact that This completes the proof of the last case, and indeed of the the whole theorem.

θ * ≤ m |w|, (36)
Roughly speaking the above theorem tells us that the minimum time function is semiconcave in a neighborhood of the target set. This is a preliminary result useful for proving the local semiconcavity in the whole controllable set R \ K. Since the most difficult work was in the Theorem 2, the local semiconcavity result is given as a consequence. Indeed by the dynamic programming principle we obtain the following result.

Corollary 1. Under the same assumptions of the Theorem 2, the minimum time function is locally semiconcave on R \ K.

Proof. Let C be any compact set of R \ K. Fix x ∈ C and ε > 0 such that x ± εB ∈ C. Moreover we fix z with z < ε and such that there exists t * in which the trajectories defined as in the Theorem 2 [START_REF] Petrov | The Bellman problem for a time-optimality problem[END_REF] and [START_REF] Hermes | Functional analysis and time optimal control[END_REF] x(t * ) = x(t * ), which is a trajectory of (2). We can consider the only case in which x(t) / ∈ K for t ∈ [t * , T (x)] (otherwise it is trivial), and set t := T (x) -t * . By the dynamic programming principle, we have

Figure 1 :

 1 Figure 1: Geometrical meaning of inequality[START_REF] Clarke | Optimization and nonsmooth analysis[END_REF]

Figure 2 :

 2 Figure 2: Trajectories up to time t * The dynamic programming principle implies T (x + z) ≤ t * , T (x -z) ≤ t * + T x -(t * ) , and T (x) = t * + T x(t * ) .

  which is true because θ * = T (y + w) ≤ m d K (y + w) ≤ m |w| since y ∈ K. Consequently, using[START_REF] Cannarsa | Interior sphere property of attainable sets and time optimal control problems[END_REF] and (36) to estimate (34), we conclude the term (I) is bounded by 2 m k |w|2 .With regard to the second term of the (33), recall that y and ȳ(θ * ) belong to K, and then note the semiconcavity hypothesis of the distance function impliesd K 2ȳ(θ * ) -y = d K 2ȳ(θ * ) -y + d K (y) -2 d K ȳ(θ * ) ≤ C ȳ(θ * ) -y 2 . (37)We have|ȳ(θ * ) -y| ≤ w + θ * 0 ẏ(t) dt ≤ |w| + kθ * ≤ (1 +km)|w| by (36). Hence (37) now implies (II) is bounded above by (1 + km) 2 C |w| 2 . Finally, combining all of the estimates, we see that (32) holds for c = m 2 m k + (1 + km) 2 C .

  belong to the set K ρ \ K with ρ > 0 fixed as in Theorem 2. By the dynamic programming principleT (x + z) + T (x -z) -2T (x) ≤ ≤ T (x + (t * )) + T (x -(t * )) -2T (x(t * )) + 2 T (x(t * )) -T (x(t * ))(38)where x(t * ) is defined as (16). Applying the semiconcavity result on a neighborhood of the target set we can find a constant c > 0 such thatT (x + z) + T (x -z) -2T (x) ≤ c|z| 2 + 2 T (x(t * )) -T (x(t * ))As in[START_REF] Cannarsa | Optimality conditions and synthesis for the minimum time problem[END_REF] let x(•) be the solution of ẋ(t) = F p(t) x(t) a.e. t ∈ [t * , T (x)]

  ) ≤ k 2 |z| (41) for all t ∈ [t * , T (x)]. Substituting (40)and (41) into (39) yieldsT (x(t * )) -T (x(t * )) ≤ m p( t), x( t) -x( t) + m k 4 C |z| 2(42)

	2 .	(40)
	using (15) twice and thanks to Proposition 4(3) we have

T (x(t * )) = t

T x(t * ) ≤ t + T x( t) ,

and therefore, by Proposition 6

T x(t * ) -T x(t * ) ≤ T x( t) ≤ md K x( t)

(39)

Applying remark 2 and proposition 8(2) with y = x( t) and x = x( t), the following estimate holds

d K x( t) ≤ -p( t), x( t) -x( t) + C x( t) -x( t) x(t) -x(t) ≤ k x(t * ) -x(t * ) = k 2 x + (t * ) + x -(t * ) -2x(t *

Using Lemma 1 again with t 1 = 0 and t 2 = t * we can find a constant c := m k

which in turn implies the semiconcavity property on the whole controllable set R \ K.


Sandra Mercier

Christèle Dubourg

Nicolas Garcelon

Boris Campillo-Gimenez

Isabelle Gicquel

Marion Belleguic

Leslie Ratié

Laurent Pasquier

Philippe Loget

Claude Bendavid

Sylvie Pr

Odent

New findings for phenotype-genotype correlations in a

Keywords: holoprosencephaly (HPE), SHH, ZIC2, SIX3, TGIF

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

INTRODUCTION

Holoprosencephaly (HPE; MIM#236100) is the most common brain malformation in humans.

