Alain Lecomte
email: alain.lecomte@upmf-grenoble.fr

Logics for Dialogue

This paper is essentially a survey of some logical approaches to dialogue. We start with Dialogical Logic, which was initiated by Lorenzen and has mainly been explored as a new foundation for logics. It continues with Hintikka's Game Theoretical Semantics, which has been more developed in contact with Natural Language. For instance, we show how to deal with generalized quantifiers by using games, after ideas taken from Ahti Pietarinen. The two perspectives, if different in their objectives, could be mixed for applicative purposes like the treatment of argumentative dialogues: this requires that they be recast in a neutral form, which consists in Dialogue Games in Extensive form. Nevertheless, to stay at one level of elementary language games is not sufficient: in every day life, games are combined. At this point, it seems that the Game-Theoretic interpretation of Linear Logic provides us with the appropriate tool for combining elementary games of various kinds.

Introduction

Logical concepts have shown to be helpful in language clarification for more than one century with the works done by Frege, Russell and their numerous followers. A little less attention has been given to dialogue during the same period. Yet dialogue is fundamental as soon as we want to understand pronouns (or shifters in Benveniste's terms) and other indexicals, but also, as we shall see further, other linguistic phenomena, like generalized quantifiers. Moreover, it has been shown by many researchers since Lorenzen [START_REF] Lorenzen | Logik und Agon[END_REF] and Hintikka [Hintikka, 1968] that logical connectives and natural language particles could be explained by referring to a dialogical interpretation of language, which is also known as Game-Theoretical Semantics in Hintikka's case. It is an open question whether those Game-Theoretic approaches are able to provide models of true dialogues, as they occur in the every day life. Solving such a question has many consequences and applications, among which are applications to Human-Machine Dialogue and to dialogue between humans via a machine. From a theoretical viewpoint, if it appears that, like some contemporary researchers point out ( [START_REF] Hurford | The Neural Basis of Predicate -Argument Structure[END_REF], [START_REF] Bickerton | Language and Species[END_REF]), predicate logic is appropriate for describing thought as pre-existing to language, it also seems that interaction structures have played a major role for humans before the emergence of language and that language received much of its features from these structures. It is therefore no surprise to find very early formulations of logic (sometimes assimilated with dialectic) which make use of dialogical concepts. We may mention here of course Aristotle, in Topica VIII, for whom dialectical situations involve two participants: Answerer, who must defend his or her thesis or positum, and Questioner, who tries to make Answerer to change his positum. In the modern times, Leibniz invented the famous 'epsilon-delta' definition of continuity and thought of it as a simple dialogue (give me one 'epsilon', I shall give you one 'delta'). In this paper, we consider different conceptions of games with regards to logic and language. A first conception, dating from Lorenzen [START_REF] Lorenzen | Logik und Agon[END_REF][START_REF] Lorenz | Aritmetik und Logik als Spiele[END_REF] amounts to use games as a foundation for logic: we call it a procedural approach because it is essentially based on rules: rules for particles and global rules. A second one, coined by Hintikka [Hintikka, 1968] amounts to use games as an alternative to the classical models used for evaluating the truth of a formula, we call it a model-based approach. There is also an other viewpoint, called geometry of interaction by Girard [START_REF] Girard | Advances in Linear Logic[END_REF] which uses games as altogether semantics and rules for defining a particular logic: Linear Logic. Our main thesis is that in order to treat real dialogues (which include language and nonlanguage games) we must altogether mix the two first approaches (the procedural one and the model-based one) and combine the elementary games by means of Linear Logic used as a "glue" logic.

Games as a foundation

Argumentation forms

Lorenzen's goal in the late fifties was to give a justification to intuitionistic logic. His purpose consisted in arguing that intuitionistic logic was the right one since it is the logic we obtain when we adopt the "natural" dialogical meaning of the connectives. This goal was not really achieved, because it was shown later on that other logics were as well obtainable simply by changing the so-called "global rules", but we may consider Lorenzen an ancestor of Girard's approach to logic that we shall look at soon. Let us simply recall here that we need a denumerable set of variables x, y, z, …, the constant connectives ∧, ∨, ⇒, ¬ and the quantifiers ∀ and ∃. Let P and O two constant symbols (respectively representing Proponent and Opponent). For every expression e, we have two distinct expressions: P:e and O:e which are said to be respectively signed by P and O. X and Y are two variables which take their values in {P, O}. We assume that in every situation, X ≠ Y. An argumentation form is a schematic presentation of how an assertion made by X can be attacked by Y and how, if possible, it can be defended by X. In order to have an argumentation form for each formula, it is enough to define argumentation forms for connectives and quantifiers. We thus have for the connectives:

∧ assertion X:w 1 ∧w 2 ∨ assertion X:w 1 ∨w 2 attack Y: ∧ i (i = 1,2) attack Y: ∨ defence X:w i defence X:w i (i = 1,2)
⇒ assertion X:w 1 ⇒w 2 ¬ assertion X: ¬w attack Y: w 1 attack Y: w defence X:w 2 defence no defence! and for the quantifiers:

∀ assertion X: ∀xw ∃ assertion X: ∃xw attack Y: t (Y chooses t) attack Y: ∃ defence X:w(t) defence X:w(t) (X chooses t)
Technically, a dialogue is a sequence δ of signed expressions, alternatively supported by P and O and satisfying at each step some argumentation form. This needs some more precision. Let also η be a function on natural numbers. Let A and D the two letters for Attack and Defence.

Definition 1 ( [START_REF] Felscher | Dialogues as a Foundation for Intuitionistic Logic[END_REF]) A dialogue is a couple (δ, η) such that:

• δ is a sequence of signed expressions,

• η is a function on the ranks of the signed expressions in δ such that η(n) is a couple [m, Z] where m ≤ n and Z = A or D, which satisfies the following three conditions (D00), (D01), (D02). andδ(p) is an answer to the attack δ(n) which corresponds to a regular argumentation form.

D00 δ(n) is P-signed if n is even and O-signed if n is odd; δ(n) is a compound formula, D01 If η(n) = [m, A], then δ(m) is a compound formula and δ(n) is an attack on δ(m) which corresponds to a regular argumentation form, D02 If η(p) = [n, D], then η(n) = [m, A],

Global rules

Global rules are used according to the logic we aim at expressing. Theorem 1: Classical theses coincide with formulae valid for the system which includes D10, D12', D13', D2

Theorem 2: Intuitionistic theses coincide with formulae valid for the system which includes D10, D11, D12, D13, D2

Examples

It is fundamental to notice here that in order to defend a formula (either in classical or in intuitionistic logic) P has not only to use more or less automatically the argumentation forms, he has also to choose some strategy which compels O to give him or her the most information as possible. This is particularly clear with regards to the global rule D10 which allows him or her to assert an atomic formula, provided that his or her partner already asserted it. In this case he or she has therefore to oblige O to concede some fact. This is clearly showed in the following cases. (The two first examples come from [START_REF] Felscher | Dialogues as a Foundation for Intuitionistic Logic[END_REF], the third one from [START_REF] Rahman | Dialogical Connexive Logic[END_REF]

) (τ P [3, D]
Notice that at step 4, P cannot choose P(τ) because it is an atomic formula, not already asserted by O. He therefore chooses ¬P(τ), which obliges O to assert P(τ) and then P can choose it, thus providing a complete argument for his or her thesis. Notice also that this is only possible in classical logic since in the intuitionistic one, P would only be allowed to answer last O's attack, which is the attack performed at step 5.

Non classical logics

Among non classical logics, we include free logics and paraconsistent ones. These logics are treated in the dialogical framework by changing global rules (or adding new ones). A good illustration for that is provided by S. Rahman's treatment of free logics ( [START_REF] Rahman | On Frege's Nightmare : A Combination of Intuitionistic, Free and Paraconsistent Logics[END_REF]). Let us suppose that in a dialogue similar to Example 3 above, only the opponent is allowed to introduce a constant when playing a move associated with a quantifier, then it becomes possible to have a logic where it is impossible to show that, say, if there is no vampire, Nosferatu is not a vampire, thus opening room to fiction: even if we know that vampires do not exist, we can accept the idea that Nosferatu is a vampire. Let us formulate the classical dialogue:

Example 4:

0. P: ) ( )) ( ( n vampire x vampire x ¬ ⇒ ¬ ∀ 1. O: ) (x vampire x¬ ∀ [ 0 , A ] 2. P: ) (n vampire ¬ [ 1 , D ] 3. O: ) (n vampire [ 2 , A ] 4. P: n [ 1 , A ] 5. O: ) (n vampire ¬ [ 4 , D ]
6. P:

) (n vampire [ 5 , A ]
At step 4, P attacks O's move at 1, thus obliging O to assert that Nosferatu is not a vampire. This statement is a compound one and P may still attack it by asserting that Nosferatu is a vampire, an atomic formula that he or she can use because it was already asserted by O at step 3. After this move, there is no possible answer by O and therefore P wins, but we can see that if we introduce the new global rule concerning the use of constants, move 4 becomes illegal, and P looses. Further, he or she has no winning strategy: this entails that P's proposition is not a thesis in such a logic. Let us express a similar dialogue in ordinary language:

I :
if there is no vampire, of course Nosferatu is not a vampire! You: OK, let us admit that there is no vampire Attack I:

then Nosferatu is not one! Defence You: I pretend it is Attack I:

take Nosferatu Attack You: I concede that Nosferatu is not vampire Defence I:

but you told me it is one! Attack You : but I did not concede it was existing! This opens the field to a conception of Fiction which involves two partners who have an asymmetric relation, that we could call Reader and Narrator. In a fiction, Narrator is the only master. She may choose any character she wishes with any kind of properties even if both (Reader and Narrator) agree on the point that these properties are not satisfied in reality. We may notice that even contradictory properties could be used in the case of a paraconsistent logic (a case we do not develop here, see [START_REF] Rahman | On Frege's Nightmare : A Combination of Intuitionistic, Free and Paraconsistent Logics[END_REF].

