Tejas Bodas

D Manjunath

On Load Balancing Equilibria in Multiqueue Systems with Multiclass Traffic

We consider a queueing system with two non identical FCFS servers together serving two classes of customers. All customers have i.i.d service requirements. One of the queues may charge an admission price, say c. Arrivals are randomly routed to one of the servers and the routing probabilities are determined centrally to optimise a global objective, or from a local mechanism minimising a local-class or individualobjective. Our interest is to analyse the use of c to achieve a target distribution of loads among the servers. We first analyse the structure of the optimal allocation and then consider (1) a system with a dispatcher for each class, (2) a non atomic system, and (3) a system where one of the classes has a dispatcher.

I. INTRODUCTION

We consider multiqueue systems, each FCFS queue with its own server, serving a multiclass population. Customers arrive according to a Poisson process with an intrinsic rate per class. The different classes have the same service requirement but different costs per unit waiting time. The servers have different service rates and may also impose an admission price on each customer joining the queue. Arrivals are randomly routed to a queue and the routing probabilities either minimize a global or a local objective.

Centralised, non competitive, probabilistic allocation of multiclass traffic to multiple M/G/1 servers to minimize the mean waiting time per customer is considered in [START_REF] Borst | Optimal probabilistic allocation of customer types to servers[END_REF], [START_REF] Sethuraman | Optimal stochastic scheduling in multiclass parallel queues[END_REF]. FCFS scheduling is analysed in [START_REF] Borst | Optimal probabilistic allocation of customer types to servers[END_REF] and optimal scheduling is analysed in [START_REF] Sethuraman | Optimal stochastic scheduling in multiclass parallel queues[END_REF]. In a more distributed system, each class has a dispatcher and Dispatcher i randomly allocates Class i customers to the servers to minimise the waiting cost per Class i customer. This leads to a competition between the dispatchers and is analysed in [START_REF] Ayesta | Price of anarchy in noncooperative load balancing[END_REF]. Also, [START_REF] Bell | Individual versus social optimization in the allocation of customers to alternative servers[END_REF] considered a single customer class and compared a centralised allocation with individually optimal schemes. Note that none of these systems use an admission price for the queues.

A decentralised system in which each customer randomly chooses a queue to individually optimise its cost is considered in [START_REF] Bodas | Load balancing and routing games with admission price[END_REF]. Here one of the queues charges an admission price to every arriving customer while the second queue does not. For this non atomic system, the equilibrium, as a function of the admission price was characterised in [START_REF] Bodas | Load balancing and routing games with admission price[END_REF]. A system in which customers of one class were routed by a dispatcher while those of a second class made individually optimal decisions was also considered in [START_REF] Bodas | Load balancing and routing games with admission price[END_REF]. Once again, the equilibrium load was analysed. The objective there was to analyse the effect Tejas Bodas is funded by the Bharti Centre for Communication at IIT Bombay of pricing as a control to achieve a specific load distribution. Some earlier literature on admission price mechanisms include the Paris Metro pricing scheme of [START_REF] Odlyzko | Paris Metro pricing for the internet[END_REF] and the Tirupati pricing schemes of [START_REF] Dube | Differential join prices for parallel queues: social optimality, dynamic pricing algorithms and application to internet pricing[END_REF], [START_REF] Borkar | Charge-based control of Diffserv-like queues[END_REF].

In this paper, we continue with the two-queue, two-class system of [START_REF] Bodas | Load balancing and routing games with admission price[END_REF]. One of the queues can charge an admission price. We first consider a centralised system in which the routing probabilities are chosen to optimise a global objective. We characterize this optimal load distribution in Section II. In Section III we analyse the equilibrium loads in a system where the allocation for each class is by a dispatcher. Here one of the queues has an admission price and we obtain the relation between the system parameters for different equilibria. In Section IV we describe a non atomic system where each customer makes an individually optimal decision and characterise the c that will make the equilibrium distribution to be the optimal distribution derived in Section II. Finally, in Section V we analyse the equilibrium structure when one of the classes has a dispatcher while the other class of customers are non atomic.

Before proceeding, we formally describe the model and the notation. Class i customers arrive according to a stationary Poisson process of rates λ i . Each customer requires a service time that is exponentially distributed with unit mean. A Class i customer has a cost of β i per unit waiting time; we assume β 1 > β 2 . Server of Queue j has a service rate of µ j ; we assume µ 1 > µ 2 . Server of Queue j also charges a per customer admission price c j . Without loss of generality, c 1 = c > 0 and c 2 = 0. p i and q i denote the fraction of Class i traffic allocated to Server 1 and Server 2 respectively; of course p i = 1q i . γ j denotes the arrival rate of the total traffic at Queue j; clearly

γ 1 = p 1 λ 1 +p 2 λ 2 , γ 2 = q 1 λ 1 +q 2 λ 2 and γ 1 + γ 2 = λ 1 + λ 2 .
For stability of the queuing system we assume that

µ 1 + µ 2 > λ 1 + λ 2 .
Let the expected waiting time in the Queue j be D j (γ j ) = 1 µj -γj for j = 1, 2. The expected total cost (sum of the expected waiting cost and the admission price) per Class i customer, denoted by ∆ i , is

∆ i = (p i (c + β i D 1 (γ 1 )) + q i β i D 2 (γ 2 )) ,
while the overall social cost when c = 0 is defined as Also define Θ = {λ 1 , λ 2 , µ 1 , µ 2 , β 1 , β 2 } to denote the set of system parameters.

∆ s = = λ 1 ∆ 1 + λ 2 ∆ 2 λ 1 + λ 2 = (p 1 λ 1 β 1 + p 2 λ 2 β 2 ) (D 1 (γ 1 ) -D 2 (γ 2 )) λ 1 + λ 2 + (λ 1 β 1 + λ 2 β 2 ) D 2 (γ 2 ) λ 1 + λ 2 (1)

II. OPTIMAL ALLOCATION IN A CENTRALISED SCHEME

We consider the centralised routing of customers to the servers to minimize the social cost ∆ s . For this centralised scheme we consider the admission price c = 0. Let (p * 1 , p * 2 ) be the routing probabilities corresponding to the optimal allocation. For the stability of the queues, we require that µ 1 > γ 1 and µ 2 > γ 2 (or equivalently,

γ 1 > λ 1 + λ 2 -µ 2 ). Thus the feasible allocation space for (p 1 , p 2 ) is obtained as S = {(p 1 , p 2 ) : γ 1 < µ 1 , γ 1 > (λ 1 + λ 2 -µ 2 )} ; of course, 0 ≤ γ 1 ≤ λ 1 +λ 2 . We also define γ + = {γ 1 : D 1 (γ 1 ) = D 2 (γ 2 )} .
It can be shown that

γ + = µ 1 -µ 2 + λ 1 + λ 2 2 (2)
While for all feasible γ 1 < γ + we have

D 1 (γ 1 ) < D 2 (γ 2 ), for γ 1 > γ + we have D 1 (γ 1 ) > D 2 (γ 2 )
. We now have the following properties. Property 1: For a fixed γ 1 = γ + , there exists a unique (p 1 , p 2 ) pair that minimizes ∆ s at the particular γ 1 .

Proof: Consider the case where 0 ≤ γ + ≤ λ 1 + λ 2 ; the arguments are similar when this is not the case. For a fixed γ 1 < γ + , D 1 (γ 1 ) and D 2 (γ 2 ) are constant and since [START_REF] Borst | Optimal probabilistic allocation of customer types to servers[END_REF] we see that ∆ s is minimized by the following linear program.

D 1 (γ 1 ) < D 2 (γ 2 ), from
arg max p1,p2 p 1 λ 1 β 1 + p 2 λ 2 β 2 subject to p 1 λ 1 + p 2 λ 2 = γ 1 (3)
In this linear program, if β 1 = β 2 , then the slope of the objective function and that of the constraint are different. Thus there exists a unique (p 1 , p 2 ) at the boundary of the feasible space which is a solution to (3). For γ 1 > γ + , the arg max in (3) is replaced by arg min and the same argument applies.

The above is illustrated in Fig. 1 where the dark dotted lines represent the solution to 3 for different values of the equality constraint γ 1 .