It results from incomplete midline division of the prosencephalon between days 18 and 28 of gestation.[1] , [START_REF] Marcorelles | Neuropathology of holoprosencephaly[END_REF] The estimated prevalence of HPE in liveborn children is less than 1/10,000, but the incidence may be as high as 1/250 in first-trimester conceptuses. [START_REF] Matsunaga | Holoprosencephaly in human embryos: epidemiologic studies of 150 cases[END_REF][START_REF] Orioli | Epidemiology of holoprosencephaly: Prevalence and risk factors[END_REF] HPE is usually associated with facial abnormalities, such as cyclopia, proboscis, or cleft lip/palate in cases of severe HPE. It has been said that "the face predicts the brain" but this rule is not applicable in all cases. [START_REF] Demyer | The Face Predicts the Brain: Diagnostic Significance of Median Facial Anomalies for Holoprosencephaly (Arhinencephaly)[END_REF][START_REF] Solomon | Mutations in ZIC2 in human holoprosencephaly: description of a novel ZIC2 specific phenotype and comprehensive analysis of 157 individuals[END_REF] Three classical anatomical classes have been described, in increasing order of severity: lobar, semi-lobar and alobar HPE. The full spectrum of HPE also includes middle interhemispheric variants (MIH) or syntelencephaly, septopreoptic HPE and microforms characterised by midline defects (e.g. single maxillary median incisor (SMMI) or hypotelorism) without the brain malformations typical of HPE. [START_REF] Hahn | Septopreoptic holoprosencephaly: a mild subtype associated with midline craniofacial anomalies[END_REF][START_REF] Barkovich | Middle interhemispheric fusion: an unusual variant of holoprosencephaly[END_REF][START_REF] Lazaro | Phenotypic and molecular variability of the holoprosencephalic spectrum[END_REF][START_REF] Simon | The middle interhemispheric variant of holoprosencephaly[END_REF] Not only is HPE phenotypically highly variable, its aetiology has also been shown to be very heterogeneous. [START_REF] Bendavid | Holoprosencephaly: An update on cytogenetic abnormalities[END_REF]12,[START_REF] Roessler | The molecular genetics of holoprosencephaly[END_REF] Indeed, HPE may be caused by chromosome abnormalities, such as trisomy 13, 18 and triploidy in particular, or may be one of the components of a multiple malformation syndrome, such as Smith-Lemli-Opitz or CHARGE syndrome. HPE may also result from exposure to maternal diabetes or hypocholesterolaemia during gestation. [START_REF] Johnson | Non-genetic risk factors for holoprosencephaly[END_REF][START_REF] Miller | Risk factors for non-syndromic holoprosencephaly in the National Birth Defects Prevention Study[END_REF] Four major genes responsible for HPE have been characterised -SHH, ZIC2, SIX3 and TGIF -and another nine seem to play a lesser role in the occurrence of HPE. [START_REF] Roessler | The molecular genetics of holoprosencephaly[END_REF] The "multiple-hit hypothesis" is now the most widely accepted model of HPE.

According to this hypothesis, combinations of mutations in major and/or minor HPE genes lead to the occurrence of HPE and may account for variability in terms of severity. [START_REF] Ming | Multiple hits during early embryonic development: digenic diseases and holoprosencephaly[END_REF] Genetic counselling thus remains very complex for families affected by HPE. [START_REF] Mercier | Genetic counseling and "molecular" prenatal diagnosis of holoprosencephaly (HPE)[END_REF] There are clearly many interactions between the various signalling pathways involved in forebrain development that remain to be elucidated.

It is therefore difficult to establish phenotype-genotype correlations for this disease, due to the large number of interacting factors. Studies of large series are therefore required. We and others have reported specific findings, including, in particular, the description of particular microforms often associated with SHH, the predominance of severe forms in subjects with SIX3 mutations and a ZIC2-specific phenotype with minor facial features. [START_REF] Solomon | Mutations in ZIC2 in human holoprosencephaly: description of a novel ZIC2 specific phenotype and comprehensive analysis of 157 individuals[END_REF][START_REF] Lazaro | Phenotypic and molecular variability of the holoprosencephalic spectrum[END_REF][START_REF] Dubourg | Molecular screening of SHH, ZIC2, SIX3, and TGIF genes in patients with features of holoprosencephaly spectrum: Mutation review and genotype-phenotype correlations[END_REF][START_REF] Lacbawan | Clinical spectrum of SIX3-associated mutations in holoprosencephaly: correlation between genotype, phenotype and function[END_REF][START_REF] Paulussen | The unfolding clinical spectrum of holoprosencephaly due to mutations in SHH, ZIC2, SIX3 and TGIF genes[END_REF][START_REF] Solomon | Analysis of genotype-phenotype correlations in human holoprosencephaly[END_REF] We report here clinical and molecular data for the largest ever European series, including 645 probands and their relatives and a high proportion of foetuses; only one larger series has been described, based solely on children with HPE (NIH, Muenke's team). [START_REF] Solomon | Analysis of genotype-phenotype correlations in human holoprosencephaly[END_REF] We aimed to identify new phenotype-genotype correlations in an updated large series of probands with at least one mutation in an HPE-related gene and/or at least one rearrangement identified on array-CGH analysis. Our statistical analyses confirm previous reports and focus, for the first time, on additional features, such as associated brain malformations, neural tube defects and extra-craniofacial defects. These new phenotype-genotype correlations should help to guide molecular analysis strategies and, thus, genetic counselling.

PATIENTS AND METHODS

In 1996, a European network for HPE was set up in Rennes, France.[1, [START_REF] Lazaro | Phenotypic and molecular variability of the holoprosencephalic spectrum[END_REF][START_REF] Dubourg | Molecular screening of SHH, ZIC2, SIX3, and TGIF genes in patients with features of holoprosencephaly spectrum: Mutation review and genotype-phenotype correlations[END_REF]22] Clinical data for about 1500 patients, including 645 HPE probands and 699 relatives, were analysed and DNA was prospectively collected. All the data were input into a specific secure online database.

This study was approved by the institutional review board. All samples and clinical data were collected after informed consent had been obtained.

Molecular analyses

The four major HPE-associated genes (SHH, ZIC2, SIX3 and TGIF) were routinely analysed by D-HPLC and/or direct sequencing in probands. The mode of inheritance was determined in cases in which samples from the parents were available. We analysed the GLI2 gene in 208 HPE patients selected on the basis of a pituitary gland defect. [START_REF] Bertolacini | Clinical findings in patients with GLI2 mutations -phenotypic variability[END_REF][START_REF] Rahimov | GLI2 mutations in four Brazilian patients: how wide is the phenotypic spectrum?[END_REF][START_REF] Roessler | Loss-of-function mutations in the human GLI2 gene are associated with pituitary anomalies and holoprosencephaly-like features[END_REF] We also searched for microcytogenetic rearrangements by multiplex ligation probe amplification (MLPA) (MRC-Holland) on the four genes. Array-CGH was performed with the Agilent Human Genome Microarray 44A and 244A kits (Agilent Technologies, Santa Clara, CA), for 260 patients. [START_REF] Bendavid | Array-CGH analysis indicates a high prevalence of genomic rearrangements in holoprosencephaly: an updated map of candidate loci[END_REF]

Clinical data

We collected data relating to facial morphology and associated features. This study focused on HPE patients without known chromosomal abnormalities that could be picked up by standard cytogenetic analysis and without identified syndromes.