Model-based games

Hintikka's Game Rules slightly differ from Lorenzen's, but these small differences entail important consequences. The main difference lies in the rule for asserting atomic statements. Because Hintikka's games are based on a model M of the language L, the only requirement on asserting atomic formulae is that these formulae be true in M. Moreover, no repetition of "attack" or "defence" is allowed: at each stage, the game-formula simplifies and the run cannot be expected to go back to an earlier stage ( [START_REF] Hintikka | The Game of Language, Studies in Game-Theoretical Semantics and Its Applications[END_REF], [START_REF] Hintikka | Game-theoretical semantics[END_REF]).

Example 5

Let us suppose we have a structure <D, R> where D = {1, 2, 3, 4} and R is a binary relation on D : R = {(1,2), (1, 4), (2,3), (3, 1), (4, 2), (4, 3)}, and let us suppose we have to evaluate the formula:

)

)) , ( ) , ( ( ) , ( ( y z R z x R z y x R y x ∧ ∃ ∨ ∀ ∀
, A possible game is the following one:

0. P: ))) , ( ) , ( ( ) , ( ( y z R z x R z y x R y x ∧ ∃ ∨ ∀ ∀ 1. O: 2 [ 0 , A ] 2. P: ))) , ( ) , 2 ( ( ) , 2 ( ( y z R z R z y R y ∧ ∃ ∨ ∀ [ 1 , D ] 3. O: 4 [ 2 , A ] 4. P: )) 4 , ( ) , 2 ( ( ) 4 , 2 ( z R z R z R ∧ ∃ ∨ [ 3 , D ] 5. O: ∨ [ 4 , A ] 6. P: )) 4 , ( ) , 2 ( ( z R z R z ∧ ∃ [ 5 , D ] 7. O: ∃ [ 6 , A ] 8. P: ) 4 , 3 ( ) 3 , 2 ( R R ∧ [ 7 , D ] 9. O: ∧2
[ 8 , A ] 10. P looses P simply looses because the only defence he could have would be to assert R(3,4), which is false in M. In fact, it is easy to check that P has no winning strategy: the formula is false in M. , the two existential quantifiers depend on x and y. Skolemization would give:

Theorem 3 ([Hintikka and Sandu, 1997]) For any first-order sentence S and model M, Tarski-type truth and GTS-truth coincide, ie: M |= Tarski S if and only if M |= GTS S, where M |= GTS S means that Proponent has a winning strategy for S with regards to M.

IF-logic

Informational independence

)

) , ( ), , ( , , ( y x g y x f y x yA x g f ∀ ∀ ∃ ∃
. Henkin was the first to point out this kind of limitation, intrinsic to first-order logic and provided a solution using what he called branching quantifiers. In a notation like:

) , , , ( u z y x A u y z x         ∃ ∀ ∃ ∀
∃z only depends on x and ∃u only depends on y, so that the skolemized form is:

)) ( ), ( , , ( y g x f y x yA x g f ∀ ∀ ∃ ∃ In Hintikka's notation, this is written: ) , , , ( ) / )( / )( )( ( u z y x uA x u y z y x ∃ ∀ ∃ ∀ ∃ ∀ ∀
Informational independence receives a convenient foundation in Game-Theoretical Semantics. It suffices to say that when defending a formula

A y z ) / ( ∀ ∃
against a ∃-attack, the player does not know the choice for y made by his partner when he chooses a value for z. Hintikka says in this case that the move prompted by ∃z is informationally independent on the move prompted by ∀y. The logic obtained by supplementing first-order formulae with those formulae in which informational independence is allowed is called IF-logic (for independence friendly logic). Hintikka proved the following:

Theorem 4: the IF first-order logic is a non recursively axiomatizable but compact extension of first-order logic.

Informational independence and incomplete information

It happens that if we play dialogical games with informational independence, that amounts in fact in playing strategic games with incomplete information. Such a property may be seen on the following example.

Example 6

Let us suppose we have a structure <D, R> where D = {s, t} and R is a binary relation on D : R = {(s, t), (t, s)}, and let us suppose we have to evaluate the formula:

) , ( y x yR x∃ ∀
, there is of course a winning strategy for the proponent P: to choose the object non selected by O. Let us suppose now we have to evaluate

) , ( ) / ( y x R x y x ∀ ∃ ∀
: there is no longer any winning strategy (neither for P nor for O). The situation is the same as in a Nash game without equilibrium.

Dialogical semantics for natural language

Language as an infinite game

Following Wittgenstein's conception on language and meaning, we can assume that interactions we have in language can always be regarded as games: they are very ordinary games played in every day life. More than that, we can assume that semantics is dialogical, in the sense that the meaning of a sentence is precisely a class of dialogues. Of course, most of them are not necessarily won by one player, but they can be won by both (or all players in case of more than two). These assumptions depend on a more general one, according to which language makes us enter into an infinite dialogue where we fortunately can isolate subdialogues which can be seen as subgames. One purpose of dialogical semantics is to study these subgames and to try to represent their combination. We will see further that linear logic provides us with good tools for expressing the combination part, but of course it is necessary to study the elementary (atomic) subgames: they are intrinsically attached to expressions in natural language. Hintikka's views on the semantics of natural language give us a natural way of solving this task. He notes that natural languages differ from formal ones mainly on some essential points:

natural sentences don't have parentheses which could disambiguate them, like it is the case in formal languages, -natural sentences don't have variables, -the domain from which individuals are extracted is not fixed once and for all, it is always evolving.

Some rules

In consequence, game rules must have a different formulation, (particularly those associated with quantifiers) : their order of application is determined by specific ordering principles (it cannot simply follow from the ordering of connectives in the tree of subformulae, because such a tree cannot be built up), and the selection of individuals is always made in a choice set which changes when a play of the game progresses. This leads Hintikka to the formulation of rules like : <R. some> If the game G(S; M) [where S is the sentence and M a model where it has to be evaluated] has reached an expression of the form: Z -some X who Y -W then the proponent may choose an individual from the appropriate domain, say b, and the game is continued as G(Z -b -W, b is an X and bY; M). The individual b is added to the choice set I S .

In such a formulation, we say that the game G reaches an expression e if e is the currently parsed sentence in a text. That e is of the form "Z -some X who Y -W" means that it contains the words some and who in the order indicated in that pattern, assuming that X does not contain a similar pattern.

Example

<R. negation>

If the game G(S; M) has reached an expression -P which is the negation of P, then the players exchange roles, i.e. the proponent will take the role of the opponent and vice versa. The game goes on as G(P;M).

Ordering principles

As said above, natural language cannot rely on a notion of specific scope for each particle. The priority scope is therefore handled by means of certain ordering principles. General principles interact with specific ones, we have for instance:

<O. Left-Right>

for any semantical game G(S; M), the game rules are applied from left to right, in a same clause (general rule).

<O. comm.>

a game rule must not be applied to an ingredient of a lower clause if it can be applied to an ingredient of a higher one (general rule).

<O. any>

the rule <R. any> has priority over <R. negation>, <R. or> and <R. conditional> (specific rule).

Example :

We haven ' According to [START_REF] Krabbe | The problem of retraction in critical discussion[END_REF], we can schematize this kind of dialogue by a "profile": 0. A: p 1. B: not p (because q) 2. A: concession(q), q' 3. B: q therefore p' 4. A: not p' (because r) 5. B: concession(r), r' 6. A: concession(r'), C 7. B: agreement(C)

where we make appear some of the fundamental status any statement of a dialogue can have: assertion, concession, reason (preceded by because), agreement, disagreement (preceded by not). E. Krabbe suggests that it is possible to deal with argumentative dialogues by using these categories together with specific rules, and notably specific retraction rules which occur here when for instance each participant abandons his or her initial statement (which amounts to a wish). We don't enter into details of this approach here but simply wish to emphasize the fact that to deal with such a dialogue not only needs a payoff function defined for terminations of histories, but a payoff function which attributes points to either participant at intermediary steps. For instance, concessions may be more or less costly, but on another hand, agreements are strongly desirable goals and therefore give high benefits to both participants. Moreover, such games may have several kinds of issues. They can have issues where one of the participants is the winner (and the other one the looser), issues where both are winners (like in this example) and also, unfortunately, issues were both are losers (imagine the conflict leads to separate choices for A and B, who yet would have preferred to share their holidays...). In such games, the extensive definition + the proper definition of the payoff functions would be the appropriate means for describing the dialogue. Let us notice here that if Dialogical Logic has mainly been explored for providing logics with new (and better?) foundations, Game Theoretical Semantics has been more developed in contact with natural language interests. We think that they can complement each other. GTS is based on the use of models: we saw that the tarskian notion of truth and the GTS's notion coincide. This explains why GTS so well suits to all applications where facts are at disposal (for instance for questioning a database here represented by hotels registrations), but in many other cases, facts are either missing or difficult to access: that does not prevent people to have debates. In those precise cases, people must have strategies in order to enforce their partners to concede facts and this is more in the spirit of Lorenzen's dialogical logic. In order to clarify these relations between two conceptions, some neutral form of logical games is required: this role is played by extensive definitions of games which provide the most general frame for developing the study of dialogues and argumentation. Further works will develop these ideas.

Geometry of interaction

Linear Logic

The study of Linear Logic was not intended to provide descriptions for dialogical situations when it was invented by J-Y. [START_REF] Girard | Linear Logic[END_REF]) at the end of the eighties: it simply came from the study of denotational semantics for programming langages, but it revealed soon to have a nice Game-Semantics, as showed primarily by Andreas Blass ( [START_REF] Blass | A game semantics for linear logic[END_REF]), and then by many other researchers like Abramsky and Jagadeesan ([Abramsky and Jagadeesan, 1992]), F. Lamarche ([Lamarche, 1992]) and others. J-Y. Girard himself recently explored this subject in depth, by providing a new game-based theory known as Ludics (but a viewpoint which will not be taken in this paper).

