Emilie † Dargaud
email: dargaud@gate.cnrs.fr

Gate

David Balan

Laurent Flochel

Michele Polo

Ashley Roughton

Mergers, cartels and leniency programs : the role of production capacities *

Keywords: Mergers, Collusion, Leniency Programs. JEL classification: L11, L41, K42

In this paper, we study the impact of a merger on collusion depending on the endowment of capital asset among firms. We show that the merger makes the collusion easier to sustain when asymmetric capital stock combines with less efficient insiders because of more symmetric conditions and closer incentive constraints. Moreover, this model allows us to determine an optimal threshold of asymmetry among insiders and outsiders such as a merger has pro-competitive effects and we compare this value with the value which would restore perfect symmetry between firms after the merger.

I. Introduction

Merger control is a preventive authorization system with the major goal to prevent mergers which diminish competitiveness. The antitrust authority analyses if a merger can make it more likely or more effective for firms to coordinate their actions (coordinated effects) and if it can make it profitable for the merging firms to reduce output and raise price unilaterally (unilateral effects) 1 .

The impact of a merger on the possibility to collude ex post is studied by several authors 2 who show that if the asymmetry between firms is increased after a merger, then the risk of coordinated actions decreases. When firms become very asymmetric, then it is more difficult to provide the right incentives to small firms not to deviate from the collusive path.

From the merger control point of view, our model reinforces these results and completes the existing literature to determine a threshold of asymmetry necessary for a merger to have pro-competitive effects.

Another strand of literature related to this paper deals with leniency programs (as [START_REF] Motta | Leniency programs and cartel prosecution[END_REF]. These programs reduce fines for cartel members who report themselves to and assist the antitrust authority, and the impact of leniency programs can be seen in the recent increase in successful cartel investigations. For example, in the period from 14 February 2002 until the end of 2005, the European Commission received 167 applications under the 2002 Leniency Notice. Of these applications, 87 were requests for immunity and 80 were requests for reduction in fines. The Commission has granted 51 decisions for conditional immunity. Over the same period, 1 Council Regulation (EC) no 139/2004 2 Compte, Jenny and Rey (2002), [START_REF] Mcafee | Horizontal mergers and antitrust policy[END_REF], [START_REF] Vasconcelos | Tacit collusion, cost asymmetries, and mergers[END_REF] 2 the Commission rejected or decided not to deal any further with 23 applications and had under scrutiny 13 more recent applications. 3 . The presence of leniency programs introduces a coordination game between the cartel members (apply for leniency or not?). [START_REF] Motta | Leniency programs and cartel prosecution[END_REF] show that leniency programs have two possible effects. The first is the deviation from the cartel agreement by firms which report information to the antitrust authority.

The second (being a negative effect) is the attenuation of the expected cost of anticompetitive behavior because of reduced fines. One key mechanism in their model is the possibility for firms to apply for leniency even if the investigation is opened.

Our model combines two streams of the literature. Firstly, from the point of view of merger control (ex ante system), we study the impact of a merger on collusion and estimate a threshold of asymmetry between firms. Secondly, we take into account leniency programs to detect and punish collusive agreements (ex post system). Taking these two streams into account allows us to develop a global model and to obtain clear policy implications.

Our principal results are as follows.

We show that when the endowment of capital asset is symmetric initially, then collusion is less feasible after the merger takes place. Indeed, the merger leads asymmetry among firms, and this determines different incentives to collude. Harder conditions for collusion arise for the same argument also when firms' capital stocks are asymmetric pre-merger and the insiders are more efficient. Ex post, the outsiders are smaller and weaker, which make their incentives harder to meet. But when asymmetric capital stock combines with less efficient insiders, then the merger makes the collusion easier to sustain because of more symmetric conditions and closer incentive constraints. This model allows us to determine an optimal threshold of asymmetry among insiders and outsiders such as a merger has pro-competitive effects. This threshold only depends on the number of active firms in the industry and on the amount of fine but does not depend on the discount factor. Finally, we compare this value with the value which would restore perfect symmetry between firms after the merger takes place.

