Grégory Bonnet

Christopher Leturc

Emiliano Lorini

Giovanni Sartor

Influencing Choices by Changing Beliefs: A Logical Theory of Influence, Persuasion, and Deception

We model persuasion, viewed as a deliberate action through which an agent (persuader) changes the beliefs of another agent's (persuadee). This notion of persuasion paves the way to express the idea of persuasive influence, namely inducing a change in the choices of the persuadee by changing her beliefs. It allows in turns to express different aspects of deception. To this end, we propose a logical framework that enables expressing actions and capabilities of agents, their mental states (desires, knowledge and beliefs), a variety of agency operators as well as the connection between mental states and choices. Those notions, once combined, enable us to capture, the notion of influence, persuasion and deception, as well as their relation.

Introduction

In many contexts, agents need the cooperation of others, to achieve their goals. To this end, they may influence others, namely execute actions leading others to perform further actions. Influence may result from impeding or enabling certain actions by others (removing or adding choices), but also from changing the mental states of others (removing or adding beliefs). In the first case we speak of regimentation, while in the second we speak of persuasion. In this paper, we focus on persuasion, i.e. on the deliberate action through which agents (persuaders) changes the beliefs of other agents (persuadees).

Persuasion in the human-machine interaction raises serious ethical and legal issues, as more and more often automated systems engage in attempts at influencing human choices through persuasive messages. Such persuasive activities may be determined by interests that are not aligned with the interests of their addressees, but are rather determined by the economic or political goals of the senders (or their principals). Automated persuaders may also be deceivers, e.g. agents which present fake information or anyway induce the persuadees into actions the latter will regret (bad economic, personal or political choices). To adequately respond to this challenge, it is important to have a clear conceptual framework, that enables us to capture the different ways in which influence, persuasion and deception can be deployed, so that adequate responses to each of such ways can be developed, through human or also computational interventions. Let us precise that, as we are mainly interested in artificial agents (automated persuaders), we will consider in this article rational agents. The anthropomorphic expression (persuader, persuadee, mental state) are just shortcuts that we use for convenience.

In the AI field, persuasion and related notions have been approached from different perspectives. Following seminal work on argumentation [START_REF] Walton | Commitment in Dialogue: Basic Concepts of Interpersonal Reasoning[END_REF], persuasion has been modelled through structured argumentation [START_REF] Prakken | Formal systems for persuasion dialogue[END_REF], abstract argumentation [START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF][START_REF] Bonzon | On the outcomes of multiparty persuasion[END_REF], probabilistic argumentation [START_REF] Hunter | Modelling the persuadee in asymmetric argumentation dialogues for persuasion[END_REF], possibilistic belief revision [START_REF] Da Costa Pereira | Changing one's mind: Erase or rewind? Possibilistic belief revision with fuzzy argumentation based on trust[END_REF], abstract argumentation combined with dynamic epistemic logic [START_REF] Proietti | Persuasive argumentation and epistemic attitudes[END_REF]. Some logical approaches have addressed notions related to persuasion, such as social influence [START_REF] Lorini | A STIT logic analysis of social influence[END_REF][START_REF] Santos | Indirect action, influence and responsibility[END_REF], manipulation (influence on choices) [START_REF] Leturc | A deliberate BIAT logic for modeling manipulations[END_REF], lying and deception [START_REF] Sakama | A formal account of dishonesty[END_REF][START_REF] Van Ditmarsch | On the logic of lying[END_REF], or changing other agents' degrees of beliefs [START_REF] Budzyńska | A logic for reasoning about persuasion[END_REF]. The original contribution of this work consists in providing a logical account of the way in which a persuader by changing the beliefs of the persuadee, influences the action of the latter. We aim indeed to provide a formal theory of the micro-foundations of deception, persuasion and influence, i.e., an account of the cognitive attitudes and agentive aspects that are involved in the persuasion and influence, and elucidate their relationship. For this purpose we provide a rich framework that expresses actions and capabilities of agents and their mental states (desires, knowledge and beliefs) as well as the connection between mental states and choices. In order to keep the framework simple while being expressive, we focus on a qualitative framework (such as Boolean games [START_REF] Van Benthem | Dependencies between players in Boolean games[END_REF]) where agents' preferences are not presented by continuous utility functions but rather by a qualitative three-valued scale desirable/undesirable/neutral. Then We express two notions of rationality (an optimistic and a pessimistic one), several agency operators (such as the so-called 'Chellas' STIT [START_REF] Horty | Agency and Deontic Logic[END_REF], the deliberative and the rational STIT operators [START_REF] Lorini | A STIT logic analysis of social influence[END_REF]) and different ways to influence agents' choices through belief change.

Relative to this rich background addressing partial aspects of persuasion processes, our model will be useful for better understanding and modelling the dynamics of social influence, especially those between artificial and human agents. It will also be relevant from a regulatory perspective, since it allows to pinpoint those instance in which online interactions, in virtue of their logical structure, can be viewed as detrimental to trustful and productive interaction, and thus call for normative limitations. It can also be relevant in the special cases where persuasion and deception can be of interest in social relationship to protect somebody [START_REF] Alistair | White lies on silver tongues: Why robots need to deceive (and how)[END_REF]. Our contribution has the advantage of providing a comprehensive model which captures the whole persuasive process including the mental states of the persuader, his persuading action, the modified mental states of persuadee and her resulting action. The model also captures the connection between influence, regimentation and persuasion, and enables to link persuasion with game theory. This article is organized as follow. Firstly, in Section 1 we introduce a running example. Then we define in Section 2 our logical framework and, in Section 3, we show how this framework can express notions of optimistic and pessimistic rationality, and a variety of agency operators. We then combine those notions in Section 4 to express deception, persuasion, regimentation and influence, and their relationships. Finally, we apply our framework to the running example.

Running example John is feeling back pain, and consequently he is consulting websites that offer medical advice as well as the opportunity to purchase drugs. A message from a bot promises, for a fair price, a drug which is said to be an excellent remedy that would eliminate all pain, and can be used for any length of time without producing any dependency. This message persuades John to buy the drug and he starts taking it. However the bot has deceived John because the bot knows that the drug creates addiction, with serious health consequences. John's wife Ann, comes to know that such pills are dangerous. She then removes all pills from the closet. As a consequence John does not take the drug.

This example shows two patterns of influence. The first is successful and misleading persuasion (deception) by the website, i.e., successful influencing by providing false information. The second stage consists in Ann successfully influencing John through regimentation, i.e., by removing a choice option.

Logical Framework

In this section, we present a modal logic language which supports reasoning about (i) actions and capabilities of agents and coalitions, (ii) agents' epistemic states and desires as well as their connection with agents' choices. We first present its syntax and its semantic interpretation (Sections 2.1 and 2.2). Then, in Section 2.3, we provide a sound and complete axiomatization of its set of validities.

Syntax

Assume a countable set of atomic propositions Atm = {p, q, . . .}, a finite set of agents Agt = {1, . . . , n}, a finite set of atomic action names Act = {a, b, . . .}. The set Act includes the (in)action skip, i.e., the action of doing nothing. We define Prop to be the set of propositional formulas, that is, the set of all Boolean combinations of atomic propositions.

The set of non-empty sets of agents, also called coalitions, is defined by 2 Agt * = 2 Agt \ {∅}. Elements of 2 Agt * are noted H, J, . . . A coalition H's joint action is defined to be a function δ H : H -→ Act. Coalition H's set of joint actions is noted JAct H . Its elements are noted δ H , δ H , . . . For notational convenience, we simply write JAct instead of JAct Agt to denote the grand coalition's set of joint actions. Its elements are noted δ, δ , . . . Moreover, we write Act i instead of JAct {i} to denote agent i's set of individual actions. Its elements are noted a i , b i , . . . We define JAct * to be the set of all finite sequences of joint actions from JAct. Elements of JAct * are noted , , . . . The empty sequence of joint actions is denoted by nil . Infinite sequences of joint actions are called histories. The set of all histories is noted Hist and its elements are noted h, h , . . . Elements of JAct * ∪ Hist are noted τ, τ , . . . For every τ 1 , τ 2 ∈ JAct * ∪ Hist, we write τ 1 τ 2 to mean that either τ 1 = τ 2 or τ 1 is an initial subsequence of τ 2 , i.e., there is τ 3 ∈ JAct * ∪ Hist such that τ 2 = τ 1 ;τ 3 . The language L is defined by the following grammar:

::= δ | ; ϕ ::= p | occ( ) | plaus i | good i | badi | neutrali | ¬ϕ | ϕ1 ∧ ϕ2 | [[δ]]ϕ | ϕ | Kiϕ
where p ranges over Atm, i ranges over Agt, δ ranges over JAct and ranges over JAct * . The other Boolean constructions , ⊥, ∨, → and ↔ are defined from p, ¬ and ∧ in the standard way. We note L -the fragment of L without operators [[δ]] defined by the following grammar:

ϕ ::= p | occ( ) | plaus i | good i | neutrali | badi | ¬ϕ | ϕ1 ∧ ϕ2 | ϕ | Kiϕ
L has special atomic formulas of four different kinds. The atomic formulas occ( ) represent information about occurrences of joint action sequences. The formula occ( ) has to be read "the joint action sequence is going to occur".

The following abbreviations capture interesting action-related concepts. For every H ∈ 2 Agt * , joint action sequence ∈ JAct * and joint action δ H ∈ JAct H we define:

choose( ,δH ) def = δ∈JAct: ∀i∈H, δ H (i)=δ(i) occ( ;δ) can( ,δH ) def = ♦choose( ,δH )
choose( ,δ H ) has to be read "the joint action sequence is going to occur and will be followed by coalition H's joint action δ H ". For convenience, when = nil , we write choose(δ H ) instead of choose(nil ,δ H ). Formula can( ,δ H ) has to be read "coalition H can choose the joint action δ H at the end of the joint action sequence ".

The atomic formula plaus i is used to identify the histories in agent i's information set that she considers plausible. It has to be read "the current history is considered plausible by agent i". The other atomic formulas good i , bad i and neutral i are used to rank the histories that an agent envisages at a given world according to their value for the agent (i.e., how much a given history promotes the satisfaction of the agent's desires). They are read, respectively, "the current history is good/bad/neutral for agent i". Let us notice it is an agent-centric point-of-view. Each good i , bad i and neutral i atoms are defined only from agent i's perspective.

