Diego Mincarelli
email: diego.mincarelli@inria.fr

Alessandro Pisano
email: pisano@diee.unica.it

Thierry Floquet
email: thierry.floquet@ec-lille.fr

Elio Usai
email: eusai@diee.unica.it

Uniformly convergent sliding mode-based

Keywords: Switched systems, observer design, second order sliding modes

come

INTRODUCTION

In the last decade, the control community has devoted a great deal of attention to the study of hybrid/switched systems [START_REF] Engell | Modelling and analysis of hybrid systems[END_REF][START_REF] Goebel | Hybrid dynamical systems[END_REF][START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF]. They represent a powerful tool to describe systems that exhibit switchings between several subsystems, inherently by nature or as a result of external control actions such as in switching supervisory control [START_REF] Morse | Supervisory control of families of linear set-point controllers -part 1: Exact matching[END_REF]. Switched systems and switched multi-controller synthesis have numerous applications in the control of mechanical systems [START_REF] Narendra | Adaptation and learning using multiple models, switching, and tuning[END_REF][START_REF] Chevallereau | Asymptotically stable running for a five-link, four-actuator, planar bipedal robot[END_REF], automotive industry [START_REF] Balluchi | Hybrid control in automotive applications: the cut-off control[END_REF], switching power converters [START_REF] Koning | Brief digital optimal reduced-order control of pulsewidth-modulated switched linear systems[END_REF], aircraft and traffic control [START_REF] Kamgarpour | Hybrid Optimal Control for Aircraft Trajectory Design with a Variable Sequence of Modes[END_REF][START_REF] Glover | A stochastic hybrid model for air traffic control simulation[END_REF], biological systems [START_REF] Vries | Hybrid system modeling and identification of cell biology systems: perspectives and challenges[END_REF][START_REF] Cinquemani | Identification of genetic regulatory networks: A stochastic hybrid approach[END_REF], and many other fields [START_REF] Branicky | Studies in hybrid systems: Modeling, analysis, and control[END_REF]. Remarkable theoretical results involving switched systems have been achieved concerning their stability and stabilizability [START_REF] Lin | Stability and stabilizability of switched linear systems: A survey of recent results[END_REF][START_REF] Liberzon | Switching in Systems and Control. Systems and Control: Foundations and Applications[END_REF], controllability and reachability [START_REF] Sun | Switched Linear Systems: Control and Design[END_REF][START_REF] Sun | Controllability and reachability criteria for switched linear systems[END_REF] and observability [START_REF] Vidal | Observability of linear hybrid systems[END_REF][START_REF] Xie | Necessary and sufficient conditions for controllability and observability of switched impulsive control systems[END_REF][START_REF] Babaali | Observability of switched linear systems in continuous time[END_REF][START_REF] Sun | Switched Linear Systems: Control and Design[END_REF]. The problem of observer design for linear switched systems has been thoroughly investigated by the control community and different approaches have been proposed. The assumptions about the knowledge of the discrete state play a crucial role. In the case of complete knowledge of the discrete state a Luenberger-like switched observer matching the currently active dynamics can be used and the problem is that of guaranteeing the stability of the switched error dynamics. In [START_REF] Alessandri | Switching observers fo continuous-time and discrete-time linear systems[END_REF] it is shown that the observer gain matrices can be selected by solving a set of linear matrix inequalities. In [START_REF] Bejarano | Switched observers for switched linear systems with unknown inputs[END_REF], the approach is generalized to cover linear switched systems with unknown exogenous inputs. In [START_REF] Tanwani | Observability implies observer design for switched linear systems[END_REF], by adopting the notion of observability over an interval, borrowed from [START_REF] Xie | Necessary and sufficient conditions for controllability and observability of switched impulsive control systems[END_REF], an observer is designed for switched systems whose subsystems are not even required to be separately observable. However, in some situations the active mode is unknown and needs to be estimated, along with the continuous state, by relying only on the continuous output measurements. Usually, in such case, the observer consists of two parts: a discrete state (or location) observer, estimating the active mode of operation, and a continuous observer that estimates the continuous state of the switched system. In [START_REF] Balluchi | Design of observers for hybrid systems[END_REF], the architecture of a hybrid observer consisting of both a discrete and continuous state identification part is presented, assuming partial knowledge of the discrete state, i.e. dealing with the case in which some discrete events causing the switchings are supposed to be measurable. When such a discrete output is not sufficient to identify the mode location, the information available from the continuous evolution of the plant is used to estimate the current mode. However, the " distinguishability" of the different modes, i.e. the property concerning the possibility to reconstruct univocally the discrete state, was not analysed. The present work intrinsically differs from [START_REF] Balluchi | Design of observers for hybrid systems[END_REF] in that we consider the case of completely unknown discrete state. In such a case the possibility to obtain an estimate of the current mode in a finite time is clearly important, not to say crucial. This is clear for instance from [START_REF] Pettersson | Designing switched observers for switched systems using multiple lyapunov functions and dwell-time switching[END_REF], where the authors focus on the continuous-time estimation problem and show that a bound to the estimation error can be given if the discrete mode is estimated correctly within a certain time. Additionally, for those switched systems admitting a dwell time, a guaranteed convergence of the discrete mode estimation taking place "sufficiently faster" that the dwell time is needed. In view of these considerations sliding mode-based observers seem to be a suitable tool due to the attractive feature of finite-time convergence charac-terizing the sliding motions [START_REF] Davila | Second-order sliding-mode observer for mechanical systems[END_REF][START_REF] Floquet | On sliding mode observers for systems with unknown inputs[END_REF][START_REF] Orlov | Finite time stabilization of a perturbed double integrator -part i: Continuous sliding modebased output feedback synthesis[END_REF][START_REF] Pisano | Sliding mode control: a survey with applications[END_REF]. As a matter of fact, sliding mode observers have been successfully implemented to deal with the problem of state reconstruction for switched systems. In [START_REF] Bejarano | State exact reconstruction for switched linear systems via a super-twisting algorithm[END_REF], an observer is proposed ensuring the reconstruction of the continuous and discrete state in finite time. In [START_REF] Davila | Continuous and discrete state reconstruction for nonlinear switched systems via high-order sliding-mode observers[END_REF], the authors present an observer, based on the high-order sliding mode approach, for nonlinear autonomous switched systems. However in the above works, though guaranteeing the finite-time convergence, the convergence time depends on the initial conditions mismatch, and, as a consequence, the estimation convergence in a certain pre-specified time can be guaranteed only upon the existence of an a-priori known admissible domain for the system initial conditions.

Main contribution and structure of the paper

In the present paper we propose a stack of observers whose output injection is computed by relying on the modified Super-Twisting Algorithm, introduced in [START_REF] Cruz-Zabala | Uniform robust exact differentiator[END_REF], that guarantees the so-called "uniform convergence" property, i.e. convergence is attained in finite-time and an upper bound to the transient time can be computed which does not depend on the initial conditions. We also show that, under some conditions, the discrete mode can be correctly reconstructed in finite-time after any switch independently of the observation error at the switching times. Using the continuous output of the switched system, the observer estimates the continuous state and, at the same time, produces suitable residual signals allowing the estimation of the current mode. We propose a residual "projection" methodology by means of which the discrete state can be reconstructed after a switching instant with a finite and pre-specified estimation transient time, allowing a more quick and reliable reconstruction of the discrete state. Additionally, we give structural "distinguishability" conditions in order to guarantee the possibility to reconstruct the discrete state univocally by processing the above mentioned residuals.

The paper structure is as follows. Section 2 formulates the problems under analysis and outlines the structure of the proposed scheme. Section 3 illustrates the design of the continuus state observers' stack by providing the underlying Lyapunov based convergence analysis. Section 4 deal with the discrete state estimation problem. Two approaches are proposed, one using the " asymptotically vanishing residuals" (Subsection 4.1) and another one, taking advantage of the above mentioned residuals' "projection" methodology ("uniform-time zeroed residuals", Subsection 4.2). Section 4.3 outlines the structural conditions addressing the "distinguishability" issues. Section 5 summarizes the proposed scheme and main results of this paper. Section 6 illustrates some simulation results and Section 7 gives some concluding remarks.

Notation

For a vector

v = [v 1 , v 2 , . . . , v n ] T ∈ R n denote sign(v) = [sign(v 1 ), sign(v 2 ), . . . , sign(v n )] T . (1)
Given a set D ⊆ R n , let vj (D) be the maximum value that can assume the j-th element of v on D. Denote as M the 2-norm of a matrix M. For a square matrix M, denote as σ(M) the spectrum of M, i.e. the set of all eigenvalues of M. Finally, denote as N ← (M) the left null space of a matrix M.