The paper is organized as follows. In section 2 we describe the basic model. Characterizations of equilibria are provided in section 3. In sections 4 and 5, we deal with the impact of a merger on collusion. Concluding remarks follow in section 6. Proofs of results appear in the appendix.

II. The model

We consider a simple framework consisting of a Cournot homogeneous product industry, with n firms, n ≥ 44 . Demand is linear, with price p = 1 -Q, where Q denotes the total quantity produced in the industry. Following [START_REF] Perry | Oligopoly and the incentive for horizontal merger[END_REF], cost is assumed to be dependant on the capital owned by the firms :

C(q i , k i ) = cq i + q 2 i 2k i , ∀i ∈ [1, n]
with 0 ≤ c < 1 and q i denotes the quantity chosen by firm i. Each firm has a share of a specific asset that affects marginal costs, which is its production capacity, denoted by k i . The total supply of capital in the industry is assumed to be fixed, which is normalized to one: n i=1 k i = 1, ∀i = 1...n. We assume an upward-sloping linear marginal cost function, where the slope is a decreasing function of the firms' share of the industry's stock of capital :

∂C(q i , k i ) ∂q i = c + q i k i > 0 ; ∂ 2 C(q i , k i ) ∂q 2 i > 0 ; ∂ 2 C(q i , k i ) ∂q i ∂k i < 0
In this context, a firm is more efficiency if it is "large" (if it owns a large fraction of the capital stock).

In a first part, we suppose that firms are ex ante symmetric :

k i = k j , ∀i, j ∈ [1, n].
In a second part, we loosen this assumption by considering two groups of firms with different production capacities. More precisely, two identical firms (firms which are supposed to merge) are to have the same production capacity, denoted by k 1 = k 2 , which is different from the production capacity of the other firms (the outsiders), denoted by: k j (∀j = 1, 2).

We consider a merger between the two firms 1 and 2 and we study its impact on the risk of collusion. The outcome of the execution of a collusive agreement is assumed to be that the colluding firms will thence reduce output below the competitive level. We suppose that the antitrust authorities use leniency programs; these programs provide the incentive of reduced fines to firms which reveal information about collusion to the antitrust authorities. [START_REF] Motta | Leniency programs and cartel prosecution[END_REF] design four parameters for the enforcement policy:

• F ∈ [0, F ] : the value of the fine if firms are detected by the antitrust authority and have not cooperated with it, F being the maximum possible fine which can be imposed.

• f ∈ [0, F ] : the value of the reduced fine for the firm which denounces the collusive agreement. It is assumed that all firms which cooperate with the antitrust authority will be granted the same reduced fines.

It is also assumed that the reduction is high enough in order for the leniency program to be really effective (if f is close to the full fine (F ), firms will not have incentive to reveal information).

• α ∈ [0, 1] : the probability that firms are reviewed by the antitrust authority.

• s ∈ [0, 1] : the probability that the antitrust authority successfully concludes the investigation when firms do not cooperate.

These parameters are assumed to be exogenous.

Type I errors (firms not colluding are still prosecuted) are assumed not to happen whereas Type II errors (colluding firms not fined) are assumed to happen.

The timing of the game is as follows (with n or n -1 active firms): At t = 1, firms decide to reach a collusive agreement or to deviate (playing non cooperatively). The collusion strategy is modelled on the basis of a trigger strategy: once a firm deviates from the collusive agreement, all the firms will play non cooperatively until the end of the game5 .

If no firm deviates until a time t then they collude again and get a profit π M . We study the case in which firms can sustain the joint profit maximization, so the subscript M indicates a collusive outcome since collusion aims at monopolizing the industry. If, at time t, the antitrust authority decides to open an investigation then there are two assumed outcomes :

• Cooperation: firms denounce the cartel to the antitrust authority; they pay the reduced fine f , play non cooperatively during one period and restart collusion after an inquiry is concluded. If however a firm deviates from the collusive agreement, firms play non cooperatively until the end of the game.