L has three kinds of modal operators: [[δ]], and K i . is the so-called historical necessity operator. The formula ϕ has to be read "ϕ is true in all histories passing through the current moment". We define ♦ to be the dual of , i.e., ♦ϕ def = ¬ ¬ϕ where ♦ϕ has to be read "ϕ is true in at least one history passing through the current moment". [[δ]] is a dynamic operator describing the fact that if the joint action δ is performed then it will lead to a state in which a given state of affairs holds. In particular, [[δ]]ϕ has to be read "if the joint action δ is performed, then ϕ will be true after its execution". Finally, K i is a modal operator characterizing the concept of ex ante (or choice-independent) knowledge [START_REF] Aumann | Rational expectations in games[END_REF][START_REF] Lorini | A STIT logic analysis of social influence[END_REF][START_REF] Roy | Epistemic logic and the foundations of decision and game theory[END_REF]. The formula K i ϕ has to be read "agent i knows that ϕ is true independently from her current choice" or "agent i thinks that ϕ is true for any choice she could have made". The dual of the operator K i is denoted by K i , i.e., K i ϕ def = ¬K i ¬ϕ. Ex ante knowledge is distinguished from ex post knowledge. Ex ante knowledge characterizes an agent's knowledge assuming that no decision has yet been made by him, whereas ex post knowledge characterizes an agent's knowledge assuming that the agent has made his decision about which action to take, but might still be uncertain about the decisions of others. The concept of ex post knowledge is expressed by the following operator K post i :

K post i ϕ def = a i ∈Act i choose(ai) → Ki(choose(ai) → ϕ)
where K post i ϕ has to be read "agent i knows that ϕ is true, given her actual choice". From the special atomic formula plaus i and the epistemic operator K i , we define a belief operator:

B i ϕ def = K i (plaus i → ϕ)
The formula B i ϕ has to be read "agent i believes that ϕ". According to this definition, an agent believes that ϕ if and only if ϕ is true at all states in the agent's information set at which ϕ is true. The dual of the operator B i is denoted by B i , i.e.,

B i ϕ def = ¬B i ¬ϕ.
Similarly to Situation Calculus [START_REF] Reiter | Knowledge in Action: Logical Foundations for Specifying and Implementing Dynamical Systems[END_REF], we describe actions in terms of their positive and negative effect preconditions. In particular, we introduce two functions γ + and γ - with domain Agt × Act × Atm and codomain L -. The formula γ + (i, a, p) describes the positive effect preconditions of action a performed by agent i with respect to p, whereas γ -(i, a, p) describes the negative effect preconditions of action a performed by agent i with respect to p. Formula γ + (i, a, p) represents the conditions under which agent i will make p true by performing action a, if no other agent interferes with i's action; while γ -(i, a, p) represents the conditions under which i will make p false by performing a, if no other agent interferes with i's action. We assume that "making p true" means changing the truth value of p from false to true, whereas "making p false" means changing the truth value of p from true to false. The reason why an action's effect preconditions range over L -and not over L is that they should be independent from the effects of the action described by dynamic formulas of type [[δ]]ϕ.

Example 1. Let us consider the story given in Section 1, and use it to illustrate the γ + and γ -functions. We shall use the following vocabulary:

Agt ={Ann, John, Bot}, Act ={take, suggest, hide, skip}, Atm ={has John,drug , ingested John,drug , addicted John , pain John }

The actions' effect preconditions can be defined as:

γ + (Bot, suggest, p) = p for all p ∈ Atm, γ -(Bot, suggest, p) = ¬p for all p ∈ Atm, γ + (John, take, ingested John,drug ) = has John,drug , γ + (i, a, ingested John,drug ) = ⊥
Moreover, if a = take or i = John, we have:

γ -(i, a, ingested John,drug ) = ¬choose(take John )
Finally, for all a ∈ Act and i ∈ Agt, we have:

γ + (i, a, has John,drug ) = has John,drug ∧ ¬choose(hideAnn ) γ -(Ann, hide, has John,drug ) =
The effect preconditions specify that the speech act of suggestion has no effect on material facts. Ingesting the drug presupposes possession of it, while not ingesting it presupposes not having taken it. Finally, John will still have the drug unless Ann hides it, while John will not have the drug if Ann hides it.

Semantics

The semantics for the language L is a possible world semantics with accessibility relations associated with each modal operator, with a function designating the history starting in a given world, a plausibility function and a trichotomous utility function relative to histories. Definition 1. A model is a tuple M = W, H, ≡, (E i ) i∈Agt , P, U, V where: (i) W is a non-empty set of worlds, (ii) H : W -→ Hist is a history function, (iii) ≡ and every E i are equivalence relations on W , (iv)

P : W × Agt -→ {0, 1} is a plausibility function, (v) U : W × Agt -→ {0, 1, -1} is a utility function, and (vi) V : W -→ 2 Atm is a valuation function.
For each binary relation R ∈ {≡, E 1 , . . . , E n }, we set R(w) = {v ∈ W : wRv}. As usual p ∈ V(w) means that proposition p is true at world w. ≡-equivalence classes are called moments. If w and v belong to the same moment (i.e., w ≡ v), then the history starting in w (i.e., H(w)) and the history starting in v (i.e., H(v)) are said to be alternative histories (viz., histories starting at the same moment). The concept of moment is the one used in STIT logic [START_REF] Belnap | Facing the future: agents and choices in our indeterminist world[END_REF][START_REF] Horty | Agency and Deontic Logic[END_REF] and, more generally, in the Ockhamist theory of time [START_REF] Thomason | Combinations of tense and modality[END_REF][START_REF] Zanardo | Branching-time logic with quantification over branches: The point of view of modal logic[END_REF]. For every world w ∈ W , H(w) identifies the history starting in w. For notational convenience, for all ∈ JAct * , i ∈ Agt, a ∈ Act and w ∈ W , we write ;a i H(w) to mean that there is δ ∈ JAct such that δ(i) = a and ;δ H(w).

We define the actual choice function C act : W × Agt -→ Act: for every w ∈ W , i ∈ Agt and a ∈ Act, we have C act (w, i) = a iff there exists δ ∈ JAct such that δ(i) = a and δ H(w). Furthermore, we define the available choice function C avail : W × Agt -→ 2 Act : for every w ∈ W , i ∈ Agt and a ∈ Act, we have a ∈ C avail (w, i) iff there exists δ ∈ JAct and v ∈≡(w) such that δ(i) = a and δ H(v).

The equivalence relations E i are used to interpret the epistemic operators K i . The set E i (w) is the agent i's information set at world w: the set of worlds that agent i envisages at w or, shortly, agent i's set of epistemic alternatives at w. As E i is an equivalence relation, if wE i v then agent i has the same information set at w and v. The function P specifies the possibility value of a history for an agent. In particular, P(w, i) = 1 (resp. P(w, i) = 0) means that the history starting in w is considered plausible (resp. not plausible) by agent i. We define agent i's belief set at world w, denoted by B i (w), as the set of worlds in i's information set at w that i considers plausible:

B i (w) = E i (w) ∩ {v ∈ W : P(v, i) = 1}.
Consequently, the complementary set E i (w) \ B i (w) is the set of worlds that agent i envisages at w but that she does not consider plausible. Since E i is an equivalence relation, the following properties for belief hold. Note that B i ⊆ E i and if

wE i v then B i (w) = B i (v), where B i = {(w, v) ∈ W × W : v ∈ B i (w)}.
Moreover, B i is transitive and Euclidean. These properties correspond to the combination of belief and knowledge studied in [START_REF] Kraus | Knowledge, belief and time[END_REF].

The function U assigns the utility value U(w, i) of the history starting in w for agent i. In particular, U(w, i) = 1, U(w, i) = -1 and U(w, i) = 0, mean respectively that the history starting in w is good/bad/neutral for agent i. We impose the following three constraints on models. For all w, v ∈ W , δ ∈ JAct, ∈ JAct * , i ∈ Agt and a ∈ Act:

(C1) if for all i ∈ Agt there is u i ∈≡ (w) such that ;δ(i) i H(w), then there is u ∈≡(w) such that ;δ H(u); (C2) if there is v ∈≡(w) such that ;a i H(v) then, for every u ∈ E i (w), there is z ∈≡(u) such that ;a i H(z); (C3) if there is v ∈≡(w) such that ;a i H(v), then there is u ∈ E i (w) such that ;a i H(u); (C4) if w ≡ v then E i (w) = E i (v); (C5) B i (w) = ∅.
According to the Constraint C1, if every individual action in a joint action δ can be chosen at the end of the joint action sequence , then the individual actions in δ can be chosen simultaneously at the end of . The Constraint C1 is a variant of the assumption of independence of agents of STIT logic. More intuitively, this means that agents can never be deprived of choices due to the choices made by other agents. The Constraint C2 is a basic assumption about agents' knowledge over their abilities: if an agent i can choose action a at the end of the joint action sequence , then he knows this. In other words, an agent has perfect knowledge about the actions he can choose at the end of a joint sequence. The Constraint C3 characterizes the basic property of ex ante knowledge: if an agent i can choose action a at the end of joint action sequence , then there is a history that the agent considers possible in which he chooses action a at the end of the joint action sequence . In other words, for every action that an agent can choose, there is a history that the agent considers possible in which he chooses this action. According to Constraint C4, an agent's knowledge is moment-determinate, i.e., it does not depend on the specific history at which it is evaluated. This assumption is justified by the fact that the only thing which can vary at a given moment are the agents' choices, but not the agents' ex ante epistemic states. Finally, Constraint C5 is a normality requirement for beliefs: there should be at least a world in an agent's information set that the agent considers possible. L-formulas are interpreted relative to a model M = W, H, ≡, (E i ) i∈Agt , P, U, V and a world w in W as follows. (We omit boolean cases as they are standard.)

M, w |= occ( ) ⇐⇒ H(w) M, w |= plaus i ⇐⇒ P(w, i) = 1 M, w |= good i ⇐⇒ U(w, i) = 1 M, w |= badi ⇐⇒ U(w, i) = -1 M, w |= neutrali ⇐⇒ U(w, i) = 0 M, w |= [[δ]]ϕ ⇐⇒ if M, w |= occ(δ) then M δ , w |= ϕ M, w |= ϕ ⇐⇒ ∀v ∈ W : if w ≡ v then M, v |= ϕ M, w |= Kiϕ ⇐⇒ ∀v ∈ W : if wEiv then M, v |= ϕ
where model M δ is defined according to Definition 2 below. Note that the belief operator B i we defined in Section 2.1 as an abbreviation has the following interpretation:

M, w |= B i ϕ if and only if ∀v ∈ W : if wB i v then M, v |= ϕ.
Definition 2 (Update via joint action). Let M = W, H, ≡, (E i ) i∈Agt , P, U, V be a model. The update of M by joint action δ is the tuple:

M δ = W δ , H δ , ≡ δ , (E δ i ) i∈Agt , P δ , U δ , V δ
where:

W δ = {w ∈ W : M, w |= occ(δ)} H δ (w) = h if H(w) = δ ;h for some δ ∈ JAct ≡ δ = ≡ ∩(W δ × W δ ) E δ i = Ei ∩ (W δ × W δ ) P δ (w, i) = P(w, i) if Bi(w) ∩ W δ = ∅ P δ (w, i) = 1 otherwise U δ (w, i) = U(w, i) V δ (w) = V(w) \ p : ∃a ∈ Act, i ∈ Agt : δ(i) = a and M, w |= γ -(i, a, p) and ∃b ∈ Act, j ∈ Agt : δ(j) = b and M, w |= γ + (j, b, p) ∪ p : ∃a ∈ Act, i ∈ Agt : δ(i) = a and M, w |= γ + (i, a, p) and ∃b ∈ Act, j ∈ Agt : δ(j) = b and M, w |= γ -(j, b, p)
The performance of a joint action δ modifies the physical facts via the positive effect preconditions and the negative effect preconditions, defined above (see the definition of V δ ). In particular, if there is an action in the joint action δ whose positive effect preconditions with respect to p hold and there is no other action in the joint action δ whose negative effect preconditions with respect to p hold, then p will be true after the occurrence of δ; if there is an action in the joint action δ whose negative effect preconditions with respect to p hold and there is no other action in the joint action δ whose positive effect preconditions with respect to p hold, then p will be false after the occurrence of δ. Besides, the occurrence of the joint action δ makes the current history advance one step forward (see the definition of H δ ). As to the equivalence relations ≡ and E i for historical necessity and ex ante knowledge, they are restricted to the set of worlds in which the joint action δ occurs (see the definitions of ≡ δ and E δ i ). The joint action δ does not modify the agents' utilities over histories (see the definitions of U δ ). Finally, as for the update of the epistemic plausibility function P, two cases are possible. If the update removes all plausible worlds from the agent's information set, then the plausibility function is reinitialized and all worlds in the agent's information set become plausible. This is a form of drastic revision which guarantees preservation of Constraint C5. Otherwise, nothing changes and the agent keeps the same beliefs as before the update. As stated by the following proposition, the update via a joint action preserves the constraints on models.