We defined four categories of facial abnormality, in descending order of medical severity: (1) very severe phenotypes, such as cyclopia, ethmocephaly (proboscis), cebocephaly; [START_REF] Marcorelles | Neuropathology of holoprosencephaly[END_REF] premaxillary agenesis, cleft lip or palate and less severe eye abnormalities (coloboma, retinal dysplasia); (3) mild midface malformations, such as pyriform sinus stenosis and choanal stenosis; (4) the mildest abnormalities, such as hypotelorism, solitary median maxillary incisor (SMMI), other mild non-specific defects and normal faces. Associated brain malformations not included in the typical spectrum of HPE were also considered. These malformations included, in particular, posterior fossa abnormalities, such as rhombencephalosynapsis (RES), cerebellar hypoplasia and agenesis, but also pituitary defects and gyral and neuronal abnormalities. Neural tube defects, such as myelomeningocele, rachischisis and spina bifida, were considered independently of the associated brain malformations.

Statistical analysis

The percentages in this report were calculated for the patients with documented data, as exhaustive clinical data were not available for all cases.

Variables are expressed as numbers and percentages. Comparisons were based on Pearson's chi² test and Fisher's exact tests, as appropriate. Correlations between ordinal variables were analysed by calculating Spearman's rank correlation coefficient and Goodman and Kruskal's gamma. We used a type I error of 0.05, by convention, and all analyses were studied carried out with R software version 2.12.2 (2011-02-25). [START_REF] Development | R: A language and environment for statistical computing[END_REF]

RESULTS

The main data, including HPE type, facial abnormality category, associated brain malformations, neural tube defects and extracraniofacial malformations, are shown in table 1.

Table 1: Clinical data for the entire series, probands with alterations to the four main HPE-associated genes (SHH, ZIC2, SIX3, TGIF) and with abnormal array-CGH data.

Percentages are expressed according to the data recorded (500 probands for HPE type and face defect category) and correspond to the ratio of each feature reported in the probands of the group concerned, with the exception of detailed extracraniofacial defects, which are expressed as a proportion of the total of extracraniofacial defects in each group. # Array-CGH was performed on 260 probands. In addition to the HPE types mentioned in the table, septooptic dysplasia and atelencephaly were reported in 1.4% and in 0.6%, respectively, of the cases in this series. Statistical analyses were carried out by comparing the SHH, ZIC2, SIX3, TGIF subgroups and probands without modifications to any of the four genes. * Significant differences were found for of the distribution of HPE type (p=3.8 x 10 10 ), face defect category (p=9 x 10 10 ) and renal/urinary defects (p=0.03) in Fisher's exact tests.

The series

The series contained 645 probands, including 51% foetuses, and 699 relatives. The sex ratio of 1.2 (F:M) was consistent with a slight female predominance. Systematic molecular analyses showed point mutations or deletions in the four major genes involved in HPE (SHH, ZIC2, SIX3 and TGIF) in 25.4% of the subjects (164 probands; Figs. 1 and2) [START_REF] Solomon | Mutations in ZIC2 in human holoprosencephaly: description of a novel ZIC2 specific phenotype and comprehensive analysis of 157 individuals[END_REF][START_REF] Dubourg | Molecular screening of SHH, ZIC2, SIX3, and TGIF genes in patients with features of holoprosencephaly spectrum: Mutation review and genotype-phenotype correlations[END_REF][START_REF] Solomon | Analysis of genotype-phenotype correlations in human holoprosencephaly[END_REF][START_REF] Roessler | The mutational spectrum of holoprosencephaly-associated changes within the SHH gene in humans predicts loss-of-function through either key structural alterations of the ligand or its altered synthesis[END_REF][START_REF] Roessler | The full spectrum of holoprosencephalyassociated mutations within the ZIC2 gene in humans predicts loss-of-function as the predominant disease mechanism[END_REF][START_REF] Traiffort | Functional characterization of sonic hedgehog mutations associated with holoprosencephaly[END_REF] More foetuses than children were found to carry an HPEassociated gene deletion (68% vs 32%). Previous array-CGH analysis showed rearrangement rates as high as 25% in an initial study of 120 patients. [START_REF] Bendavid | Array-CGH analysis indicates a high prevalence of genomic rearrangements in holoprosencephaly: an updated map of candidate loci[END_REF] We describe here an additional 140 undergoing screening, with rearrangements identified in 22% (n=56) of the total of 260 probands. ( Most severe cases of HPE were assigned to facial defect category 1 or 2, whereas most of microforms were classified as category 3 or 4 (Fig. 3). As expected, category 4 included all cases of MIH and, surprisingly, a relatively high proportion of the cases of alobar or semilobar HPE. RES, the equivalent of HPE in the cerebellum, was described in six HPE cases and schizencephaly was reported as potentially associated in only three cases. [START_REF] Hehr | Heterozygous mutations in SIX3 and SHH are associated with schizencephaly and further expand the clinical spectrum of holoprosencephaly[END_REF] Extracraniofacial features were observed in 70 cases, including 13 cases of multiple congenital malformations mostly concerning abnormalities of the extremities, such as club hand or foot, simian crease, postaxial polydactyly (5 cases) and visceral malformations, frequently with lung defects. Furthermore, three cases of situs inversus were observed.

An HPE spectrum disorder was reported in 79 of the 699 relatives (29% in parents and 71% in other relatives). All affected parents were diagnosed with brain microforms, which accounted for 44% of the abnormalities in these 79 relatives. More severe forms were reported in other relatives, such as siblings or cousins: 26% alobar HPE, 13% semi-lobar HPE and 7% lobar HPE. Facial abnormalities were correlated with HPE severity: 26% category 1, 13% category 2, 10% category 3 and 51% category 4. Associated brain malformations were observed in 6% of the affected relatives and neural tube defects were found in 8%. Extracraniofacial malformations were reported in 16% of relatives, including only two cases of multiple congenital malformations.