• No cooperation: at t + 1 (firms do not cooperate with the antitrust authority so one other period is needed to conclude the investigation), if firms are not condemned with probability (1 -s) then they continue to play cooperatively; if they are found guilty, then they pay the total fine F and have to play non cooperatively during one period. As in the cooperation case, in one firm deviates from the collusive agreement, firms play non cooperatively until the end of the game.

Consequently, in this game, there are two types of deviation : the deviation from the collusive agreement and the reporting to the antitrust authority.

The equilibria of the game can now be analysed.

III. Equilibria

This section describes all the various types of equilibria. We describe the perfect Nash equilibria of the game and we have to resolve this game considering merger or not in the first period (game with n or n -1 active firms).

Firstly, we consider the case of symmetric firms. In this context, [START_REF] Motta | Leniency programs and cartel prosecution[END_REF] suggest that there are three types of equilibria: no collusion (NC), collusion and revelation (CR) and collusion and no revelation (CNR).

The existence of these equilibria depends on the value of the relevant parameters. CR equilibrium exists if the profit of firm colluding and revealing to the antitrust authority is greater than the profit of a firm which deviates.

CNR equilibrium exists if firm which colludes and does not reveal information has no incentive to reveal information to the antitrust authority and also to deviate from the collusive agreement.

Let δ ∈ (0, 1) be the common discount factor.

The results obtained by [START_REF] Motta | Leniency programs and cartel prosecution[END_REF] are as follows:

lemma 1. [START_REF] Motta | Leniency programs and cartel prosecution[END_REF] for low values of f ) 6

• for s ∈ [0, s CN R [ and α < α N C (s) : CNR and CR exist and CNR is Pareto dominant.

• for s ∈ [s CN R , 1] and α ∈ [0, α CR [ : CNR and CR exist and CR is Pareto dominant.

• in other cases: only NC equilibrium exists.

with:

         s CN R = (1+δ)(π M -π N +f ) δ(π M -π N +F ) α CR = π M -(1-δ)π D -δπ N (π M -π N +f ) α N C = (1+δ)(π M -(1-δ)π D -δπ N ) δs(π M -π N +F )
The following graphic represents the Subgame Perfect Equilibria. 7 In order for α CR to be strictly positive, it is necessary that δ must be higher than δ = π D -π M π D -π N . δ is the usual critical discount factor when firms collude with no threat of prosecution. The case α CR < 0 means that firms never make any collusive agreement.

IV. The impact of a merger on collusion

The two firms which propose to merge are identical (k 1 = k 2 ). The others have another same production capacity :

k i = k j , ∀(i, j) ∈ n, (i, j) = (1, 2).
The impact of a merger on collusion is captured in our framework by comparative statics analysis on each border line of the Figure 1. In this section, we present the method and the results for the border line α CR . In the next section, the two other border lines are studied.

To obtain comparative statics results on α CR , we compute the value of φ = α bm CR -α am CR , where the superscripts bm and am denote the threshold value α CR before and respectively after the merger. If φ is strictly positive, then the border line α CR in the Figure 1 moves down and so, there exist some values of α and s for which firms collude and reveal before merger whereas they do not collude if the merger is accepted by the antitrust authority. In other words, a merger attenuates the scope of collusion.

To simplify the calculation, we assume that c = 0.

Individual profit functions of the firms (∀i = 1...n) are given by :

         π M = 1 6 k i π D = k i (-2-k i ) 2 18(1+2k i ) π N = β i (2k i -β i ) 2k i (1+β) 2 with β i = k i k i +1 and β = n i=1 β i .
Proof. See appendix B.

As regards collusion profit, we study perfect collusion in the sense that all firms make a collusive agreement and we suppose that monopoly aggregate output is simply divided in proportion to capital endowments. This is a key element in the model. Concerning deviation profit, we suppose that deviation takes the form which is conventionally assumed in this type of models: the deviant takes the allocated cartel outputs of all other firms as given and selects the output which maximizes its own profits.

The previous expressions of profits are calculated assuming that n firms are active (no merger situation). We do the same for the other case, where n-1 firms stay active. In this case, the production capacity of merged entity is equal to:

k m = k 1 + k 2 = 2k 1 .