Proposition 1. If M is a model then M δ is a model too.
Notions of validity and satisfiability for formulas in L relative to models is defined in the usual way. The fact that a formula ϕ is valid is noted |= ϕ.

Axiomatization

We call EVAL (Epistemic Volitional Action Logic) the extension of propositional logic by the principles in Figures 1, 2 and 3 and the following rule of replacement of equivalents:

ϕ 1 ↔ ϕ 2 ψ ↔ ψ[ϕ 1 /ϕ 2 ] (RE)
They consist in (i) a theory for the special atomic formulas, (ii) S5-principles for the epistemic and historical necessity operators, and (iii) reduction axioms which allow to eliminate all the dynamic operators [[δ]] from formulas.

As the next theorem indicates, they provide an axiomatics.

Theorem 1. The logic EVAL is sound and complete for the class of models of Definition 1.

Regarding complexity, we believe that checking satisfiability of formulas in the fragment L -can be polynomially reduced to satisfiability checking for star-free PDL with converse of atomic programs that, by adapting the technique in [START_REF] Giacomo | Eliminating "converse" from converse PDL[END_REF], can be proved to be in PSPACE. Given the polysize satisfiability preserving reduction from L-formulas to L --formulas based on the reduction axioms of Proposition 3, this guarantees that checking satisfiability of formulas in L is also in PSPACE. As for PSPACE-hardness, it follows from the fact that EVAL is a conservative extension of multi-agent epistemic logic S5 n , whose satisfiability problem is known to be PSPACE-hard [START_REF] Halpern | A guide to completeness and complexity for modal logics of knowledge and belief[END_REF]. Future work will be devoted to prove this conjecture. Nevertheless, it is of interest to have a decidable logic to deal with artificial agents.

occ( ) → δ∈JAct occ( ;δ) (OneJAct) occ(nil ) (EmptySeq) occ( ;δ) → ¬occ( ;δ ) if δ = δ (UniqueJAct) occ( ) → occ( ) if (SubSeqJAct) good i ∨ neutrali ∨ badi (ComplUtil)
xi → ¬yi if x, y ∈ {good, neutral, bad} and x = y (UniqueUtil)

( i∈Agt ♦choose( ,ai)) → ♦choose( ,δAgt ) (IndepAgt) can( ,ai) → Kican( ,ai) (KnowCan) can( ,ai) → Kichoose( ,ai) (ExAnteKnow) Kiϕ → Kiϕ (MomDetKnow) Kiplaus i (NormBel)
Fig. 1: Theory for the atomic formulas

ϕ ∧ (ϕ → ψ) → ψ (K ) ϕ → ϕ (T ) ϕ → ϕ (4 ) ¬ ϕ → ¬ ϕ (5 ) ϕ ϕ (Nec )

Agency and Types

We now represent agency operators and two opposite rationality types, namely, the optimistic (or risk seeking) agent Rat opt i and the pessimistic (or risk averse) agent Rat pess i :

Rat opt i def = a i ∈Act i choose(ai) ∧ b i ∈Act i : b i =a i can(bi) → ( Bi(neutrali ∧ choose(bi)) → Bi((neutrali ∨ good i ) ∧ choose(ai)))∧ ( Bi(good i ∧ choose(bi)) → Bi(good i ∧ choose(ai))) [[δ]]¬ϕ ↔ (occ(δ) → ¬[[δ]]ϕ) [[δ]](ϕ ∧ ψ) ↔ ([[δ]]ϕ ∧ [[δ]]ψ) [[δ]]p ↔ (occ(δ) → (( i∈Agt γ + (i, δ(i), p)∧ j∈Agt:j =i ¬γ -(j, δ(j), p))∨ (p ∧ i∈Agt ¬γ -(i, δ(i), p))∨ (p ∧ i∈Agt γ + (i, δ(i), p)))) [[δ]]occ( ) ↔ (occ(δ) → occ(δ; )) [[δ]]plaus i ↔ occ(δ) → Bi¬occ(δ)∨ ( Biocc(δ) ∧ plaus i ) [[δ]]good i ↔ (occ(δ) → good i ) [[δ]]neutrali ↔ (occ(δ) → neutrali) [[δ]]badi ↔ (occ(δ) → badi) [[δ]] ϕ ↔ (occ(δ) → (occ(δ) → [[δ]]ϕ)) [[δ]]Kiϕ ↔ (occ(δ) → Ki(occ(δ) → [[δ]]ϕ))
Fig. 3: Reduction axioms for the dynamic operators

Rat pess i def = a i ∈Act i choose(ai) ∧ b i ∈Act i : b i =a i can(bi) → ( Bi(neutrali ∧ choose(ai)) → Bi((neutrali ∨ badi) ∧ choose(bi)))∧ ( Bi(badi ∧ choose(ai)) → Bi(badi ∧ choose(bi)))
As Proposition 2 highlights, a rationally optimistic agent makes a certain choice from her set of available choices, if she believes that its best possible outcome is at least as good as the best possible outcome of the other available choices. A rationally pessimistic agent makes a certain choice, if she believes that its worse possible outcome is at least as good as the worse possible outcome of the other available choices. Proposition 2. Let M = W, H, ≡, (E i ) i∈Agt , P, U, V be a model and let w ∈ W . Then, M, w |= Rat i iff:

Cact (w, i) ∈ arg max b∈C avail (w,i) max v∈B i (w): C act (v,i)=b U(v, i) if = opt Cact (w, i) ∈ arg max b∈C avail (w,i) min v∈B i (w): C act (v,i)=b U(v, i), if = pess
In the language L, we can express a variety of agency operators from STIT theory [START_REF] Belnap | Facing the future: agents and choices in our indeterminist world[END_REF][START_REF] Horty | Agency and Deontic Logic[END_REF]. Indeed, EVAL can be seen as a variant of STIT with explicit actions: while in STIT an action is identified with the result brought about by a coalition (i.e., in STIT one can only express that a given coalition H sees to it that ϕ), in EVAL an action is identified both with the result brought about by the coalition and with the means used by the coalition to bring about the result. For instance, the so-called 'Chellas' operator [H:cstit] of STIT is definable in our language as follows:

[H:cstit]ϕ def = δ H ∈JAct H choose(δH ) ∧ δ ∈JAct: ∀i∈H,δ H (i)=δ (i) choose(δ ) → ϕ
This means that the coalition H sees to it that ϕ if and only if, the agents in H choose some joint action δH such that, no matter what the agents outside H choose, if the agents in H choose δH then ϕ will be true. The 'deliberative' STIT operator is also definable:

[H:dstit]ϕ def = [H:cstit]ϕ ∧ ¬ ϕ.
Deliberative STIT adds the negative condition ¬ ϕ to Chellas STIT. It captures the fact that for a coalition H to see to it that ϕ, ϕ should not be inevitable (according to deliberative STIT, action is not compatible with necessity). Having formalized rationality types, we can define two rational STIT operators, [H:rstit] opt and [H:rstit] pess :

[H:rstit] ϕ def = i∈H Rat i → [H:cstit]ϕ
with ∈ {opt, pess}. Formula [H:rstit] opt ϕ (resp. [H:rstit] pess ϕ) has to be read "coalition H sees to it that that ϕ in an optimistic (resp. pessimistic) rational way". The latter means that if all agents in H are optimistically (resp. pessimistically) rational, then then they see to it that ϕ. Note that we could also define deliberative STIT counterparts of the previous (Chellas STITbased) rational STIT operators, in which operator [H:cstit] is replaced by operator [H:dstit]. Our language also integrates a temporal dimension allowing us to express the LTL operator 'next':

Xϕ def = δ∈JAct δ ϕ.

Influence, Persuasion and Deception

In this section, we define influence, persuasion, and deception. We also highlight the relationship between influence and persuasion, namely when an agent is persuaded to do an action a, it means that she may have acted differently but found rational to do a. Firstly, let us define influence. Influencing consists in an agent i (the influencer) intentionally seeing to it that another agent j (the influencee) rationally sees to it that a proposition ϕ. As we have two kinds of rationality types, we can define two kinds of influence with ∈ {opt, pess}.

Influences (i, j, ϕ) def = K post i [{i}:dstit]X[{j}:rstit] ϕ
Let us now define persuasion as the intentional action of changing another agent's mental state [START_REF] O'keffe | Persuasion: Theory and Research[END_REF][START_REF] Poggi | The goals of persuasion[END_REF]. Persuasion consists in an agent i (the persuader) knowingly seeing to it that another agent j (the persuadee) believes that a certain fact ϕ is true.

Persuades(i, j, ϕ) def = K post i [{i}:dstit]XBjϕ.
As the persuader knowingly sees to it that the persuadee will have a given belief, this definition expresses two different kinds of persuasion. One agent i may persuade another agent j either acquire a new belief that she does not have or to maintain a belief that she already has. Interestingly, a relationship between persuasion and influence can be deduced. An agent i influences an agent j to make a given choice a if i persuades j that choosing a is good for j and that all other choices are not good while knowing that j will be optimistically rational and can possibly choose a.

Proposition 3. Let i, j ∈ Agt and aj, bj ∈ Actj. Then,

|= (Persuades(i, j, Bj(choose(aj) ∧ good j )∧ b =a (choose(bj) → ¬good j )) ∧ KiX(Rat opt j ∧ ♦choose(aj))) → Influences opt (i, j, choose(aj))
Firstly, by using (Kiϕ → Ki ϕ), (♦ϕ∧ ϕ → ♦(ϕ∧ψ)), (Bjϕ → Bjϕ), (Xϕ∧Xψ → X(ϕ∧ ψ)) and previous definition of persuasion, we easily prove that (Persuades(i, j, choose(aj)

→ good j ∧ b =a (choose(bj) → ¬good j )) ∧ KiX(Rat opt j ∧ ♦choose(aj))) → Ki[{i}:dstit] X(Rat opt j ∧ ♦choose(aj) ∧ Bj(choose(aj) → good j ∧ b =a (choose(bj) → ¬good j ))))). Sec- ondly, since (Ki[{i}:dstit]X(Rat opt j ∧♦choose(aj)∧ Bj(choose(aj) → good j ∧ b =a (choose(bj) → ¬good j ))) → Ki[{i}:dstit]X(Rat opt j → [{j}:cstit]choose(aj))
), -intuitively this tautology means that since the only one good action for agent j is aj and since all other actions are either badj or neutralj (because of ¬good j ), necessarily the only optimistic rational choice for j is to choose aj -, and since we have the following equivalence : (Ki[{i}:dstit]X(Rat opt j → [{j}:cstit]choose(aj)) ≡ Influences opt (i, j, choose(aj))) we then immediately prove by modus ponens the theorem.