A brief history

Girard's Linear Logic is now well known. Let us simply recall that it comes from problems of theoretical computer science. Deep insights into denotational semantics showed that if we want to have a more fine-grained theory of functions, suitable to computer science problems, we have to take into account not only a general notion of function like it was already expressed in topological terms by Dana Scott, but a notion of function which is sensitive to the number of times it accesses its input. Aiming at this, Girard found particular spaces and particular functions from and to these spaces which were intended to become the mathematical objects needed for this semantics, they were called coherent spaces and linear functions. Intuitively, a linear function is such that when computed from an input, its argument is accessed only once. If we want to speak of an argument which could be accessed an arbitrary number of times, we make use of another type of data, written !A instead of A. If the type of a linear function is denoted by A-o B, the type of an ordinary function (which could access to its input an arbitrary number of times) is therefore !A-o B, in such a way that, remembering of the well known Curry-Howard isomorphism, according to which the space A→B corresponds to the logical formula A⇒B, !A-o B is equivalent to A⇒B. This is the corner stone of Linear Logic, based on a decomposition of the implication into more primitive operators, ! (usually called bang and treated as a kind of modality, that Girard prefers to call an exponential by reference to its mathematical properties), and --o now known as the linear implication (or lollypop!).

Starting from this possible logical reading of this denotational semantics, Girard expressed it in a sequent calculus, which is very similar to the one Gentzen provided for classical logic, but with an important difference. Because Linear Logic is sensitive to the amount of resources, structural rules, usually called Weakening and Contraction are removed. The reason for that is clear. When translated into functional terms, the Weakening rule would say that we should be allowed to give more resources to the computation than required (and therefore "useless resources"), and the Contraction rule would say that we should be allowed to consider two accesses to the same input as being the same one, thus violating the limitation to the number of accesses. Such a restriction has a very strong consequence. Now, two rules which can be very simply shown equivalent in classical logic (by means of Weakening and Contraction) like the following ones :

Γ→A, Γ' ∆→B, ∆' Γ→A,∆ Γ→B,∆ Γ, ∆→ A∧B, Γ',∆' Γ → A∧B,∆
are no longer equivalent, thus giving room for two different connectives, which behave according to the two following rules:

([⊗, R] and [&, R]) Γ→A, Γ' ∆→B, ∆' Γ→A,∆ Γ→B,∆ Γ,∆→ A⊗B,Γ',∆' Γ → A&B,∆
By symmetry, we get four distinct connectives, and their rules for left and right introduction :

(respectively [℘, L], [⊕, L], [⊗, L], [&, L], [℘, R], [⊕, R]) Γ,A→Γ' ∆,B →∆' Γ,A→∆ Γ,B →∆ Γ,∆,A℘B →Γ',∆' Γ, A⊕B → ∆ Γ, A, B → Γ' Γ,A [resp. B]→Γ' Γ, A⊗B → Γ' Γ, A&B → Γ' Γ→ A, B, Γ' Γ →A[resp. B], Γ Γ→ A℘B, Γ' Γ→ A⊕B, Γ'
This formulation also needs to express two identity rules: the axiom and the cut rule.

Axiom : A → A Cut-rule: Γ→A, Γ' ∆, A→ ∆' Γ,∆→ Γ',∆'
Girard showed that the latter is eliminable, and thus, that Gentzen's Hauptsatz applies to Linear Logic. In all these formulations, block Greek letters indicate finite lists of formulas. The only structural rule is the Permutation rule, which allows us not to take care of the order of formulas. Intuitively, the linear implication should have as its rules:

Θ→A, Γ' Γ,B,∆→ ∆' Γ,A →B, ∆ Γ, Θ, A-oB,∆→ Γ',∆' Γ → A-oB,∆
In order to have the same property as in classical logic concerning the implication, i.e. the fact that it can be expressed in terms of disjunction (A⇒B = ¬A∨B), we have to introduce a special negation, called linear negation, denoted by ⊥ which has the following rules:

Γ → A, ∆ Γ,A → ∆ Γ,A ⊥ → ∆ Γ → A ⊥ ,∆
by means of which it is possible to show the analogues of the De Morgan laws:

(A⊗B) ⊥ = A ⊥ ℘B ⊥ (A℘B) ⊥ = A ⊥ ⊗B ⊥ (A&B) ⊥ = A ⊥ ⊕B ⊥ (A⊕B) ⊥ = A ⊥ &B ⊥

Game theoretical aspects

Andreas Blass ( [START_REF] Blass | A game semantics for linear logic[END_REF]) gave an appealing interpretation of Linear Logic, based on the client-server metaphor. It is this interpretation that can help us for our purpose. He says:

"Until now, we have regarded types as essentially synonymous with sets. Computationally, a type could be regarded as a server from which a client can get, in one access, an element of that type; the client need not do anything more than show up". This kind of type would be said simple... simply because the client has nothing to do! But we could also build complex types. In fact, we can understand A&B as a choice, the right rule given above can be read: the resources you have can provide you with A as well as with B, therefore they provide you a choice between the two kinds of data. We may therefore have the complex type A&B which is a server from which the client can get either one A or one B: he must only make a choice. Continuing in this vein, because ⊕ has a symmetric law with regards to &, it seems natural to understand it as a dual choice : the client has nothing to do because the server makes the choice. De Morgan laws interpret as a simple exchange of roles: the server becomes the client and reciprocally! Let us think, say, of a client who phones to a booking office in order to make reservations in a hotel: she asks for a room. The man or woman in the office gives her some choice : bath or shower? TV or not TV? The client makes her choice (the type is bath & shower or TV & not-TV}). The client may also choose the price. After that, the office makes his own choice concerning, say, the location, according to the rooms which are still free in the hotel. Having made her choice, the man or the woman in the office can still give the floor to the client and she may say whether she will come by car or not, so telling whether she needs a garage or not.

Taking again from Blass, we shall say that "we regard a data type as consisting not only of a set of possible data elements but also as the access protocol that is to be run before a data element is provided". Formally, a protocol is a pair consisting of: -a non empty set H of finite sequences, the possible histories of the protocol up to any point in its execution, such that every initial segment of a sequence in H is also in H and such that no infinite sequence has all its finite initial segments in H, and a function N from H to {c, s, t} such that a node h with N(h) = t cannot be a proper initial segment of any other node in H (N(h) = c means that the next move is by the client, N(h) = s that the next move is by the server and N(h) = t that the protocol is terminated).

The dialogical meaning of the multiplicatives

It remains to interpret the other connectives in this light. It is suggested that ⊗, being cumulative, expresses several tasks to be performed necessarily before reaching the terminating point. Because Linear Logic is commutative, it seems that the order between these tasks does not matter: in fact, the dialogue will reach the same point if the man or woman in the office asks first whether the client comes by car or whether she wants a shower or a bathroom... But it can be otherwise in other cases, simply because some choices can depend on others, for instance it is unlikely for the server to propose a garage before knowing whether the client comes by car. A reasonable amount of non-commutativity seems to be required in this case (cf. [START_REF] Retoré | Pomset logic, a non-commutative extension of classical linear logic[END_REF]). [START_REF] Blass | Is game semantics necessary?[END_REF], [START_REF] Blass | Logic and Games : close encounters of the third kind[END_REF]) proposes a more elaborate interpretation of the multiplicatives. "A play of A⊗B consists of interleaved runs of the two constituents A and B." Whenever it is the proponent's turn (= server's turn) to move, he or she must move in the same component in which Opponent (= client) last moved, while Opponent is free to switch components. Let us suppose A and B be two games, like two different chess games, they are played "in parallel". At each move, Opponent chooses the game where he wants to play, and Proponent must reply in the same game. The complex game expressed by A⊗B is terminated when both games A and B are. If we take the dual formula A ⊥ ℘B ⊥ , we simply interchange the two roles: Proponent can switch and Opponent has to reply in the game Proponent imposes to her. This situation is now familiar in the literature on Game-semantics, from the famous example of the game against Karpov and Kasparov: I have a winning strategy (called the copy-cat strategy) against Karpov or Kasparov. I simultaneously play against both (who cannot communicate) and I copy each move made by one of them on the game played against the other. So, if one game is A, the other (obtained by the role exchange) is A ⊥ , and because Proponent (here I) can switch, the whole game is expressed by A℘A ⊥ , which is a provable formula in LL (simply coming from the axiom A→A, which gives →A ⊥ , A and therefore →A ⊥ ℘A).

In any case, A⊗B means a complex game with the impossibility for the Proponent to make use of information obtained from one component in playing in the other, while A℘B means such a game where she can. This of course reflects in the behaviour of the linear implication. As we know, A-o B is equal to A ⊥ ℘B. Therefore players simultaneously play A and B, but in A, players' roles are reversed, and of course information coming from A can be used in playing B. Blass comments: "I could win B if I were shown how to win A", and further: "In effect, the players are playing B but the proponent may, whenever he wishes, temporarily suspend the play of B to consult an expert oracle about how to play A. The consultation consists of the proponent acting as the opponent in a play of A while the oracle, acting as proponent in that play, shows him how to win A". The proponent for A is also the opponent for B, thus really entering into dialogue. We can take as a funny example of such a situation the well known following joke: two persons sit before two unequal pieces of meat, one big and one small. One of them, B takes the biggest one, they have the following dialogue: In this piece of dialogue, B acts exactly like we suggest in the previous paragraph, i.e. by making A playing a reverse game and by deducing its move from the one made by A. In fact, there are four games according to the player who plays first and whether he or she plays with politeness rules or not. B plays the game where she begins and where she does not use politeness rules, but she asks A to play the game where A begins and uses them.