PROBLEM STATEMENT

Consider the linear autonomous switched dynamics

ẋ (t) = A σ x (t) y(t) = C σ x(t) (2)
where x (t) ∈ R n represents the state vector, and y(t) ∈ R p represents the output vector. The so-called switching law or discrete state σ(t) : [0, ∞) → {1, 2, ..., q} determines the actual system dynamics among the possible q "operating modes" which are represented, for system (2), by the set of matrix pairs {A 1 , C 1 }, {A 2 , C 2 }, ...{A q , C q }. Without loss of generality, it is assumed that the output matrices C i , ∀i = 1, 2, ..., q are full row rank matrices. The switching law is a piecewise constant function with discontinuities at the switching time instants:

σ(t) = σ k , t k ≤ t < t k+1 , k = 0, 1, ..., ∞ (3)
where t 0 = 0 and t k (k = 1, 2, . . .) are the time instants at which the switches take place.

Definition 1

The dwell time is a constant ∆ > 0 such that the switching times fulfill the inequality t k+1t k ≥ ∆ for all k = 0, 1, ....

The objective is to design an observer able to simultaneously estimate the discrete state σ(t) and the continuous state x(t) of system (2), by relying on the availability for measurements of the output vector y(t).

We propose a design methodology based on a stack of sliding mode observers producing estimates of the evolution of the continuous state of the switched system. At the same time, suitable residuals are provided for identifying the actual value of the discrete state and, consequently, the specific observer of the stack that is producing an asymptotically converging estimate of the continuous state.

The observer structure, depicted in Fig. 1, mainly consists of two parts: a location observer and a continuous state observer. The location observer is devoted to the identification of the discrete state, i.e. the active mode of operation of the switched system, on the basis of some residuals signals produced by the continuous observer.

The continuous state observer receives as input the output vector y(t) of the switching system and, using the location information provided by the location observer, produces an estimation of the continuous state of the system. Finally, the proposed approach requires each subsystem to be observable:

Assumption 3 The pairs (A i , C i ) are observable ∀i = 1, 2, ..., q.
3 Continuous state observer design

Let us preliminarily consider, as suggested in [START_REF] Edwards | Sliding Mode Control: Theory and applications[END_REF], a family of nonsingular coordinate transformations such that the output vector y(t) is a part of the transformed state z(t), i.e.

z(t) =    ξ(t) y(t)    = T σ x(t) (4)
where ξ(t) ∈ R (n-p) and the transformation matrix is given by

T σ =    (N σ ) T C σ    , (5)
where the columns of

N i ∈ R n×(n-p) span the null space of C i , i = 1, 2, ..., q.
By Assumption 2, the trajectories z(t) will evolve into some known compact domain D z . The transformation ( 4) is always nonsingular, and the switched system dynamics (2) in the transformed coordinates are:

ż (t) = Āσ z (t) (6)
where

Āσ = T σ A σ (T σ ) -1 =    Āσ11 Āσ12 Āσ21 Āσ22    (7)
A stack of q dynamical observers, each one associated to a different mode of the switched system, is suggested as follows:

żi (t) = Āi ẑi (t) + Li ν i (t), if ẑi ∈ D z i = 1, 2, . . . , q, ẑij (t) = zj (D z ), if ẑij ≥ zj (D z ) j = 1, 2, . . . , n (8)
where ẑi = [ ξi , ŷi ] T is the state estimate provided by the i-th observer, ν i ∈ R p represents the i-th observer injection input, yet to be designed, and Li ∈ R n×p takes the form

Li =    L i -I    (9)
where L i ∈ R (n-p)×p are observer gain matrices to be designed and I is the identity matrix of dimension p. In the second of [START_REF] Chevallereau | Asymptotically stable running for a five-link, four-actuator, planar bipedal robot[END_REF], which corresponds to a saturating projection of the estimated state, according to the a-priori known domain of evolution D z of vector z(t), the notation ẑij denotes the j-th element of the vector ẑi .

By introducing the error variables

e i ξ (t) = ξi (t) -ξ(t), e i y (t) = ŷi (t) -y(t), i = 1, 2, ..., q, (10)
in view of ( 6) and of the first equation of ( 8) the next error dynamics can be easily obtained:

ėi ξ (t) = Āi11 e i ξ (t) + Āi12 e i y (t) + ( Āi11 -Āσ11 )ξ(t) + ( Āi12 -Āσ12 )y(t) + L i ν i (t) (11) ėi y (t) = Āi21 e i ξ (t) + Āi22 e i y (t) + ( Āi21 -Āσ21 )ξ(t) + ( Āi22 -Āσ22 )y(t) -ν i (t) (12)
Then, by defining

ϕ iσ (e i ξ , e i y , ξ, y) = Āi21 e i ξ (t) + Āi22 e i y (t) + ( Āi21 -Āσ21 )ξ(t) + ( Āi22 -Āσ22 )y(t) (13) one can rewrite (12) as ėi y (t) = ϕ iσ (e i ξ , e i y , ξ, y) -ν i (t) (14)
Let us consider now the second equation of [START_REF] Chevallereau | Asymptotically stable running for a five-link, four-actuator, planar bipedal robot[END_REF]. We know that the trajectories z(t) evolve inside a compact domain D z and the "reset" relation ẑij (t) = zj (D z ) , applied when the corresponding estimate leaves the domain D z , forces them to remain in the set D z . Such a "saturation" mechanism in the observer guarantees that the estimation errors e i ξ and e i y are always bounded. Consequently the functions ϕ ij are smooth enough, that is there exist a constant Φ such that the following inequality is satisfied

d dt ϕ ij (e i ξ , e i y , ξ, y) ≤ Φ, ∀i, j = 1, 2, ..., q (15)
Following [START_REF] Cruz-Zabala | Uniform robust exact differentiator[END_REF], the observer injection term ν i is going to be specified within the next Theorem which establishes some properties of the proposed observer stack that will be instrumental in our next developments.

Theorem 1 Consider the linear switched system (2), satisfying the Assumptions 1-3 along with the stack of observers [START_REF] Chevallereau | Asymptotically stable running for a five-link, four-actuator, planar bipedal robot[END_REF], and the observer injection terms set according to

ν i = k 1 φ 1 (e i y ) -ν 2i (16) ν2i = -k 2 φ 2 (e i y ) (17)
where

φ 1 (e i y ) = |e i y | 1/2 sign(e i y ) + µ|e i y | 3/2 sign(e i y ) (18)
φ 2 (e i y ) = 1 2 sign(e i y ) + 2µe i y + 3 2 µ 2 |e i y | 2 sign(e i y ) (19)
with µ > 0 and the tuning coefficients k 1 and k 2 selected in the set:

K = (k 1 , k 2 ) ∈ R 2 |0 < k 1 < 2 √ Φ, k 2 > k 2 1 4 + 4Φ 2 k 2 1 ∪ (k 1 , k 2 ) ∈ R 2 |k 1 < 2 √ Φ, k 2 > 2Φ (20)
Let L i be chosen such that

Re σ Āi11 + L i Āi21 ≤ -γ, γ > 0 (21)
Then, for sufficiently large Φ and γ, there exists an arbitrarily small time T * << ∆ independent of e i y (t k ) such that, for all k = 0, 1, ..., ∞ and for some α > 0, the next properties hold along the time intervals t ∈ [t k + T * , t k+1 ):

e i y (t) = 0 ∀i (22)
ν i (t) = ϕ iσ (e i ξ , 0, ξ, y) ∀i (23)
e σ k ξ (t) ≤ αe -γ(t-t k -T * ) (24)
Proof. The theorem can be proven by showing the uniform (i.e. independent of the initial condition) time convergence of e i y to zero for all the q observers, after each switching, and analyzing the dynamics of the error variables e i ξ once the trajectories are restricted on the surfaces

S i o = { e i ξ , e i y : e i y = 0}. ( 25
)
Considering the input injection term ( 16) into ( 14) the output error dynamics are given by ėi

y = ϕ iσ -k 1 φ 1 (e i y ) + ν 2i (26)
By introducing the new coordinates

ψ i = ν 2i + ϕ iσ (27)
and considering (17) one obtains the system

ėi y = -k 1 φ 1 (e i y ) + ψ i (28) ψi = -k 2 φ 2 (e i y ) + d dt ϕ iσ (29)
In light of [START_REF] Engell | Modelling and analysis of hybrid systems[END_REF], system (28)-( 29) is formally equivalent to that dealt with in [START_REF] Cruz-Zabala | Uniform robust exact differentiator[END_REF], where suitable Lyapunov analysis was used to prove the uniform-time convergence to zero of e i y and ψ i , i.e. e i y = 0 and ψ i = 0 on the interval t ∈ [t k + T * , t k+1 ), where T * is an arbitrarily small transient time independent of e i y (t k ). Consequently ėi y = 0 and, from equation ( 14), condition ( 23) is satisfied too. By substituting ( 23) into ( 11) with e i y = 0, it yields the next equivalent dynamics of the error variables:

ėi ξ (t) = Āi11 e i ξ (t)+( Āi11 -Āσ11 )ξ(t)+( Āi12 -Āσ12 )y(t)+L i ϕ iσ (e i ξ , 0, ξ, y) (30)
where

ϕ iσ (e i ξ , 0, ξ, y) = Āi21 e i ξ (t) + ( Āi21 -Āσ21 )ξ(t) + ( Āi22 -Āσ22 )y(t) (31)
Finally, by defining the following matrices