We can now analyses the structure of (p * 1 , p * 2 ). Denote the

γ 1 corresponding to (p * 1 , p * 2 ) as γ * 1 . If Θ is such that γ * 1 = γ + , then the optimal allocation (p * 1 , p * 2 )
is at the boundary of the feasible space. This follows from Property 1 and is illustrated in Fig. 1

. If Θ is such that γ * 1 = γ + , then D 1 (γ 1 ) = D 2 (γ 2 ) and any (p 1 , p 2 ) satisfying p 1 λ 1 + p 2 λ 2 = γ + is a valid(p * 1 , p * 2 ). We also remark that if γ * 1 = γ + , then choosing a (p * 1 , p * 2 )
that is not at the boundary would result in a higher variance of the waiting cost than that from the boundary. This is because in the former case, the traffic mix at each of the server will be more heterogeneous than that at the boundary. Thus even when γ

* 1 = γ + , a (p * 1 , p * 2 )
= {(p 1 , p 2 ) : p 1 = 1, p 2 ∈ [0, 1]}, B 2 = {(p 1 , p 2 ) : p 1 ∈ [0, 1], p 2 = 0}, B 3 = {(p 1 , p 2 ) : p 1 = 0, p 2 ∈ [0, 1]}, and B 4 = {(p 1 , p 2 ) : p 1 ∈ [0, 1], p 2 = 1}. First consider B 1 . Since p 1 = 1, ∆ 1 = β1 µ1-λ1-p2λ2 . It can be shown that ∆ 1 is convex in p 2 . Similarly ∆ 2 = β2p2 µ1-λ1-p2λ2 + β2q2
µ2-q2λ2 is a sum of two convex functions and hence is convex in p 2 . Since ∆ s is a weighted sum of two convex functions in p 2 , ∆ s is also convex in p 2 with a unique minimizer. Arguing along similar lines, ∆ s is convex in p 2 in set B 3 , while it is convex in p 1 in sets B 2 and B 4 . This completes the proof.

We can make the following remarks from the above properties.

• Extensive numerical investigation indicates that γ * < γ + implying D 1 (γ 1 ) < D 2 (γ 2 ). Further, we also see that (p * 1 , p * 2 ) is unique. However we do not have a formal proof and we leave that as an open problem for future work.

• At the optimal allocation, if ) Class 1 traffic utilises the server with a lower expected cost while the Class 2 traffic is forced to use a server with a higher expected waiting time.

γ * 1 < γ + then D 1 (γ 1 ) < D 2 (γ 2 ) and either p * 1 = 1 and p * 2 ∈ [0, 1] or p * 1 ∈ (0, 1] and p * 2 = 0. • If γ * 1 > γ + then D 1 (γ 1 ) > D 2 (γ 2 )

III. EQUILIBRIUM LOADS WITH TWO DISPATCHERS

We now consider a decentralised system in which each traffic class has a dispatcher. Queue 1 charges an admission price c > 0 per customer and Queue 2 does not. Dispatcher i chooses a strategy p i to minimize ∆ i , the expected total cost of a Class i customer. The optimal strategy for each dispatcher depends on the strategy of the other dispatcher. A Nash equilibrium strategy pair for the dispatchers is then defined as the allocation (p 1 , p2 ) from which there is no incentive for either dispatcher to change its strategy. From Theorem 2.1 in [START_REF] Orda | Competitive routing in multi-user communication networks[END_REF], we see that ∆ 1 and ∆ 2 satisfy the conditions for the existence of a Nash equilibrium. The equilibrium allocation, (p 1 , p2 ), can be obtained as the solution to the following set of equations derived from the first order condition on ∆ 1 and

γ 1 = 0 γ 1 = λ 1 β 2 c 2 γ 1 = λ 1 + λ 2 β 1 c 2 price c c 3 c 1
∆ 2 . c + µ 1 -p 2 λ 2 (µ 1 -γ 1 ) 2 - µ 2 -q 2 λ 2 (µ 2 -γ 2 ) 2 = 0. c + µ 1 -p 1 λ 1 (µ 1 -γ 1 ) 2 - µ 2 -q 1 λ 1 (µ 2 -γ 2 ) 2 = 0
An explicit solution to the above set of equations is messy.

Rather than elaborate on that, we now find conditions on the parameter set Θ for specific types of equilibria. First consider the equilibrium p1 = p2 = 1. For this we require that at

p 1 = p 2 = 1, ∂∆1 ∂p1 ≤ 0 and ∂∆2 ∂p2 ≤ 0. Thus c + µ 1 -λ 2 (µ 1 -λ 1 -λ 2 ) 2 - 1 µ 2 ≤ 0. c + µ 1 -λ 1 (µ 1 -λ 1 -λ 2 ) 2 - 1 µ 2 ≤ 0.
This implies that if the queue stability condition

µ 1 > (λ 1 + λ 2 ) is satisfied, (p 1 = 1, p2 = 1) if µ 2 ≤ (µ 1 -λ 1 -λ 2 ) 2 c(µ 1 -λ 1 -λ 2 ) 2 + (µ 1 -max(λ 1 , λ 2 )) (4)
Similarly for p1 = p2 = 0 we require

µ 1 ≤ (µ 2 -λ 1 -λ 2 ) 2 c(µ 1 -λ 1 -λ 2 ) 2 + (µ 2 -max(λ 1 , λ 2 ))
which implies µ 1 < µ 2 . Recall that we have assumed that µ 1 > µ 2 ; hence this equilibrium is not possible.

For p1 = 1, 0 < p2 < 1 to be a valid equilibrium the necessary condition on the parameters are obtained from the conditions ∂∆1 ∂p1 ≤ 0 and ∂∆2 ∂p2 = 0 at (p 1 = 1, 0 < p 2 < 1). Thus we need p 2 that solves

µ 1 -λ 1 (µ 1 -λ 1 -p 2 λ 2 ) 2 = µ 2 -q 1 λ 1 (µ 2 -q 2 λ 2 ) 2 and satisfies c + µ 1 -p 2 λ 2 (µ 1 -λ 1 -p 2 λ 2 ) 2 - µ 2 -q 2 λ 2 (µ 2 -q 2 λ 2 ) 2 ≤ 0.
The necessary conditions on system parameters for other equilibrium can be obtained similarly. Numerical results to illustrate the equilibria will be provided in the final version.

IV. EQUILIBRIUM LOADS IN A NON ATOMIC SYSTEM

The equilibrium load distribution of the non atomic system with two classes was studied in [START_REF] Bodas | Load balancing and routing games with admission price[END_REF]. Recall that in such a system, each customer makes an individually optimal queuejoin decision by joining a queue that minimizes its expected total cost. Of course the total cost is the sum of the admission price and the waiting cost. In [START_REF] Bodas | Load balancing and routing games with admission price[END_REF], for a fixed Θ, the equilibrium traffic distribution, (p 1 , p2 ), as a function of c was characterized. It was shown that the traffic at equilibrium is in one of the following five regimes. (1) Regime 1 for which p1 = p2 = 1, (2) Regime 2 for which p1 = 1 and 0 < p2 < 1, (3) Regime 3 for which p1 = 1 and p2 = 0, (4) Regime 4 for which 0 < p1 < 1 and p2 = 0, and (5) Regime 5 for which p1 = 0 and p2 = 0. This is illustrated in Fig. 2. Note that in Fig. 2, c 1 , c 2 and c 3 are a function of Θ, the system parameters. Since we assume that c > 0, the different regimes are feasible if 0 < c 1 < c 2 < c 3 . For example, it is shown in [START_REF] Bodas | Load balancing and routing games with admission price[END_REF] that c 1 > 0 requires µ 1 > µ 2 + λ 1 + λ 2 . Also, note that the five regimes of the non atomic model correspond to the boundary sets B 1 and B 2 defined in Section II. We thus have the following.

Lemma

1: If µ 1 > µ 2 + λ 1 + λ 2 ,
there exists an admission price c a corresponding non atomic traffic equilibrium (p 1 , p2 ) which is also the minimizer of social cost

∆ s i.e. (p 1 , p2 ) = (p * 1 , p * 2 ). Proof: Now as µ 1 > µ 2 + λ 1 + λ 2 , from Eq. 2 it is clear that γ + > λ 1 + λ 2 . Refer Figure 1. As γ * 1 < γ + , (p * 1 , p * 2 ) lies in either B 1 or B 2 .
As Regimes 1 through 5 correspond to regions on the boundaries B 1 and B 2 , we can charge an admission price c to customers joining Server 1 such that the non atomic equilibrium distribution will be (p * 1 , p * 2 ) with the corresponding

γ 1 = γ * 1 . Lemma 1 says that if Θ is such that γ * 1 < γ + , then a (p * 1 , p * 2 )
minimizing the social cost can be achieved in the non atomic system by charging an appropriate admission price c as suggested by Fig. 2. Our numerical results suggest that if µ 1 > µ 2 and β 1 > β 2 , then there exists a unique γ * 1 ≤ γ + and hence in the non atomic system, the social cost ∆ s can be minimized by charging an appropriate admission price to achieve γ * 1 . At this time, the proof is an open problem. Note that if c = 0, then the equilibrium (p 1 , p2 ) is not unique. In fact any (p 1 , p 2 ) that satisfies p 1 λ 1 + p 2 λ 2 = γ + is a valid non atomic equilibrium.