SHH

SHH mutations were found in 67 probands (10.4%) and in 47 relatives from 59 different families, confirming that SHH was the gene most frequently mutated in HPE. An HPE spectrum disorder was described in 13 relatives: 1 lobar case and 11 microforms (1 non documented). We found that 23% of the parents with SHH mutations presented a microform.

Brain MRI revealed the presence of midline defects, such as an abnormal corpus callosum in the mother of one proband.

The sex ratio (F:M) was more balanced in SHH probands than in the series as a whole (0.9).

The four categories of facial features were evenly distributed and microforms were, above all, associated with pyriform sinus and choanal stenosis (74% category 3). The proportion of coloboma cases was relatively high (15% of the series as a whole). Associated brain malformations mostly concerned posterior fossa abnormalities (7%

SIX3

SIX3 mutations were identified in 33 probands (5.1%) and 21 relatives. Microforms were found in three of the 17 parents. The sex ratio was in favour of a higher proportion of females. Very severe HPE, such as alobar HPE and atelencephaly, was described in 57% of cases. In one family, a missense mutation was found in a mother and her three foetuses with alobar HPE or atelencephaly. [32,[START_REF] Pasquier | First occurrence of aprosencephaly/atelencephaly and holoprosencephaly in a family with a SIX3 gene mutation and phenotype/genotype correlation in our series of SIX3 mutations[END_REF] Probands with SIX3 mutations mostly had severe facial features (category 1 or 2). Severe ophthalmological defects were often reported with SIX3 mutations, as shown by a high percentage of category 1 abnormalities, and coloboma was found in two category 2 cases.

Associated brain malformations were mentioned in only 6% of the probands with SIX3 mutations, including one case of gyrus abnormality associated with heterotopia. Finally, only one case of multiple congenital malformations was reported, with no heart defects among the extracraniofacial features.

TGIF

TGIF mutation was found in only 11 probands (1.7%) and eight relatives, including five parents with no reported HPE spectrum disorder. The sex ratio (1.3) was similar to that for the series as a whole. HPE tended to be of the alobar type, and facial defects were mostly severe: 82% of probands were classified as category 1 or 2. Associated brain malformation was reported in only one case of ectopic posterior pituitary gland. The extracraniofacial abnormalities were not specific (one case of multiple congenital malformations).

GLI2

Evidence for a minor role of GLI2 was provided by the very low rate of mutations in this gene in HPE spectrum disorders. Indeed, GLI2 mutation was found in only three probands (1.4% in selected patients) and three relatives. Two of these missense mutations may be involved in the disorder but the pathogenicity of the third mutation could not been established with certainty. Samples from the parents were available for one patient, and analysis of these samples showed the mutation to be inherited. The brain malformations observed were lobar, semi-lobar and alobar HPE with facial abnormalities of category 2 or 4 (one undocumented case).

Interestingly, one foetus presented the combination of a GLI2 missense mutation, inherited from an asymptomatic mother, with a de novo 13q deletion including the ZIC2 gene. The proband had alobar HPE with complex heart malformations, bilobulated lung, gallbladder agenesis, hypospadias, no thumb or first metacarpal and ungual hypoplasia.

Microrearrangements

Array-CGH analysis identified rearrangements in 22% (56 cases) of the 260 probands.

These rearrangements had arisen de novo in 14% and were inherited in 8%. Finally, we report seven cases in which a deletion or duplication was associated with a single mutation in an HPE-related gene (SHH, ZIC2 and SIX3) (Table 2). The three foetuses and four children concerned had either isolated or syndromic HPE, with very various phenotypes in terms of HPE type and facial features.

Statistical analyses

We carried out statistical analyses to identify significant phenotype-genotype correlations.

The population of probands with rearrangements was highly heterogeneous. We therefore decided to focus on probands with mutations in SHH, ZIC2, SIX3, TGIF, comparing these probands with those with no mutation in these genes.

Proportions of foetuses and children

As expected, the foetuses were significantly more likely to have severe types of HPE, whereas the liveborn children tended to have microforms (p<0.0001). However, although there were more foetuses with HPE-associated gene deletions or array-CGH rearrangements, these differences were not statistically significant (p=0.09 and p=0.2, respectively).

HPE types

Studies of the distribution of HPE between the five subgroups provided evidence for a highly significant correlation (p<0.0001). Alobar HPE tended to be associated with SIX3, semilobar HPE with ZIC2 and microform with SHH mutations. Lobar HPE did not seem to be particularly associated with any specific HPE gene in these analyses.

Facial defect categories

Facial features were studied as a function of the severity of the brain malformation. The findings for the ZIC2 subgroup differed significantly from those for the other genes, as no correlation between the HPE severity type and the severity of facial defects category was found (Fig. 3).

We also analysed the facial defects categories independently. A significant difference in the distributions of categories 1 and 4 was found, with a low frequency of probands with ZIC2 mutations in category 1 and a very high frequency of such probands in category 4. A significant correlation was also found between SHH microforms and category 3 (p=0.01).

Brain-associated malformations; neural tube defects

No significant overall correlation was found between associated brain malformations or neural tube defects and molecular data, although these defects did appear to be more frequent in individuals with ZIC2 and TGIF mutations (p=0.18).

Extracraniofacial defects

A correlation between renal/urinary malformations and certain mutations (p=0.03) was found.

Renal/urinary defects were significantly more frequent in probands with SHH and ZIC2 than in the probands of other subgroups. No significant difference was found in the distribution of the other extracraniofacial defects.

DISCUSSION

HPE is a very complex disorder in terms of both clinical variability and genetic signalling pathways. We report here the clinical and molecular data for the largest ever European series, encompassing 645 probands and 699 relatives, with a higher proportion of foetuses (51%) than in previous series. [START_REF] Lacbawan | Clinical spectrum of SIX3-associated mutations in holoprosencephaly: correlation between genotype, phenotype and function[END_REF][START_REF] Paulussen | The unfolding clinical spectrum of holoprosencephaly due to mutations in SHH, ZIC2, SIX3 and TGIF genes[END_REF][START_REF] Solomon | Analysis of genotype-phenotype correlations in human holoprosencephaly[END_REF] The large number of patients included in this series resulted in sufficient power for the detection of statistically significant phenotype-genotype correlations.