Ãi = ( Āi11 + L i Āi21 ) ∆A ξ iσ = ( Āi11 -Āσ11 ) + L i ( Āi21 -Āσ21 ) ∆A y iσ = ( Āi12 -Āσ12 ) + L i ( Āi22 -Āσ22 ) (32)
equation ( 30) can be rewritten as ėi ξ (t) = Ãi e i ξ (t) + ∆A ξ iσ ξ(t) + ∆A y iσ y(t)

It is worth noting that for the correct observer (i.e., that having the index i = σ k which matches the current mode of operation σ(t)) one has ∆A ξ iσ = ∆A y iσ = 0. Hence, along every time intervals t ∈ [t k + T * , t k+1 ), with k = 0, 1, ..., ∞, the error dynamics of the correct observer are given by

ėσ k ξ (t) = Ãσ k e σ k ξ (t) (34)
which is asymptotically stable by [START_REF] Liberzon | Switching in Systems and Control. Systems and Control: Foundations and Applications[END_REF]. Since by Assumption 3 the pairs (A i , C i ) are all observable, the pairs ( Āi11 , Āi21 ) are also observable, which motivates the tuning condition [START_REF] Liberzon | Switching in Systems and Control. Systems and Control: Foundations and Applications[END_REF]. The solution of (34) fulfills the relation [START_REF] Narendra | Adaptation and learning using multiple models, switching, and tuning[END_REF].

Remark 1 Equation (34) means that the estimation ξi provided by the "correct" observer, i.e. the observer associated to the mode σ k activated during the interval t ∈ [t k , t k+1 ), at time t k + T * starts to converge exponentially to the real continuous state ξ. By appropriate choice of L i the desired rate of convergence can be obtained.

Remark 2 It is worth stressing that the time T * is independent of the initial error at each time t k and can be made as small as desired (and in particular such that T * << ∆) by tuning the parameters of the observers properly.

Discrete state estimation

In the previous section it was shown that there is one observer in the stack that provides the asymptotic reconstruction of the continuous state of the switched system (2). However, the index of such "correct" observer is the currently active mode, which is still unknown to the designer, hence the scheme needs to be complemented by a discrete mode observer. In the next subsections we present two methods for reconstructing the discrete state of the system by suitable processing of the observers' output injections.

Asymptotically vanishing residuals

Along the time intervals [t k + T * , t k+1 ) the observers' output injection vectors [START_REF] Morse | Supervisory control of families of linear set-point controllers -part 1: Exact matching[END_REF] satisfy the following relationship:

ν i (t) =      Āσ k 21 σ k ξ (t) for i = σ k Āi21 e i ξ (t) + ( Āi i21 -Āσ k 21 )ξ(t) + ( Āi22 -Āσ k 22 )y(t) for i = σ k (35)
It turns out that along the time intervals [t k +T * , t k+1 ) the norm of the injection term of the correct observer will be asymptotically vanishing in accordance with

ν σ k (t) ≤ A M 21 αe -γ(t-t k -T * ) → 0, (36)
where

A M 21 = sup i∈{1,2,...,q}
Āi21 .

The asymptotic nature of the convergence to zero of the residual vector corresponding to the correct observer is due to the dynamics (34) of the error variable e σ k ξ (t), which in fact tends asymptotically to zero.

Uniform-time zeroed residuals

By making the injection signals insensitive to the dynamics of e i ξ (t) it is possible to obtain a uniform-time zeroed residual for the correct observer, i.e. a residual which is exactly zeroed after a finite time T * following any switch, independently of the error at the switching time. Let us make the next assumption.

Assumption 4 For all i = 1, 2, ..., q, the submatrices Āi21 are not full row rank.

The major consequence of Assumption 4 is that N ←

Āi21 is not trivial. Let U i be a basis for N ← Āi21 (i.e. U i Āi21 = 0) and denote

νi (t) = U i ν i (t) (38)
Clearly, by (35), on the interval [t k + T * , t k+1 ) one has that

νi (t) =      0 for i = σ k -U i A σ k 21 ξ(t) + U i ( Āi22 -Āσ k 22 )y(t) for i = σ k (39)
It turns out that starting from the time t k + T * the norm of the injection term of the correct observer will be exactly zero, i.e. νσ k (t) = 0 ∀t ∈ [t k +T * , t k+1 ).

In order to reconstruct univocally the discrete state, it must be guaranteed that the "wrong" residuals cannot stay identically at zero. In the following section we shall derive a structural condition on the system matrices guaranteeing that the uniform-time zeroed residuals associated to the "wrong" observers stay at the origin.

Uniqueness of the reconstructed switching law

The main property allowing the discrete mode reconstruction is that, ter a finite time T * following any switch, the residual corresponding to the "correct" observer converges to zero, according to (35), or it is exactly zero if the uniform-time zeroed residuals (38) are used. In order to reconstruct the discrete state univocally, all the other residuals (i.e. those having indexes corresponding to the non activated modes) must be separated from zero. In what follows the uniform-time zeroed residuals (38) are considered as they provide better performance and faster reconstruction capabilities. Nevertheless, analogue considerations can be made in the case of the asymptotically vanishing residuals (35).

Definition 2 Given the switched system (2), a residual νi (t) is said to be non vanishing if x(t) = 0 almost everywhere implies νi (t) = 0 almost everywhere ∀i = σ k , that is the residuals corresponding to the "wrong" observers cannot be identically zero on a time interval unless the state of the system is identically zero in the considered interval.

Next Lemma establishes an "observability-like" requirement guaranteeing that that the uniform-time zeroed residuals (39) are non vanishing.

Lemma 1

The uniform-time zeroed residuals νi (t) in (39) are non vanishing if and only if the pairs ( Āj , Cij ) are observable ∀i = j, where Āj (j = 1, 2, ..., q)

are the state matrices of system ( 6) and Cij = -U T i Āj21 U T i ( Āi22 -Āj22 ) .

Proof. Along the time interval [t k + T * , t k+1 ), the "wrong" residuals νi (t) in (39), i.e. those with index i = σ k , are related to the state z(t) of ( 6) as

νi (t) = Ciσ k z(t) (40)
where z(t) is the solution of

ż (t) = Āσ k z (t) (41)
It is well known that if the pair ( Āσ k , Ciσ k ) of system (40)-( 41) is observable, then νi (t) is identically zero if and only if z(t) (and, thus, x(t)) is null. Therefore, to extend this property over all the intervals t k + T * ≤ t < t k+1 with k = 0, 1, ..., ∞, all the pairs ( Āj , Cij ) ∀i = j have to be observable.

Assumption 5

The state trajectories x(t) in (2) are not zero almost everywhere and all the residuals νi (t) in (38) are non vanishing according to Definition (2).

Thus if Assumption 5 is fulfilled, only the residual associated to the correct observer will be zero, while all the others will have a norm separated from zero. It is clear that if such a situation can be guaranteed, then the discrete mode can be univocally reconstructed by means of simple decision logic, as discussed in Theorem 2 of the next Section.

Remark 3 If Assumption 4 is not satisfied the residual (38) cannot be considered. Nevertheless, by considering the asymptotically vanishing residuals (35), similar structural conditions guaranteeing that the "wrong" residuals cannot be identically zero can be given. Consider the following extended vector:

z i e (t) =        e i ξ (t) ξ(t) y(t)        (42)
From ( 6), ( 33) and (35) the following system can be considered on the interval

[t k + T * , t k+1 ): żi e (t) = A iσ k z i e (t) ν i (t) = C iσ k z i e (t) (43)
where

A iσ k =    Ãi ∆A ξ iσ k ∆A y iσ k 0 Āσ k    C iσ k =        Āi21 Āi21 -Āσ k 21 Āi22 -Āσ k 22        T (44)
The asymptotically vanishing residuals (35) are non vanishing if the pairs (A ij , C ij ) are observable ∀i = j.

Continuous and discrete state observer

The proposed methodology of continuous and discrete state estimation is summarized in the next Theorem 2 Consider the linear switched system (2), fulfilling the Assumption 1-5, and the observer stack [START_REF] Chevallereau | Asymptotically stable running for a five-link, four-actuator, planar bipedal robot[END_REF] described in the previous Theorem 1.

Consider the next evaluation signal

ρi (t) = t t-ǫ νi (τ ) dτ . ( 45
)
where ǫ, is a small time delay, and the next active mode identification logic:

σ(t) = argmin i∈{1,2,...,q} ρi (t) (46)
Then, the discrete state estimation will be such that

σ(t) = σ(t), t k + T * + ǫ ≤ t ≤ t k+1 , k = 0, 1, ..., ∞ (47)
and the continuous state estimation given by

x(t) = (T σ ) -1    ξσ (t) ŷσ (t)    (48)
will be such that

x(t) -x(t) ≤ αe -γ(t-t k -T * ) ∀t ∈ [t k + T * , t k+1 ) (49)
Proof.