The previous validity shows how an optimistically rational agent can be influenced through persuasion. Similar theorems can be proved for pessimistically rational agents but are omitted due to space constraints. The idea in this case is simply to persuade agent j that action a has no bad consequence while all other actions have it. Thank to persuasion we can now define deception. Deception consists in persuasion of a proposition ϕ under the assumption that the persuader believes that ϕ is false. For instance, consider the student that tells the professor that he could not study for family commitments (when he had no such commitments).

Deceives(i, j, ϕ) def = Persuades(i, j, ϕ) ∧ BiX¬ϕ Let us notice that we only capture successful deception, and we do not explicitly model deception by truthfully telling. Indeed, truthfully telling is simply captured by persuasion, as we do not make assumption on the persuader's intention (an agent can simply persuade another one in a malevolent intention, which capture truthfully telling deception). Smooth-talking is weaker than deceiving, since it only requires that the persuader is uncertain whether ϕ is true or false. Consider the journalist spreading the news that Obama was a muslim, without having any clue on the matter. Formally, it is equivalent to what Sakama et al. called "bullshitting" [START_REF] Sakama | A formal account of dishonesty[END_REF].

PersuadesBySmoothTalking(i, j, ϕ) def = Persuades(i, j, ϕ) ∧ ¬K post i X¬ϕ ∧ ¬K post i ¬X¬ϕ
Let us notice we use ex post knowledge in this definition as we want to represent to complete uncertainty about ϕ. We do not want the persuader being able to belief ϕ being either true or false.

We can also distinguish three types of belief deception, a benevolent, a malevolent and a reckless form. In the malevolent form, the persuader i deceives the persuadee j into believing a proposition ϕ, given that i believes that believing ϕ will have bad consequences for j. An agent z will accomplish an action if z believes that the action has good consequences (and no bad ones). Consider for instance the case of the charlatan offering a miraculous cure for boldness. Or consider the website ensuring gamblers that they are going with certainty to gain a lot of money.

MalevolentDeception(i, j, ϕ) def = Deceives(i, j, ϕ) ∧ BiX(Bjϕ → badj)

A benevolent deception consists for the deceiver to transmit a false proposition, believing that believing that proposition is good for the deceived. Consider for instance the atheist philosopher, who persuades the credulous citizens that if they act morally, they are going to heaven, in order to induce them to behave well.

BenevolentDeception(i, j, ϕ) def = Deceives(i, j, ϕ) ∧ BiX(Bjϕ → good j )
The last form is reckless deception. It consists for the deceiver to transmit a false proposition, while not knowing whether that proposition is good or bad for the deceived. Consider for instance the case of Boris Johnson, who did not know (or did not care) whether Brexit would be good or bad for Britain, but induced people to believe that Brexit would provide money for the NHS, a belief that would lead them to vote for Brexit. He did not know whether this belief (which led to Brexit) would be good or bad for them.

RecklessDeception(i, j, ϕ) def = Deceives(i, j, ϕ) ∧ ¬K post i X(Bjϕ → good j ) ∧ ¬K post i X(Bjϕ → badj)
Application to the running example. Let us consider the actions' effect preconditions in Example 1, given the following hypotheses on agents' knowledge: (1, 2) the bot knows that its suggestion will persuade john that taking the pill will remove his pain and he will not get addicted, (3) the Bot and Ann know that John will also believe that this is good for him, (4) the bot knowingly makes the suggestion and it knows if it makes the suggestion, John will be aware of it, [START_REF] Belnap | Facing the future: agents and choices in our indeterminist world[END_REF] the Bot knows that John has the drug, that Ann does not hide the drug and that John can choose to not take the drug, (6) Ann will knowingly hide the drug, [START_REF] Bonzon | On the outcomes of multiparty persuasion[END_REF] Ann knows that John will possibly have the drug and can choose to take it, (8) the Bot knows that John will be optimistically rational. Finally, (9) means the bot knows that ingesting the drug will create addiction, and being addicted is a bad thing. From premises ϕ1, . . . ϕ8, we can then deduce Proposition 4 which means that, in a first step, the Bot influences John to ingest the drug by persuasion, i.e. suggesting him that his unique good option is to take the drug. In the next step, Ann influences John to not ingest the drug by removing the choice to take the drug. With ϕ9, we can also deduce Proposition 5 which means that the bot malevolently deceives John about the fact that ingesting the drug will not make him addict.

ϕ1 def = KBot B John (choose(suggest Bot ) → X(¬pain John ↔ choose(take John ))) ϕ2 def = KBot B John (choose(suggest Bot ) → X(choose(take John ) → ¬addicted John )) ϕ3 def = KBot B John X(good John ↔ (¬addicted John ∧ ¬pain John ))∧ KAnn B John X(good John ↔ (¬addicted John ∧ ¬pain John )) ϕ4 def = KBot (choose(suggest Bot ) ∧ (choose(suggest Bot ) → K John (choose(suggest Bot )))

|=

(ϕ1 ∧ ϕ2 ∧ ϕ3 ∧ ϕ4 ∧ ϕ5 ∧ ϕ6 ∧ ϕ7 ∧ ϕ8) → Influences opt (Bot, John, Xingested John,drug ) ∧ X Influences opt (Ann, John, X¬ingested John,drug )
Firstly, we can prove that the bot knows that John knows it suggests him to take drug by applying axiom K on KBot i.e.: ϕ4 → KBotK John choose(suggestBot) We also consider the following theorem, that can be easily proved, ∀a ∈ Act, ∀i, j ∈ Agt: (Bj(choose(ai) → Xϕ) ∧ Kjchoose(ai)) → [{i}:cstit]XBjϕ. The proof relies on the fact that knowledge implies beliefs and BjXϕ → XBjϕ . By generalization with and since Kjchoose(ai) → choose(ai), we immediately prove the STIT and so the theorem. This theorem means that if one agent j believes that an action made by another agent i will imply a consequence and j knows i does this action, then the agent i sees to it that it will imply the agent j believes this consequence to be true. The theorem allows us to prove that if the bot suggests John to take the drug then, the bot knows John will believe taking the drug implies something good for him i.e. the bot persuades John of it. Starting from the assumptions, we prove this by substitution, augmentation, generalization (with KBot) and normal properties of modalities K, B and X, the following theorem. Note that the robot has an ex-post knowledge of the consequences of its action of suggesting. We have :

(ϕ1 ∧ ϕ2 ∧ ϕ3 ∧ ϕ4) → K post Bot (B John (choose(suggestBot) → X((¬pain John ↔ choose(take John )) ∧ (choose(take John ) → ¬addicted John ))) ∧ K John choose(chooseBot)) → [Bot:cstit]XB John (choose(take John ) → good John )
By contraposition on ϕ3, we prove that it is not good for John to be addicted or having pain and allows us to prove that John believes that the only good option for him is to take the drug :

(ϕ1 ∧ ϕ2 ∧ ϕ3) → KBot B John ((addicted John → ¬choose(take John ))
∧ (pain John → ¬choose(take John )) ∧ (¬choose(take John ) ≡ choose(skip John )))

→ KBot B John (¬choose(take John ) ≡ ¬good John ) Since John is assumed to be rationally optimistic with the hypothesis ϕ8 and as the unique good option for John is to take the drug, then John takes the drug i.e. we have the following validity:

(choose(take John ) → good John ) ∧ Rat opt John ∧ b =take (choose(b John ) → ¬good John ) → choose(take John )

As John is rationally optimistic, he rationally sees to it that he ingests drugs. Finally, let us notice that since John takes the drug, then John is also able to take the drug due to the theorem choose(take John ) → ♦choose(take John ). Furthermore, John can either do the action skip or take drug . Thus by generalization, all agents know the following validity:

(♦choose(take John ) ∧ b =take ♦choose(b John )) → ♦¬choose(take John ) Furthermore since the preconditions for taking drugs is to have drugs in regard to the frame, i.e. γ + (John, take, ingested John,drug ) = has John,drug , and having drugs implies that Ann does not hide drugs i.e. ∀a ∈ Act and i ∈ Agt γ + (i, a, has John,drug ) = has John,drug ∧ ¬choose(hideAnn ) and we have these preconditions by ϕ5, we deduce that John is able to take drugs or skipping. Then, as γ -(i, a, ingested John,drug ) = ¬choose(take John ), i.e. not taking the drugs would imply ¬Xingested John,drug , we have that it is not necessary for John to take drugs, with the previous validity. Thus, by applying axioms in Fig 3, we deduce the negative part of deliberative STIT operator, i.e. the bot deliberately sees to it that John is going to ingest drugs:

(ϕ1 ∧ ϕ2 ∧ ϕ3 ∧ ϕ4 ∧ ϕ5 ∧ ϕ8) → K post
Bot [Bot:dstit]X(Rat opt John → Xingested John,drug )

Consequently:

(ϕ1 ∧ ϕ2 ∧ ϕ3 ∧ ϕ4 ∧ ϕ5 ∧ ϕ8) → Influences opt (Bot, John, Xingested John,drug )

For the second part of the implication, let us notice that since Ann hides drugs, John has only one possible action which is to skip. It is necessarily a rationally pessimistic choice but also an optimistic one, because of the following theorem:

♦choose(skip John ) ∧ ¬♦¬choose(skip John ) → Rat opt John ∧ Rat pess

John

With the same method and with hypothesis ϕ6 and ϕ7 we can prove the second part, i.e.:

(ϕ1 ∧ ϕ2 ∧ ϕ3 ∧ ϕ4 ∧ ϕ5 ∧ ϕ6 ∧ ϕ7 ∧ ϕ8) →
XInfluences opt (Ann, John, X¬ingested John,drug ) Proposition 5.

|= (ϕ1 ∧ ϕ2 ∧ ϕ3 ∧ ϕ4 ∧ ϕ5 ∧ ϕ8 ∧ ϕ9) → MalevolentDeception(Bot, John, ingested John,drug ∧ ¬addicted John,drug ))

We can prove it with the same method as previously. By adding hypothesis ϕ9, it allows to prove a MalevolentDeception from the bot since it persuades John that he will not be addicted if he takes drugs while the bot knows the contrary, and being addicted is bad for John.

Conclusion

We have modelled influence on choices through belief change. To the end, we have introduced a logical framework covering capabilities, choices and mental states, and have expressed, through the combination of these notion, rationality and agency operators. We have also expressed formally the way in which persuasion leads to influence, i.e. the way in which by modifying the beliefs of agents, the latter can be induced to act accordingly. This has enabled us to distinguish two ways in which agents can be influenced: (a), through persuasion, i.e. by changing their beliefs, or (b) though regimentation, i.e. by changing the options that are available to them. Moreover, it allows us to express different kind of deception, such as malevolent and benevolent deception.

In the future, we would like to extend the framework towards a quantitative model in order to represent graded beliefs [START_REF] Budzyńska | A logic for reasoning about persuasion[END_REF]. Other perspectives are to reformulate our framework into a concurrent game structure [START_REF] Alur | Alternating-time temporal logic[END_REF] to deal with a richer notion of time, and to incorporate emotions as in the OCC model [START_REF] Adam | OCC's emotions: a formalization in a BDI logic[END_REF] to express richer notions of rationality.

  A history is good for an agent if the agent obtains what she likes and avoids what she dislikes along it. It is bad for the agent if the agent does not avoid what she dislikes and does not obtain what she likes along it. Finally, it is neutral for the agent if either the agent does not obtain what she likes and avoids what she dislikes along it, or the agent obtains what she likes and does not avoid what she dislikes along it. Our simplified account of utility presupposes that every agent is identified with a single appetitive desire (i.e., what the agent likes) and a single aversive desire (i.e., what the agent dislikes).