V. A DISPATCHER FOR ONLY ONE OF THE CLASSES

We now consider a model where Class 1 traffic has an associated dispatcher while Class 2 traffic is non atomic. First consider the case of c = 0. A Class 2 arrival makes an individually optimal join decision, but the dispatcher chooses a strategy p 1 that minimizes the cost ∆ 1 of the Class 1 customer. Clearly, ∆ 1 is a function of the traffic distribution of Class 2 traffic; hence for any p 1 chosen by the dispatcher, there is a corresponding equilibrium p2 for Class 2 traffic. Once again we have a competitive situation and we are interested in the equilibrium pair (p 1 , p2 ).

p 1 1 1 0 0 p 2 γ 1 > γ + γ 1 < γ + p 1a p 1b
Line satisfying γ 1 = p 1 λ 1 + p 2 λ 2 = γ + Fig. 3. Feasible (p 1 , p2 ) for a particular parameter set Θ. For certain Θ it may be true that p 1a / ∈ [0, 1], p 1b and / ∈ [0, 1]. In that case γ1 = γ + is the only feasible region.

Let γ 1 corresponding to (p 1 , p2 ) be denoted by γ1 = p1 λ 1 + p2 λ 2 . For all γ1 < γ + , we see that D 1 (γ 1 ) < D 2 (γ 2 ). This requires that p2 = 1. Similarly for γ1 > γ + , as D 1 (γ 1 ) > D 2 (γ 2 ) we have p2 = 0. For γ1 = γ + we have p2 ∈ [0, 1] and p1 λ 1 + p2 λ 2 = γ + .

Note that the existence of p 1a ∈ [0, 1] and p 1b ∈ [0, 1] as shown in Fig. 3 may not hold for some Θ. For p1 < p 1a , D 1 (γ 1 ) < D 2 (γ 2 ) and hence p2 = 1. Similarly For p1 > p 1b , D 1 (γ 1 ) > D 2 (γ 2 ) and hence p2 = 0. The dark line of Figure 3 is the feasible region for (p 1 , p2 ). In [START_REF] Bodas | Load balancing and routing games with admission price[END_REF], we have considered the 1-Dispatcher model with Server 1 charging an admission price c per customer. We characterize the feasible (p 1 , p2 ) for different values of c charged by Server 1. For a particular Θ and c, the method to obtain the equilibrium is given in [START_REF] Bodas | Load balancing and routing games with admission price[END_REF]. We now provide a numerical example comparing the non atomic model with this 1-Dispatcher model.

Example: Let λ 1 = λ 2 = 2, β 1 = 2, β 2 = 1, µ 1 = 12, and µ 2 = 5. For this Θ, we have c 1 = 0.0750, c 2 = 0.233 and c 3 = 1.833. Fig. 4 shows ∆ s , the the system cost, for the two models for different values of the admission price c. Observe that for lower values of c the system cost in two models is the same but as the admission price c is increased, the one dispatcher system has a lower per customer cost as compared to the non atomic system. Fig. 5 shows the equilibrium p1 in the two models. Clearly as c is increased, the expected cost per Class 1 customer is smaller with the dispatcher than in the non atomic model.

Finally, in Fig. 6 we see that although the system equilibrium p2 in both the models is the same, for higher admission price the per Class 2 customer expected cost also decreases due to the dispatcher for Class 1.

VI. SUMMARY

We began by analysing a centralised allocation of multiclass traffic to heterogeneous servers to minimise the system cost. We analysed the structure of optimal routing probabilities (p * 1 , p * 2 ). Decentralisation by having dispatchers for each class yields a competitive system for which we have analysed the equilibrium allocations. A further decentralisation gives us the non atomic system in which each arrival makes its own routing decision. This system was analysed in some detail in [START_REF] Bodas | Load balancing and routing games with admission price[END_REF] and here we described how price could be used to make the equilibrium allocation to be socially optimal. We briefly considered a system where only one of class has a dispatcher. Thus, we have investigated, in some detail, the use of differential admission pricing as a means of decentralised control of load distribution.

2 Fig. 1 .

 21 Fig. 1. Structure of (p * 1 , p * 2 ) minimizing ∆s for arbitrary γ +

Fig. 2 .

 2 Fig. 2. The operating regimes as c is increased from 0.

Fig. 4 .

 4 Fig. 4. Comparison of the per customer expected total cost in the two models.

1 in

 1 Non-atomic model equilibrium p 1 in Dispatcher -Non atomic model δ Non-atomic model δ 1 in Dispatcher -non atomic model

Fig. 5 .

 5 Fig. 5. Comparison of p 1 at system equilibrium for the two models

Fig. 6 .

 6 Fig.6. Comparison of p 2 at system equilibrium in the two models.

  at the boundary of the feasible space may be preferred. Property 2: There are at most four minimizers (p * 1 , p * 2 ). Proof: The boundary of the set of feasible (p 1 , p 2 ) consists of four parts-B 1

  From the preceding two remarks, we see that Class 1 traffic 'has priority over Class 2' in the choice of server.

	At (p * 1 , p * 2

and hence p * 2 = 1 and p * 1 ∈ [0, 1] or p * 2 ∈ (0, 1] and p * 1 = 0. •


Nathalie Sans
email: nathalie.sans@inserm.fr

Christelle M Durand

Julie Perroy

Francois Loll

David Perrais

Laurent Fagni

Thomas Bourgeron

Mireille Montcouquiol

François Loll

SHANK3 mutations identified in autism

Keywords: Shank3, autism, hippocampus, spine, actin, axonal outgrowth

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

Autism spectrum disorders (ASD) are heterogeneous neurodevelopmental disorders diagnosed on the basis of three behaviorally altered domains: social deficits, impaired language and communication, and stereotyped behaviors. ASD affect 1% of the population, with males being affected four times more often than females [START_REF] Chakrabarti | Pervasive developmental disorders in preschool children: confirmation of high prevalence[END_REF][START_REF] Newschaffer | The epidemiology of autism spectrum disorders[END_REF][START_REF] Baird | Prevalence of disorders of the autism spectrum in a population cohort of children in South Thames: the Special Needs and Autism Project (SNAP)[END_REF] . Family and twin studies indicate a strong genetic basis for ASD, but the mechanisms of transmission remain largely unknown. Recently, several synaptic genes have been associated with ASD, providing a better idea of the pathways involved [START_REF] Toro | Key role for gene dosage and synaptic homeostasis in autism spectrum disorders[END_REF] . The 22q13 deletion syndrome is characterized by developmental delay, moderate to profound language delay, delay/absence of expressive speech, hypotonia and autistic features that are caused by the loss of the SHANK3 gene located at the tip of this chromosome [START_REF] Bonaglia | Disruption of the ProSAP2 gene in a t(12;22)(q24.1;q13.3) is associated with the 22q13.3 deletion syndrome[END_REF] . The SHANK3 gene can also be altered in patients with ASD, and recently, we have shown that mutations, or the loss of one copy of SHANK3, are associated with autism, whereas the presence of an extra copy of SHANK3 might be associated with Asperger syndrome [START_REF] Durand | Mutations in the gene encoding the synaptic scaffolding protein SHANK3 are associated with autism spectrum disorders[END_REF] . The variations identified included de novo deleterious mutations and inherited nonsynonymous variations affecting highly conserved amino acids in the ankyrin domain [START_REF] Durand | Mutations in the gene encoding the synaptic scaffolding protein SHANK3 are associated with autism spectrum disorders[END_REF][START_REF] Moessner | Contribution of SHANK3 mutations to autism spectrum disorder[END_REF] . Although these mutations were inherited from healthy parents, they could contribute to the disorder in combination with other unidentified SHANK3 mutations or in combination with other mutated genes.