By considering (39) which, specified for the correct observer (i = σ k ), guarantees that νσ k (t) = 0,

t k + T * ≤ t ≤ t k+1 (50)
along with the Assumption 5 , whose main consequence is that νi (t) cannot be identically zero over an interval when i = σ(t), it follows that it is always possible to find a threshold η such that for the evaluation signals ρi (t) in (45) one has ρi (t) > η,

t k + T * + ǫ ≤ t < t k+1 , i = σ k (51) ρσ k (t) ≤ η, t k + T * + ǫ ≤ t < t k+1 . (52)
Thus the mode decision logic (46) provides the reconstruction of the discrete state after the finite time T * + ǫ to any switching time instant, i.e.

σ(t) = σ k , t k + T * + ǫ ≤ t < t k+1 (53)
The second part of the Theorem 2 concerning the continuous state estimation can be easily proven by considering the coordinate transformation (4) and Theorem 1, which imply (49).

Remark 4

We assumed that the state trajectories are not zero almost everywhere (Assumption 5). As a result the wrong residuals can occasionally cross the zero value. This fact motivates the evaluation signal introduced in (45): considering a window of observation for the residuals, all the wrong residuals will be separated from zero, while only the correct one can stay close to zero during a time interval of nonzero length.

The architecture of the observer is shown in Fig. 2.

Remark 5 It is possible to develop the same methodology if the residual (35) is considered instead of (38). The evaluation signal can be used to identify the discrete state. However, the time to identify the discrete mode will be bigger than the one of Theorem 2 since the vanishing transient of the error variable e i ξ is needed to last for a while, starting from the finite time t k + T * .

ρ i (t) = t t-ǫ ν i (τ ) dτ . (54)

SIMULATION RESULTS

In this section, we discuss a simulation example to show the effectiveness of our method. Consider a switched linear system as in (2) with q = 3 modes by the matrices

A 1 =        0.1 0.6 -0.4 -0.5 -0.8 1 0.1 0.4 -0.7        , A 2 =        -0.2 0.3 -0.8 -0.2 -0.4 0.8 1 0.6 -0.3        , A 3 =        -0.8 -0.5 0.2 -0.5 -0.1 -0.5 -0.3 -0.2 0.3        (55)
C 1 =    1 0 0 0 1 0    , C 2 =    1 0 0 0 0 1    , C 3 =    0 0 1 0 1 0    (56)
The system starts from the mode 1 with the initial conditions x(0) = [-3, -1, 6] T and evolves switching between the three modes according to the switching law shown in Fig. 3. After the coordinate transformation (4) obtained by the transformation matrices

T 1 =        0 0 1 1 0 0 1 0        , T 2 =        0 -1 0 1 0 0 0 0 1        , T 3 =        -1 0 0 0 0 1 0 1 0        ( 57
)
the system is in the proper form to apply our estimation procedure. Since the pairs (A i , C i ) are all observable for i = 1, 2, 3, the stack of observers ( 8) can be implemented. By properly tuning the parameters of the STA-based observers according to [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF], the components of the vector error e y are exactly zero for each observer after a time T * subsequent to any switch (Fig. 4) as proven in Theorem 1. On the contrary, the error e ξ at time t k + T * starts to converge exponentially to zero only for the correct observer, as shown in Fig. 5. Notice that the three signals occasionally cross the zero, but only the one corresponding to the correct observer remains zero on a time interval. The gains of the observers L i are chosen such that the eigenvalues of the matrices Ãi in [START_REF] Vries | Hybrid system modeling and identification of cell biology systems: perspectives and challenges[END_REF] governing the error dynamics ( 33) are located at -5.

Since Assumption 5 is satisfied, the discrete mode can be univocally identified.

To this end let us consider the asymptotically vanishing residuals (35) and the uniform-time zeroed residuals (39). The simulations confirm that both signals stay at zero only for the correct observer. Moreover, the signal corresponding to the asymptotically vanishing residual is in general "slower" than the signal corresponding to the uniform-time zeroed residual. In order to highlight the different behaviours of the two signals, we reported in Fig. 6 the dynamics of the two residuals provided by the third observer when the mode 3 becomes active at t = 20. The evaluation signal obtained with the uniform-time zeroed residuals allows us a faster estimation of the switching law, as compared to their asymptotic counterparts. In Fig. 7 the actual and the reconstructed switching law are depicted by using the two different evaluation signals.

CONCLUSIONS

The problem of simultaneous continuous and discrete state reconstruction has been tackled for linear autonomous switched systems. The main ingredient of the proposed approach is an appropriate stack of high-order sliding mode observers used both as continuous state observers and as residual generators for discrete mode identification. As a novelty, a procedure has been devised to algebraically process the residuals in order to reconstruct the discrete state after a finite time that can be arbitrarily small, and, additionally, conditions ensuring the identifiability of the system modes are derived in terms of the original system matrices. An original "projection" procedure has been proposed, leading to the so-called uniform-time zeroed residuals, which allows a faster reconstruction of the active mode as compared to the (however feasible) case when such projection is no adopted. The same observer can be designed in the case of forced switched systems, too. However, further investigation is needed concerning the underlying conditions to univocally reconstruct the discrete state, which would be affected by the chosen input too.

ACKOWLEDGMENTS

A. Pisano gratefully acknowledges the financial support from the Ecole Centrale de Lille under the 2013 ECL visiting professor program.

Fig. 1 .Assumption 2

 12 Fig. 1. Observer structure

Fig. 2 .

 2 Fig. 2. Continuous and discrete state observer

Fig. 3 .

 3 Fig. 3. Actual switching signal.

3 y2Fig. 4 . 1 ξFig. 5 .

 3415 Fig. 4. Estimation vector error e y corresponding to the three observers.

17

 17

3 Fig. 6 .

 36 Fig. 6. Evaluation signal used to estimate the discrete state.

Fig. 7 .

 7 Fig. 7. Actual and reconstructed switching signals.


Favier M Pd

Bordet Jc Phd

MD Alessi Mc

Nurden P Md

PhD Nurden At

Remi Favier
email: remi.favier@aphp.fr

Heterozygous mutations of the integrin αIIbR995/β3D723 intracytoplasmic salt bridge cause macrothrombocytopenia, platelet functional defects and enlarged α-granules

Keywords: inherited macrothrombocytopenia, integrin a2IIbb3, platelet function defects, enlarged a -granules

Rare gain-of-function mutations within the ITGA2B or ITGB3 genes have been recognized to cause macrothrombocytopenia (MTP). Here we report three new families with autosomal dominant (AD) MTP, two harboring the same mutation of ITGA2B, αIIb R995W, and a third family with an ITGB3 mutation, β3D723H. The two mutated amino-acids are directly involved in the salt bridge linking the intra-cytoplasmic part of IIb to 3 of the integrin αIIbβ3.

For all affected patients, the bleeding syndrome and MTP was mild to moderate. Platelet aggregation tended to be reduced but not absent. Electron microscopy associated with a morphometric analysis revealed large round platelets; a feature being the presence of abnormal large -granules with some giant forms showing signs of fusion. Analysis of the maturation and development of megakaryocytes reveal no defect in their early maturation but abnormal proplatelet formation was observed with increased size of the tips.

Interestingly, this study revealed that in addition to the classical phenotype of patients with αIIbβ3 intracytoplasmic mutations, an abnormal maturation of -granules. It will be interesting to determine if this feature is a characteristic of mutations disturbing the IIb R995/3 D723 salt bridge.