Problems of a complete semantics for linear logic

Connectives of linear logic have thus been shown to be naturally interpreted by game semantics, but if compared with dialogues, linear logic quickly shows its limitations. A classical theorem of Zermelo asserts that if a game of perfect information, between two players, in which every play is a win for exactly one player, is such that it always ends up in a finite number of moves, then the game is determined in the sense that one of the players has a winning strategy [START_REF] Blass | Logic and Games : close encounters of the third kind[END_REF]. This has an important consequence for linear logic: if we identify valid formulae with those for which Proponent has a winning strategy and invalid ones with those for which Opponent has a winning strategy, Zermelo's theorem implies that if we limit ourselves to this particular class of games, all the formulae which express them are either valid or invalid, and it can be shown then that the distinctions between ℘ and ⊕ on one hand and between ⊗ and & on the other collapse, thus leading us back to classical logic. In order to save the completeness of Game semantics for Multiplicative Linear Logic, Abramsky and Jagadeesan ([Abramsky and Jagadeesan, 1992]) are therefore led to admit infinite games and to restrict admissible strategies (to those which are "uniform" and "history-free"). But there could be as well another issue. Zermelo's theorem says that games of perfect information, between two players, in which every play is a win for exactly one player are in some sense trivial, but these are not actually plays we enter into in real life, simply because we play games with imperfect information or games in which every play is not necessarily a win for one of two players, but perhaps a win for both or... a win for none! We shall therefore suggest further that games can be composed by linear logical operators, say in protocols, but that at some stage of the decomposition, we have particular (imperfect information) games which have no necessary winning strategy for one player or the other and which are not necessarily of complete information.

Dialogical acts and combinations of games

If elementary sentences can be understood as simple dialogical games using specific rules together with ordering principles, as soon as they enter into real dialogical acts, they are parts of processes which are analogous to protocols. By a dialogical act, we mean any use of a verbal expression in a context given by two or more participants. The clearest examples we have of such acts are given by requests. We may have very simple requests, like:

(1) (in a library) A: I would like a novel by Zola or more complex ones like:

(2) I would like a particular room in your hotel, such that somebody that I know slept in it… I would like the same room.

We make the assumption that such acts are made by combining elementary atomic games by means of the linear connectives. In a simple request like (1), the interaction may be represented by a game between a client and a server such that after the client asserts (1), the server embeds the verbal material in a linear formula which expresses the fact that "if you give me the title of the novel by Zola that you want to get, then I will (try to) provide you with it". Such an exchange can be represented by a linear implication A -o B where A is the game consisting for the server in asking to the client what novel she has in mind, and B is the game played by the server trying to find it in the library. A request like (2) is more complex because it contains more elementary pieces to combine. We would like to associate with it a class of dialogues exemplified by the following one: This dialogue shows that before the server playing the consequent (the game in which the server answers the request), two elementary games are necessarily played: one where the client has to give the name of somebody she knows who stayed in that hotel some times ago, and another one where the server has to find the right number of the room, depending on the name of the person. These two games are connected by a multiplicative connective (because there is no choice, the two pieces of information are needed), and they are dual to each other because in the first one, the client has to give an information, and in the second one, it is server's turn to give a correct information. Expressing the exact combination is not easy because of the quantifiers. We shall use the same trick as in Hintikka & Sandu, 1997, concerning the difference between priority scope and binding scope. Let us recall that the priority scope (indicated by brackets) indicates the relative priority of different logically active ingredients of the sentence and that the binding scope (indicated by parentheses) indicates for each quantifier the segment where pronouns have that particular quantifier as their heads. Following these distinctions, we shall write the formula (3) as representing the combination of elementary games which enter into (2):

(3) This formula implies that the first game inside brackets is played first, with an existential (A asserts that she knows somebody who slept in that hotel), then the second one is played, also with an existential, but played by B (this is expressed by the linear negation). The values of x and y are kept all over the segment delimitated by parentheses having same index as their corresponding binder. Of course, in such a formula, the order of quantifiers is relevant (even if they are of same type). If we had: (4)

( ) 2

(

) the first game to be played would be the second one, represented by [slept-in(x, y) ⊥ ]. Winning strategies would be different. This complex game would suppose that the client actually does not a priori know the person who slept there before, she only knows that there has been someone, and the server tries to solve the problem by enumerating a list of persons who slept recently in that hotel. Therefore (4) would be associated with a non specific reading of the same request. We can see that this has something to do with the ambiguities of the indefinite a(n), when inserted into a request (cf. Peter wishes to marry a Swiss girl…).

2

Conclusion

This paper was mainly a survey of existing approaches which mix logic and dialogue. We think that they complement each other. Concerning their application to language, we saw that GTS leads to interesting views, expressed in the IF-logic, according to which a proper semantics for elementary sentences in ordinary language is dialogical (cf. the examples of any and of the generalized quantifiers like "as many B's as A's"). But speech acts, which are in fact dialogical ones, require several such elementary expressions be put together: this seems to require another kind of logic, a glue logic. We showed that this role could be played by linear logic. Finally, we sketched an approach of argumentative dialogues, where Lorenzen's concepts seem to be helpful, because in many debates, points are scored not by asserting some atomic facts which happen to be true in some model, but by forcing the partner to concede facts. This kind of debate is in fact closer to Lorenzen's procedural approach than to Hintikka's model based one, but those two approaches can be mixed in the neutral form of Extensive Games.

  : P : I know somebody who came here O : (who?) P : I know Mr Nightingale, (Mr Nightingale is a person), Mr Nightingale came here. <R. every> If the game G(S; M) has reached an expression of the form: Z -every X who Y -W then the opponent may choose an individual from the appropriate domain, say b, and the game is continued as G(Z -b -W, b is an X and bY; M). The individual b is added to the choice set I S . <R. any> like <R. every> Example : P : Everybody wants peace. O : George W. Bush? … P fails!

A

  -you are not fair, you took the biggest one B -what should have you done if you had to choose? A -I should have taken the smallest one, of course! B -So you are happy, because you have it!

  Let us envisage the following rules (some of which are not compatible with others):

D10 P may assert an atomic formula only if O asserted it previously, D11 If at a position p -1, several attacks are still not answered, it is only to the last not already defended attack that it can be answered at p, D12 An attack can have at most one answer, D12' P may repeat an attack or a defence if and only if O has introduced a new atomic formula (which can now be used by P), D13 A P-signed formula can be attacked at most once, D13' In any move, each player may attack a complex formula asserted by his partner or he may defend himself against any attack (including those which have already been defended) Playing this kind of games obviously needs a winning rule: D2 X wins if and only if it is Y's turn but Y cannot move Definition 2: A formula is said to be valid in a dialogical system (defined by the previous definition + some of the global rules) if and only if P has a winning strategy for it, that is, P can win whatever moves are done by O.

  .)

	Example 1
	0. P: (a∧b)⇒(a∧b)
	1. O: (a∧b)					[0, A]
	2. P: ∧ 1											[ 1 , A ]
	3. O : a											[2, D]
	4. P: ∧ 2											[ 1 , A ]
	5. O: b											[4, D]
	6. P: (a∧b)					[1, D]
	7. O: ∧ 1									[ 6 , A ]	7 ' .	O: ∧ 1	[6, A]
	8. P: a											[ 7 , D ]	8 ' .	P: b	[7' D]
	Example 2
	0. P: ((a⇒a)⇒b)⇒b
	1. O: (a⇒a)⇒b	[0, A]
	2. P: (a⇒a)				[1, A]
	3. O: b											[ 2 , D ]	3 . O : a	[2, A]
	4. P : b											[1, D]	4. P : a	[3, D]
	5. O : a											[2, A]	5. O : b	[2, D]
	6. P : a											[5, D]	6. P : b	[1, D]
	Example 3
	0. P:	x ∀	(	P	(	x	)	∨	¬	P	(	x	))
	1. O: τ 2. P: P	( τ	)	∨	¬	P	( τ	)	[ 0 , A ] [1, D]
	3. O: ∨ 4. P: ¬	P	(τ	)		[ 2 , A ] [3, D]
	5. O:	P	(τ	)				[4, A]
	6. P:

  goal G (to go to Paris this summer), 1. B expresses a disagreement on G and she gives a reason (Paris too hot in summer), 2. A agrees with this last statement but A thinks he has a better argument, 3. B suggests another goal G' in substitution to G, 4. A objects to that goal, 5. B objects to the objection, 6. A partially agrees, he retracts and suggests a compromise, 7. B also retracts and agrees with this compromise

		0. A expresses a
		t got any bananas
	0	P : We haven't got any bananas
	1	O : b?	Attack 0 <R. any>
	2	P : we haven't got b, b is a banana	Defence 1
	3	O : you have got b, b is a banana	Attack 2 <R. negation>

because O is asserting an atomic fact, it is either true or false, if it is true, P is wrong, if it is false, P is right. We must notice on that example that the normal order would be: <R. negation> and then <R. any> because of <O. LeftRight>, but <O. LeftRight> is overruled by <O. any>. If we had not this specific rule ordering, the game would have been: which is actually a win for O if P has got any particular banana and is a win for P if she did not get a particular banana, which is obviously not the expected meaning of the sentence. (The winning strategies are obviously not the same in the two games).

On generalized quantifiers

It is known that generalized quantifiers (like most A's, more A's than B's, as many A's as B's and so on) are not definable in first order logic [START_REF] Van Benthem | Higher Order Logic[END_REF]). A.

Pietarinen [START_REF] Sandu | Informationally independent connectives[END_REF] demonstrates that they are in IF-logic. That shows that some verbal expressions can (must?) be interpreted dialogically. What is the reasoning? Let us consider the generalized quantifier "at least as many B's as A's" and let us try to represent the expression "there are at least as many B's as A's". Mathematically speaking, such an expression means that there is an injection from A to B, or said differently that there is some mapping from A to B and that there is some inverse mapping from a subpart of B to A. This therefore can be expressed by the following second order formula:

But this formula has an equivalent in IF-logic, which is:

which means that the expression may be evaluated by means of a game where Opponent chooses at first an x in A, then Proponent chooses a y in B, moreover to ensure that it is not always the same y which is chosen, Opponent chooses a value in B, and then, ignoring what values were chosen previously for x and y, Proponent has to choose a value for u in A, in such a way that if u x ≠ , then z y ≠ must hold.