Fig. 2 :

 2 Fig. 2: S5-system for knowledge and historical necessity with ∈ { } ∪ {K1} ∪ . . . ∪ {Kn}

ϕ5 def===

 ϕ5 KBot ( (has John,drug ) ∧ ¬choose(hideAnn ) ∧ ♦¬choose(take John )) ϕ6 def XKAnn choose(hideAnn ) ϕ7 def = KAnn X♦(choose(take John ) ∧ has John,drug ) KBot X((ingested John,drug → addicted John ) ∧ (addicted John ↔ bad John ))


Antheor-Ceyreste Antheor-Danio

8 different fragments of L. humile colonies from Provence (France).

Behavioural assays :

-We assessed the intraspecific aggression rate using a test which measure the level of agression between two groups of 100 individuals.

-For each test, workers were placed in closed artificial nests (6 x 4.5 x 5 cm) with soil.

-Artificial nest were connected via plastic tubes to a common foraging arena (6 x 9x 5 cm).

-Workers were kept in artificial nests for one hour of acclimation.

-Behavioural tests were conducted during 20 min, using a scale :

-3 or 4 replicates were conducted for each pair of colonies.

Chemical analysis:

We perform gas chromatography of 1 worker for each colony, 10 replicates were conducted for each colony.

  In introduced areas, Argentine ant (Linepithema humile) is known to form huge colony structured called supercolony with free exchange of individuals among nests. Study goal : Here, we present analyzes of Argentine ant recognition cues (cuticular hydrocarbons) and aggressive interactions between 8 coastal sites in Provence-Côte d'Azur (France). on all sites revealed a marked discrimination between colonies. Discrimination was achieved throught 65 peaks of their major cuticular hydrocarbons. Chemical profiles between the colonies differed qualitatively and quantitatively. the results of behavioural tests between two sites belonging to MaIn Europeen Supercolony (Giraud & al., 2002). Behavioral tests between Giens and the other supercolonies are represented in pink. Red dots represent the results of behavioral tests between two sites belonging to a New Supercolony (Blight & al., 2009). The behavioral test between the New Supercolony and the MaIn Supercolony are represented in yellow. Results of behavioral test of 200 workers. Dots show the maIn aggression level between workers coming from different location. The difference * in maIn aggression level between two groups is statistically significant (Mann-Whitney test , p<0.001). The Giens population share a part of the chemical characteristics of each of the two groups. Giens profiles present peaks which were also found on chemical profiles of the Mean supercolony (eg. 90A). Furthermore, others peaks (eg. 67) were found only on the profile of the New supercolony (Blight. & al, 2009).


Marion Thauvin

Irène Amblard

Christine Rampon

Aurélien Mourton

Isabelle Queguiner

Chenge Li

Arnaud Gautier

Alain Joliot

Michel Volovitch

Sophie Vriz
email: vriz@u-paris.fr

Reciprocal regulation of Shh trafficking and H2O2 levels via a noncanonical BOC-Rac1 pathway

Among molecules that bridge environment, cell metabolism, and cell signaling, H2O2 recently appeared as an emerging but central player. Its level depends on cell metabolism and environment and was recently shown to play key roles during embryogenesis, contrasting with its long-established role in disease progression. We decided to explore whether the secreted morphogen Sonic hedgehog (Shh), known to be essential in a variety of biological processes ranging from embryonic development to adult tissue homeostasis and cancers, was part of these interactions. Here, we report that H2O2 levels control key steps of Shh delivery and that physiological in vivo modulation of H2O2 levels changes Shh distribution and tissue patterning. A feedback loop exists in which Shh trafficking controls H2O2 synthesis via a non-canonical BOC-Rac1 pathway, leading to cytoneme growth. Our findings reveal that Shh directly impacts its own distribution, thus providing a molecular explanation for the robustness of morphogenesis to both environmental insults and individual variability.

Introduction

Hedgehog proteins (Hh in Invertebrates and Shh or paralogs in Vertebrates, collectively Hhs) are secreted morphogens playing key roles in biological processes ranging from embryonic development, proliferation, adult tissue homeostasis, and cancers [START_REF] Briscoe | The mechanisms of Hedgehog signalling and its roles in development and disease[END_REF][START_REF] Ingham | Hedgehog signaling in animal development: paradigms and principles[END_REF][START_REF] Mcmahon | Developmental roles and clinical significance of hedgehog signaling[END_REF]. Signaling by Hhs implies a complex protein journey that includes several steps incompletely characterized, and possibly depending on physiological context [START_REF] Guerrero | Hedgehog and its circuitous journey from producing to target cells[END_REF][START_REF] Hall | Dispatching Sonic Hedgehog: Molecular Mechanisms Controlling Deployment[END_REF].

Recently, we observed that Shh compartimentation was modified ex vivo by H2O2 (Gauron et al., 2016), suggesting that H2O2 could regulate Shh secretion process. Besides, preliminary data suggested that Shh controls H2O2 levels during cell plasticity and tissue remodeling in adults [START_REF] Meda | Nerves Control Redox Levels in Mature Tissues Through Schwann Cells and Hedgehog Signaling[END_REF]. Hydrogen peroxide (H2O2) has long been exclusively considered as a deleterious molecule damaging cellular integrity and function. It is now becoming evident that H2O2 also contributes to bona fide physiological processes notably through protein cysteine targeting [START_REF] Holmstrom | Cellular mechanisms and physiological consequences of redox-dependent signalling[END_REF][START_REF] Sies | Hydrogen peroxide as a central redox signaling molecule in physiological oxidative stress: Oxidative eustress[END_REF][START_REF] Sies | Reactive oxygen species (ROS) as pleiotropic physiological signalling agents[END_REF].

The physiological role of H2O2 in living systems has gained increased interest and has been investigated in different models of development and regeneration [START_REF] Breus | Genetically encoded thiol redox-sensors in the zebrafish model: lessons for embryonic development and regeneration[END_REF][START_REF] Coffman | Redox regulation of development and regeneration[END_REF][START_REF] Rampon | Hydrogen Peroxide and Redox Regulation of Developments[END_REF][START_REF] Timme-Laragy | Redox stress and signaling during vertebrate embryonic development: Regulation and responses[END_REF]. Several studies have revealed strong spatio-temporal variation of H2O2 levels in live animals of various species (fly, nematode, zebrafish, Xenopus) during these processes [START_REF] Albadri | Redox Signaling via Lipid Peroxidation Regulates Retinal Progenitor Cell Differentiation[END_REF][START_REF] Albrecht | In vivo mapping of hydrogen peroxide and oxidized glutathione reveals chemical and regional specificity of redox homeostasis[END_REF][START_REF] Amblard | H2O2 and Engrailed 2 paracrine activity synergize to shape the zebrafish optic tectum[END_REF][START_REF] Bazopoulou | Developmental ROS individualizes organismal stress resistance and lifespan[END_REF][START_REF] Gauron | Hydrogen peroxide (H2O2) controls axon pathfinding during zebrafish development[END_REF][START_REF] Gauron | Sustained production of ROS triggers compensatory proliferation and is required for regeneration to proceed[END_REF][START_REF] Han | Ca(2+)-Induced Mitochondrial ROS Regulate the Early Embryonic Cell Cycle[END_REF][START_REF] Katikaneni | Lipid peroxidation regulates long-range wound detection through 5-lipoxygenase in zebrafish[END_REF][START_REF] Knoefler | Quantitative in vivo redox sensors uncover oxidative stress as an early event in life[END_REF][START_REF] Love | Amputation-induced reactive oxygen species are required for successful Xenopus tadpole tail regeneration[END_REF][START_REF] Meda | Nerves Control Redox Levels in Mature Tissues Through Schwann Cells and Hedgehog Signaling[END_REF][START_REF] Mendieta-Serrano | NADPH-Oxidase-derived reactive oxygen species are required for cytoskeletal organization, proper localization of E-cadherin and cell motility during zebrafish epiboly[END_REF][START_REF] Niethammer | A tissue-scale gradient of hydrogen peroxide mediates rapid wound detection in zebrafish[END_REF][START_REF] Pak | Ultrasensitive Genetically Encoded Indicator for Hydrogen Peroxide Identifies Roles for the Oxidant in Cell Migration and Mitochondrial Function[END_REF][START_REF] Tao | Genetically encoded fluorescent sensors reveal dynamic regulation of NADPH metabolism[END_REF]. The correlation between mild oxidative bursts and developmental events led to early suggestions of a mechanistic link between them [START_REF] Covarrubias | Function of reactive oxygen species during animal development: passive or active?[END_REF][START_REF] Hernandez-Garcia | Reactive oxygen species: A radical role in development?[END_REF][START_REF] Sies | Role of metabolic H2O2 generation: redox signaling and oxidative stress[END_REF]. Since the patterning of a developing embryo relies on the graded activity of morphogens [START_REF] Lander | Morpheus unbound: reimagining the morphogen gradient[END_REF][START_REF] Rogers | Morphogen gradients: from generation to interpretation[END_REF][START_REF] Turing | The Chemical Basis of Morphogenesis[END_REF], we decided to precisely determine by which mechanism H2O2 could regulate Shh trafficking. We set up a quantitative assay to measure the efficiency of each step of Shh's journey and demonstrate that H2O2 inhibits Shh secretion but enhances Shh internalization. Shh internalization per se enhances endogenous H2O2 levels via a Rac1/NADPH oxidase pathway that induces filopodia growth and regulates Shh trafficking in an H2O2/Shh feedback loop.

Results and Discussion

H2O2 affects Shh trafficking ex vivo

To analyse the mechanism by which H2O2 levels could affect Shh trafficking, we used an ex vivo system enabling accurate quantification of this process. HeLa cells were chosen to avoid complex feedback loops: these cells do not express endogenous Shh or respond to the canonical Shh signaling pathway but otherwise express pathway components known to be involved in Shh trafficking (lipidation, access to the extracellular space, endocytosis, and delivery to receiving cells, Supplementary Table S1). Shh trafficking from producing to receiving cells is a circuitous journey. While some aspects are still a matter of debate, there is no doubt that cis-endocytosis in producing cells can be part of the process [START_REF] Guerrero | Hedgehog and its circuitous journey from producing to target cells[END_REF][START_REF] Petrov | Sending and Receiving Hedgehog Signals[END_REF]. Making use of Shh constructs (mouse sequence) tagged according to [START_REF] Chamberlain | Notochordderived Shh concentrates in close association with the apically positioned basal body in neural target cells and forms a dynamic gradient during neural patterning[END_REF] to preserve the qualitative properties of the protein, we first verified that HeLa cells recapitulated the overall traffic of Shh (Fig. 1A-D). To distinguish different steps of the Shh journey (Fig. 1A), we exploited a double tag combining a fluorogen-activating peptide (YFAST) and a classical fluorescent protein (mCherry).

Contrary to mCherry, which chromophore takes time to mature, YFAST fluoresces instantly after fluorogen addition [START_REF] Plamont | Small fluorescence-activating and absorption-shifting tag for tunable protein imaging in vivo[END_REF] , allowing Shh detection from the beginning of its journey in the ER (step 1 in Fig. 1A) too early for mCherry detection. However, YFAST is sensitive to pH and cannot fluoresce in endosomes (step 2 in Fig. 1A), contrary to mCherry.