The three members of the Shank family (Shank1, 2 and 3) are core components of the postsynaptic density (PSD), a highly organized cytoskeletal structure found adjacent to the postsynaptic membrane of excitatory synapses. Shank proteins have ankyrin repeats at their N-terminus, followed by a SH3 domain, a PDZ domain, a proline-rich region and a SAM domain at their C-terminal region 8- 10 . All of these domains are involved in protein-protein interactions, linking different glutamate receptors, scaffolding proteins and intracellular effectors to the actin cytoskeleton. Indeed, Shank proteins are associated with NMDA receptors via the GKAP/PSD-95 complex and with type I metabotropic glutamate receptors via Homer in the proline-rich domain [START_REF] Sheng | The Shank family of scaffold proteins[END_REF][START_REF] Boeckers | Prolinerich synapse-associated proteins ProSAP1 and ProSAP2 interact with synaptic proteins of the SAPAP/GKAP family[END_REF][START_REF] Naisbitt | a novel family of postsynaptic density proteins that binds to the NMDA receptor/PSD-95/GKAP complex and cortactin[END_REF] . In addition, Shank proteins can bind to several actin-regulatory molecules, such as cortactin [START_REF] Hering | Activity-dependent redistribution and essential role of cortactin in dendritic spine morphogenesis[END_REF][START_REF] Uruno | Activation of Arp2/3 complex-mediated actin polymerization by cortactin[END_REF] , Sharpin [START_REF] Lim | a novel postsynaptic density protein that directly interacts with the shank family of proteins[END_REF] , α-fodrin [START_REF] Bockers | Synaptic scaffolding proteins in rat brain. Ankyrin repeats of the multidomain Shank protein family interact with the cytoskeletal protein alpha-fodrin[END_REF] , Abp1 [START_REF] Qualmann | Linkage of the actin cytoskeleton to the postsynaptic density via direct interactions of Abp1 with the ProSAP/Shank family[END_REF] , β-Pix [START_REF] Park | The Shank family of postsynaptic density proteins interacts with and promotes synaptic accumulation of the beta PIX guanine nucleotide exchange factor for Rac1 and Cdc42[END_REF] and SPAR [START_REF] Wendholt | ProSAPinteracting protein 1 (ProSAPiP1), a novel protein of the postsynaptic density that links the spine-associated Rap-Gap (SPAR) to the scaffolding protein ProSAP2/Shank3[END_REF] . The actin cytoskeleton of the dendritic spine is particularly dynamic and plays a pivotal role in the formation and the maturation of spines and is also involved in the morphological changes that occur in spines during synaptic plasticity [START_REF] Sekino | Role of actin cytoskeleton in dendritic spine morphogenesis[END_REF] . Thus, Shank proteins and their binding partners are involved in vitro in the regulation of the size and the shape of dendritic spine [START_REF] Sala | Regulation of dendritic spine morphology and synaptic function by Shank and Homer[END_REF][START_REF] Roussignol | Shank expression is sufficient to induce functional dendritic spine synapses in aspiny neurons[END_REF][START_REF] Grabrucker | Concerted action of zinc and ProSAP/Shank in synaptogenesis and synapse maturation[END_REF] .

To further assess the functional consequences of SHANK3 mutations, we used an overexpression approach in cultured neurons to investigate the molecular mechanisms modulated by Shank3 in synapse formation and axonal outgrowth. We first examined the subcellular localization of the wild-type and mutated Shank3 proteins in fibroblasts and embryonic primary neuronal cultures.

Our results show that the truncating mutation strongly affected spine development and morphology, as well as growth cone motility, whereas the de novo mutation Q321R preferentially had an effect in early stages of development. Finally, inherited mutations (R12C and R300C) displayed an intermediate phenotype, with lowered spine induction and maturation and growth cone motility compared with wild-type Shank3.

Materials and methods

DNA constructs

Full-length rat Shank3 cDNA has been used previously [START_REF] Durand | Mutations in the gene encoding the synaptic scaffolding protein SHANK3 are associated with autism spectrum disorders[END_REF] . Mutated forms (GFP-Shank3 R12C , GFP-Shank3 R300C and GFP-Shank3 Q321R ) of GFP-Shank3 were generated using the QuikChange Site-Directed Mutagenesis Kit (Stratagene). Deletion variants of GFP-Shank3 STOP were made by

Antibodies

Primary antibodies were anti-actin (Cytoskeleton), β-tubulin (Tuj-1, Covance), MAP2 (Chemicon), anti-DsRED (Clontech), anti-GFP (Invitrogen), anti-phospho-paxillin (Cell Signaling), anti-Shank3 (NeuroMab Facility), anti-cortactin and anti-vinculin (Sigma-Aldrich). Alexa Fluor 546 and 647 phalloidin were used to stain F-actin (Invitrogen). Secondary antibodies were conjugated to Alexa Fluor 488, 546 and 647 (Invitrogen).

Cell culture and transfection

HEK293T and COS-7 cells used for transient transfection were maintained in DMEM supplemented with fetal calf serum and antibiotics. Cells were transfected using calcium phosphate as previously described [START_REF] Sans | mPins modulates PSD-95 and SAP102 trafficking and influences NMDA receptor surface expression[END_REF] . Hippocampal neuronal cultures were prepared and transfected with the different constructs using the calcium phosphate method [START_REF] Sans | mPins modulates PSD-95 and SAP102 trafficking and influences NMDA receptor surface expression[END_REF][START_REF] Sans | NMDA receptor trafficking through an interaction between PDZ proteins and the exocyst complex[END_REF] .

Immunocytochemistry

After four days, cells were fixed with 4% PFA, washed, permeabilized with 0.1% Triton/PBS or 0.05% Saponin/PBS, blocked for 1 hr with 10% NGS or 3% BSA before incubating for 1.5 hr with primary antibodies at room temperature in 3% NGS/PBS or in 0.2% BSA/PBS, then washed in PBS, and incubated with secondary antibodies in 3% NGS/PBS or in 0.2% BSA/PBS. Then, phalloidin was incubated 1 hr in 0.2% BSA/PBS.

Protein, western blots, immunoprecipitation

Protein preparation, western blots and immunoprecipitation were performed as described [START_REF] Sans | A developmental change in NMDA receptor-associated proteins at hippocampal synapses[END_REF] and detailed in the supplementary note.

Ratio of F-to G-actin

The amount of F-actin and G-actin was measured with an Actin Polymerization Assay kit (BK037-Cytoskeleton). HEK293T cells were co-transfected with RFP-actin and GFP-C3 (control) or GFP-Shank3 WT or GFP-Shank3 STOP . Two days after transfection, the cells were resuspended in F-actin stabilization buffer with ATP (1 mM) and protease inhibitors (10X). To preserve F-actin, experiments were performed at 37°C. F-actin was pelleted by a 1-hr centrifugation at 100,000 x g, and the supernatant (G-actin) was separated. The pellet was resuspended in the same volume as the supernatant using ice-cold Milli-Q water and 10 μM cytochalasin D for 2 hr at 4°C. Actin was quantified by western blot using an anti-actin antibody (Cytoskeleton). The ratio of F-actin to G-actin was determined using an Odyssey infrared imaging system (LI-COR Biosciences).

Microscopy and quantitative analyses of spine morphology

Confocal images were acquired with a Leica DMR TCS SP2 AOBS confocal microscope, using the 60x objective. Each image was a z-series projection taken at 0.02-μm-deep intervals. The morphometric measurements (width and length of spine) were made using Volocity analysis software.

Each experiment was performed on five to seven independent neuronal preparations. Cells were cotransfected with mRFP to visualize detailed morphology and to outline the spines. To determine spine size, 1500-1800 spines (from 16-28 neurons) were measured for each condition. For spine length, the distance from the base of the neck to the spine head was measured, and for spine width, the maximal width of the spine head perpendicular to the length was measured. To determine spine density, 10 regions of 20 μm were averaged per neuron.

To quantify F-actin and cortactin in spines, immunofluorescence images were acquired on a DM6000 Leica microscope equipped with MetaMorph software (Universal Imaging). The fluorescence intensity of F-actin or cortactin was measured as the average intensity in the manually area-represented head of the spine. Each experiment was performed on four to six independent neuronal preparations, and 20-38 neurons were measured for each condition.