Dialogue Games in Extensive Form

Extensive form of a game

In the general setting issued from early works by von Neumann, extensive-form games are defined by five-tuples G A = <H, Z, P, N, (u i ) i∈N > where:

• A is set of possible actions,

• H is a set of finite sequences of actions h = <a i > {i=1…n} from A, called histories such that: o the empty sequence <> is in H, o if h∈H then any initial segment of h is in H too • Z is the set of maximal histories,

• P:H -Z → N is the player function which assigns to every non terminal history a player in N whose turn is to move, • each u i where i∈N is the payoff function, that is the function which gives for each maximal history the payoff for player i. In such games, strategies can be defined as functions f i which determine for any history where player i is to move a proper action in A. [START_REF] Sandu | Informationally independent connectives[END_REF]) propose to recast dialogical or semantic games inside extensive-form games. For that, the set of players is restricted to two players: V(erifier) and F(alsifier) (our previously called Proponent and Opponent) and the definition of an extensive-form game is enriched with a sixth component L which is a labelling function L:H → Sub(Φ) where Φ is the game formula and Sub(Φ) is its set of subformulae. L has to verify the following properties:

• L(<>) = Φ,

• for every maximal history h, L(h) is an atomic formula or its negation,

• if L(h) = ¬Φ and P(h)=V, then hΦ∈ H, L(hΦ) = Φ, P(hΦ)=F,

• if L(h) = ¬Φ and P(h)=F, then hΦ∈ H, L(hΦ) = Φ, P(hΦ)=V, ..t m false in the model, then u V (h) = -1 and u F (h) = 1, A winning strategy for X∈{V, F} is a set of strategies f X which leads X to u X (h) = 1 no matter how the player Y decides to act.

Argumentative dialogues

Such definitions are interesting in that they allow more variability in the definition of games. One of the most obvious points where it is possible to make a change is the payoff function. In particular, we can expect now an account of non strictly competitive games, that argumentative dialogues often exemplify. Let us take for instance a small dialogue like the following: We can analyse this dialogue like this:


Lucian Zamfir

Philippe Fortgang

Carole Farre

Micaël Ripert

Gabriel De Crozals

Nicole Jaffrezic-Renault

Camelia Bala

Pierre Temple-Boyer

Carole Chaix
email: carole.chaix@isa-lyon.fr

Synthesis and electroactivated addressing of ferrocenyland azido-modified stemloop oligonucleotides on an integrated electrochemical device

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

Synthesis and electroactivated addressing of ferrocenyl and azido-modified stem-loop oligonucleotides on an integrated electrochemical device

Lucian-Gabriel Zamfir, Philippe Fortgang, Carole Farre, Micaël Ripert, Gabriel de Crozals, Nicole Jaffrezic-Renault, Camelia Bala, Pierre

Temple-Boyer, Carole Chaix

To cite this version:

Introduction

Nucleic acids are precious biological tools that are frequently used to elaborateanalytical devices dedicated to medical diagnostics. Their sequence-specific binding properties are exploited tocontrol either theirarchitecture, for instance by the way of a stem-loop structure, or thepatterning on a support by self-assembly such as for DNA origami [START_REF] Pibiri | Single-molecule positioning in zeromode waveguides by DNA origami nanoadapters[END_REF].The conformational changeof a DNA probe induced by the binding process is frequently exploited to developelectrochemical DNA sensors. Fan et al. [START_REF] Fan | Electrochemical interrogation of conformational changes as a reagentless method for the sequence-specific detection of DNA[END_REF] and Immoos et al. [START_REF] Immoos | Conformationally gated electrochemical gene detection[END_REF] pioneered the use of ferrocene(Fc) modified stem-loop DNA as capture probes for sensors. Probe binding with its complementary DNA target induces a conformationalchange which is followed by electrochemical methods. In our laboratory, we developed strategies for modifying oligonucleotide sequences with many ferrocene derivatives, directly during DNA automated solid-phase synthesis [START_REF] Navarro | Automated synthesis of new ferrocenyl-modified oligonucleotides: study of their properties in solution[END_REF][START_REF] Brisset | The first automated synthesis of ferrocene-labelled phosphorothioate DNA probe: A new potential tool for the fabrication of DNA microarrays[END_REF][START_REF] Chatelain | Synthesis of electrochemical probes for nucleic acid detection[END_REF][START_REF] Chatelain | A thermodynamic study of ferrocene modified hairpin oligonucleotides upon duplex formation: applications to the electrochemical detection of DNA[END_REF][START_REF] Chatelain | Electrochemical detection of nucleic acids using pentaferrocenyl phosphoramidate alpha-oligonucleotides[END_REF]. A stem-loop DNA labeled with four Fc molecules was synthesized and used as a probe in a gold electrode microsensor to accomplish DNA target detection [START_REF] Chatelain | A "four-ferrocene" modified stem-loop structure as a probe for sensitive detection and singlebase mismatch discrimination of DNA[END_REF].A detection limit of 3.5 pM was reached by this DNA sensor.

The electrochemical methods can be easily adapted to a multiplexed format. The addressing of probes on a multi-detection system is a crucial step that directly impacts biochip performance.Devaraj et al. first described the addressing of a ferrocene derivative on independent gold microelectrodes by electrochemical activation of the alkyne/azide cycloaddition (Sharpless "click" reaction) [START_REF] Devaraj | Selective functionalization of independently addressed microelectrodes by electrochemical activation and deactivation of a coupling catalyst[END_REF].This group introduced a method by which the active Cu(I) catalyzing the 1,2,3-triazole formation between a terminal alkyne and an organic azidewas selectively and locally generated under a negative potential on the electrode. They demonstrated that this technique can provide a spatial resolution of the grafting reaction [START_REF] Devaraj | Copper catalyzed azide-alkyne cycloadditions on solid surfaces: applications and future directions[END_REF].Canete and Lai reported the elaboration of a multi-detection system by addressing stemloop oligonucleotides (ON) onto an electrode array through the Cu(I) electrocatalyzed "click" reaction [START_REF] Canete | Fabrication of an electrochemical DNA sensor array via potential-assisted "click" chemistry[END_REF]. A three-probe DNA sensor was developed that could simultaneously detected three different DNA targets at 1 µM concentration.Furst et al. recently reported the same grafting method for DNA patterning ona two-electrode platform [START_REF] Furst | Electrochemical patterning and detection of DNA arrays on a two-electrode platform[END_REF].

In our laboratory, we described a new approach to produce robust sensor arraysviathe Cu(I) electrocatalyzedazide/alkyne cycloaddition [START_REF] Ripert | Selective functionalization of Au electrodes by electrochemical activation of the "click" reaction catalyst[END_REF].It consists ofusinga tetrathiol anchor grafted on gold electrodes to enhance the stability of the tethered entities in buffer and allow efficient surface coverage. A bis(dithiolphosphate)hexynyl derivative was synthesized. The compound provideda tetrathiolbound on gold electrode. The alkyne function of the linker allowed achievingthe "click" reaction. The gain of stability provided by thismultidental anchor was confirmed, allowing several regenerations of the sensor.

In this paper, we focus on addressing ferrocene-modified stem-loop oligonucleotides on gold electrodes viathe Cu(I) electrocatalyzedazide/alkyne "click" reaction. Electrochemical characterization of the functionalized electrodes is presented. Hybridization study permitted to determine a detection limit of the complementary DNA target at 10 fM that is much better than the sensitivity reached by previously described stem-loop DNA sensors developed with different grafting strategies [START_REF] Chatelain | A "four-ferrocene" modified stem-loop structure as a probe for sensitive detection and singlebase mismatch discrimination of DNA[END_REF][START_REF] Yu | A reagentless and reusable electrochemical DNA sensor based on target hybridization-induced stem-lopp probe formation[END_REF].

We also confirmed the robustness of the sensitive layer on the sensor that allowed several regenerations of the system. Finally, we investigated the electrocatalyzed "click" reaction on an integrated electrochemical device, constructed with a gold working microelectrode of 19.6 x 10 -4 mm 2 , a platinum counter electrode of 1 mm 2 and a Ag/AgClreference electrode of 2 x 10 -2 mm 2 . A fluorescent azide-modified oligonucleotide was electroaddressed following our protocol. Fluorescence recorded by microscopy confirmed the selective functionalization of the gold surface.

Experimental

Chemicals

Potassium phosphate monobasic (≥ 98%), sodium perchlorate and tris[(1-benzyl-1H-1,2,3triazol-4-yl)methyl]amine (TBTA), tris(2-carboxyethyl)phosphine hydrochloride (TCEP/HCl), all other solvents and chemicals were purchased from Sigma-Aldrich. Sodium phosphate dibasic dihydrate (≥ 99%), celite and silica gel were obtained from Fluka. Sodium hydroxide was obtained from Laurylab.Phosphoramidites, dT-CPG column, reagents and solvents used for DNA synthesis were purchased from Glen Research (Sterling, Virginia). The salmon sperm DNA (10 mg/mL) was purchased from Invitrogen. The non-complementary target ( 5' TTT TTA TTG AGA TTC CCG AGA TTG ATT TTT 3' ) and the complementary influenza target ( 5' TTT TTTTTT TAG TTT TTG GAC GTC TTC TCC TTT TTTTTT T 3' ) were obtained from Eurogentec.The fluorescent (Cyanine5) oligonucleotide (5'C 6 H 12 N 3 -TTT TTT TTT T Cy5 G 3') was synthesized in the laboratory using standard DNA solid-phase synthesis methods. ESI Mass Spectrometry was conducted on a Bruker micrOTOF-Q II and MALDI-TOF MS on an Applied Biosystems Voyager DE-PRO (Foster city, USA), using 2,4,6-trihydroxyacetophenone monohydrate (THAP) as the matrix. 1 H NMR spectra were recorded on a DRX 300 Bruker spectrometer.