When finally sent for receiving cells (step 3 in Fig. 1A) Shh should be detectable via both tags. Transfection of HeLa cells with such a construct indeed confirmed the usefulness of this cell line for our purposes (Fig. 1B). At a steady state, the majority of cells display diffuse green staining, including the ER, as well as large red spots about the size of endosomes (Fig. 1B). HeLa cells with cellular protrusions associated with yellow puncta were found, indicating detection via both tags of Shh en route to receiving cells (Fig. 1B). The end of the journey was also easy to image by co-culturing transfected and untransfected HeLa cells. As shown in Fig. 1C, when transfected and untransfected cells were nearby, Shh-mCherry could be detected in the untransfected cells, and the arrangement of cytoplasmic protrusions between the two cells suggested a transfer occurring via filopodia.

To reach an adequate level of precision in the analysis of H2O2 effect on Shh trafficking, we needed to synchronize its secretion. We thus made use of the RUSH system [START_REF] Boncompain | Synchronization of secretory protein traffic in populations of cells[END_REF], where Shh fused to the streptavidin binding peptide (SBP) is retained in the ER of cells expressing streptavidin fused to the KDEL retrieving signal until the addition of biotin.

As shown in Fig. 1D, before biotin addition (t=0), Shh-SBP-AFP (mCherry here) was hooked in the ER, and biotin addition allowed us to calibrate the timing of the Shh journey using realtime imaging. It takes approximately 30 to 40 min for the bulk of Shh to reach the Golgi (very similar to many secreted proteins in these conditions, e.g., Wnt3a [START_REF] Moti | Wnt traffic from endoplasmic reticulum to filopodia[END_REF])), secretion is abundant from 1 h, localization in endosomes is conspicuous between 2 and 3 h, and detection in (or at the surface of) non-producing cells lasts up to 5 h and then disappears. HeLa cells thus display typical features of Shh trafficking and can be used to analyse the potential effects of modifying H2O2 levels if we have means to rigorously quantify Shh in different compartments at different steps of the process.

In addition, we recently developed (Amblard et al., 2020a) quantitative assays to track the different steps of a protein journey by combining the RUSH [START_REF] Boncompain | Synchronization of secretory protein traffic in populations of cells[END_REF] and HiBit systems [START_REF] Dixon | NanoLuc Complementation Reporter Optimized for Accurate Measurement of Protein Interactions in Cells[END_REF] (HiBit assay, Promega) (Fig. 2). The Hibit system is constituted by a split luciferase: when in the same compartment, the two luciferase moieties, HiBiT (Sbi in plasmid names) and LgBiT (G/Lbi in plasmid names) may spontaneously assemble and restore the luciferase activity that can be measured by substrate addition. These assays are inducible, quantitative, and specifically adapted to protein trafficking. We then set up this quantitative assay to analyse separately each step of Shh's journey. This strategy allowed us to determine the optimal time frame for analysing each step of Shh trafficking in cell culture, i.e. secretion, endocytosis, and delivery via filopodia. These quantitative results are consistent with our direct fluorescence microscopy observations (Fig. 1D) and were combined with redox modulation tools to test the hypothesis of the redox regulation of Shh trafficking.

First, we studied the effects of H2O2 on primary Shh secretion. To increase H2O2 levels, we expressed inducible D-aminoacid oxidase (DAO), which produces H2O2 in the presence of D-Ala and is not expressed in HeLa cells [START_REF] Haskew-Layton | Controlled enzymatic production of astrocytic hydrogen peroxide protects neurons from oxidative stress via an Nrf2-independent pathway[END_REF][START_REF] Matlashov | How much H(2)O(2) is produced by recombinant D-amino acid oxidase in mammalian cells?[END_REF].

For cells expressing DAO and treated with 10 mM D-Ala, we observed a specific reduction in Shh secretion, not observed with a secGFP construct taken as a control (Fig. 3A-B). This result is in close agreement with our previous observations: after oxidative treatment, cells showed reduced Shh secretion, and a pool of Shh was trapped in the Golgi apparatus [START_REF] Gauron | Hydrogen peroxide (H2O2) controls axon pathfinding during zebrafish development[END_REF]. Conversely, reduction of H2O2 levels by the direct addition of Catalase (CAT) in cell culture [START_REF] Amblard | H2O2 and Engrailed 2 paracrine activity synergize to shape the zebrafish optic tectum[END_REF], enhanced Shh but not SecGFP secretion (Fig. 3A,C). Next, we applied the same treatments to study Shh endocytosis on cells incubated with conditioned media of Shh expressing cells. Compared to control conditions (secreted mCherry conditioned media), enhancing H2O2 levels with DAO (by addition of D-Ala) stimulated Shh endocytosis (Fig. 3D-E). Conversely, the reduction in H2O2 levels with CAT treatment reduced Shh endocytosis (Fig. 3D,F). Finally, we studied the effects of H2O2 level modulation on Shh delivery to recipient cells in the co-culture assay. Cells expressing DAO and treated with D-Ala (but not untreated cells) showed increased Shh delivery to recipient cells (Fig. 3G-H), while a reduction in H2O2 levels with extracellular catalase had the opposite effect (Fig. 3G,I). Altogether, these results indicate that physiological variations in H2O2 levels impact Shh trafficking ex vivo. This raises the interesting possibility that the heterogenous H2O2 distribution could polarize Shh secretion and endocytosis in vivo.

H2O2 levels are dynamic in time and space in the embryonic spinal cord

Between 25 and 45 h post fertilization (hpf) in zebrafish, Shh morphogen is first expressed in the medial floor plate (MFP) consisting of a single row of cells at the central midline, then extends to the flanking lateral floor plate (LFP), playing an important role in the neuroglial switch [START_REF] Danesin | Moving the Shh Source over Time: What Impact on Neural Cell Diversification in the Developing Spinal Cord?[END_REF]. Using the improved H2O2 ratiometric probe HyPer7 [START_REF] Pak | Ultrasensitive Genetically Encoded Indicator for Hydrogen Peroxide Identifies Roles for the Oxidant in Cell Migration and Mitochondrial Function[END_REF], we first measured H2O2 levels in Shh-expressing cells in the spinal cord of live zebrafish embryos during this time window (Fig. 4). Quantitative analysis of the H2O2 signal demonstrated a regular and significant decrease (approximately 25%) in H2O2 levels in the MFP between 25 and 45 hpf (Fig. 4A-B). A close-up view of the MFP showed that, in addition to an overall decrease in concentration, the spatial distribution of H2O2 levels varies over time (Fig. 4C-D). Time-lapse analysis of these cells revealed a dynamic intracellular distribution of the HyPer7 signal between 25 and 45 hpf. At the beginning and end of this period, H2O2 levels are homogeneously spread throughout the cell. In between, however, a distinct gradient is transiently established from higher concentrations apically to lower concentrations basolaterally, and most marked at 31 hpf (Fig. 4C-D). Thus, during this neurogenesis period, when Shh induces oligodendrocyte precursor cells (OPCs), H2O2 levels decrease over time and exhibit a transiently polarized distribution within MFP cells.

H2O2 impacts filopodial formation and Shh cellular targets in the embryonic spinal cord

Shh can be delivered to the receiving cells via filopodia, and it has been proposed that the length of filopodia impacts Shh distribution [START_REF] Fairchild | Specialized filopodia: at the 'tip' of morphogen transport and vertebrate tissue patterning[END_REF][START_REF] Gonzalez-Mendez | The cytoneme connection: direct long-distance signal transfer during development[END_REF][START_REF] Gradilla | From top to bottom: Cell polarity in Hedgehog and Wnt trafficking[END_REF][START_REF] Hall | Dispatching Sonic Hedgehog: Molecular Mechanisms Controlling Deployment[END_REF][START_REF] Hall | Cytoneme delivery of Sonic Hedgehog from ligand-producing cells requires Myosin 10 and a Dispatched-BOC/CDON coreceptor complex[END_REF][START_REF] Kornberg | Cytonemes as specialized signaling filopodia[END_REF][START_REF] Sanders | Specialized filopodia direct long-range transport of SHH during vertebrate tissue patterning[END_REF]. We used a double-transgenic fish 2.4Shha-ABC:Gal4FF; Shh-mChe<UAS>GFP-farn with bidirectional UAS-driven expression to visualize both Shh and plasma membranes in the MFP. Shh-mCherry was indeed detected along and at the tip of filopodia as shown at 48 hpf (Fig. 5A-A'). To test whether physiological modifications of H2O2 levels could impact filopodia, we expressed DAO at the level of the plasma membrane in spinal cord canal at 48 hpf. We then visualized the filopodia using the fluorescence of the membrane-bound mCherry (Fig. 5B) and quantified the number of filopodia per cell (Fig. 5C) as well as the length of the filopodia (Fig. 5D). Interestingly, enhancement of H2O2 levels induced increases in both the number and the length of filopodia (Fig. 5C-D) in vivo, suggesting that mild modifications of H2O2 levels could modulate Shh functioning in the neural tube. To test this hypothesis, we treated zebrafish larvae (2.4Shha:Gal4; UAS:HyPer7) with NOX-i (NADPH oxidase pan inhibitor) to reduce H2O2 levels (Supplementary Fig. S1) from 30 hpf to 48 hpf. This reduction in H2O2 levels induces a decrease in filopodia number in MFP (Supplementary Fig. S1). We then analysed the distribution of OPCs in the embryonic spinal cord at 72 hpf in (Olig2:EGFP) larvae (Fig. 5E-F). A small reduction in H2O2 levels (approximately 10%) was sufficient to enhance by a factor of 2 the generation of OPC (Fig. 5F), known to dependt on Shh activity (Al [START_REF] Oustah | Dynamics of sonic hedgehog signaling in the ventral spinal cord are controlled by intrinsic changes in source cells requiring sulfatase 1[END_REF], without affecting shh expression (not shown). Thus, small modifications in H2O2 levels not only modified the number and length of filopodia in vivo but also altered a Shhdependent switch.

Shh regulates H2O2 levels and filopodial growth via a non-canonical route

After demonstrating that H2O2 impacts Shh trafficking (Fig. 3) and signaling outcome (Fig. 5), we wondered whether this was integrated into a larger H2O2-Shh feedback loop, as first observed during adult zebrafish caudal fin regeneration [START_REF] Meda | Nerves Control Redox Levels in Mature Tissues Through Schwann Cells and Hedgehog Signaling[END_REF]. If this were the case, Shh's interaction with cells should itself impact H2O2 levels. We first observed that HeLa cells co-expressing HyPer and Shh-mCherry exhibited higher levels of H2O2 than cells expressing HyPer and mCherry (Fig. 6A), suggesting that Shh per se affected H2O2 balance.