The case of ex ante symmetric firms

We suppose that :

k i = k j , ∀i = j, ∀(i, j) ∈ [1, n].
We have to compute the value of α CR before merger (which is the same for insider or outsider because we consider ex ante symmetric firms).

We consider an effective leniency program and a high reduction of fine in case of revelation of information. The reduced fine must satisfy the following inequality:

f < δ(π D -π N )
We choose to apply the total immunity for firms which reveal the information to the antitrust authority (f = 0), because it is optimal from the authority point of view8 .

We compute the difference: α bm CR -α am CR for the insiders and the outsiders and we obtain: lemma 2.

• For the insiders: α bm CR < α am CR , ∀n

• For the outsiders: α bm CR > α am CR , ∀n

Proof. See appendix C.

After merger, the border line α CR moves up for the insiders and moves down for the outsider firms.

The most restricting border line α CR is the one which is, graphically, the lowest down (because we consider perfect collusion in the industry, if only one firm has no incentive to make a collusive agreement, the NC equilibrium takes place). It is given by:

min i=ins,out α i CR
where i designs insiders or outsiders.

We obtain the following proposition:

proposition 1. If each firm has initially the same production capacity, then a merger will increase (resp. decrease) the space of parameters under which firms tend to collude for the insiders (resp. the outsiders). The merger will then reduce the space of parameters under which firms tend to make perfect collusion in the industry.

The following graphic summarizes the previous results:

Figure 2: The impact of a merger on α CR in the symmetric case.

Post-merger firms are asymmetric, this decreases the possibility to make collusive agreements because outsiders have incentives to deviate from the collusive agreement. In fact, outsiders have smaller production capacity than merged entity, which creates relative inefficiency. Recall that the collusive profit is computed with a key element of the model which is that monopoly aggregate output is divided in proportion to capital endowments, and so asymmetry causes incentives to deviate form this collusive agreement. This result reinforces the results obtained by [START_REF] Cabral | Introduction to industrial organization[END_REF], [START_REF] Rothschild | Cartel stability when costs are heterogeneous[END_REF] or [START_REF] Penard | Choix de capacités et comportements stratégiques : une approche par la théorie des jeux répétés[END_REF] who show that it is easier for firms to make collusive agree-ments when they are symmetric.

The case of ex ante asymmetric firms

In this section, we suppose that, initially, the insiders have the same production capacity (k 1 = k 2 ), the outsiders have another same production capacity

too (k i = k j , ∀(i, j) = (1, 2)), which is different from the production capacity of the insiders (k i = k 1 , ∀i = (1, 2)).
We design by ∆ the parameter measuring the ex ante relative asymmetry of firms. More precisely, we suppose that the production capacity of a merging firm is:

k 1 = k 2 = 1 + ∆ n
The production capacity of an outsider is then 9 :

k j = 1 -2∆ n-2 n , ∀j = (1, 2)
We have to analyse two cases. Whether we consider an ex ante relative efficiency of insiders: ∆ > 0, or an ex ante relative efficiency of outsiders:

∆ < 0.
For the first case:

∆ ∈]0, n 2 -1[
For the second case:

∆ ∈] -1, 0[ Proof. See appendix D.
9 The total production capacity in the industry is equal to 1, so : 2k1 + (n -2)kj = 1, ∀j = (1, 2). Then we check that :

2 1+∆ n + (n -2) 1-( 2∆ n-2 ) n = 1.
We compute the value of φ (α bm CR -α am CR )for each category of firms and we obtain the following lemma: lemma 3.

• For the insiders :

α bm CR < α am CR , ∀n, ∀∆ ∈] -1, n 2 -1[
• For the outsiders :

α bm CR > α am CR , ∀n, ∀∆ ∈] -1, n 2 -1[ Proof. See appendix E.
Whatever the sign of ∆ is, a merger between firms 1 and 2 will reduce at one and the same time the space of no collusion for insiders and the space of collusion for outsiders. We have to determine the total effect of the merger.

Firstly, we compare the border line α CR before merger for each category of firms. We design by α CR (Out) the border line for outsider firms, and by α CR (Ins) the border line for insiders. Lemma 4 follows: lemma 4.