Electrophysiology

Neurons (DIV17) co-transfected with GFP and the plasmids of interest were selected based on their fluorescence, and miniature EPSCs (mEPSCs) were recorded as previously published [START_REF] Roussignol | Shank expression is sufficient to induce functional dendritic spine synapses in aspiny neurons[END_REF] . Briefly, the recording pipettes had a resistance of 5-10 MW when filled with the following medium (in mM): 140 CsCl, 0.5 CaCl 2 , 20 EGTA, 10 HEPES, 10 D-glucose, pH 7.2 and osmolarity of 300 mOsm. The high concentration of EGTA avoided slow Ca 2+ -dependent desensitization of NMDARs [START_REF] Medina | Calcium-induced inactivation of NMDA receptor-channels evolves independently of run-down in cultured rat brain neurones[END_REF] . Neurons were perfused continuously with the following external medium (in mM): 140 NaCl, 2 CaCl2, 3 KCl, 10 HEPES, 10 D-glucose, 0.01 glycine, 0.01 bicuculline, 0.0003 tetrodotoxin, pH 7.4 and osmolarity of 330 mOsm.

Live Imaging

Four days after transfection, approximately 10 growth cones of neurons transfected at DIV2 were analyzed by imaging every 2 seconds for 10 minutes with an Olympus IX71 inverted microscope (Olympus) using a 60x oil immersion objective (n = 35-67 growth cones from five independent experiments). The growth cone motility index was calculated as previously described [START_REF] Ibarretxe | Fast regulation of axonal growth cone motility by electrical activity[END_REF] .

Statistical significance

Data are shown as the mean ± SEM. The statistical significance of the results was analyzed using oneway ANOVA in all experiments. For electrophysiological studies, we used the non-parametric Kruskal & Wallis test for more than two independent small samples.

Results

One of the aims of this study was to evaluate the contribution of Shank3 gene mutations to the actin dynamics-dependent formation of dendritic spines. For this purpose, we studied SHANK3 mutations identified in patients with ASD [START_REF] Durand | Mutations in the gene encoding the synaptic scaffolding protein SHANK3 are associated with autism spectrum disorders[END_REF][START_REF] Moessner | Contribution of SHANK3 mutations to autism spectrum disorder[END_REF] . Among these mutations (Figure 1a), one is a frameshift mutation at position 3680 in human SHANK3 cDNA (3915 in rat), which introduces a premature STOP codon at position 1227 in humans (1304 in rat) leading to a truncated form of Shank3 (Shank3 STOP ). The remaining three mutations are non-synonymous mutations located in or near the highly conserved ankyrin domain (R12C, R300C and Q321R, referred to as Shank3 R12C , Shank3 R300C and Shank3 Q321R , respectively) [START_REF] Durand | Mutations in the gene encoding the synaptic scaffolding protein SHANK3 are associated with autism spectrum disorders[END_REF][START_REF] Moessner | Contribution of SHANK3 mutations to autism spectrum disorder[END_REF] . An immunoblot analysis of the different constructs revealed that the fusion proteins carrying the point mutations Shank3 R12C , Shank3 R300C and Shank3 Q321R had similar levels of expression and size as the GFP-Shank3 WT , whereas the frameshift mutation (Shank3 STOP ) led to a shorter protein (Figure 1b).

Decreased clustering of mutated Shank3 at the tips of actin filaments

We first analyzed the intracellular localization of Shank3 WT in COS-7 cells (Figure 1c). Shank3 WT was found at the plasma membrane and in large intracellular clusters, as previously reported in HEK cells [START_REF] Quitsch | Postsynaptic shank antagonizes dendrite branching induced by the leucine-rich repeat protein Densin-180[END_REF] . Interestingly, we noticed that Shank3 clustered at the tips of actin filaments and co-localized with vinculin or phospho-paxillin, two focal adhesion proteins (Figure 1c). Overexpression of Shank3 R12C , Shank3 R300C and Shank3 Q321R in COS-7 showed that mutations in the ankyrin domain did not affect the localization of Shank3 at the membrane or in intracellular clusters (Figure 1d). However, the distribution of Shank3 STOP was restricted to small aggregates around the nucleus, suggesting that the lack of the C-terminus in Shank3 affected its targeting and its interaction with the actin cytoskeleton in COS-7 cells. The analysis of the co-localization of Shank3 with actin stress fibers showed that 60±2% of actin filaments per cell had clusters of Shank3 WT . Mutations in the ankyrin domain (R12C, R300C, and Q321R) significantly reduced this localization of Shank3 (43±3% for Shank3 R12C , 42±3% for Shank3 R300C , ***p<0.001, 48±3% for Shank3 Q321R , *p<0.05; Figure 1e).

Shank3 STOP was almost completely absent at the tips of actin filaments (10±7% for Shank3 STOP , ***p<0.001).

Shank3 binds actin and positively affects actin polymerization

Because Shank proteins and their partners are involved in actin regulation [START_REF] Hering | Activity-dependent redistribution and essential role of cortactin in dendritic spine morphogenesis[END_REF][START_REF] Park | The Shank family of postsynaptic density proteins interacts with and promotes synaptic accumulation of the beta PIX guanine nucleotide exchange factor for Rac1 and Cdc42[END_REF][START_REF] Haeckel | The actin-binding protein Abp1 controls dendritic spine morphology and is important for spine head and synapse formation[END_REF] and Shank3 colocalizes with actin fibers, we performed a co-immunoprecipitation using hippocampal tissue to demonstrate the existence of Shank3-actin complexes in vivo (Figure 1f). In addition, we used an actin polymerization assay to measure and compare the F-to G-actin ratio in GFP-Shank3-or GFP-C3transfected cells (Figure 1g-h). In this assay, RFP-actin was co-transfected with each construct to stimulate actin polymerization. We analyzed the Shank3 STOP construct because this mutated protein does not co-localize with actin filaments. Overexpression of Shank3 WT significantly increased the amount of F-actin (1.399±0.039) compared with the control (***p<0.001; Figure 1g-h), showing the role of Shank3 WT in in vitro actin polymerization. In contrast, no significant difference in actin polymerization was observed with Shank3 STOP protein (0.971±0.062; p>0.05) compared with the control. These results suggest that the C-terminus of Shank3 is crucial for actin dynamics.

Shank3 mutations affect spine induction and spine morphology

To analyze the role of Shank3 in actin cytoskeleton regulation in dendritic spines, we overexpressed the different constructs in hippocampal neurons and studied their effects on the organization and shape of postsynaptic spines. First, we analyzed the subcellular localization of the different Shank3 mutants co-transfected with an RFP plasmid in hippocampal neurons. Shank3 WT overexpression led to an accumulation of the protein in dendritic spines (Figure 2a), as previously reported [START_REF] Boeckers | C-terminal synaptic targeting elements for postsynaptic density proteins ProSAP1/Shank2 and ProSAP2/Shank3[END_REF] . Point mutations (Shank3 R12C , Shank3 R300C and Shank3 Q321R ) did not affect this synaptic targeting. As observed in COS-7 cells, Shank3 STOP accumulated in the soma, was weakly localized in the dendrites and was absent from dendritic spines, indicating that the C-terminus was required for the dendritic targeting of Shank3. These results are consistent with previous data showing that the Homer binding site is involved in the synaptic localization of Shank3 [START_REF] Roussignol | Shank expression is sufficient to induce functional dendritic spine synapses in aspiny neurons[END_REF][START_REF] Boeckers | C-terminal synaptic targeting elements for postsynaptic density proteins ProSAP1/Shank2 and ProSAP2/Shank3[END_REF] .

In addition, Shank3 WT significantly increased the number of spines compared with the control (Figure 2b) and reduced the number of filopodia (Figure 2c) in rat hippocampal neurons. The number of spines per 20 μm of dendrite length was 6.3±0.4 spines when Shank3 WT was overexpressed, compared with 3.4±0.3 spines in control GFP-positive neurons (***p<0.001). Overexpression of Shank3 R12C , Shank3 R300C or Shank3 Q321R also increased the number of spines, but not as much as Shank3 WT (4.6±0.3, 4.7±0.3, or 5.0±0.2 spines per 20 μm, respectively; **p<0.01 and ***p<0.001 compared with control). The mutations in the ankyrin domain therefore had moderate effects on spine induction, but still significantly lower than Shank3 WT ( o p<0.05; °°p<0.01 and ooo p<0.001, respectively). The overexpression of Shank3 STOP had a stronger effect, with a reduction in the number of spines compared with Shank3 WT and with the control (2.6±0.2 spines per 20 μm; °°°p<0.001 compared with Shank3 WT and *p<0.05 compared with the control). Moreover, a Sh-resistant form of Shank3 STOP did not rescue the spine phenotype in hippocampal neurons where Shank3 was downregulated (supplementary figure 1). Altogether, our results suggest that Shank3 STOP has a dominantnegative effect on spine induction in a wild-type background, probably by affecting the targeting and/or the stability of the wild-type protein at synapses. Finally, we investigated the impact of Shank3 mutations on spine maturation and measured spine head enlargement and spine length from the base to the top of the head in rat hippocampal neurons transfected with each construct (Figure 2d). The number of spines with head widths > 0.5 μm was significantly increased in neurons transfected with Shank3 WT compared with neurons transfected with GFP (63±3% of spines > 0.5 μm head width compared to 51±3% of spines; *p<0.05; Figure 2e-f). No difference in spine length was observed (44±3% of spines > 1.0 μm length for the control and 41±3% of spines for Shank3 WT , Figure 2g). Thus, in mature neurons, Shank3 overexpression promoted the morphological maturation of spines, as previously observed with Shank3 WT expressed in aspiny neurons [START_REF] Roussignol | Shank expression is sufficient to induce functional dendritic spine synapses in aspiny neurons[END_REF] or with Shank1 in hippocampal cells [START_REF] Sala | Regulation of dendritic spine morphology and synaptic function by Shank and Homer[END_REF] .