In the series as a whole, the sex ratio (F:M) was 1.2, in favour of a slight predominance of female subjects. Female subjects were in the minority among probands with SHH mutations, for which the sex ratio was 0.9, whereas a high predominance of female subjects (sex ratio of 1.8) was observed among those with ZIC2 and SIX3 mutations. The SIX3 sex ratio was consistent with that reported by Lacbawan. [START_REF] Lacbawan | Clinical spectrum of SIX3-associated mutations in holoprosencephaly: correlation between genotype, phenotype and function[END_REF] This suggests that mutations in ZIC2 or SIX3 may be embryonic-lethal in males.

Screening for mutations in the major genes SHH, ZIC2, SIX3 and TGIF and the minor gene GLI2 in this large series gave an estimated global mutation rate of 25.8%, with specific mutation rates of 10.4% (SHH), 8.2% (ZIC2), 5.1% (SIX3), 1.7% (TGIF) and 0.4% (GLI2).

These results are consistent with our previous findings for smaller series.[1, [START_REF] Lazaro | Phenotypic and molecular variability of the holoprosencephalic spectrum[END_REF][START_REF] Dubourg | Molecular screening of SHH, ZIC2, SIX3, and TGIF genes in patients with features of holoprosencephaly spectrum: Mutation review and genotype-phenotype correlations[END_REF][START_REF] Dubourg | Clinical utility gene card for: Holoprosencephaly[END_REF] Point mutations in SHH, SIX3 and TGIF were highly heritable, with heritabilities of 73%, 88% and 100%, respectively, whereas most ZIC2 mutations were de novo (70% of cases). [START_REF] Solomon | Mutations in ZIC2 in human holoprosencephaly: description of a novel ZIC2 specific phenotype and comprehensive analysis of 157 individuals[END_REF][START_REF] Paulussen | The unfolding clinical spectrum of holoprosencephaly due to mutations in SHH, ZIC2, SIX3 and TGIF genes[END_REF] This may be due to a higher penetrance and the high frequency of severe HPE in individuals with ZIC2 mutations, resulting in the identification of a larger number of sporadic cases. By contrast to what was reported by Lacbawan et al., no predominance of maternal inheritance was observed for SIX3 mutations in our series, whereas such a pattern of inheritance was observed for SHH mutations. [START_REF] Lacbawan | Clinical spectrum of SIX3-associated mutations in holoprosencephaly: correlation between genotype, phenotype and function[END_REF] Overall, more severe types of HPE (atelencephaly, alobar or semilobar HPE) were reported in individuals with SIX3 (74%) and ZIC2 (75%) mutations than in individuals with SHH (62%) and TGIF (57%) mutations. Moreover, atelencephaly/aprosencephaly was associated exclusively with SIX3 mutations. Pasquier et al. reported this particular association and other teams have reported a high proportion of severe cases in subjects with SIX3 mutations, including ophthalmological defects in particular. [START_REF] Lacbawan | Clinical spectrum of SIX3-associated mutations in holoprosencephaly: correlation between genotype, phenotype and function[END_REF][START_REF] Paulussen | The unfolding clinical spectrum of holoprosencephaly due to mutations in SHH, ZIC2, SIX3 and TGIF genes[END_REF][START_REF] Pasquier | First occurrence of aprosencephaly/atelencephaly and holoprosencephaly in a family with a SIX3 gene mutation and phenotype/genotype correlation in our series of SIX3 mutations[END_REF] Thus, alobar HPE tends to be associated with SIX3, semilobar HPE with ZIC2 and microforms with SHH, consistent with the findings of Solomon. [START_REF] Solomon | Analysis of genotype-phenotype correlations in human holoprosencephaly[END_REF] Furthermore, our findings confirmed the specific association of MIH or syntelencephaly with ZIC2, as the mutations identified in four of the 11 cases of MIH in the series concerned the ZIC2 gene. [START_REF] Solomon | Mutations in ZIC2 in human holoprosencephaly: description of a novel ZIC2 specific phenotype and comprehensive analysis of 157 individuals[END_REF] This study highlights the considerable intra-and interfamilial phenotypic variability in SHH probands. [START_REF] Hehr | Wide phenotypic variability in families with holoprosencephaly and a sonic hedgehog mutation[END_REF] We report here, for the first time, the high prevalence of brain microforms (74%) in probands with SHH mutations presenting, essentially, pyriform sinus and choanal stenosis (category 3). Associations with midline defects, coloboma and sacral agenesis (systematically linked to a large 7q36 deletion) were also observed. By contrast, facial morphology in subjects with ZIC2 mutations was generally characterised by moderate facial dysmorphia or even a normal face (category 4), associated with alobar or semilobar HPE.

This feature seems to be specific to ZIC2 mutations and has also been reported by other teams. [START_REF] Solomon | Mutations in ZIC2 in human holoprosencephaly: description of a novel ZIC2 specific phenotype and comprehensive analysis of 157 individuals[END_REF][START_REF] Paulussen | The unfolding clinical spectrum of holoprosencephaly due to mutations in SHH, ZIC2, SIX3 and TGIF genes[END_REF] Solomon et al. mentioned a particular facial phenotype found in some of the cases in our series, with a high proportion of abnormal noses (9 cases) usually short, sometimes flat or large, and ear dysplasia (4 cases). Furthermore, statistical analyses revealed a lack of correlation between the severity of HPE brain malformation and facial features in probands with ZIC2 mutations, but not in those with SHH, SIX3 and TGIF mutations.