INTRODUCTION

Integrin αIIbβ3 is the platelet receptor for fibrinogen (Fg) and other adhesive proteins and mediates platelet aggregation playing a key role in hemostasis and thrombosis. It circulates on platelets in a low-affinity state becoming ligand-competent as a result of conformational changes induced by "inside-out" signaling following platelet activation [START_REF] Coller | The GPIIb/IIIa (integrin αIIbβ3) odyssey: a technology-driven saga of a receptor with twists, turns, and even a bend[END_REF]. Inherited defects of αIIbβ3 with loss of expression and/or function are causal of Glanzmann thrombasthenia (GT), an autosomal recessive bleeding disorder [START_REF] George | Glanzmann's thrombasthenia: The spectrum of clinical disease[END_REF][START_REF] Nurden | Glanzmann thrombasthenia: a review of ITGA2B and ITGB3 defects with emphasis on variants, phenotypic variability, and mouse models[END_REF]. Rare gain-of-function mutations of the ITGA2B or ITGB3 genes encoding αIIbβ3 also cause macrothrombocytopenia (MTP) with a low platelet count and platelets of increased size [START_REF] Nurden | Glanzmann thrombasthenia: a review of ITGA2B and ITGB3 defects with emphasis on variants, phenotypic variability, and mouse models[END_REF][START_REF] Nurden | Glanzmann thrombasthenia-like syndromes associated with macrothrombocytopenias and mutations in the genes encoding the αIIbβ3 integrin[END_REF]. Mostly heterozygous with autosomal dominant (AD) expression these include D621_E660del*, L718P, L718del and D723H mutations in β3, and G991C, G993del, R995Q or W in αIIb (Table I) [START_REF] Hardisty | A defect of platelet aggregation associated with an abnormal distribution of glycoprotein IIb-IIIa complexes within the platelet: The cause of a lifelong bleeding disorder[END_REF][START_REF] Peyruchaud | R to Q amino acid substitution in the GFFKR sequence of the cytoplasmic domain of the integrin αIIb subunit in a patient with a Glanzmann's thrombasthenia-like syndrome[END_REF][START_REF] Ghevaert | A nonsynonymous SNP in the ITGB3 gene disrupts the conserved membrane-proximal cytoplasmic salt bridge in the αIIbβ3 integrin and cosegregates dominantly with abnormal proplatelet formation and macrothrombocytopenia[END_REF][START_REF] Jayo | L718P mutation in the membrane-proximal cytoplasmic tail of β3 promotes abnormal αIIbβ3 clustering and lipid microdomain coalesce, and associates with a thrombasthenia-like phenotype[END_REF][START_REF] Kunishima | Heterozygous ITGA2B R995W mutation inducing constitutive activation of the αIIbβ3 receptor affects proplatelet formation and causes congenital macrothrombocytopenia[END_REF][START_REF] Kashiwagi | Demonstration of novel gain-of-function mutations of αIIbβ3: association with macrothrombocytopenia and Glanzmann thrombasthenia-like phenotype[END_REF][START_REF] Kobayashi | Identification of the integrin β3 L718P mutation in a pedigree with autosomal dominant thrombocytopenia with anisocytosis[END_REF][START_REF] Nurden | An intracytoplasmic β3 Leu718 deletion in a patient with a novel platelet phenotype[END_REF]. While Asp621_Glu660del affects the extracellular cysteine-rich βA domain of β3, the others affect transmembrane or intracellular cytoplasmic domains and in particular the salt bridge linking the negatively charged D723 of β3 with the positively charged R995 of the much studied GFFKR sequence of αIIb [START_REF] Hughes | The conserved membraneproximal region of an integrin cytoplasmic domain specifies ligand-binding affinity[END_REF][START_REF] Hughes | Breaking the integrin hinge. A defined structural constraint regulates integrin signaling[END_REF][START_REF] Kim | Interactions of platelet integrin αIIb and β3 transmembrane domains in mammalian cell membranes and their role in integrin activation[END_REF]. These mutations permit residual or even total αIIbβ3 expression but give rise to conformation changes that propagate through the integrin and which are recognized by binding of the monoclonal antibody, PAC-1 [START_REF] Shattil | Changes in the platelet membrane glycoprotein IIb.IIIa complex during platelet activation[END_REF]. The MTP appears related to cytoskeletal changes during the late stages of megakaryocyte (MK) development and altered proplatelet formation [START_REF] Bury | Outside-in signaling generated by a constitutively activated integrin αIIbβ3 impairs proplatelet formation in human megakaryocytes[END_REF][START_REF] Bury | Cytoskeletal perturbation leads to platelet dysfunction and thrombocytopenia in variant forms of Glanzmann thrombasthenia[END_REF][START_REF] Hauschner | Abnormal cytoplasmic extensions associated with active αIIbβ3 are probably the cause for macrothrombocytopenia in Glanzmann thrombasthenia-like syndrome[END_REF]). Yet, while most of the above variants combine MTP with a substantial loss of platelet aggregation and a GT-like phenotype, the D723H β3 substitution had no effect on platelet aggregation and was called a non-synonymous single nucleotide polymorphism (SNP) by the authors [START_REF] Peyruchaud | R to Q amino acid substitution in the GFFKR sequence of the cytoplasmic domain of the integrin αIIb subunit in a patient with a Glanzmann's thrombasthenia-like syndrome[END_REF]. This was surprising as another cytoplasmic domain mutation involving a near-neighbour Arg724Ter truncating mutation in β3, while not preventing αIIbβ3 expression gave a full GT phenotype [START_REF] Wang | Truncation of the cytoplasmic domain of beta3 in a variant form of Glanzmann thrombasthenia abrogates signaling through the integrin alphaIIbbeta3 complex[END_REF]. We recently reported a heterozygous intracytoplasmic β3 Leu718del that resulted in loss of synchronization between the cytoplasmic tails of β3 and αIIb; changes that gave moderate MTP, a reduced platelet aggregation response and, unexpectedly, enlarged α-granules [START_REF] Nurden | An intracytoplasmic β3 Leu718 deletion in a patient with a novel platelet phenotype[END_REF]. It is in this context that we now report our studies on a second European family with a heterozygous β3 D723H variant as well as the first two families to be described outside of Japan with a heterozygous αIIb R995W substitution. Significantly, not only both of these variants of the αIIbR995/β3D723 salt bridge give rise to moderate MTP and platelet function defects; their platelets also contained enlarged α-granules.

CASE HISTORIES

We now report three families (A from Reunion island; B and C from France) with inherited MTP transmitted across 2 or 3 generations suggestive of autosomal dominant (AD) inheritance.

The family pedigrees are shown in Fig. 1 and the three index cases (AII.1 an adult female, BI.1 and CI.1 adult males) identified. Other family members known to have MTP and significant subpopulations of enlarged platelets are also highlighted. All showed moderate to mild thrombocytopenia and often a higher proportion of immature platelets when analyzed with the Sysmex XE-5000 automat (Sysmex,Villepinte, France) (Table I). Increased mean platelet volumes were observed for BI.1, CI.1, and C1.2 (Table I) but values are not given when the large diameter of many of the platelets meant that they were not taken into account by the machine (particularly so for members of family A with MTP). Other blood cell lineages were usually present for affected family members and all routinely tested coagulation parameters were normal. As quantitated by the ISTH-BAT bleeding score, members of family A with MTP were the most affected (Table I). For example, AII.1 suffered from severe menorrhagia and severe post-partum bleeding requiring platelet and red blood cell transfusions after her second childbirth although two other children were born without problems (including a cesarean section). AII.1 also experienced occasional spontaneous bruising and episodes of iron-deficient anemia of unknown cause. An affected sister also has easy bruising and childbirth was under the cover of platelet transfusion. In family B the index case BI.1 suffered epistaxis but no bleeding has been reported for other family members. No bleeding was seen for the index case (CI.1) in family C despite major surgery following a bomb explosion while working as a war photographer. His daughter (CII.1) however experiences mild bleeding with frequent hematomas. Our study was performed in accordance with the declaration of Helsinki after written informed consent, and met with the approved protocol from INSERM (RBM-04-14).

METHODS

Platelet aggregation

Platelet aggregation was tested in citrated platelet-rich plasma (PRP) according to our standard protocols [START_REF] Bluteau | Thrombocytopenia-associated mutations in the ANKRD26 gene regulatory region induce MAPK hyperactivation[END_REF] and compared to PRP from healthy control donors without adjustment of the platelet count. The following agonists were used: 10µM adenosine diphosphate (ADP); 1mM arachidonic acid (AA); 1M U46619, (all from Sigma Aldrich, L'isle d'Abeau, Chesnes, France); 20 µM thrombin receptor activating peptide (TRAP) (Polypeptide Group, Strasbourg, France); 1µg/mL collagen (COL) (Chronolog Corporation, Havertown, USA); 5 M, epinephrine (Sigma), 1.5 mg/mL, and 0.6 mg/mL ristocetin (Helena Biosciences Europe, Elitech, Salon-en-Provence, France). Results were expressed as percentage maximal intensity.

Flow cytometric analysis (FCM)

Glycoprotein expression on unstimulated platelets was assessed using citrated PRP according to our standard protocols [START_REF] Nurden | An intracytoplasmic β3 Leu718 deletion in a patient with a novel platelet phenotype[END_REF][START_REF] Bluteau | Thrombocytopenia-associated mutations in the ANKRD26 gene regulatory region induce MAPK hyperactivation[END_REF]. On occasion, platelet surface labelling for αIIb, β3, GPIbα and P-selectin was quantified using the PLT Gp/Receptors kit (Biocytex, Marseille, France) at room temperature before and after stimulation with 10µM ADP and 50µM TRAP using the Beckman Coulter Navios flow cytometer (Beckman Coulter, Villepinte, France). Platelets were identified by their light scatter characteristics and their positivity for a PC5 conjugated plateletspecific monoclonal antibody (MoAb) (CD41). An isotype antibody was used as negative control. To study platelet αIIbβ3 activation by flow cytometry, platelets were activated with either 10 μΜ ADP or 20 μΜ TRAP in the presence of FITC-conjugated PAC-1. A fluorescence threshold was set to analyze only those platelets that had bound FITC-PAC1. In brief, an antibody mixture consisting of 40 μl of each MoAb (PAC-1 and CD41) was diluted with 280 μl of PBS. Subsequently 5μl of PRP were mixed with 40μl of the antibody mixture and with 5 μl of either saline or platelet activator. After incubating for 15 min at room temperature in the dark, 1ml of isotonic PBS buffer was added and samples were analyzed. Antibody binding was expressed either as the mean fluorescence intensity or as the percentage of platelets positive for antibody.