Compared with the control, Shank3 STOP did not change the width of spine heads (46.6±2% of spines > 0.5 μm: p>0.5) but reduced the length of the spines (34±2% of spines > 1.0 μm length; *p < 0.05). Shank3 Q321R showed a slight increase in the width of the spine head (59±2% of spines > 0.5 μm, ***p<0.001) and a reduction in spine length (34±2% of spines > 1.0 μm, *p<0.05). This suggests that the Q321R mutation does not abolish the effect of Shank3 on spine maturation but does affect neck retraction. Finally, Shank3 R12C and Shank3 R300C did not increase the size of spine-heads (54±4% and 51±3% of spines > 0.5 μm; p>0.05 compared with control), suggesting that these mutations disrupt the effect of Shank3 on spine morphology. Altogether, our data suggest that Shank3 WT overexpression promotes spine induction and maturation and that all mutations identified in ASD modify the function of Shank3.

Shank3 mutations affect F-actin content but not cortactin level in dendritic spines

As actin dynamics play key roles in dendrite development and specifically in spine formation and maturation [START_REF] Caceres | Immunocytochemical localization of actin and microtubule-associated protein MAP2 in dendritic spines[END_REF][START_REF] Tada | Molecular mechanisms of dendritic spine morphogenesis[END_REF] , we further examined the effects of Shank3 mutations on the synaptic levels of F-actin and cortactin (Figure 3). Cortactin is an F-actin binding protein enriched in cell matrix contact sites that facilitates the nucleation of new actin branches at the tips of actin filaments. In mature neurons, Factin appeared in the form of patches and puncta along the dendrites, and it was enriched in the heads of dendritic spines (Figure 3a). We quantified the level of actin in spine heads as a ratio of the average intensity of transfected neurons to untransfected neurons. We observed an increase in F-actin levels in the spines of Shank3 WT -transfected neurons compared with the control (147±8% for Shank3 WT and 109±6% for the control; ***p<0.001; Figure 3b), as previously described for Shank1 [START_REF] Sala | Regulation of dendritic spine morphology and synaptic function by Shank and Homer[END_REF] . Neurons transfected with Shank3 R12C and Shank3 R300C constructs displayed intermediate levels of F-actin between the control and Shank3 WT levels (123±7% for Shank3 R12C , 125±9% for Shank3 R300C ). The absence of an increase in F-actin as large as that induced by Shank3 WT suggests that mutations in the ankyrin domain affect F-actin content in dendritic spines. Neither the truncating mutation nor Shank3 Q321R increased F-actin level (93±6% for Shank3 STOP and 115±8% for Shank3 Q321R ; °°°p<0.001 and °°p<0.01 compared to Shank3 WT ). In the case of the truncating mutation, this was most likely due to a lack of targeting to the spines. Our data show that enlargement of dendritic spines induced by Shank3 WT was correlated with increases in F-actin level. The mutations in the ankyrin domain and the truncated form of Shank3 all affected spines maturation via an actin-dependent mechanism, as suggested by our fibroblast experiments (Figure 1). Because Shank3 binds to cortactin and the loss of cortactin causes a decrease in spine density [START_REF] Naisbitt | a novel family of postsynaptic density proteins that binds to the NMDA receptor/PSD-95/GKAP complex and cortactin[END_REF][START_REF] Hering | Activity-dependent redistribution and essential role of cortactin in dendritic spine morphogenesis[END_REF][START_REF] Sala | Regulation of dendritic spine morphology and synaptic function by Shank and Homer[END_REF] , we examined the effect of each mutation on cortactin content in dendritic spines. Despite Shank3 mutations significantly reducing the level of F-actin in spines, cortactin level was not significantly different between the control and Shank3 WT (Figure 3c, p>0.05).

The Shank3 mutations decrease spontaneous neuronal activity

Because the number and the morphology of spines might affect the strength of synaptic transmission, we further explored the functional consequences of Shank3 mutations by recording mEPSCs in hippocampal neurons. Shank3 WT had no significant effect on mEPSC amplitude but markedly increased mEPSC frequency compared with GFP-only neurons (1.48±0.08 and 0.74±0.08 Hz, respectively, ***p<0.001; Figure 4). The present correlation between Shank3 WT -mediated changes in spine morphology and electrical activity is in agreement with previous results [START_REF] Sala | Regulation of dendritic spine morphology and synaptic function by Shank and Homer[END_REF] . Shank3 R12C , Shank3 R300C and Shank3 Q321R modestly increased mEPSC frequency, but not as much as Shank3 WT (0.98±0.16, 0.97±0.23, and 1.1±0.24 Hz, respectively; °p<0.05). In contrast, Shank3 STOP strongly reduced the frequency (0.50±0.17 Hz; °°°p<0.001) as well as the amplitude of mEPSCs compared with the control (Shank3 STOP , 15.90±0.72 pA; GFP-only 31.49±2.48 pA; ***p<0.001), extending the dominant-negative effect displayed by Shank3 STOP to the modulation of synaptic transmission.

The Q321R and STOP mutations affect growth cone motility induced by Shank3 WT

Q321R is a de novo mutation identified in a girl with autism, language delay and developmental delay [START_REF] Moessner | Contribution of SHANK3 mutations to autism spectrum disorder[END_REF] . Here we found that this mutation was associated with abnormalities in spine morphology (even if less pronounced than for Shank3 STOP ). We then assessed whether Shank3 could play a role earlier in neuronal development. Indeed, Shank3 is expressed in the brain at early stages [START_REF] Petralia | Ontogeny of postsynaptic density proteins at glutamatergic synapses[END_REF] , and Shank and cortactin are enriched within the lamellipodia of motile cells and in neuronal growth cones [START_REF] Naisbitt | a novel family of postsynaptic density proteins that binds to the NMDA receptor/PSD-95/GKAP complex and cortactin[END_REF][START_REF] Du | Identification of a novel cortactin SH3 domain-binding protein and its localization to growth cones of cultured neurons[END_REF] . We analyzed the localization of endogenous or overexpressed Shank3 in cultured hippocampal neurons by immunocytochemistry in early stages (Figure 5a-b). Endogenous Shank3 was detected in young neurons (day 2), and labeling with actin revealed the accumulation of the protein in ruffles. We also confirmed the co-localization of Shank3 and cortactin in axonal growth cones. Then, we tested whether Shank3 regulated growth cone dynamics by co-transfecting GFP-C3 with GFP-Shank3 WT or with the mutated forms in young hippocampal neurons. Four days after the transfection, live cell imaging was performed as described [START_REF] Ibarretxe | Fast regulation of axonal growth cone motility by electrical activity[END_REF] . The axonal growth cone motility was quantified as a dimensionless motility index (MI). We quantified the mean value of MI for 10-15 axonal growth cones for each construct (Figure 5c-d). We observed an increase of MI with Shank3 WT compared with the control (0.129±0.005 for Shank3 WT to 0.101±0.005 for GFP-C3; ***p<0.001). This suggests that full-length Shank3 induces higher axonal growth cone motility. We did not observe such an increase for GFP-Shank3 Q321R or GFP-Shank3 STOP (0.106±0.003 and 0.112±0.004; p<0.001). Together, these data suggest that these two de novo mutations, Shank3 Q321R and Shank3 STOP , affect neuronal motility during development.