We also considered other associated brain malformations, focusing in particular on RES, which was associated with HPE in six of the 645 HPE probands. This extremely rare malformation is the histological equivalent of HPE for the cerebellum. The signalling pathways involved in this disease has yet to be identified, but we suggest that there may be interactions with HPE signalling pathways, as a ZIC2 mutation has been reported in two siblings with partial RES and HPE. [START_REF] Pasquier | Rhombencephalosynapsis and related anomalies: a neuropathological study of 40 fetal cases[END_REF][START_REF] Ramocki | Recurrent partial rhombencephalosynapsis and holoprosencephaly in siblings with a mutation of ZIC2[END_REF] SHH and SIX3 mutations have recently been associated with schizencephaly, but only three cases of schizencephaly were reported in our series, with no mutation identified in the probands concerned. [START_REF] Hehr | Heterozygous mutations in SIX3 and SHH are associated with schizencephaly and further expand the clinical spectrum of holoprosencephaly[END_REF] We also identified two other previously unknown associations with ZIC2 mutations: rachischisis and neuronal migration abnormalities. Neural tube defects are frequently reported in HPE, but rachischisis seems to be more specifically associated with ZIC2 mutation, as such mutations were found in the only two cases of rachischisis in our series. In addition, neuronal migration abnormalities have already been described, but without phenotype-genotype correlations. [START_REF] Marcorelles | Neuropathology of holoprosencephaly[END_REF][START_REF] Hahn | Evaluation and management of children with holoprosencephaly[END_REF][START_REF] Jellinger | Holoprosencephaly and agenesis of the corpus callosum: frequency of associated malformations[END_REF] We focus here on neuronal migration abnormalities, such as gyral pattern abnormalities and heterotopia, which were almost exclusively restricted to probands with ZIC2 mutations (8 cases). Hahn et al. found subcortical heterotopic grey matter or cortical dysplasia in two thirds of MIH cases. [START_REF] Hahn | Evaluation and management of children with holoprosencephaly[END_REF] In our series, neuronal migration abnormalities were observed in two of 11 cases of MIH, but no ZIC2 mutation was identified in these cases.

We report here, for the first time, the frequency and variety of extracraniofacial malformations in a large series, together with the corresponding molecular data. Molecular defects were more frequent in ZIC2 than in the other three main genes. We found a higher frequency of extracraniofacial defects than reported by Solomon, probably because his study focused on children, rather than foetuses. [START_REF] Solomon | Mutations in ZIC2 in human holoprosencephaly: description of a novel ZIC2 specific phenotype and comprehensive analysis of 157 individuals[END_REF] Renal/urinary defects were significantly associated with SHH or ZIC2 mutations and no such defects were found in probands with SIX3 or TGIF mutations. SIX3 was the only gene for which no heart defect was reported.

The data for TGIF and, above all, for GLI2 revealed no significant phenotype-genotype correlations, but alobar HPE with severe facial features (category 1) seemed to be more frequent in probands with TGIF mutations, although phenotypic variability was high. We suggest that phenotype-genotype correlations for TGIF and GLI2 should be analysed in larger series. Another disease spectrum, characterised by anterior pituitary insufficiency and polydactyly, seems to be more specifically linked to GLI2 mutations [START_REF] Bertolacini | Clinical findings in patients with GLI2 mutations -phenotypic variability[END_REF][START_REF] Franca | Novel heterozygous nonsense GLI2 mutations in patients with hypopituitarism and ectopic posterior pituitary lobe without holoprosencephaly[END_REF] (Dubourg, unpublished data).

Several signalling pathways are known to be involved in HPE, and the various mechanisms underlying HPE may account for the observed phenotypic variability. Indeed, SIX3 is involved in eye development, consistent with the eye defects observed in probands with SIX3 mutations, and in ventral forebrain development, through regulation of the SHH signalling pathway. Furthermore, ZIC2 significantly differs from the other HPE-related genes, as shown in phenotype-genotype correlations, and its product also acts by different mechanisms. Indeed, ZIC2 is known to act both early stage, in upstream SHH signalling during midgastrulation, and at later stages, in the development of the dorsal telencephalon, potentially accounting for the specific occurrence of MIH and neural tube defects in ZIC2 probands.

Array-CGH studies in this series confirmed the high frequency of rearrangements reported in a preliminary study. [START_REF] Bendavid | Array-CGH analysis indicates a high prevalence of genomic rearrangements in holoprosencephaly: an updated map of candidate loci[END_REF] Indeed, rearrangements were identified in 22% of the 260 patients studied (14% de novo and 8% inherited). The proportion of foetuses and the severity of HPE phenotype (extracraniofacial malformations included) were not significantly higher in probands with rearrangements than in the series as a whole, as might have been expected.

The higher frequency of category 2 and 4 probands may reflect more common features, such as cleft lip or palate (category 2) or mild non-specific features (category 4). This population seemed to be heterogeneous and an understanding of the molecular basis of this heterogeneity should facilitate the definition of more homogeneous subgroups. Indeed, the high frequency of rearrangements was consistent with there being a large number of loci corresponding to new candidate genes. The recurrent regions included the MACROD2 and PATCHD3 loci, in particular. Furthermore, recurrent deletion 6qter led to the identification of DLL1 as a new HPE-associated gene and of NOTCH as a new signalling pathway involved in HPE. [START_REF] Dupé | NOTCH, a new signaling pathway implicated in holoprosencephaly[END_REF] It has already been suggested that HPE involves a multiple-hit process. [START_REF] Bendavid | Array-CGH analysis indicates a high prevalence of genomic rearrangements in holoprosencephaly: an updated map of candidate loci[END_REF]42] Animal models have provided evidence of digenism, by implicating either the same or two different signalling pathways. Genetic background also played an important role in determining the severity of the phenotype. [START_REF] Cole | Microform holoprosencephaly in mice that lack the Ig superfamily member Cdon[END_REF][START_REF] Geng | Haploinsufficiency of Six3 fails to activate Sonic hedgehog expression in the ventral forebrain and causes holoprosencephaly[END_REF][START_REF] Pei | Identification of common and unique modifiers of zebrafish midline bifurcation and cyclopia[END_REF] In humans, low penetrance or variable levels of expression in HPE multiplex families and reports of patients with two different mutations support this hypothesis. [START_REF] Ming | Multiple hits during early embryonic development: digenic diseases and holoprosencephaly[END_REF][START_REF] Rahimov | GLI2 mutations in four Brazilian patients: how wide is the phenotypic spectrum?[END_REF][START_REF] Nanni | The mutational spectrum of the sonic hedgehog gene in holoprosencephaly: SHH mutations cause a significant proportion of autosomal dominant holoprosencephaly[END_REF][START_REF] Wannasilp | Holoprosencephaly in a family segregating novel variants in ZIC2 and GLI2[END_REF] Our series included up to 15 cases of double mutation: eight cases of double rearrangements and seven cases of rearrangement associated with a mutation in an HPE-related gene (6 rearrangements associated with a mutation in SHH, ZIC2 or SIX3 and one ZIC2 deletion associated with a GLI2 mutation). This findings provide solid support for the multiple-hit hypothesis in HPE.