Synthesis of the (DTPA) 2 hexynyl linker

The (DTPA) 2 hexynyl linker was synthesized by a solid-phase approach according to a protocol previously described [START_REF] Ripert | Selective functionalization of Au electrodes by electrochemical activation of the "click" reaction catalyst[END_REF]. The purity was controlled by high performance liquid chromatography and a global yield of 80% was estimated by UV spectroscopy. The oligonucleotides were synthesized with an automated instrument via the phosphoramidite chemistry, according to a standard protocol. A dT-CPG column was used to start each synthesis.All the resulting oligonucleotidic sequences ended with a thymidine at the 3'extremity. The synthesis was achieved via the standard 1µM phosphoramidite cycle on the synthesizer. The bifunctionalferrocenephosphoramidite

(1-[3-O-dimethoxytritylpropyl]-1-[3-O-(2-cyanoethyl-N,N- diisopropylphosphoramidityl)propyl]
ferrocene) was used in the DNA synthesizer to incorporate Fc directly into the oligonucleotide sequence during solid-phase synthesis. A coupling time of 30 seconds was applied for the incorporation of the ferrocenesynthon (0.1M in Acetonitrile (AcCN)). For the (DTPA) 2 In4Fc, two successive incorporations of the dithiolphosphoramiditesynthon (DTPA) were achieved at the end of synthesis, following the protocol recommended by Glen Research. For the N 3 In'1Fc, the 5'-bromohexyl phosphoramidite was incorporated at the 5' extremity with a coupling yield > 98%.

For the Tb(N 3 ) 3 In'4Fc, the treblersynthon (Tris-2,2,2-[3-(4,4'dimethoxytrityloxy)propyloxymethyl]ethyl-[(2-cyanoethyl)-(N,N-diisopropyl)]phosphoramidite) was incorporated at the 5' extremity with a coupling yield of 85% before coupling the bromohexylphosphoramidite. The N 3 In'1Fc and Tb(N 3 ) 3 In'4Fc functionalized CPG were then suspended in 2 mL of a solution of sodium azide (100 mM), sodium iodide (100 mM) in anhydrous DMF in order to substitute the bromo functions to azides. The reaction was performed for 2 hours at 65°C. After the cleavage and deprotection step in 1mL of concentrated NH 4 OH (30%/H 2 O), 16 hours at 60°C, the crude materials were concentrated to dryness in a speed-vacuum instrument. The oligonucleotides were purified by HPLC using a DeltaPak C18 15 µm 300 Å (300 x 7.8 mm) column with an acetonitrile gradient from 0 to 50% in 0.05 M triethylammonium acetate buffer (TEAAc), pH 7. Oligonucleotide purity was controlled by HPLC using an X-terra MS C18 2.5 µm (50 x 4.6) column from Waters (Versailles, France). Analyses were carried out with an acetonitrile gradient from 5 to 50% in 0.05 M TEAAc, pH 7, in 65 min at 60°C. Then, the oligonucleotide was characterized by MALDI-TOF MS analysis (M-H) -(g mol -1 ): 10185.2 for(DTPA) 2 In4Fc (Calculated : 10187.49); 10771.0 for N 3 In'1Fc (Calculated : 10773.8); 12641.9 for Tb(N 3 ) 3 In'4Fc (Calculated : 12646.4).

Electrochemical materials

Aqueous buffers were made with deionized water purified through a Milli-Q system (Millipore, Bedford, MA). All analyses were conducted in PBE (phosphate buffer electrolyte: sodium phosphate dibasic and potassium phosphate monobasic (20 mM), NaClO 4 (250 mM), pH 6.4). The solutions were deoxygenated under argon before use. Electrochemical measurements were carried out in a conventional one compartment-three electrode cell with an internal volume of 5 mL (VerreEquipements, Collonges au Mont d'Or, France), hermetically closed on one side with a planar gold electrode (active surface of 7 mm 2 ) used as the working electrode and, on the other side, a planar platinum electrode (active surface of 38.5 mm 2 ) used as the counter electrode. A KCl saturated calomel electrode (SCE) from Radiometer Analytical (Villeurbanne, France) was used as a reference for aqueous solution and an Ag/Ag+ reference electrode from ALS Japan (Ag wire in acetonitrile containing 0.01M AgNO 3 and 0.1M tetrabutylammonium perchlorate) for non-aqueous solvents. This system was connected to a Bio-Logic potentiostat VMP2 (Bio-Logic Science Instruments, Pont de Claix, France). Results were recorded using EC-Lab software from Bio-Logic Science Instruments. In this work, we addressed and recorded the responses of the gold electrode by cyclic voltammetry (CV).

(Au -Pt -Ag) electrochemical microcells (ElecCell) devices were fabricated using silicon technology [START_REF] Christophe | Elaboration of integrated microelectrodes for the detection of antioxidant species[END_REF]. This electrochemical microdeviceintegrates on an oxidized (silicon oxide thickness: ~ 1 m) silicon substrate a gold working microelectrode, a platinum counter microelectrode and a silver microelectrode to be used as an integrated reference one. The different metallic layers were deposited by evaporation techniques, and patterned using a bilayer lift-off process. Three depositions were performed in a row: firstly, a 120 nm platinum layer was deposited on a 20 nm titanium underlayer in order to ensure platinum adhesion on silicon oxide SiO 2 , followed by a 400 nm gold and a 400 nm silver layer. Finally, a biocompatible Si 3 N 4 passivation layer (thickness: 100 nm) was deposited at the wafer level and patterned using photolithography techniques [START_REF] Vanhove | Final capping passivation layers for long-life microsensors in real fluids[END_REF].According to this final wafer-level passivation process, the different metallic layers were insulated electrically and their active surfaces were defined precisely.The gold working microelectrode was defined as a disk of 19.6 10 -4 mm 2 (diameter: 50 m). In contrast, a very large silver/silver chloride reference microelectrode (0.02 mm 2 ) and platinum counter microelectrode (1 mm 2 ) were co-integrated. The whole silicon chip was then wire bonded and packaged on a specifically coated printed circuit to be compatible with liquid phase measurement (see Supplementary information).The gold squares and ElecCell devices were gratefully furnished by the LAAS-CNRS (Toulouse -France).

Activation of the gold surface

The gold electrode was sonicated in acetone for 15 minutes. The surface was cleaned with ethanol and ultrapure water and dried with argon. The electrode was positioned on the electrochemical cell and 4 mL of NaOH solution 0.5 M were added. The gold surfaces were activated by cycling 10 times between -0.4 and -1.4 Vvs.SCE, with a scan rate of 0.05 V s -1 . The gold electrode was washed with ultrapure water and dried with an argon flow.

Electrode functionalization with the tetrathiol-modified oligonucleotide (DTPA) 2 In4Fc

Prior to thiolgrafting onto the gold surface, 1µL of tris(2-carboxyethyl) phosphine hydrochloride (TCEP/HCl, 160 mM in water) was added to 1 nmol of (DTPA) 2 In4Fc in 5µL of water and the reduction was carried out for 2 hours. In parallel, gold surfaces were activated using the method previously described. Then, (DTPA) 2 In4Fc functionalized electrode was prepared by soaking the freshly activated gold surface at room temperature under argon in 96µL of phosphate buffer solution (PB, 100 mM), pH8 containing reduced (DTPA) 2 In4Fc (10 µM) and TCEP/HCl (1.6 mM) during 3 days, according to the protocol described by Ricci et al. [START_REF] Ricci | Effect of Molecular Crowding on the Response of an Electrochemical DNA Sensor[END_REF]. Then, after rinsing the electrode with water, the system was passivated by soaking for 1 h in 60 µL of a solution of mercaptopropanol (1 mM) in water and finally rinsed with water.

Electrode functionalization with the azideferrocene oligonucleotides N 3 In'1Fc and Tb(N 3 ) 3 In'4Fc

Prior to thiolgrafting onto the gold surface, 1 µL of TCEP/HCl (160 mM in water) was added to 5 µL of (DTPA) 2 hexynyl 1 (0.8 mM in water) to reduce the thiol functions [START_REF] Burns | Selective reduction of disulfides by tris(2carboxyethyl)phosphine[END_REF]. The gold surfaces were activated as described in the precedent part. The (DTPA) 2 hexynyl functionalized electrodes were prepared by soaking the freshly activated gold surface at room temperature under argon in 96 µL of phosphate buffer solution (100 mM), pH 8 containing reduced (DTPA) 2 hexynyl 1 (40 µM) and TCEP/HCl (1.6 mM) during 3 days. Then, after rinsing the electrode with water, the system was passivated by soaking for 1 h in 60 µL of a solution of mercaptopropanol (1 mM) in water and finally rinsed with water. The electrode was used immediately after preparation in order to "click" the azideferrocene oligonucleotides. The "click" reaction procedure used was similar to that described by Devaraj et al. [START_REF] Devaraj | Selective functionalization of independently addressed microelectrodes by electrochemical activation and deactivation of a coupling catalyst[END_REF]and Canete and Lai [START_REF] Canete | Fabrication of an electrochemical DNA sensor array via potential-assisted "click" chemistry[END_REF]. The modified electrode was thoroughly rinsed with water, and incubated in 4 mL of 10 µM TBTA, 10 µM CuSO 4 and 1 µM of the azideferrocene oligonucleotide in a mixed solution of DMSO/water (50/50). The "click" reaction proceeded in chronoamperometryfor 2h with an applied potential of -300 mV vs. Ag/Ag + . The electrode was rinsed thoroughly with water and then with PBE to remove physically adsorbed substances.

Electrochemical measurements and characterization

Before analysis, the grafted electrode was washed with water and finally with PBE. Electrochemical measurements were performed in 5 mL of PBE. Cyclic voltammograms were recorded within a potential range from 0.1 V to 0.9 V vs. SCE. For each measurement, 3 successive cycles were carried out to ensure signal stabilization and the third cycle was kept as the result. To determine the electron transfer constant k et , the same cyclic voltammetry (CV) measurements were conducted at increasing scan rates, from 0.05 V s -1 to 5 V s -1 .