Transmission electron microscopy (EM)

PRP from blood taken into citrate or ACDA anticoagulant was diluted and fixed in PBS, pH 7.2, containing 1.25 %(v/v) glutaraldehyde for 1h as described [START_REF] Nurden | An intracytoplasmic β3 Leu718 deletion in a patient with a novel platelet phenotype[END_REF]. After centrifugation and two PBS washings, they were post-fixed in 150 mM cacodylate-HCl buffer, pH 7.4, containing 1% osmium tetroxide for 30 min at 4°C. After dehydration in graded alcohol, embedding in EPON was performed by polymerization at 60°C for 72 h. Ultrathin sections 70-80 nm thick were mounted on 200-mesh copper grids, contrasted with uranyl acetate and lead citrate and examined using a JEOL JEM1400 transmission electron microscope equipped with a Gatan Orius 600 camera and Digital Micrograph software (Lyon Bio Image, Centre d'Imagerie Quantitative de Lyon Est, France). Morphometric measurements were made using Image J software (National Institutes of Health, USA).

Genetic analysis and mutation screening.

DNA from AII.1, BI.1 and CI.1 was subjected to targeted exome sequencing (v5-70 Mb) as part of a study of a series of families with MTP due to unknown causes organized within the Paris Trousseau Children's Hospital (Paris, France). Single missense variants known to be pathological for MTP in the ITGA2B and ITGB3 cytoplasmic tails were highlighted and their presence in other family members with MTP was confirmed by Sanger sequencing (primers are available on request). The absence of other potentially pathological variants in genes known to be causal of MTP in the targeted exome sequencing analysis was confirmed.

In silico models to investigate αIIbβ3 structural changes induced by the mutations were obtained using the PyMOL Molecular Graphics System, version 1.3, Schrödinger, LLC (www.pymol.org) and 2k9j pdb files for transmembrane and cytosolic domains as described in our previous publications [START_REF] Nurden | Glanzmann thrombasthenia: a review of ITGA2B and ITGB3 defects with emphasis on variants, phenotypic variability, and mouse models[END_REF][START_REF] Nurden | Glanzmann thrombasthenia-like syndromes associated with macrothrombocytopenias and mutations in the genes encoding the αIIbβ3 integrin[END_REF][START_REF] Nurden | An intracytoplasmic β3 Leu718 deletion in a patient with a novel platelet phenotype[END_REF]. Amino acid changes are visualized in the rotamer form showing side change orientations incorporated from the Dunbrack Backbone library with maximum probability.

In vitro MK differentiation, ploidy analyses, quantification of proplatelets and immunofluorescence analysis

Plasma thrombopoietin (TPO) levels were measured as previously described [START_REF] Bluteau | Thrombocytopenia-associated mutations in the ANKRD26 gene regulatory region induce MAPK hyperactivation[END_REF]. Patient or control CD34 + cells were isolated using an immunomagnetic beads technique (Miltenyi, Biotec, France) and grown supplemented with 10 ng/mL TPO (Kirin Brewery, Tokyo, Japan) and 25 ng/mL Stem Cell Factor (SCF; Biovitrum AB, Stockholm, Sweden).

(i) Ploidy analyses. At day 10, Hoechst 33342 dye (10 μg/mL; Sigma-Aldrich, Saint Quentin Fallavier, France) was added to the medium of cultured MKs for 2 h at 37°C. Cells were then stained with directly coupled MoAbs: anti-CD41-phycoerythrin and anti-CD42aallophycocyanin (BD Biosciences, Le Pont de Claix, France) for 30 min at 4°C. Ploidy was measured in the CD41 + CD42 + cell population by means of an Influx flow cytometer (BD; Mountain View,USA) and calculated as previously described [START_REF] Bluteau | Thrombocytopenia-associated mutations in the ANKRD26 gene regulatory region induce MAPK hyperactivation[END_REF].

(ii) Quantification of proplatelet-bearing MKs. To evaluate the percentage of MKs forming proplatelets (PPTs) in liquid medium, CD41 + cells were sorted at day 6 of culture and plated in 96-well plates at a concentration of 2000 cells per well in serum-free medium in the presence of TPO (10 ng/mL). MKs displaying PPTs were quantified between day 11 and 13 of culture by enumerating 200 cells per well using an inverted microscope (Carl Zeiss, Göttingen, Germany) at a magnification of ×200. MKs displaying PPTs were defined as cells exhibiting ≥1 cytoplasmic process with constriction areas and were analyzed in triplicate in two independent experiments for each individual.

(iii) Fluorescence microscopy. Primary MKs grown in serum-free medium were allowed to spread for 1 h at 37 °C on 100 g/mL fibrinogen (Sigma Aldrich, Saint Quentin Fallavier, France) coated slides, then fixed in 4% paraformaldehyde (PFA), washed and permeabilized for 5 min with 0.2% Triton-X100 and washed with PBS prior to being incubated with rabbit anti-VWF antibody (Dako, Les Ulis, France) for 1 h, followed by incubation with Alexa 546-conjugated goat anti-rabbit immunoglobulin G (IgG) for 30 min and Phalloidin-FITC (Molecular Probes, Saint Aubin, France). Finally, slides were mounted using Vectashield with 4, 6 diamidino-2-phenylindole (Molecular Probes, Saint Aubin, France). The PPT-forming MKs (cells expressing VWF) were examined under a Leica DMI 4000, SPE laser scanning microscope (Leica Microsystems, France) with a 63×/1.4 numeric aperture oil objective.

RESULTS

Molecular genetic analysis:

We describe 3 previously unreported families, one based in Reunion Island and the others in France, with inherited MTP and mild to moderate bleeding. Targeted exome sequencing revealed heterozygous missense mutations of residues that compose the platelet αIIbR995/β3D723 intracytoplasmic salt bridge whose loss is integral to integrin signaling.

Probands AII.1 and BI.1 have the αIIbR955W variant previously identified in Japanese families with MTP [START_REF] Kunishima | Heterozygous ITGA2B R995W mutation inducing constitutive activation of the αIIbβ3 receptor affects proplatelet formation and causes congenital macrothrombocytopenia[END_REF]. In contrast CI.1 possesses β3D723H originally described as a nonsynonymous SNP and associated with MTP in a UK family [START_REF] Ghevaert | A nonsynonymous SNP in the ITGB3 gene disrupts the conserved membrane-proximal cytoplasmic salt bridge in the αIIbβ3 integrin and cosegregates dominantly with abnormal proplatelet formation and macrothrombocytopenia[END_REF]. Sanger sequencing confirmed the presence of both variants and showed that their expression segregated with MTP in the family members available for genetic analysis (see Fig.

Platelet aggregation and flow cytometry analysis:

Citrated PRP from each index case was stimulated with ADP, TRAP, AA and collagen and platelet aggregation measured using standard procedures (Fig. 2A). Results were variable, and taking the curves obtained for ristocetin as a control of the low platelet count for each index case platelets from each family with αIIbR995 and β3D723 variants retained at least a partial aggregation response with family AII.1 showing the largest lossparticularly for TRAP. The response to epinephrine was also much reduced or absent for all samples (data not illustrated).

Striking was the low response to AA for patients CI.1and CII.2, a finding reversed on addition of the thromboxane A2 analog U46619 (data not shown). Otherwise, the platelets retained a rapid response to ADP and collagen. Flow cytometry and MoAbs recognizing determinants specific for αIIb, β3 or the αIIbβ3 complex (data not shown) gave comparable results for each index case with surface levels for the 3 index cases ranging from 48 to 75% of those on normal platelets (Fig. 2B). Taking into account the increased platelet size, such intermediate levels would suggest that both mutations have a direct influence on αIIbβ3 expression. Enigmatically, the platelet expression of GPIb was particularly increased for the 4 tested family members (AII.1, AII.2, BI.1, BII.2) with the αIIbR995W mutation a finding only partially explained by the increased platelet volume of these patients with MTP. Binding of PAC-1 recognizing an activation-dependent epitope on αIIbβ3 was analyzed as a probe of the activation state of the integrin. Spontaneous binding of PAC-1 was seen for the platelets of index case AII.1 with the αIIbR995W mutation suggesting signs of activation but was not seen for the index case of the second family with this mutation or for the index case of the family with β3D723H (Fig. 2C).

Studies were extended to platelets stimulated with high doses of ADP and TRAP; increased binding was seen for AII.1 consistent with further activation of the residual surface αIIbβ3 of the platelets. However, no binding was seen for BI.1 or C.I.1. suggesting that for these patients the residual αIIbβ3 was refractory to stimulation under the non-stirred conditions of this set of experiments (Fig. 2B).