In conclusion, even if the identification of an HPE gene mutation cannot provide a precise prognosis, the new phenotype-genotype correlations identified here should facilitate the definition of a better molecular analysis strategy. We propose a new algorithm based on these correlations (Fig. 3). Evidence of double mutations in HPE probands provides support for the HPE multiple-hit hypothesis underlying the complexity of genetic counselling and research on this disorder. The known clinical variability of HPE also suggests an overlap between environmental and genetic mechanisms. A combination of clinical and functional studies should help to improve our understanding of the complex interactions between the various signalling pathways involved in HPE and brain development. nonsense or frameshift, 13% polyalanine tract, 3% splice alterations) with 70% occurring "de novo" and two cases of probable germline mosaicism. SIX3: 18 different point mutations (78% missense, 22% nonsense or frameshift). TGIF: 3 missense mutations and 1 nonsense mutation. The pathogenicity of these alterations has been confirmed by predictive or functional studies. [START_REF] Solomon | Mutations in ZIC2 in human holoprosencephaly: description of a novel ZIC2 specific phenotype and comprehensive analysis of 157 individuals[END_REF][START_REF] Dubourg | Molecular screening of SHH, ZIC2, SIX3, and TGIF genes in patients with features of holoprosencephaly spectrum: Mutation review and genotype-phenotype correlations[END_REF][START_REF] Roessler | The mutational spectrum of holoprosencephaly-associated changes within the SHH gene in humans predicts loss-of-function through either key structural alterations of the ligand or its altered synthesis[END_REF][START_REF] Roessler | The full spectrum of holoprosencephalyassociated mutations within the ZIC2 gene in humans predicts loss-of-function as the predominant disease mechanism[END_REF][START_REF] Traiffort | Functional characterization of sonic hedgehog mutations associated with holoprosencephaly[END_REF] GenBank accession numbers are NM_00193.2 for SHH cDNA, NM_007129.2 for ZIC2 cDNA, NM_005413.2 for SIX3 cDNA and NM_003244.2 for TGIF cDNA. In cases of familial HPE, analyses of SHH, SIX3 or TGIF should be given priority over ZIC2 analysis. Very severe forms of HPE, such as alobar and semilobar HPE, should lead to analyses of the SIX3 (particularly in cases of atelencephaly/aprosencephaly) and ZIC2 genes, but facial features may provide useful information. Indeed, probands with category 1 or 2 facial features associated with severe HPE, such as alobar or semilobar HPE, are more likely to display mutations in SIX3 or SHH. By contrast, probands with severe brain malformations and category 4 facial features are likely to have mutations in ZIC2. No particular association with lobar HPE was found. Thus, if associated brain malformations are described, particularly in cases of neuronal migration abnormalities, and in cases of neuronal tube defects (rachischisis in particular), ZIC2 gene analysis should be performed as a matter of priority. Moreover, ZIC2 mutations are the principal cause of extracraniofacial malformations. Renal/urinary malformations are found preferentially in patients with ZIC2 or SHH mutations and coloboma tends to be found in patients with SHH or SIX3 mutations.

Figure 1 :

 1 Figure 1: Schematic representation of point mutations and deletions in the SHH, ZIC2, SIX3, TGIF genes. Point mutations are represented in yellow and gene deletions are shown in red, with the corresponding number of probands. The percentage of probands (/645 probands) with a particular point mutation or a deletion is noted below each gene.

Figure 2 :

 2 Figure 2: Point mutations and inheritance of the SHH, ZIC2, SIX3 and TGIF genes. SHH: 41 different variations (73% missense, 27% nonsense or frameshift mutations); 73% inherited point mutations, predominantly of maternal (67%) rather than paternal (33%) origin. ZIC2: 44 different point mutations (29% missense, 55%nonsense or frameshift, 13% polyalanine tract, 3% splice alterations) with 70% occurring "de novo" and two cases of probable germline mosaicism. SIX3: 18 different point mutations (78% missense, 22% nonsense or frameshift). TGIF: 3 missense mutations and 1 nonsense mutation. The pathogenicity of these alterations has been confirmed by predictive or functional studies.[START_REF] Solomon | Mutations in ZIC2 in human holoprosencephaly: description of a novel ZIC2 specific phenotype and comprehensive analysis of 157 individuals[END_REF][START_REF] Dubourg | Molecular screening of SHH, ZIC2, SIX3, and TGIF genes in patients with features of holoprosencephaly spectrum: Mutation review and genotype-phenotype correlations[END_REF][START_REF] Roessler | The mutational spectrum of holoprosencephaly-associated changes within the SHH gene in humans predicts loss-of-function through either key structural alterations of the ligand or its altered synthesis[END_REF][START_REF] Roessler | The full spectrum of holoprosencephalyassociated mutations within the ZIC2 gene in humans predicts loss-of-function as the predominant disease mechanism[END_REF][START_REF] Traiffort | Functional characterization of sonic hedgehog mutations associated with holoprosencephaly[END_REF] GenBank accession numbers are NM_00193.2 for SHH cDNA, NM_007129.2 for ZIC2 cDNA, NM_005413.2 for SIX3 cDNA and NM_003244.2 for TGIF cDNA.

Figure 3 :

 3 Figure 3: Face defect category, as a function of HPE type, for the series as a whole, probands with abnormal array-CGH findings and with altered SHH, ZIC2, SIX3 and TGIF genes. The results presented are based on the data for 369 informative probands. Considering a positive gradient of severity from category 1 to category 4 in Spearman's rank correlation test, we found a significant correlation between the severity of HPE type and facial features for the entire series (p<0.001) and for the SHH (p<0.001), SIX3 (p<0.001) and TGIF (p<0.001) subgroups, but not for the ZIC2 subgroup (p=0.5).