Hybridization assays in complex medium

The functionalized electrodes were immerged in water (milliQ grade) and incubated at 70°C for one minute. After a rapid wash in water, the electrochemical cell was filled with buffer and the current intensity of the oxidation peak was measured every 15 minutes for 2 hours.Then, PBE buffer was changed to complex medium made of PBE implemented with 20 nMof non-complementary target, 50 µg mL -1 of salmon sperm DNA and deoxygenated under argon before use. The signal stabilization was followed by CV for 3 hours. Hybridization assays were performed by recording CV (from 0 V to 0.9 V vs. SCE, scan rate of 0.1 V s -1 ) each 15 minutes after adding in the solution the non-complementary target (NCT) or the complementary target (CT) at different concentrations.

Results and discussion

3.1. Grafting strategy of the (DTPA) 2 hexynyl or (DTPA) 2 In4Fc derivative Fig. 1.Gold electrode functionalization with the tetrathiol derivatives ((DTPA) 2 hexynyl or (DTPA) 2 In4Fc) Gold electrode functionalization was achieved withtetrathiol derivatives. Thetetrathiololigonucleotide (DTPA) 2 In4Fcwas used for direct grafting. Thiols were first reduced with the tris(2carboxyethyl) phosphine hydrochloride (TCEP) prior to react with freshly activated gold surfaces, as described in the experimental part. The probe addressing via the "click" reaction was achieved in two steps. The first step was the gold surface functionalization by alkyne functions using the (DTPA) 2 hexynyllinker. A passivation step was performed with mercaptopropanol to forma mixed compact layer of (DTPA) 2 derivative and mercaptopropanol.The latter was used as a diluent of the hexynyl functions and for passivation of the electrode. The second step was the probe immobilizationvia the electroactivated "click" reaction.

3.2.Design and synthesis of the ferroceneazideoligonucleotides

The stem-loop sequences described in this article (Table 1) were designed by bioMerieux for Influenza B virus RT-PCR product analysis [START_REF] Chatelain | A "four-ferrocene" modified stem-loop structure as a probe for sensitive detection and singlebase mismatch discrimination of DNA[END_REF]. Previous thermodynamic studies demonstrated that the folded structure was not destabilized by incorporating numerous Fc molecules inside or at the extremities of the hairpin [START_REF] Chatelain | A thermodynamic study of ferrocene modified hairpin oligonucleotides upon duplex formation: applications to the electrochemical detection of DNA[END_REF].Ferrocenes bound to oligonucleotide probes act as electrochemical markers, affording a sharp electrochemical response which is highly sensitive to ionic and steric surrounding media. In the literature, Fc modified stem-loop structures have already been described as efficient probes for electrochemical DNA detection after anchoring on electrode surfaces [START_REF] Fan | Biosensors based on binding-modulated donor-acceptor distances[END_REF][START_REF] Radi | Reagentless, Reusable, ultrasensitive electrochemical molecular beacon aptasensor[END_REF][START_REF] Han | A regenerative electrochemical sensor based on oligonucleotide for the selective determination of mercury(II)[END_REF].In comparison to the papers already published in this field, the main advantage of our work lies in the easy incorporation of numerous ferrocenes in the probes to enhance the electrochemical response of the DNA sensor. The melting temperature (Tm) values were calculated directly from the sequences. This parameter gives relevant information on the thermodynamics of the DNA systems. Considering the sequences, two different equilibria were observed. With a Tm of 60.4°C, the hairpin structure is favored in the case of In sequence. On the other hand, duplex gives a higher Tm than hairpin for the In' sequence. The two sequences have been chosen to compare the influence of the hairpin structure on the performance of the DNA electrochemical assay. In'4FcImages could be on one row N 3 In'1Fc and Tb(N 3 ) 3 In'4Fc probes are 32 mer hairpin oligonucleotides bearing one or four successive Fc-phosphates at the 3' end, respectively. As described in Fig. 2, N 3 In'1Fc is functionalized at the 5'end by an azidohexyl arm. Tb(N 3 ) 3 In'4Fc bears a tridental azido arm. The oligonucleotides were synthesized using standard phosphoramidite chemistry. The ferrocenephosphoramidite used in the study was synthesized according to the protocol described by Chatelain et al. [START_REF] Chatelain | A "four-ferrocene" modified stem-loop structure as a probe for sensitive detection and singlebase mismatch discrimination of DNA[END_REF].The three azidearms of Tb(N 3 ) 3 In'4Fcwereincorporated at the 3'endby reaction on a tridental "trebler" linker directly during oligonucleotide synthesis, as described in the experimental part.Oligonucleotides were purified by HPLC and analyzed by mass spectrometry (MALDI-MS) (see Supplementary information).

Grafting strategy of the N 3 In'1Fc and the Tb(N 3 ) 3 In'4Fc

The different steps of the gold electrode functionalization via the "click" reaction are described in Fig. 1 and Fig. 3. Firstly, the (DTPA) 2 hexynyl linker was grafted on the surface. Then, the oligonucleotides were tethered by reactingazide oligonucleotides to alkyne functionalized layer, in presence of CuSO 4 and TBTA as a stabilizing ligand of Cu(I) catalysis [START_REF] Canete | Fabrication of an electrochemical DNA sensor array via potential-assisted "click" chemistry[END_REF][START_REF] Chan | Polytriazoles as copper(I)-stabilizing ligands in catalysis[END_REF]. Two hours under -300 mV vs. Ag/Ag + was the optimized protocol as described in previous papers [START_REF] Canete | Fabrication of an electrochemical DNA sensor array via potential-assisted "click" chemistry[END_REF][START_REF] Ripert | Selective functionalization of Au electrodes by electrochemical activation of the "click" reaction catalyst[END_REF].The coupling yield was estimated around 55% with the N 3 In'1Fc and 45% with the Tb(N 3 ) 3 In'4Fc by comparison with the surface coverage reached with the (DTPA) 2 In4Fc (see Table 2). It is worth noting that the resulting link between the gold surface and the hairpin probe is much longer via the "click" grafting than via the DTPA grafting. This difference will impact the sensitivity of the hybridization assay.

Fig. 3.Grafting strategy of the N 3 In'1Fc probe via the Cu(I) electrocatalyzed "click" reaction

The hairpin design has been optimized with the In' sequence. It corresponds to a hairpin with seven base pairs in the stem to keep a good stability of the folded structure and eighteen bases in the loop instead of fourteen presented in the In sequence. This longer part of the complementary strand will favor the hairpin opening upon binding with the complementary target.

Electrochemical properties of the electrodes grafted with the oligonucleotidic probes

The functionalized electrodes were characterized with a standard electrochemical cell using conventional equipment (see Experimental part).The cyclic voltammogramswere recorded from 0 V to 0.9 V vs. SCE at a scan rate of 0.1 V s -1 in order to characterize our systems. Firstly, the system's oxidation and reduction potentials were determined. CV was performed after stabilization of the systemwhen the hairpin probes were perfectly folded under a stem-loop structure on the gold electrode.Fig. 4 shows CV of the electrodes functionalized with N 3 In'1Fc and Tb(N 3 ) 3 In'4Fc. The current response of the Tb(N 3 ) 3 In'4Fc electrode was clearly enhanced by the fourferrocene tag compared to the N 3 In'1Fc electrode. The CV profiles were similar to the one obtained from the (DTPA) 2 In4Fc functionalized electrode elaborated by a one-step grafting (see Supplementary information). The increased current observed by CV with the (DTPA) 2 In4Fc and Tb(N 3 ) 3 In'4Fc electrodes confirms the interest of using a polyferrocene label to improve sensor performance.

An E 1/2 of 0.55 V vs. SCE was measured for (DTPA) 2 In4Fc electrode, E 1/2 of 0.55 V and 0.57 V were recorded forN 3 In'1Fcand Tb(N 3 ) 3 In'4Fc electrodes respectively (Fig. 4). Low ΔE (≤ 0.06V) were measured that confirms the good reversibility of our system. We recorded CV signals at different scan rates (ʋ). A linear curve was obtained when plotting I max vs.ʋfor the three electrodes that allowed us to demonstrate the covalent attachment of the probeson surface (see Supplementary information). The electron transfer rate constants (k et ) were calculated as described by Yu et al. [START_REF] Yu | Voltammetric procedure for examining DNA-modified surfaces: quantitation, cationic binding activity, and electron-transfer kinetics[END_REF]. The obtained values were similar to those reported by our group with comparable stem-loop structures bond to the electrode through a different linker [START_REF] Chatelain | A "four-ferrocene" modified stem-loop structure as a probe for sensitive detection and singlebase mismatch discrimination of DNA[END_REF]. The k et values calculated for N 3 In'1Fc and Tb(N 3 ) 3 In'4Fc probes were slightly lower than the (DTPA) 2 In4Fc k et . This result can be relatedto the greater length of the linker used for the "click" reaction that impact theelectron transfer of ferrocenedue to its higher distance to the electrode.

Surface coverage of the functionalized gold electrodes

By integrating the oxidation and reduction peaks observed by CV, the charge transferred to the electrode (Q) can be calculated as described previously [START_REF] Farre | Synthesis and electropolymerisation studies of water-soluble pyrrole-ferrocene derivatives towards biochip device application[END_REF].Then, the total amount of bound ferrocene units onto the electrode Γcan be quantitatively evaluated by using the equation Γ= Q ox /nFA, where n is the number of electrons transferred (n = 1), F the Faraday constant (coulombs per equivalent), and A is the effective surface area (square centimeters).That permits to estimate the amount of probes boundto the electrode dividing Γ by the number of ferrocene per oligonucleotide (Table 3). In a general manner, the surface coverages indicated in the table 3 are similar to those obtained with ferrocene-modified oligonucleotides [START_REF] Brisset | The first automated synthesis of ferrocene-labelled phosphorothioate DNA probe: A new potential tool for the fabrication of DNA microarrays[END_REF]. By comparing with other electroactivated grafting strategies reported in the literature, it is worth noting that the oligonucleotide grafting density obtained through our strategy of "click" reaction is higher than the one reported by Canete and Lai (from 6.88 10 11 to 1.88 10 12 molecules cm -2 ) on an azide-modified SAM made with C6 alkyl chains. In our case, the (DTPA) 2 hexynyllinker probably offers a better accessibility for terminal alkynes to optimize the "click" reaction with azide oligonucleotides.