α bm CR (Out) ≤ α bm CR (Ins) ⇔ ∆ ≥ 0
To prove this, we do exactly the same than for the lemma 3.

When there is a relative efficiency which favours insiders (∆ > 0), we obtain the following result:

α am CR (Ins) > α bm CR (Ins) > α bm CR (Out) > α am CR (Out)
When the asymmetry favours insiders, outsiders have higher incentives to deviate from a collusive agreement because the merger reinforces the relative inefficiency. For the insiders, the merger reinforces their relative efficiency and increases the space of parameters under which firms tend to collude.

We then obtain the following result: proposition 2. When the ex ante asymmetry favours insiders, then the space of parameters under which firms make a collusive agreement is reduced after the merger.

The following graphic summarizes our results when we consider an ex ante asymmetry favouring insiders (we represent only the area which is interesting for us, which is the North-East of the Figure 1). We now look into the case in which there is an ex ante relative inefficiency of insiders (∆ < 0). We know that the space of collusion under outsiders tend to collude decreases after the merger, whereas it increases for insiders10 .

Moreover, before merger, α CR for outsiders is above α CR for insiders11 . In order to evaluate the global impact of the merger, we have to compare α CR for insiders before merger with α CR for outsiders after merger. We obtain the following lemma.

lemma 5. α bm CR (Ins) > α am CR (Out) for ∆ > ∆(n).

proposition 3. Given ∆(n) ( ∆(n) < 0) such as : α bm CR (Ins) = α am CR (Out). For ∆ < ∆(n), the merger increases the space of parameters under which firms collude. For ∆ > ∆(n), the merger reduces the space of parameters under which firms collude.

The main originality of the paper is that we can compute the exact value of asymmetry ex ante in order for the merger to have pro-competitive effects, taking into account the possibility for firms to denounce the cartel agreement (apply for leniency).

The variable ∆(n), which satisfies the equality: α bm CR (Ins) = α am CR (Out), is not the parameter which establishes a perfect symmetry of firms ex post.

As a matter of fact, the parameter which establishes an ex post perfect symmetry of firms is given by:

∆(n) = - (n -2) 2(n -1) (1)
Proof. See the appendix G.

Moreover, we prove that: ∆(n) < ∆(n)

Proof. See the appendix H.

The following graphic summarizes the effect of a merger, depending of the degree of asymmetry between firms. V. Comparative statics analysis on the other border lines

NC versus CNR

In this section, we make comparative statics on the border line α N C before and after merger. The parameters δ and s are assumed to be constant (they do not have any impact of the moving of the border line α N C ) and consequently, the only difference considering the sign of α bm CR -α am CR and α bm N C -α am N C is that in one case we consider reduced fine (f is assumed to be equal to 0), whereas in the other case, we consider the full fine F .

The previous propositions 1 and 2 remain valid considering α N C .

For the proposition 3, the only difference is the value of the threshold ∆(n) for which the merger has procompetitive effects. We obtain the following proposition.

proposition 4. Given ∆ ( ∆ < 0) such as : α bm N C (Ins) = α am N C (Out). For ∆ < ∆, the merger increases the space of parameters under which firms collude without revelation. For ∆ > ∆, the merger reduces the space of parameters under which firms collude without revelation.

Proof. The values of ∆(n) are reported in the appendix I.

These results reinforce our previous results obtained for the border line α CR .

CR versus CNR

Comparative statics results on s CN R allow to know the impact of a merger on the effectiveness of leniency.

Firstly, we suppose that firms are ex ante symmetric. Then we obtain the following proposition : proposition 5. If each firm has initially the same production capacity, then a merger will increase (resp. decrease) the space of parameters under which firms tend to collude without revelation for the insiders (resp. the outsiders).

The merger will then reduce the space of parameters under which firms tend to collude without revelation in the industry.

Proof. See Appendix J.