Figure 4 :

 4 Figure 4: Algorithm for molecular analysis in HPE.In cases of familial HPE, analyses of SHH, SIX3 or TGIF should be given priority over ZIC2 analysis. Very severe forms of HPE, such as alobar and semilobar HPE, should lead to analyses of the SIX3 (particularly in cases of atelencephaly/aprosencephaly) and ZIC2 genes, but facial features may provide useful information. Indeed, probands with category 1 or 2 facial features associated with severe HPE, such as alobar or semilobar HPE, are more likely to display mutations in SIX3 or SHH. By contrast, probands with severe brain malformations and category 4 facial features are likely to have mutations in ZIC2. No particular association with lobar HPE was found. Thus, if associated brain malformations are described, particularly in cases of neuronal migration abnormalities, and in cases of neuronal tube defects (rachischisis in particular), ZIC2 gene analysis should be performed as a matter of priority. Moreover, ZIC2 mutations are the principal cause of extracraniofacial malformations. Renal/urinary malformations are found preferentially in patients with ZIC2 or SHH mutations and coloboma tends to be found in patients with SHH or SIX3 mutations.

  Extracraniofacial defects were essentially classified into six groups: (i) heart defects; (ii)

	visceral malformations (abnormal lung lobulation, lung hypoplasia, thymus hypoplasia,
	adrenal hypoplasia, adrenal cytomegaly, common mesentery, single umbilical artery,
	diaphragmatic hernia, exomphalos, inguinal hernia, oesophageal atresia, intestinal stenosis);
	(iii) genital malformations (micropenis, hypospadias, external and internal genital organs); (iv)
	renal/urinary abnormalities (ureteral dilatation, renal hypoplasia); (v) skeletal abnormalities
	(sacral agenesis and costovertebral abnormalities); (vi) abnormalities of the extremities (club
	hand or foot, postaxial polydactyly, brachydactyly, clinodactyly or simian crease).

Table 1 )

 1

				HPE type				Face defect category		Associated	Neural			Extracraniofacial defects
		Probands	alobar	semilobar	lobar	MIH	microform	Cat I	Cat II	Cat III	Cat IV	brain malformations	tube defect	% of probands	Heart	Visceral	Genital	Renal/urinary	Skeletal	Extremities
	Total cohort	645	25% atelencephaly + 1%	28%	19%	2%	25%	27%	19%	14%	40%	13%	4%	27%	19%	23%	11%	11%	8%	27%
	SHH	67 (10.4%)	28%*	34%*	4%*	0%*	34%*	38%*	22%*	26%*	14%*	9%	5%	24%	8%	24%	8%	28%*	16%	16%
	ZIC2	53 (8.2%)	30%*	45%*	15%*	8%*	2%*	8%*	8%*	3%*	81%*	21% migration) (15% neuronal	9%	40%	14%	20%	17%	14%*	9%	26%
	SIX3	33 (5.1%)	52%* atelencephaly + 7%	15%*	11%*	0%*	15%*	45%*	24%*	10%*	21%*	7%	7%	24%	0%	22%	11%	0%*	22%	45%
	TGIF	11 (1,7%)	43%*	14%*	29%*	0%*	14%*	55%*	27%*	0%*	18%*	9%	9%	27%	25%	50%	12%	0%*	0%	13%
	CGH	56 # (22%)	19%	37%	17%	0%	27%	8%	38%	8%	46%	11%	4%	26%	19%	22%	17%	11%	11%	20%

  Both the cases of rachischisis in the overall series were found to have a ZIC2 mutation. A high proportion of extracraniofacial abnormalities, mostly visceral abnormalities and abnormalities of the extremities, was associated with ZIC2 mutations, including eight cases of multiple congenital malformations.

	). Extracraniofacial
	abnormalities included five cases of multiple congenital malformations, mostly visceral and
	renal/urinary defects. Sacral agenesis was also observed (associated in one case with
	costovertebral abnormalities) in several probands with large 7q36 deletions, including the
	HLXB9 gene in particular (Currarino syndrome, MIM#176450).

ZIC2

ZIC2 mutations were identified in 53 probands (8.2%) and 14 relatives. Microforms were reported in 36% of the parents with ZIC2 mutations (4/11). The sex ratio (F:M) of 1.8 was consistent with a marked predominance of female subjects. HPE tended to be alobar or semilobar (72%). Furthermore, only two HPE-related gene mutations were found in the 11 MIH cases in the total series, and both were ZIC2 mutations.

The probands of the ZIC2 group tended to have a combination of severe HPE with few facial features. Other specific findings were obtained for associated brain malformations, neural tube defects and extracraniofacial malformations. The neuronal migration abnormalities (15%) detected were lissencephaly, polymicrogyria or pachygyria in four cases and heterotopia in white matter and/or in the cerebellum in another four cases. One foetus with lobar HPE had heterotopia and gyration abnormalities among other multiple congenital anomalies (rachischisis, cleft palate, visceral and extremity abnormalities). Neural tube defects were reported in five cases, including two cases of rachischisis: one combined with pseudoanencephaly and the other with anencephaly with facial clefting in a monozygous twin previously reported by Lazaro et al.

[START_REF] Lazaro | Phenotypic and molecular variability of the holoprosencephalic spectrum[END_REF]

Table 2 :

 2 Clinical and molecular data for the 8 probands with double mutations:

	Rearrangement

rearrangement detected on array-CGH and HPE gene mutation. The inheritance of rearrangements and mutations is reported in all cases in which molecular analysis of the parents of the proband was possible. The GenBank accession numbers are NM_00193.2 for SHH cDNA, NM_007129.2 for ZIC2 cDNA, NM_005413.2 for SIX3 cDNA, NM_003244.2 for TGIF cDNA and NM_005270.4 for GLI2 cDNA.

ACKNOWLEDGEMENTS

We would like to thank the families for their participation in the study and the geneticists, pathologists (SOFFOET), obstetricians and paediatricians all over Europe who referred them. We particularly thank all members of the Molecular Genetics Laboratory (CHU, Rennes) and of the Department of Genetics and Development (UMR6061 CNRS, Université Rennes1) for their help and advice.