Interestingly, untransfected neighbours of cells expressing Shh-mCherry also displayed increased H2O2 levels compared to neighbours of cells expressing only mCherry (Fig. 6A). This observation suggests a paracrine effect of Shh on H2O2 levels. Moreover, when HyPer was addressed to the plasma membrane (Lck-HyPer), filopodia bearing Shh were readily identified, and high levels of H2O2 consistently co-localized with Shh (Fig. 6B). To quantitively assess the impact of Shh on H2O2 levels, we treated a stable cell line expressing Lck-HyPer with Shh-containing conditioned medium. This treatment induced a global increase in H2O2 levels (Fig. 6C-D), which was accompanied by filopodia growth, as quantified by their cumulative length (Fig. 6E). Finally, we investigated the molecular pathway used by Shh to increase H2O2 levels. Smoothened involvement was highly unlikely since its mRNA was not detected in our RT-qPCR experiments (Supplementary Table S1). Indeed, cyclopamine (Shh-i), a Shh antagonist that specifically binds Smo, did not affect basal H2O2 levels and did not inhibit the Shh-induced H2O2 burst (Fig. 6F). In contrast, treatment with Nox-i efficiently inhibited Shh-mediated H2O2 increase, demonstrating the involvement of NOX enzymes (Fig. 6F). Rac1 was an attractive candidate to bridge Shh and NOX in our system. It is ubiquitous, activates NOX2 (the only NOX complex present in HeLa cells, Supplementary Table S1), and is well known to regulate cytoskeleton dynamics [START_REF] Acevedo | Crosstalk between Rac1-mediated actin regulation and ROS production[END_REF]; in addition, its interplay with Shh and NOX1 was previously observed in a different system [START_REF] Polizio | Sonic Hedgehog activates the GTPases Rac1 and RhoA in a Gli-independent manner through coupling of smoothened to Gi proteins[END_REF]. We used a Rac1 inhibitor (Rac1-i) to evaluate its involvement in H2O2 production induced by Shh treatment. As shown in Figure 6G, Rac1-i itself had no effect on H2O2 levels in untreated cells but was able to block the effect of Shh treatment. Rac1 thus sits at the crossroads of Shh activity on HeLa cells, leading both to H2O2 production via the NOX/SOD complex and to cytoskeleton remodeling that ultimately enables filopodium formation. A potential pathway to mediate Rac1 activation by Shh was recently published [START_REF] Makihara | Polarized Dock Activity Drives Shh-Mediated Axon Guidance[END_REF]: Shh-mediated axon guidance in the spinal cord depends on BOC receptor stimulation, leading to ELMO-Dock release and Rac1 activation. Since BOC, ELMO2 and 3, as well as several DOCK proteins, are expressed in HeLa cells (Supplementary Table S1), we tested this hypothesis using a pan-Dock inhibitor (CPYPP, Dock-i). As shown in Fig. 6G, DOCK-i per se did not affect H2O2 levels but blocked the H2O2 increase induced by Shh. Moreover, simultaneous treatment with DOCK-i and Rac-i had no cumulative effect (Fig. 6G), suggesting that they belong to the same pathway. In summary, these experiments show that Shh induces a local increase in H2O2 levels in HeLa cells via a non-canonical route involving its receptor Boc and the Elmo-Dock pathway to activate Rac1.

As thoroughly discussed in Acevedo and Gonzalez-Billault [START_REF] Acevedo | Crosstalk between Rac1-mediated actin regulation and ROS production[END_REF], this leads to NOX2 activation along the redox branch of Rac1 activity and membrane protrusion growth along the actin cytoskeleton branch. Very recently, Shh cytoneme formation in rodent cells was demonstrated to depend on another feedback loop, including a Disp-BOC co-receptor complex [START_REF] Hall | Cytoneme delivery of Sonic Hedgehog from ligand-producing cells requires Myosin 10 and a Dispatched-BOC/CDON coreceptor complex[END_REF].

Signaling filopodia or cytonemes are a common route for the secretion of morphogens [START_REF] Gradilla | From top to bottom: Cell polarity in Hedgehog and Wnt trafficking[END_REF][START_REF] Ramirez-Weber | Cytonemes: cellular processes that project to the principal signaling center in Drosophila imaginal discs[END_REF] and it was recently shown that several morphogens (Fgf2, Wnt8a, Wnt3a, Shh) are involved in a regulatory loop with cytoneme formation and growth, independent of their canonical pathway [START_REF] Hall | Cytoneme delivery of Sonic Hedgehog from ligand-producing cells requires Myosin 10 and a Dispatched-BOC/CDON coreceptor complex[END_REF][START_REF] Mattes | Wnt/PCP controls spreading of Wnt/beta-catenin signals by cytonemes in vertebrates[END_REF]. This happens during embryonic development, at a time when H2O2 levels are highly dynamic [START_REF] Rampon | Hydrogen Peroxide and Redox Regulation of Developments[END_REF].

We report here a mechanism by which physiological levels of H2O2 adjust the trafficking of Shh, which, in turn, enhances H2O2 levels. As a consequence, local and timely H2O2 production may polarize the journey of Shh by modifying the rates of Shh secretion and endocytosis as well as the regulation of filopodia. We speculate that H2O2 could integrate the spreading of morphogens in a developing tissue, with different morphogens responding differentially to specific concentrations of H2O2 and, in turn, modifying H2O2 levels. Indeed, we already know that H2O2 impact on trafficking differs between Engrailed [START_REF] Amblard | H2O2 and Engrailed 2 paracrine activity synergize to shape the zebrafish optic tectum[END_REF] and Shh (this study). This interplay between morphogens and redox signaling is also likely to have a role in various pathologies [START_REF] Sies | Hydrogen peroxide as a central redox signaling molecule in physiological oxidative stress: Oxidative eustress[END_REF].

In conclusion, we propose that environmental insults or individual genetic variations inducing very subtle differences in H2O2 levels could impact morphogen distribution, resulting in interindividual differences in organism patterning or disease susceptibility.

Materials and methods

Fish husbandry and pharmacological treatments

Zebrafish were maintained and staged as previously described [START_REF] Amblard | H2O2 and Engrailed 2 paracrine activity synergize to shape the zebrafish optic tectum[END_REF].

Experiments were performed using the standard AB wild-type strain. The embryos were incubated at 28°C. Developmental stages were determined and indicated as hours postfertilization (hpf). The animal facility obtained permission from the French Ministry of Agriculture for all the experiments described in this manuscript (agreement no. C 75-05-12).

To decrease H2O2 levels, embryos were incubated in VAS-2870 (NOX-i) (100 nM) from Enzo Life Sciences (#BML-El395-0010, Enzo Life Sciences, Inc.; Farmingdale, NY, USA) or an equivalent amount of DMSO as a control for the duration of the time-lapse analysis. To enhance H2O2 levels D-Ala (Sigma-Aldrich, #A7377) (10 mM) was injected in zebrafish ventricles one hour prior H2O2 levels or filopodia analysis.

Expression constructs, permanent cell lines, fish transgenic lines

All recombinant DNA (Supplementary Table 3) were prepared by standard cloning methods.

Plasmids and sequences are available on request.

Stable cell lines (Supplementary Table 2) were prepared under Hygromycin selection using the HeLa FlpIn-TREX cell line kindly provided by Stephen Taylor [START_REF] Tighe | Mps1 kinase activity restrains anaphase during an unperturbed mitosis and targets Mad2 to kinetochores[END_REF] expressing the tetracycline repressor (T-Rex, Life Technologies).

Transgenic lines used in this study were: 2.4Shha-ABC:Gal4 (this article); UAS:Igkmb5DAOmCherry (this article); UAS:Igkmb5CATmCherry (this article); UAS:Shh-mCherry, GFP-farn (this article); UAS:HyPer7 (this article); olig2:GFP [START_REF] Shin | Neural cell fate analysis in zebrafish using olig2 BAC transgenics[END_REF]. The transgenic fish lines were constructed as described [START_REF] Urasaki | Functional dissection of the Tol2 transposable element identified the minimal cis-sequence and a highly repetitive sequence in the subterminal region essential for transposition[END_REF] using pTol2 derivatives containing the appropriate promoter/enhancer and the SV40 late polyadenylation signal (SVLpA). Shh:Gal4 contains the Gal4 DNA binding domain fused to 2 minimal activator sequences (Gal4BD-FF [START_REF] Seipel | Different activation domains stimulate transcription from remote ('enhancer') and proximal ('promoter') positions[END_REF]) inserted between the -2.4Shha promoter and ar-A, ar-B, ar-C Shha enhancers (a kind gift of R. Ertzer [START_REF] Ertzer | Cooperation of sonic hedgehog enhancers in midline expression[END_REF]). UAS:HyPer7 contains HyPer7 sequence [START_REF] Pak | Ultrasensitive Genetically Encoded Indicator for Hydrogen Peroxide Identifies Roles for the Oxidant in Cell Migration and Mitochondrial Function[END_REF] down 5xUAS (derived from [START_REF] Akitake | Transgenerational analysis of transcriptional silencing in zebrafish[END_REF]). UAS:igkmb5-DAOmCherry and UAS:igkmb5-CATmCherry contain, downstream the same enhancer and provided with a signal peptide (Igk) from kappa light chain and a minimal transmembrane domain (mb5) from CD4, fusions of mCherry (in Cterminal position) with either D-Aminoacid oxidase (DAO [START_REF] Matlashov | How much H(2)O(2) is produced by recombinant D-amino acid oxidase in mammalian cells?[END_REF]) or mouse Catalase deprived from its lysosome-targeting signal (CAT). For bidirectional expression the 5xUAS regulatory element was flanked by minimal promoter/5' UTR sequences from pCS2 on one side and CMV on the other side [START_REF] Anton | Src-transformed cells hijack mitosis to extrude from the epithelium[END_REF].

Embryo live imaging and image processing

The larvae were anesthetized in tricaine solution and embedded in low-melting agarose (0.8 %). Imaging was performed with a CSU-W1 Yokogawa spinning disk coupled to a Zeiss Axio Observer Z1 inverted microscope equipped with a sCMOS Hamamatsu camera and a 10× (Zeiss 0.45 Dry WD: 2.1mm) or a 25× (Zeiss 0.8 Imm DIC WD: 0.57 mm) oil objective. DPSS 100 mW 405 nm and 150 mW 491 nm lasers and a 525/50 bandpass emission filter were used for HyPer7 imaging, and DPSS 100mW 561 nm laser and a BP 595/50 was used for mCherry imaging. Floor plate cells imaging was performed using a Zeiss LSM 980 -AiryScan 2 confocal equipped with a AiryScan detector GaAsP-PMT and a 25× (Zeiss 0.8 Imm WD: 0.57mm) or a 40× (Zeiss 1.3 Oil DIC (UV) WD: 0.22mm) oil objectives. DPSS 10 mW 488 nm and 10 mW 561 nm lasers and respectively 517 nm and 610 nm AiryScan emission filters were used for GFP and mCherry acquisition. AiryScan SR mode was used and AiryScanprocessed images were analysed. To calculate the HyPer ratio, images were treated as previously described [START_REF] Mishina | Visualization of intracellular hydrogen peroxide with HyPer, a genetically encoded fluorescent probe[END_REF]. For filopodia analysis, 48 hpf Tg(2.4Shha-ABC:Gal4-FF) larvae expressing Igkmb5-DAO-mCherry were used for fluorescence acquisition as described above, and one slice of each Zstack was extracted. Slices presenting the maximum filopodia number were selected for FiloQuant processing and analysis [START_REF] Jacquemet | Filopodia Quantification Using FiloQuant[END_REF].

Pharmacological treatments

To decrease H 2 O 2 levels, cells were treated with extracellular Catalase (Sigma-Aldrich, #C1345, 400 U/mL). To increase H 2 O 2 levels, cells expressing D-amino acid oxidase (DAO) were treated with 10 mM D-alanine (Sigma-Aldrich, #A7377) before the internalization or secretion assays were performed. To inhibit NOX activity, cells were pre-treated for 1 h with 10 μM VAS-2870 (NOX-i; #BML-El395-0010, Enzo Life Sciences, Inc.; Farmingdale, NY, USA) or an equivalent amount of DMSO as a control. To inhibit Rac1, cells were pre-treated for 6 h with 20 μM NSC23766, a Rac1-inhibitor (Rac1-i; No2161, Tocris). To inhibit Dock release from Elmo, cells were pre-treated for 6 hours with 100 µM CPYPP (DOCK-i; No4568, Tocris). To inhibit Shh signaling, cells were pretreated for 24 h with 10 μM cyclopamine (Shhi; #239803, Millipore).