Discussion

Three important conclusions can be drawn from this study. First, we found that Shank3 increases actin polymerization, promoting spine formation and increasing the frequency of mEPSPs. Second, we demonstrated that Shank3 mutations identified in ASD disrupt these processes, providing a cellular readout for Shank3-associated pathologies. Third, we described two de novo mutations, Shank3 Q321R and Shank3 STOP , associated with severe loss of function, and two inherited mutations, Shank3 R12C and Shank3 R300C , which produced an intermediate phenotype and might represent a risk factor for ASD.

Shank3 promotes spine formation and shape via an actin-dependent mechanism.

Spine structure is regulated by various molecular mechanisms, and actin polymerization is a crucial process for synaptic structure and function [START_REF] Cingolani | Actin in action: the interplay between the actin cytoskeleton and synaptic efficacy[END_REF] . Actin filaments in the spine head are highly dynamic and regulate the molecular organization of the PSD and modulate postsynaptic signal transduction [START_REF] Kuriu | Differential control of postsynaptic density scaffolds via actin-dependent and -independent mechanisms[END_REF][START_REF] Star | Rapid turnover of actin in dendritic spines and its regulation by activity[END_REF][START_REF] Honkura | The subspine organization of actin fibers regulates the structure and plasticity of dendritic spines[END_REF] . Our data on neuronal cultures show that Shank3 overexpression induces an increase in spine density associated with an F-actin accumulation and an increase in mEPSP frequency. These effects could be mediated in part by a regulation of capping proteins, as we demonstrated that Shank3 co-localized with vinculin and phospho-paxillin at the tip of actin stress fibers in fibroblasts. These capping proteins control the growth of actin filaments in lamellipodia, resulting in filopodia formation and a reduction in lamellipodia [START_REF] Mejillano | Lamellipodial versus filopodial mode of the actin nanomachinery: pivotal role of the filament barbed end[END_REF] . In mature neurons, they localize throughout dendritic spines [START_REF] Korobova | Molecular architecture of synaptic actin cytoskeleton in hippocampal neurons reveals a mechanism of dendritic spine morphogenesis[END_REF] , similar to Shank3. Furthermore, the Shank complex could be linked to several other proteins regulating actin cytoskeleton dynamics, such as β-Pix, a guanine nucleotide exchange factor for Rac1 and cdc42 small GTPases 18 ; Abp1, a member of the Arp2/3 complex 17 ; and cortactin [START_REF] Du | Identification of a novel cortactin SH3 domain-binding protein and its localization to growth cones of cultured neurons[END_REF] . Indeed, depletion of the Arp2/3 complex, including cortactin, Abi2, WAVE-1, N-WASP, and Abp1, alters the morphology and number of spines [START_REF] Hering | Activity-dependent redistribution and essential role of cortactin in dendritic spine morphogenesis[END_REF][START_REF] Grove | ABI2deficient mice exhibit defective cell migration, aberrant dendritic spine morphogenesis, and deficits in learning and memory[END_REF][START_REF] Kim | Interaction of SPIN90 with the Arp2/3 complex mediates lamellipodia and actin comet tail formation[END_REF][START_REF] Soderling | A WAVE-1 and WRP signaling complex regulates spine density, synaptic plasticity, and memory[END_REF][START_REF] Wegner | N-wasp and the arp2/3 complex are critical regulators of actin in the development of dendritic spines and synapses[END_REF] . Altogether, these results suggest that Shank3 could directly regulate F-actin level in spines, resulting in a modification of the shape and the function of these spines.

In this study, we observed increased spine density and morphology with Shank3, which is consistent with previous results obtained in aspiny neurons with Shank3 22 . Shank1B induces an enlargement of spine heads independent of spine density, and this enlargement is enhanced by coexpression of the long form of Homer [START_REF] Sala | Regulation of dendritic spine morphology and synaptic function by Shank and Homer[END_REF][START_REF] Kim | Interaction of SPIN90 with the Arp2/3 complex mediates lamellipodia and actin comet tail formation[END_REF][START_REF] Sala | Inhibition of dendritic spine morphogenesis and synaptic transmission by activity-inducible protein Homer1a[END_REF] . Recent data on Shank1 knock-out mice suggest that Shank1 is not required for spine formation but promotes the growth and stability of large spines, which is consistent with in vitro results [START_REF] Hung | Smaller dendritic spines, weaker synaptic transmission, but enhanced spatial learning in mice lacking Shank1[END_REF] . Shank2 also affects spine head maturation, depending on the Arp2/3 complex [START_REF] Haeckel | The actin-binding protein Abp1 controls dendritic spine morphology and is important for spine head and synapse formation[END_REF] . Therefore, it is now clearly established that Shank proteins are involved in spinogenesis, especially in spine maturation, and we highlighted in this study that this occurs via an actin-dependent mechanism.

Shank3 mutations associated with ASD disrupt its cellular functions

We showed that de novo or inherited Shank3 mutations associated with ASD can affect its localization at the tip of the F-actin fibers, subsequently disrupting spine formation and morphology. This effect was especially strong for Shank3 STOP and was similar to that observed in hippocampal neurons where Shank3 was downregulated (this study and ref23) or in cerebellar neurons transfected with Shank3 14 bearing a mutation in the Homer binding site [START_REF] Roussignol | Shank expression is sufficient to induce functional dendritic spine synapses in aspiny neurons[END_REF] . This mutation appears to have a dominant-negative effect on the induction and the maturation of spines in a wild-type background (Figure 2b). This could be due to interference with the localization of other binding partners and/or to PDZ domain dimerization of the mutated form with the endogenous Shank3 [START_REF] Iskenderian-Epps | Modulation of Shank3 PDZ domain ligand-binding affinity by dimerization[END_REF] .

Our results highlight the potential impact of the two de novo mutations of Shank3 on early stages of neuronal development. We showed that the endogenous as well as the overexpressed Shank3 was localized in the growth cones of young neurons, suggesting that Shank3 could participate in neurite extension. Consistent with this idea, cortactin mediates neurite outgrowth [START_REF] Hering | Activity-dependent redistribution and essential role of cortactin in dendritic spine morphogenesis[END_REF] , and the depletion of Abp1, a linker of Shank3 to the actin complex, results in increased axon length in early stages of neuronal development [START_REF] Pinyol | Regulation of N-WASP and the Arp2/3 complex by Abp1 controls neuronal morphology[END_REF] . The Shank3 STOP and Shank3 Q321R de novo mutations decreased growth cone motility, which could lead to disrupted connectivity in patients carrying either mutation. These results are consistent with a recent study showing that the lack of C-terminal domains (including Homer and cortactin binding sites and the SAM domain) inhibits the induction of neurite outgrowth [START_REF] Gauthier | De novo mutations in the gene encoding the synaptic scaffolding protein SHANK3 in patients ascertained for schizophrenia[END_REF] .

Inherited mutations represent risk factors for ASD

Mutations in SHANK3, including de novo mutations, de novo copy number variations and inherited mutations, have been frequently reported in patients [START_REF] Durand | Mutations in the gene encoding the synaptic scaffolding protein SHANK3 are associated with autism spectrum disorders[END_REF][START_REF] Moessner | Contribution of SHANK3 mutations to autism spectrum disorder[END_REF][START_REF] Gauthier | De novo mutations in the gene encoding the synaptic scaffolding protein SHANK3 in patients ascertained for schizophrenia[END_REF] . The importance of the Shank family for ASD was further strengthened by the recent identification of mutations in SHANK2 in ASD patients [START_REF] Berkel | Mutations in the SHANK2 synaptic scaffolding gene in autism spectrum disorder and mental retardation[END_REF][START_REF] Pinto | Functional impact of global rare copy number variation in autism spectrum disorders[END_REF] . If the impact of de novo deleterious mutations is well established, the role of inherited mutations remains unclear. Here, we showed that the two inherited mutations, Shank3 R12C and Shank3 R300C , have intermediate effects, mainly on spine formation. These results suggest that despite their transmission from healthy parents, these mutations could cause subtle synaptic defects increasing the risk of developing ASD. SHANK3 mutations can also be involved in other neurologic disorders, such as intellectual disability and schizophrenia [START_REF] Gauthier | De novo mutations in the gene encoding the synaptic scaffolding protein SHANK3 in patients ascertained for schizophrenia[END_REF] . In the case of schizophrenia, the de novo mutation R1117X with germline mosaicism has been observed in three affected brothers. They presented mild to moderate intellectual disability and schizoaffective disorder, but no evidence of autistic features.