Fig. 2. Selected biological platelet findings for the three index cases. In A) Light transmission

aggregometry performed in citrated platelet-rich plasma (PRP) compares typical responses of platelets from the index cases (AII.1, BI.1, CI.1) to that of a typical control donor. For AII.1 aggregation with high doses of ADP, Col, TRAP and AA were reduced compared to ristocetin-induced platelet agglutination whose intensity reflected the low platelet count of the patient. For BI.1 platelet aggregation was moderately reduced with Col and TRAP while for CI.1 platelet aggregation was reduced essentially for TRAP and AA (it should be noted that it was restored with the thromboxane receptor agonist U46619, not shown). In B) Spontaneous PAC1 binding evaluated by flow cytometry on resting platelets was marginally increased for AII.1 but not for the other index cases. Binding increased for AII.1 after platelet activation with ADP and TRAP but remained low compared to the control. In contrast, PAC-1 binding was basal for BI.1 and CI.1 even after addition of ADP or TRAP. In C) we illustrate the levels of GPIb and αIIbβ3 receptors evaluated by flow cytometry not only for the probands but also for other selected family members. A decreased surface expression of αIIb and β3were found for all affected patients, values ranging between 43% (AII.2) and 70% (BII.2) of the control mean. Interestingly, levels of GPIb were increased for the patients and particularly so for families A and B with sometimes values beyond 150% of normal values.

Electron microscopy:

Platelets from the index cases of all 3 families were examined by transmission EM and for each subject a significant subpopulation of the platelets were larger than normal (Fig 3). Overall, the platelets showed wide size variations; many tended to be round in contrast to the discoid shape of controls (control platelets are illustrated by us in ref [START_REF] Nurden | An intracytoplasmic β3 Leu718 deletion in a patient with a novel platelet phenotype[END_REF]. Patients from all index cases possessed platelets with large variations in the numbers of vacuoles. Striking was a heterogeneous distribution of α-granules with the presence of giant forms particularly evident for patient AII.1 and what appears to represent granule fusion was seen (highlighted in panel 3b). All of the morphological changes were analyzed quantitatively; statistical significance was achieved for all measurements except for those concerning α-granule numbers of patient CI.1 with the β3D723H substitution (Fig. 3). Please note that the platelets of the patients with the αIIbR995W mutation tended to be larger and show more ultrastructural changes. Interestingly, the greater the frequency of giant granules, the lower is their concentration/µm 2 . The presence of giant α-granules repeats what we have recently seen for a patient with a heterozygous β3718del and is reminiscent of the feature that we have much earlier described in Paris Trousseau syndrome (12, 22, and 23). Targeted exome sequencing failed to show mutations in the FlI1 gene of the three index cases.

Megakaryopoiesis:

Plasma TPO levels were within normal range for each of the index cases (Table I). Analysis of MK maturation and development did not reveal any defect in early megakaryocyte maturation.

Ploidy measured in theCD41 + CD42 + cell population at day 10 of culture by means of an Influx flow cytometer with the proportion of 2N-32N MKs being within the normal range (Fig. 4).

Proplatelet formation was examined on days 11 and 12 of culture using an inverted microscope and no difference in percentage of proplatelet bearing MKs was detected. Proplatelet morphology was analyzed at the same time using a SPE laser scanning microscope after dual fluorescent labeling of PFA-permeabilized cells with Phalloidin and antibody to VWF (green) (Fig. 4). While the mature MKs basically showed normal morphology proplatelet numbers tended to be lower and some extensions appeared swollen and with decreased branching.

Another finding was that the size of the tips and bulges occurring at intervals along the proplatelets tended to be larger than for control MKs and especially so for the two index cases with αIIbR995W (AII.1 and BI.1); an image of a giant granule can be observed in an illustrated extension of AII.

DISCUSSION

In the resting state, the trans-membrane and intra-cytoplasmic segments of the two subunits of αIIbβ3 interact, an interaction that is key to maintaining the extracellular domain of the integrin in its bent resting state [START_REF] Kim | Interactions of platelet integrin αIIb and β3 transmembrane domains in mammalian cell membranes and their role in integrin activation[END_REF][START_REF] Litinov | Activation of individual alphaIIbbeta3 integrin molecules by disruption of transmembrane domain interactions in the absence of clustering[END_REF]. One area of contact between the cytoplasmic tails involves π interactions and aromatic cycle stacking of consecutive F residues within the highly conserved αIIb GFFKR (aa991-955) sequence with W713 of β3 (shown in Fig. 1). A second interaction principally involves a salt bridge between the positively charged αIIbR995 and negatively charged β3D723 [START_REF] Hughes | The conserved membraneproximal region of an integrin cytoplasmic domain specifies ligand-binding affinity[END_REF][START_REF] Hughes | Breaking the integrin hinge. A defined structural constraint regulates integrin signaling[END_REF][START_REF] Kim | Interactions of platelet integrin αIIb and β3 transmembrane domains in mammalian cell membranes and their role in integrin activation[END_REF]. Early studies including site-directed mutagenesis, truncation models and charge reversal mutations showed that loss of this intra-molecular clasp led to integrin activation and modified function [START_REF] Hughes | The conserved membraneproximal region of an integrin cytoplasmic domain specifies ligand-binding affinity[END_REF][START_REF] Hughes | Breaking the integrin hinge. A defined structural constraint regulates integrin signaling[END_REF]. Hence the mutations described in our patients are of high significance for integrin biology. Enigmatically, the β3D723H change has the more pronounced structural effect resulting in (i) repulsive electrical charge forces with the positively charged H now facing the positively charged R995 and (ii) steric encumbrance due to the larger H. The net result is a widening of the interval between R995 and H723 and a weakening of the salt bridge, changes that accompany the acquisition of a higher affinity state [START_REF] Adair | Three-dimensional model of the human platelet integrin alphaIIbbeta3 based on electron cryomicroscopy and x-ray crystallography[END_REF]. Of similar consequence but milder in nature is the replacement of αIIb995R by the neutral W while both mutations potentially also interfere with π interactions involving αIIbF992.

A novel feature of our study is the presence of enlarged α-granules in the platelets of all 3 index cases. This is of interest for we have recently reported enlarged α-granules for a patient with MTP associated with a β3 L718del resulting in loss of synchronization between opposing amino acids of the αIIb and β3 cytoplasmic tails and a weakening of the αIIbR995/β3D723 salt bridge (Table SI) [START_REF] Nurden | An intracytoplasmic β3 Leu718 deletion in a patient with a novel platelet phenotype[END_REF]. The presence of enlarged α-granules in platelets of patients from 3 unrelated families with MTP linked to cytoplasmic tail mutations in ITGA2B or ITGB3 in our current study strongly suggests that they have a role in the ultrastructural changes with emphasis on the αIIbR995W mutation. But further studies will be required to define this role and rule out secondary genetic variants in linkage disequilibrium with the primary mutations. Classically, enlarged α-granules are a consistent feature of the Paris-Trousseau syndrome first being seen on stained blood smears and then confirmed by EM [START_REF] Breton-Gorius | A new congenital dysmegakaryopoietic thrombocytopenia (Paris-Trousseau) associated with giant platelet α-granules and chromosome 11 deletion at 11q23[END_REF][START_REF] Favier | Paris-Trousseau syndrome: clinical, hematological, molecular data of ten cases[END_REF]. Paris-Trousseau syndrome results from genetic variants and haplodeficiency of the FLI1 transcription factor [START_REF] Favier | Progress in understanding the diagnosis and molecular genetics of macrothrombocytopenia[END_REF], variants that were absent from our families when studied by targeted exome sequencing. For our previously studied patient with the L718del, immunogold labeling and EM clearly showed the association of P-selectin, αIIbβ3 and fibrinogen with the giant α-granules suggesting a normal initial granule biosynthesis [START_REF] Nurden | An intracytoplasmic β3 Leu718 deletion in a patient with a novel platelet phenotype[END_REF]. Whether the giant granules are formed as part of the secretory pathway, as has been proposed for normal platelets [START_REF] Eckly | Respective contributions of single and compound granule fusion to secretion by activated platelets[END_REF], or perhaps are the consequences of premature apoptosis remain subjects for further study. In this context it would be interesting to know if they are more abundant in aging platelets and at what stage they appear in maturing MK and/or during platelet biogenesis. Preliminary immunofluorescence studies showed what appeared to giant granules in the proplatelets of cultured megakaryocytes from AII.1. It is also important to know if enlarged α-granules have been overlooked in previously published cases of cytoplasmic tail mutations affecting αIIbβ3 or are restricted to certain cases. In parallel, we also keep in mind that FLI-1 with others transcription factors coregulate directly ITGA2B and ITGB3 [START_REF] Tijssen | Genome-wide analysis of simultaneous GATA1/2, RUNX1, FLI1, and SCL binding in megakaryocytes identifies hematopoietic regulators[END_REF].