Quantitative secretion assay

Cells (13,000 per well) were plated on 96-well plates (Greiner Bio-one) coated with polyornithine (10 μg/mL). After 10 h, the cells were co-transfected with a plasmid expressing Shh-SBP-HiBiT (or a secreted control, SecGFP-SBP-HiBiT) under doxycycline control and a plasmid constitutively expressing the hook (core streptavidin-KDEL provided with a signal peptide). Some of these cells received an additional plasmid constitutively expressing a membrane-bound fusion between mCherry and D-Aminoacid Oxidase (Lck-Che-DAO). After 24 h medium was removed and cells were incubated with fresh medium containing doxycycline for 2 h (to induce HiBiT fusion expression). Medium was changed and secretion was induced with biotin (100 μM final) and, after purified LgBiT protein addition in the medium, luciferase activity was measured 1 h later with a 96-well plate luminometer (Tristar, Berthold) as described in the HiBiT assay kit (Promega). The cells were then lysed to measure intracellular protein expression. Normalization with biotin-untreated wells enabled us to calculate the secretion index and report the secretion efficiency.

Quantitative internalization assay

Cells (90,000 per well) stably expressing LgBiT were plated in 24-well culture dishes. After 24 h, medium was changed and cells were incubated for 30 min with medium containing HiBiT fusions (taken from cells expressing Shh-HiBiT or SecCh-HiBiT for 48 h, and adjusted on protein Luciferase activity), before incubating the cells with trypsin and removing them.

After centrifugation cells were lysed and the luciferase activity of endocytosed protein was measured with a 96-well plate luminometer (Tristar, Berthold) with a HiBiT assay kit (Promega).

Quantitative intercellular transfer assay

Cells (90,000 per well) were plated in 24-well culture dishes. After 10 h, the cells were cotransfected with a plasmid expressing Shh-SBP-HiBiT under the control of doxycycline and a plasmid constitutively expressing Lck-Che-DAO. After 24 h, cells were removed using trypsin and co-cultured on 96-well plates (Greiner Bio-one) coated with polyornithine (10μg/ml) with cells stably expressing LgBiT anchored to the extracellular side of the cell surface (siL-LgBiT-mb5). After 5 h, cells were incubated with fresh medium containing doxycycline for 2 h (to induce Shh-SBP-HiBiT expression). Medium was then changed and secretion was induced with biotin (100 μM final), and luciferase activity was measured every 1 h over 4 h with a 96well plate luminometer (Tristar, Berthold) as described in the HiBit assay kit (Promega).

H 2 O 2 imaging with the HyPer probe in HeLa cells

HyPer fluorescence was excited with DPSS 10 mW 488 nm and 10 mW 405 nm lasers, and the corresponding YFP emission was measured using a 525/50 bandpass emission filter.

Spinning-disk images were acquired using a 63× objective (63×/1.4 oil WD: 0.17 mm) on a Spinning-Disk CSU-W1 (Yokogawa) equipped with a sCMOS Hamamatsu 2048×2048 camera. To calculate the HyPer ratio, images were treated as previously described [START_REF] Mishina | Visualization of intracellular hydrogen peroxide with HyPer, a genetically encoded fluorescent probe[END_REF].

Quantification and statistics

Data were analyzed using GraphPad Prism 6 and expressed as the mean ± standard error of the mean (SEM). Statistical significance was calculated using a 2-sided paired Student's ttest. For multiple conditions, ordinary one-way ANOVA followed by Tukey's multiple comparison test was performed to evaluate the significant differences. For filopodia analysis, statistical errors (SD) were estimated as √p(1-p)/n, where p is the percentage in a class and n is the total number investigated (or √1/n when p=0 or 1).The degree of significance was represented as follows: * p-value ≤ 0.05; ** p-value ≤ 0.01; *** p-value ≤ 0.001; and **** pvalue ≤ 0.0001. Sample sizes and number of replicates are given in Supplementary Table 4.

FiguresFig. 1

 1 Figures

Fig. 2

 2 Fig. 2 Quantitative monitoring of the journey of Shh ex vivo. (A-C) Quantifying Shh secretion by HeLa cells. (A, B) Schematic representation of the tools (A) and assay (B). (A) Strepta: core streptavidin linked to the KDEL hook; SBP: streptavidin binding protein domain linked to Shh or secGFP; SBi: small fragment of split nanoluciferase (HiBiT) linked to Shh or secGFP; inactive nanoLuc: large fragment of split nanoluciferase (LgBiT); active nanoLuc: reconstituted nanoluciferase. (B) Synchronized HeLa cells constitutively express the hook, and they express Shh fused to SBP and HiBiT (Shh-SBP-HiBiT) under doxycycline control. Purified LgBit added to the medium binds HiBiT when Shh reaches the extracellular space, and nanoLuc activity gives a measure of secretion. (C) Time course of Shh synchronized secretion compared to the control (secGFP fused to SBP and HiBiT). (D-E) Quantitating Shh internalization by HeLa cells. (D) Schematic representation of the assay. Shh fused to HiBiT (Shh-HiBiT), purified from the conditioned medium of producing HeLa cells and calibrated in vitro, was added to HeLa cells expressing inactive nanoLuc (LgBiT) in the cytosol. Upon cell lysis, nanoluciferase is reconstituted from endocytosed Shh-HiBiT and cytoplasmic LgBiT, allowing quantification of internalization. (E) Time course of Shh internalization compared to the control (secCherry fused to HiBiTI, prepared in parallel to Shh). (F-G) Quantitating Shh delivery to recipient HeLa cells. (F) Schematic representation of the assay. Two HeLa cell populations are co-cultured. In the first one, synchronization of Shh release is achieved as in the secretion assay but for a longer period of time. The second population expresses inactive nanoLuc (LgBiT) at the cell surface (anchored via the CD4 transmembrane domain). Active nanoluciferase is reconstituted by the transport of tagged Shh (Shh-SBP-HiBiT) from a donor cell to the surface of a receiving cell. (G) Time-course analysis of Shh delivery to recipient cells becomes strongly significant approximately 3 h after biotin addition. Ctrl: without biotin. Details on statistics in Methods.

Fig. 3 :

 3 Fig. 3: H2O2 affects Shh secretion, endocytosis and delivery in HeLa cells. (A, D, G) Schematic steps (Fig. 3): secretion (A), endocytosis (D), delivery (G). A-C, Effects of increased (B) or decreased (C) H2O2 levels on secretion of Shh or secGFP from cells expressing Lck-DAO supplemented (or not) with D-Ala (B) or cells treated (or not) with extracellular catalase (C). (D-F) Effects of increased (E) or decreased (F) H2O2 levels on Shh or secCherry endocytosis by cells expressing Lck-DAO supplemented (or not) with D-Ala (E) or cells treated (or not) with extracellular catalase (F). G-I, Effects of increased (H) or decreased (I) H2O2 levels on delivery to recipient cell surface of cells expressing Lck-DAO supplemented (or not) with D-Ala (H) or cells treated (or not) with extracellular catalase (I). Details on statistics in Methods.

Fig. 4 :

 4 Fig. 4: H2O2 levels are dynamic in time and space in the floor plate. (A-B) Variation in time. (A) H2O2 levels in Tg(2.4Shha-ABC:Gal4-FF; UAS:Hyper7) zebrafish transgenic embryos at different times after fertilization as indicated (lateral view, anterior to the left). Scale bar, 200 µm. (B) Variations in H2O2 levels at different stages. Statistics are presented compared to 30 hpf. (C-D) Transient dorso-ventral gradient. (C) Embryos injected at the one-cell stage with mRNA for farnesylated mCherry were imaged for membranes (left panels: mCherry-F) and H2O2 (right panels: HyPer7; bottom: same field as in the left panel; top: enlarged view as indicated by the dotted line). fp: floor plate; nc: notochord. Scale bar: 20 µm. (D) Variations in H2O2 levels along the apico-basal axis of MFP cells. Details on statistics in Methods.

Fig. 5

 5 Fig. 5 H2O2 modulates filopodia production and OPC bihaviour in the zebrafish spinal cord. (A) Shh-mCherry visualization in MFP filopodia in live embryos at 48 hpf (Scale bar: 5 µm). (A') Enlarged view (scale bar: 1 µm). (B) Visualization of filopodia at 48 hpf in Tg(2.4Shha-ABC:Gal4-FF;UAS:Igk-mb5-DAO-mCherry) before or after D-Ala (10 mM) injection (scale bar: 5 µm). (C) Quantification of filopodia in MFP cells (see Methods) in Ctrl and D-Alainjected larvae. (D) Proportion of filopodia longer than 1 µm in MFPs of Ctrl and D-Alainjected larvae. (E) Detection of GFP at 72 hpf in Tg(olig2:EGFP) larvae incubated from 30 hpf in control solution or in NOX-i at 10 μM. (F) Quantification of OPCs generated at 72 hpf. Details on statistics in Methods. sc: spinal cord; vmn: ventral motor nerves.

Fig. 6 :

 6 Fig. 6: Shh regulates H2O2 levels and filopodial growth. (A-E) Cells stably expressing membrane-bound HyPer (A-B) either transfected with constructs coding for Shh-Cherry (A, B) or Cherry (A), were (C-E) treated (Shh) or not (Ctrl) with Shh, or were (F, G) treated with various inhibitors. (A) H2O2 was measured separately in cells expressing (Expr+) and non-expressing (Expr-) the transfected construct, coding for either Shh-mCherry (right) or Cherry (left). H2O2 levels in the non-expressing cells transfected with mCherry were taken as 100%. (B) Merging the red (Shh-mCherry) and green (Lck-Hyper488nm) signals highlights Shh-bearing filopodia (left), where H2O2 levels are distinctly elevated (right). (C-E) Treated cells (Shh) display higher H2O2 levels (imaged in C, quantified in D) and have longer filopodia (imaged in C, quantified in E) than untreated cells (Ctrl). (F) Cells treated (or not) with Shh were treated (or not) with either cyclopamine (Shh-i) or VAS2870 (NOX-i). (G) Cells treated (or not) with Shh were treated (or not) with either NSC23766 (Rac1-i) or CPYPP (DOCK-i) alone or in combination. In D and F-G, untreated cells were taken as 100%. Details on statistics in Methods. Scale bar: 50 µm.

Acknowledgements and funding sources

We thank Helmut Sies (Heinrich Heine University Düsseldorf, D) for helpful discussions, Sylvie Schneider-Maunoury (Sorbone Université, F) and Christine Vesque (Sorbone Université, F) for fruitful comments on the manuscript. This work was performed at the Collège de France and benefited from Alain Prochiantz's constant support. The authors are grateful to Stephen Taylor (University of Manchester, UK) for providing the HeLa Flp-In cell and to Ariel Ruiz i Altaba (Université de Genève, CH) for providing mouse Shh and mouse Shh::GFP expressing plasmids, the second one being derived from an original construct from Andrew McMahon's laboratory (Harvard, USA). This work received support under the program "Investissements d'Avenir" launched by the French Government and implemented by the ANR with the following references: ANR-10-LABX-54 MEMO LIFE -ANR-11-IDEX-0001-02 PSL* Research University.