Hybridization assays in complex medium

Hairpin design were optimized in order to maintain hairpin folding stability in PBE buffer and complex medium together with an efficient opening towards complementary target binding. In and In' designs were compared. Assays achieved with the In sequence (resulting from (DTPA) 2 In4Fc grafting) revealed a strong instability of the electrochemical signal, avoiding a clear variation upon binding. On Fig. 4, we can observe that the binding assay achieved at 100 pM of DNA target led to an average variation of 16%. The other experiments run at lower target concentrations failed to give significant results.One explanation of this lack of sensitivity could be that at low target concentration, the In hairpin preferred to stay folded on itself as the thermodynamic equilibrium favored selfassembling,instead of hybridizeto the complementary DNA target. On the other hand, In' hairpin gave significant and much reproducible variations of the electrochemical response upon binding with the complementary strand (see Fig. S5 in the Supplementary information). The electrodes functionalized with N 3 In'1Fc and Tb(N 3 ) 3 In'4Fc probes were assessed for DNA detection. A notable variation of current was observed upon binding with the complementary target. Signal turn-off was never total, even at a target concentration higher than 100 pM (data not shown). To explain the residual signal observed upon hybridization, Anne et al. hypothesized a certain flexibility of the duplex formed by short-DNA fragments [START_REF] Anne | Dynamics of electron transport by elastic bending of short DNA duplexes. Experimental study and quantitative modeling of the cyclic voltammetric behavior of 3'-ferrocenyl DNA end-grafted on Gold[END_REF]. In our case, the 40-base DNA target is probably too short to ensure total inhibition of the duplex elastic bending towards the electrode. A significant residual signal is observed even after duplex formation. The maximum variation observed at the plateau was around 25% (between 100pM and 100fM) and 30% (between 100 pM and 10 pM) for N 3 In'1Fc and Tb(N 3 ) 3 In'4Fc functionalized electrodes respectively. Negative controls achieved with the non-complementary DNA did not show any significant variation (less than 5% variation) confirming the good selectivity of the sensor. The detection threshold (5%) was determined as the average value of three negative assays. A signal variation superior to this value was considered as a positive response. The detection limit was determined at 100 fM for the Tb(N 3 ) 3 In'4Fc and 10fM for the N 3 In'1Fc electrodes respectively. The longer length of the linker used for Tb(N 3 ) 3 In'4Fc grafting could explain the more important variation recorded upon binding. On the other hand, the flexibility of the link is probably increased. At very low target concentration (≤ 10fM), the duplex formed has more room on the surface that permits elastic bending towards the electrode. The best detection limit was determined for N 3 In'1Fc electrode.Both N 3 In'1Fc andTb(N 3 ) 3 In'4Fc sensors exhibit very good sensitivity by comparison with previous studies from our group performed with hairpin probes with a C6-thiol linker [START_REF] Chatelain | A "four-ferrocene" modified stem-loop structure as a probe for sensitive detection and singlebase mismatch discrimination of DNA[END_REF]. The electroactivated "click" reaction is a performant grafting strategy to elaborate ultra-sensitive sensors.

3.7.Electrode stability in buffer

The stability of the functionalized electrodes was studied in PBE. The functionalized gold electrodes were immersed in PBE and maintained at room temperature. The maximumintensity at Epa was recorded by CV and monitored over a period of 24 days (see Supplementary information). The PBE solution was changed every couple of days, immediately before CV analysis. Phares et al. reported a similar stability study with a monothiol oligonucleotide compared to atrithiol anchoring groups [START_REF] Phares | Improving the Stability and Sensing of Electrochemical Biosensors by Employing Trithiol-Anchoring Groups in a Six-Carbon Self-Assembled Monolayer[END_REF]. When stored for 24 days in an aqueous buffer at room temperature, the monothiol sensor exhibited a ∼70% loss in signaling current. In our case, the recorded signal exhibited a very slight decrease (less than 10%) for both electrodes over the period. The results confirmed the excellent stability of the electrode in buffer and the pertinence of using a tetradental anchoring in order to increase the stability of the sensor over time.

3.8.ElecCell integrated device functionalization by "click" reaction with a fluorescent oligonucleotide probe

The ElecCell integrated device was used for the immobilization of a fluorescent oligonucleotide using the electrocatalyzed "click" reaction previously described. The system was characterized using a fluorescence microscope with an excitation wavelength of 525 ± 25nm. After grafting, the average fluorescence recorded at the emission wavelength of 605 ± 70 nm on the gold working electrode was 8 fold superior to those of the reference and counter electrodes considered asfluorescence background signal (see Supplementary information). The whole integrated device was treated under the same protocol of activation and grafting (as described in the experimental part). This experiment proved the selectivity of the grafting approach. The electrocatalyzed "click" reaction is pertinent for the specific addressing of a gold electrode in an integrated sensor without significant cross-reaction on the other electrodes of the device.

Conclusion

Ferrocenyl and azido-modified stem-loop oligonucleotides were successfully synthesized by solid-phase synthesis. Their immobilization by the potential-assisted "click" reaction on an electrochemical system was studied. This method permitted selective functionalization of gold electrode on an electrochemical microcell (ElecCell) device that integrated the working, counter and reference electrodes. Specific detection of the complementary DNA target was demonstrated in complex medium at 10 fMlevel. Perspectives of this work will be to demonstrate the addressing efficiency on a multi-electrode system, opening the route of the multiplex detection.

  Lucian-Gabriel Zamfir, Philippe Fortgang, Carole Farre, Micaël Ripert, Gabriel de Crozals, et al.. Synthesis and electroactivated addressing of ferrocenyl and azido-modified stem-loop oligonucleotides on an integrated electrochemical device. Electrochimica Acta, 2015, 164, pp.62-70. 10.1016/j.electacta.2015.02.167. hal-01227342

2. 3 .

 3 Synthesis of the oligonucleotides (DTPA) 2 In4Fc , N 3 In'1Fc and Tb(N 3 ) 3 In'4Fc

Fig. 2 .

 2 Fig. 2.Structures of the probes a) N 3 In'1Fc and b) Tb(N 3 ) 3 In'4FcImages could be on one row

Fig. 4 .

 4 Fig.4.Cyclic voltammograms recorded on a gold electrode functionalized with a) N 3 In'1Fc; b) Tb(N 3 ) 3 In'4Fc(scan rate of 0.1 V s -1 vs. SCE)

Fig. 5 .

 5 Fig.5.Hybridization assays with the complementary target (+) (sequence from Influenza virus genome) and the non-complementary target (-) on the functionalized electrodes.Peak current (I ox ) variation on CVs (in percentage) after 1 hour of incubation with the DNA target at different concentrations (either 100pM, 10pM, 100fM, 10fM or 1fM). The bare errors are the average of two or three experiments. The background signal is determined to be ≤ ±5% of Iox variation.

Table 1 .

 1 Theoretical T m values of In and In' hairpins calculated from the free software http://eu.idtdna.com(250 mM [Na + ], 1 µM [oligonucleotide], 20°C). The gray overlay corresponds to the binding part with the complementary target. *The T m duplex was calculated with the complementary strand of the loop.

	ON probe	sequence	T m hairpin self-folding (°C)	T m duplex* (°C)
	In	5' CGA TCG GAA GAC GTC CAA AAC GAT CG 3'	60.4	54.1
	In'	5' CTG CTA GGG AGA AGA CGT CCA AAA ACT AGC AG 3'	56.7	63.8

Table 2 .

 2 Electrochemical properties of the functionalized gold electrodes in a conventional device; reference electrode: KCl saturated calomel electrode (SCE), scan rate: 0.1 Vs -1 , electrolytic medium (PBE): phosphate buffer (20mM), NaClO 4 (250 mM), pH 6.4.

	hairpinprobes	E pc (V)	E pa (V)	E 1/2 (V)	ΔE (V)	k et (s -1 )
	(DTPA) 2 In4Fc	0.53	0.58	0.55	0.05	6.3
	N 3 In'1Fc	0.53	0.58	0.55	0.05	5.5
	Tb(N 3 ) 3 In'4Fc	0.54	0.60	0.57	0.06	5.2

Table 3 .

 3 Surface coverage of the functionalized gold electrodes.

	Oligonucleotides	Surface coverage (molcm -2 )	Surface coverage (moleculescm -2 )
	(DTPA) 2 In4Fc N 3 In'1Fc Tb(N 3 ) 3 In'4Fc	1.4 10 -10 ± 4.6 10 -11 0.78 10 -10 ± 1.4 10 -11 0.63 10 -10 ± 1.7 10 -11	8.5 10 13 ± 2.8 10 13 4.7 10 13 ± 8.6 10 12 3.8 10 13 ± 1.0 10 13

Acknowledgments

The authors are grateful to project Marie Curie PIRSES-GA-2012-318053.L-G. Zamfir was supported by the strategic grant POSDRU/159/1.5/S/137750, "Project Doctoral and Postdoctoral programs support for increased competitiveness in Exact Sciences research" cofinanced by the European Social Found within the Sectorial Operational Program Human Resources Development 2007-2013.MicaëlRipert and Gabriel De Crozals PhD theses were financed by the French Ministry of Research. The work was partly supported by CAMPUS-France through PHC Project BRANCUSI No29504QL. Finally, the technological realizations and associated research works were partly supported by the French RENATECH network.