The genotypes and phenotypes of previously published cases associating cytoplasmic tail mutations of αIIb or β3 and MTP are compared in Table S1. There is much phenotypic variability but as in our cases all give rise to a mild to moderate thrombocytopenia and platelet size variations including giant forms. Inheritance is AD when family studies permit this conclusion, although two reports associate single allele missense mutations of the αIIb cytoplasmic tail with a second and different mutation causing loss of expression of the second allele [START_REF] Peyruchaud | R to Q amino acid substitution in the GFFKR sequence of the cytoplasmic domain of the integrin αIIb subunit in a patient with a Glanzmann's thrombasthenia-like syndrome[END_REF][START_REF] Kashiwagi | Demonstration of novel gain-of-function mutations of αIIbβ3: association with macrothrombocytopenia and Glanzmann thrombasthenia-like phenotype[END_REF]. Such a loss may exaggerate the effect of single allele missense mutations in these cases. In all but one of the published cases bleeding was mild to moderate or was absent and our cases follow this pattern. Platelet aggregation was never totally abrogated but tended to occur more slowly with a reduced final intensity as was largely the case for families A and B in our study. Platelets of family C (β3D723H) retained a good aggregation response, a finding in agreement with the report on the UK family with the same mutation [START_REF] Ghevaert | A nonsynonymous SNP in the ITGB3 gene disrupts the conserved membrane-proximal cytoplasmic salt bridge in the αIIbβ3 integrin and cosegregates dominantly with abnormal proplatelet formation and macrothrombocytopenia[END_REF]. For families A, B and C intermediate levels of αIIbβ3 were shown at the surface, results again consistent with many of the literature reports (Table S1). It is noteworthy that the platelet aggregation response of obligate heterozygotes for classic type I GT is normal [START_REF] George | Glanzmann's thrombasthenia: The spectrum of clinical disease[END_REF] suggesting that despite the influence of the low platelet count, the cytoplasmic domain mutations have a direct effect on the platelet aggregation response. Interestingly, a low platelet surface αIIbβ3 expression was shown to be associated with normal internal pools of αIIbβ3 in patients with αIIbR995Q and αIIbR99W substitutions suggesting defects in integrin recycling [START_REF] Hardisty | A defect of platelet aggregation associated with an abnormal distribution of glycoprotein IIb-IIIa complexes within the platelet: The cause of a lifelong bleeding disorder[END_REF][START_REF] Kashiwagi | Demonstration of novel gain-of-function mutations of αIIbβ3: association with macrothrombocytopenia and Glanzmann thrombasthenia-like phenotype[END_REF]. Our results for family C with intermediate platelet surface levels of αIIbβ3 differed from the results for the UK family where αIIbβ3 expression was normal [START_REF] Ghevaert | A nonsynonymous SNP in the ITGB3 gene disrupts the conserved membrane-proximal cytoplasmic salt bridge in the αIIbβ3 integrin and cosegregates dominantly with abnormal proplatelet formation and macrothrombocytopenia[END_REF]. Expression of the adhesion receptor GPIb was increased on the platelets of our index cases and especially so for the two with the αIIbR995W variant a finding that was previously observed for Japanese cases with the same mutation [START_REF] Kunishima | Heterozygous ITGA2B R995W mutation inducing constitutive activation of the αIIbβ3 receptor affects proplatelet formation and causes congenital macrothrombocytopenia[END_REF].

The reason for this is not known but could reflect altered megakaryopoiesis.

A feature of cytoplasmic domain mutations causal of MTP is that long-range conformational changes extend to the functional domains of the integrin and give what is often termed a partially activated state (6-12) (Table S1). This was indeed shown by Hughes et al [START_REF] Hughes | The conserved membraneproximal region of an integrin cytoplasmic domain specifies ligand-binding affinity[END_REF][START_REF] Hughes | Breaking the integrin hinge. A defined structural constraint regulates integrin signaling[END_REF] who expressed αIIbβ3 in CHO cells after modifying residues of the salt bridge through site-directed mutagenesis. While the changes permit binding of the activation-dependent IgM MoAb PAC-1, only for one report has spontaneous binding of Fg been observed for this class of mutation [START_REF] Kobayashi | Identification of the integrin β3 L718P mutation in a pedigree with autosomal dominant thrombocytopenia with anisocytosis[END_REF]. These results therefore differ from the C560R mutation in the β3 cysteinerich β (A) extracellular domain as reported for a French patient whose platelets circulated with αIIbβ3-bound Fg [START_REF] Ruiz | A point mutation in the cysteine-rich domain of glycoprotein (GP) IIIa results in the expression of a GPIIb-IIIa (alphaIIbbeta3) integrin receptor locked in a high-affinity state and a Glanzmann thrombasthenia-like phenotype[END_REF]. The conformational changes permitting spontaneous PAC-1 binding but only rarely Fg binding remain to be defined although αIIbβ3 clustering remains a potential explanation [START_REF] Jayo | L718P mutation in the membrane-proximal cytoplasmic tail of β3 promotes abnormal αIIbβ3 clustering and lipid microdomain coalesce, and associates with a thrombasthenia-like phenotype[END_REF]. The activation state of αIIbβ3 is also often greater in transfected heterologous cells than for platelets of the patients themselves perhaps due to abnormal recycling and concentration of the mutated integrin in internal pools. Unexpectedly, variable or no PAC-1 binding in our patients was seen after stimulation with TRAP despite these patients showing a residual aggregation response in citrated PRP. This apparent contradiction is possible related to the non-stirred conditions of the in vitro PAC-1 binding experiments. As patients from family C showed a markedly abnormal response to AA, a role for thromboxane A2 generation in αIIbβ3 activation merits investigation.

Previous studies have examined MK maturation in culture but have largely been performed for patients with a 40 amino acid del (p.647-686) in the β3 extracellular β-tail domain causal of MTP [START_REF] Bury | Outside-in signaling generated by a constitutively activated integrin αIIbβ3 impairs proplatelet formation in human megakaryocytes[END_REF][START_REF] Bury | Cytoskeletal perturbation leads to platelet dysfunction and thrombocytopenia in variant forms of Glanzmann thrombasthenia[END_REF][START_REF] Hauschner | Abnormal cytoplasmic extensions associated with active αIIbβ3 are probably the cause for macrothrombocytopenia in Glanzmann thrombasthenia-like syndrome[END_REF]. Among the changes that were noted were (i) fewer proplatelets and (ii) tips of larger size; changes associated with abnormal MK spreading on Fg and a disordered actin distribution and cytoskeletal defects seemingly linked to a sustained "outside-in" signaling induced by the constitutively active αIIbβ3 [START_REF] Bury | Outside-in signaling generated by a constitutively activated integrin αIIbβ3 impairs proplatelet formation in human megakaryocytes[END_REF][START_REF] Bury | Cytoskeletal perturbation leads to platelet dysfunction and thrombocytopenia in variant forms of Glanzmann thrombasthenia[END_REF][START_REF] Hauschner | Abnormal cytoplasmic extensions associated with active αIIbβ3 are probably the cause for macrothrombocytopenia in Glanzmann thrombasthenia-like syndrome[END_REF]. Analysis of megakaryopoiesis for our patients did not reveal a defect in MK maturation or in ploidy but confirmed the above studies and previous studies on MKs from Japanese families with αIIbR995W (9) or the UK family with β3D723H defect in that abnormal proplatelet formation with decreased branching and with bulges of increased size at their tips. This defect was quite similar between our patients even if the size of tips seemed larger for the patients with αIIbR995W. The relationship between defects in αIIbβ3 complexes to changes in α-granule size remains to be determined. VWF-labelled granules with increased size were detected already in proplatelets in AII.1 and interestingly for the three patients studied, the larger the platelet surface area the larger the α-granule diameter suggesting that the defect responsible for increased platelet size contributes also to determination of α-granule size. The cause of the apparent fusion or coalescing of granules as a mechanism of forming the giant granules not only observed for this patient but also for the β3 Leu718 deletion ( 12) merits further study. The fact that mutations modifying the salt bridge between the positively charged αIIbR995 and negatively charged β3D723 is particularly intriguing.

Our results therefore support a generalized hypothesis where mutations within αIIb or β3 cytoplasmic domains somehow lead to a facilitated MK surface interaction with stromal proteins in the marrow medullary compartment that in turn promote cytoskeletal changes that not only lead to altered proplatelet formation and platelet biogenesis but also, at least on occasion, an altered α-granule maturation.

Footnote

* Human Genome Variation Society nomenclature for the αIIb and β3 mature proteins are used in this study.

Acknowledgment: the authors thank N.Saut for her technical expertise for sequencing.

  1 for families A, B and C) and absent from the subjects AIII.1, BI.2, CII.2 A who have a normal platelet count. The structural effect of the mutations was studied using the sculpting function incorporated in the PyMol in silico modeling program (see Methods); the images in Fig. 1 show the transmembrane and cytoplasmic domain segments of αIIb (blue) and β3 (green). The interactions creating the inner membrane association clasp are highlighted for wild type αIIbβ3 in dashed circles with the positive αIIbR995 and negative β3D723 represented as sticks. Both substitutions result in steric interference, especially when β3D723 is replaced by the larger H. The substitutions of αIIbR995 with neutral W or β3D723 with the positive H necessarily weaken or abrogate the salt bridge potentially leading to a separation of the subunit tails. Secondary influences also extend to other membrane proximal amino acids in π interactions shown as sticks and transparent spheres (see Discussion).

Fig. 1 .

 1 Fig. 1. Genetic analysis and structural in silico modeling of the αIIb R995W and the β3

Fig. 3 .

 3 Fig. 3. Transmission electron microscopy of platelets from the index cases of each family

1 (

 1 Fig 4, yellow arrow).

Fig. 4 .

 4 Fig. 4. In vitro derived MK differentiation. MK differentiation was induced from control (Cont1,