If we consider ex ante asymmetric firms, we prove that, as previously, there exists a threshold under which a merger has procollusive effects in As [START_REF] Vasconcelos | Tacit collusion, cost asymmetries, and mergers[END_REF], we suppose that the collusive outcomes are based on an allocation of the monopoly output proportional to the individual capital stocks. This is the profit maximizing allocation of a monopolist with decreasing returns.

The maximization program when firms play cooperatively is:

M ax Q M (1 -Q M )Q M -(Q M ) 2 2 k i with n i=1 k i = 1
We obtain:

Q M = 1
3 and q M = k i * 1 3 and we replace Q M and q M in the profit function to obtain the equilibrium collusion profit.

• Deviation profit.

As in [START_REF] Rothschild | Cartel stability when costs are heterogeneous[END_REF], we suppose here that deviation takes the form which is conventionally assumed in models of this type : the deviant takes the allocated cartel outputs of all other firms as given and selects the output which maximizes its own profits.

The maximization program for a firm which deviates from the collusive agreement is:

M ax q i (1 -q i -(1 -k i )( 1 3 )) * q i - q 2 i 2k i q D i = 1-(1-k i )( 1 3 ) 2+k -1
i and we replace in the profit function to obtain the deviation profit.

6

  Figure 1: Subgame Perfect Equilibria

  merger reduces the space of collusion and revelation for firms because outsiders have incentives to deviate from the collusive agreement. Proposition 2 follows:

Figure 3 :

 3 Figure 3: The impact of a merger on α CR in the case: ∆ > 0.

Figure 5 :

 5 Figure 5: Summary diagram

European Commission, MEMO/06/357.

The minimal number of firms (n = 4) is made clear in the appendix A.

These strategies have been initially suggested by[START_REF] Friedman | A non-cooperative equilibrium for supergames[END_REF]

See Motta and Polo (2003).

See lemma 3.

See lemma 4.

(2+n+2 ∆) 2 (n 2 -4 (1+∆)) (n-2 (1+∆)) (n+2 n 2 +4 n ∆-2 (1+∆) (5+2 ∆)) 2

Proof. Values of ∆(n) are reported in the appendix F.

We then obtain the following results:

• if ∆ > ∆(n) : α am CR (Ins), α bm CR (Out) > α bm CR (Ins) > α am CR (Out)

The comparison of α am CR (Ins) and α bm CR (Out) is not useful, only the border lines which are the lowest down are restricting. If we take into account all the firms in the industry, then, for the case ∆ > ∆(n), the merger will reduce the possibility of collusion ex post. But if there is a high degree of asymmetry between firms (∆ < ∆(n)), the space of parameters under which firms tend to collude increases after the merger takes place. The following graphic summarizes the results: We then obtain the following proposition: the sense that under collusion, a merger decreases the incentives for firms to reveal the information to the antitrust authority and so to apply for leniency.

We then obtain the following proposition : proposition 6. Given ∆ ( ∆ < 0) such as : s bm CN R (Ins) = s am CN R (Out). For ∆ < ∆, the merger reduces the incentives for colluding firms to apply for leniency. For ∆ > ∆, the merger increases the incentives for colluding firms to apply for leniency.

Proof. The values of ∆(n) are reported in the appendix K.

VI. Concluding remarks

This paper deals with the impact of a merger on the possibility for firms to collude. The major contribution of this article is to consider an ex ante merger control associated with an ex post detection of explicit collusion, allowing for leniency. Our results confirm the results obtained by several authors dealing with this subject [START_REF] Compte | Capacity constraints, mergers and collusion[END_REF], [START_REF] Vasconcelos | Tacit collusion, cost asymmetries, and mergers[END_REF], ...) and we reinforce these by computing the optimal threshold of asymmetry among insiders and outsiders such as a merger has procompetitive effects.

Of course, antitrust authority does not evaluate only pro-collusive effect but also the possibility of abuse of dominant position and then faces a tradeoff between decreasing the risk of pro-collusive effects (when two efficient firms merge for example) and increasing the market power of merged entity.

Appendix A : Justification of n > 3

For n ≤ 2, then a merger reduces the competition to only one firm, which is not an interesting case to study.