Functional studies have confirmed that this mutation causes a loss of SHANK3 function in zebrafish and decreases neurite outgrowth in mice [START_REF] Gauthier | De novo mutations in the gene encoding the synaptic scaffolding protein SHANK3 in patients ascertained for schizophrenia[END_REF] . It is unclear how similar mutations can lead to different clinical conditions. The same is true for mutations in NLGN3 and NLGN4 described in autism but also in intellectual disability, Asperger syndrome, and Tourette syndrome with hyperactivity disorder [START_REF] Jamain | Mutations of the X-linked genes encoding neuroligins NLGN3 and NLGN4 are associated with autism[END_REF][START_REF] Laumonnier | X-linked mental retardation and autism are associated with a mutation in the NLGN4 gene, a member of the neuroligin family[END_REF][START_REF] Lawson-Yuen | Familial deletion within NLGN4 associated with autism and Tourette syndrome[END_REF] .

Mutations in NRXN1 have also been observed in patients with schizophrenia and in asymptomatic carriers [START_REF] Zweier | CNTNAP2 and NRXN1 are mutated in autosomal-recessive Pitt-Hopkins-like mental retardation and determine the level of a common synaptic protein in Drosophila[END_REF] . These different outcomes could be due to mutations altering either all or specific roles of these proteins (e.g., axonal guidance, synapse formation). We can also expect that additional genetic/epigenetic and environmental risk factors acting at different levels of brain development could bias the phenotype towards different sets of disorders [START_REF] Cook | Copy-number variations associated with neuropsychiatric conditions[END_REF] . The identification of such factors would allow us to better understand the nature of the biological mechanisms underlying these different disorders and to provide insight for new treatments. ratio is normalized to GFP-C3 and is represented as the mean ± SEM. Shank3 WT enhances the polymerization of actin (***p<0.001; n = 6 independent experiments). Shank3 STOP inhibits the effect of Shank3 on actin polymerization.

  introducing a STOP codon by direct mutagenesis to generate Shank3 mutants lacking the C-terminal part of the protein, corresponding to the mutation frameshift 3915 identified in patients. The monomeric red fluorescent protein (mRFP) construct was generously provided by I. Macara (University of Virginia, Charlottesville, VA). C. Gauthier-Rouviere (Centre de Recherche en Biochimie Macromoléculaire, Centre National de la Recherche Scientifique Montpellier, France) generously provided the red fluorescent protein RFP-actin construct. The SureSilencing short-hairpin RNA (shRNA) plasmids cloned in p-RFP-C-RS were purchased from Origene. Details of the downregulation experiments are described in the supplementary note.

Figure 2 :

 2 Figure 2: Effect of Shank3 WT and mutants on dendritic spine induction and maturation. (a) Hippocampal neurons were transiently co-transfected with GFP-Shank3 WT or mutated forms (green) and RFP (red) and processed for immunofluorescence after 4 days (days in vitro 21, DIV21), and the subcellular localization of each construct was analyzed. Neurons are stained with anti-GFP (green) and anti-DsRED (red). Scale bars represent 20 µm. On the bottom side, magnification of 20 μm of the dendrite transfected with each construct. (b) Comparison of spine density in neurons transfected with the different Shank3 constructs (number of spines per 20 µm of dendrite length) in co-transfected neurons. (c) Comparison of the density of filopodia (number of filopodia per 20 μm of dendrite length) in co-transfected neurons. Shank3 WT leads to an increase in the number of spines and a decrease in the number of filopodia. All mutations reduce the ability of Shank3 to induce spines. (***p<0.001, *p<0.05 compared with the control; °°°p<0.001, °°p<0.01, °p<0.05 compared with Shank3 WT ). (d) Examples of spines from neurons transfected with GFP-C3 or GFP-Shank3 WT . Spine length (green line) and head width (red line) were computed using Volocity software. (e) Truncating mutation causes a decrease in the number of larger spines and an increase in the number thin spines. Relative abundance of four different spine classes according to spine head size (0 to 0.25; 0.25 to 0.5; 0.5 to 0.75; and > 0.75 µm). (f) Percentage of spine heads > 0.5 μm within the same population. Spine heads are significantly larger in GFP-Shank3 WT -and GFP-Shank3 Q321R -transfected neurons compared with the control (***p<0.001, **p<0.01). Spine heads are significantly smaller in GFP-Shank3 STOP . (g) Percentage of spines of > 1.0 μm within the same population. Spine length is significantly reduced with GFP-Shank3 Q321R and GFP-Shank3 STOP constructs compared with the control (*p<0.05; n= 17-28 neurons and 1500-1800 spines).

Figure 3 :

 3 Figure 3: Effects of Shank3 wt and mutants on F-actin and cortactin levels in spines. (a) Immunofluorescence microscopy of hippocampal neurons at DIV21 transiently co-transfected with GFP-Shank3 WT or mutated forms and RFP. Neurons are stained with anti-GFP and anti-DsRED. Factin was visualized with phalloidin 647. Cortactin is visualized with anti-cortactin (blue). Scale bars represent 6 µm. (b-c) Quantification of changes in the synaptic staining intensity of F-actin or cortactin induced by the overexpression of Shank3 WT or the mutated forms. (**p<0.01 compared with the control; °°°p<0.001, °°p<0.01, °p<0.05 compared with Shank3 WT ).

Figure 4 :

 4 Figure 4: The Shank3 mutations decreased spontaneous neuronal activity. (a) mEPSCs recorded at -60 mV on hippocampal neurons (DIV21) transiently co-transfected with GFP-C3, GFP-Shank3 WT or mutated forms. (b) Quantification of mEPSCs frequency and amplitude under different transfection conditions. In Shank3 WT , there is a significant increase in mEPSC frequency. Each bar of the histogram is the mean ± SEM of 10 experiments. ***p<0.001 and *p<0.05 compared with control; °°°p<0.001, °°p<0.01, °p<0.05 compared with Shank3 WT .

Figure 5 :

 5 Figure 5: Effects of Shank3 WT and mutants on growth cone dynamics. (a) Shank3 is found in the growth cone of neonatal rat cultured hippocampal neurons on DIV2. Neurons are labeled for Shank3 (green), F-actin (phalloidin; blue), and either tubulin (anti-Tuj1; red) or cortactin (anti-cortactin, red). Scale bars represent 20 µm. Insets represent high magnifications of the axonal growth cone. Shank3 co-localized with F-actin (blue) and with cortactin (red) in axonal growth cones. (b) Localization of Shank3 overexpressed in young neurons. GFP-Shank3 WT or Shank3 STOP or GFP-C3 (green) was transfected in young neurons (DIV0). Immunostaining for MAP2 (red) and actin (phalloidin; blue) revealed that Shank3 WT accumulates in growth cones, whereas Shank3 STOP is restricted to the cell body. (c) Time-lapse video microscopy of growth cones from rat hippocampal neurons co-transfected with the control (GFP-C3) and Shank3 WT or mutated forms. Merge represents three frames with different colors (0 min in green, 5 min in red and 10 min in blue). Scale bars represent 5 µm. (d)

Acknowledgements

We thank I. Macara for providing the mRFP construct and C. Gauthier-Rouviere for the mRFP-actin construct. We thank Julien Bensalem, Marie-Claude Donat, Lea Lasvaux, and Chantal Medina for technical assistance; we thank Drs Jerome Ezan, Roberto Toro and Claudia Racca for helpful comments and discussion. This work was supported by INSERM AVENIR grant to N.S. and M.M., Conseil Regional d'Aquitaine (NS, MM), La Fondation pour la Recherche Medicale (NS, MM), La Fondation Jerome Lejeune (NS), FRM and Orange fellowships (CD) and the European Commission Coordination Action ENINET (contract number LSHM-CT-2005-19063; NS & MM). The sequencing was performed at the Sequencing facility of Bordeaux (grants from the Conseil Régional d'Aquitaine n° 20030304002FA and 20040305003FA and from the European Union, FEDER n° 2003227). The confocal microscopy was done at the Bordeaux Imaging Center of the Neurosciences Institute of the University of Bordeaux II. The monoclonal antibody against Shank3 was developed by and obtained from the UC Davis/NINDS/NIMH NeuroMab Facility, supported by NIH grant U24NS050606 and maintained by the Department of Pharmacology, School of Medicine, University of California, Davis, CA 95616.

Supplementary information

Supplementary information is available at Molecular Psychiatry's website

Conflict of interest

The authors declare no conflict of interest.