For n = 3, we can easily prove that, in the case of symmetric firms before merger (k i = 1/3), then the post-merger profit of an outsider is less in the case of collusive equilibrium than in a case of Nash equilibrium 12 . Also, the discount factor under which collusion is sustainable for the outsider is higher than 1, there is never collusion after merger, which is not an interesting case to study also.

Appendix B : Profit functions

• Nash profit.

The individual maximization program when firms play non coopera-

tively is:

Then, equilibrium values are : [START_REF] Mcafee | Horizontal mergers and antitrust policy[END_REF].

• Collusion profit.

12 Profit functions are reported in Appendix B

Appendix C : Proof of lemma 2

In order to compute the value of φ, we have to compute the values of α CR before and after the merger.

Profits functions 13 for the insiders before and after merger (with k i = 1 n ) are given by:

13 We compute the individual profit function after merger.

We then replace these expressions in α bm CR and α am CR in order to compute the value of: α bm CR -α am CR for the insiders. We obtain:

It is straightforward to prove that φ Ins < 0.

Now we have to compute the value of φ for the outsiders.

Premerger profit are the same for insiders and outsiders because firms are perfectly symmetric. Post-merger profits are given by:

We infer that:

φ Out is strictly positive (by assumptions n > 3).

Appendix D : Domain of definition of ∆

When we consider a relative efficiency of insiders (∆ > 0), the sum of production capacities of insiders can not exceed 1, then:

and ∆ < n 2 -1. For the opposite case (∆ < 0), the production capacity of a merged firm must be greater than 0, then:

Appendix E : Proof of lemma 3

After simplifications, we obtain :

We compute the value of α bm CR (Ins) -α am CR (Ins). To sign this difference depending on n, δ and ∆, we plot (in three dimensions) this function varying n and we obtain that this difference is always negative ∀n

For the outsiders, we obtain:

We prove that:

Appendix F : Proof of lemma 5

We obtain that α bm CR (Ins) -α am CR (Out) > 0 for ∆ > ∆(n).

The derivative of this expression function of δ does not depend of δ and so we can fix δ = 0.9 for example because δ does not affect the values of ∆(n).

The values of ∆(n) are given by resolving:

We then obtain:

Appendix G : Values of ∆(n)

In order to establish perfect symmetry ex post, ∆(n) must satisfy:

Appendix H : Proof of: ∆(n) < ∆(n)

We compute the value of ∆(n) for each value of n:

By making a comparison between the two previous tables, we infer that :

Appendix I : values of ∆(n).

Appendix J : Proof of proposition 5

For the insiders :

It is straightforward to prove that this expression is strictly negative.

For the outsiders :

s bm CN R is the same than for insiders.

It is sraightforward to prove that this expression is strictly positive.

Appendix K : Values of ∆(n).

We obtain that s bm CN R (Ins) -s am CN R (Out) > 0 for ∆ > ∆(n).

The values of ∆(n) are given by resolving: s bm CN R (Ins) -s am CN R (Out) = 0 for each value of n. After having set values for n, we can prove that F has no impact of the value of ∆(n), and we obtain the following table (for every value of F > 0) :


Prof Lucy L Coria-Oriundo

Dr Helena Ceretti

Yoann Roupioz

Fernando Battaglini
email: battagli@qi.fcen.uba.ar

Fernando Battaglini

Redox Polyelectrolyte Modified Gold Nanoparticles Enhance the Detection of Adenosine in an Electrochemical Split-Aptamer Assay

Keywords: aptamers, biosensors, gold nanoparticle, metabolite, multiple binding site

We present an electrochemical sandwich-type assay based on the splitting of an aptamer into two fragments. Gold nanoparticles are modified with one of the fragments and a redox polyelectrolyte. The first is used as the recognition element, while the other for the electrochemical signal generation. The splitaptamer used here can detect adenosine, used as a model system for recognizing small molecules. The multiple binding sites on the nanoparticle, along with the high number of redox probes, yield a selective and sensitive assay for adenosine, achieving a limit of detection of 3.1 nM and a linear range up to 75 nM. The obtained results are analyzed in terms of the nanoparticle and electrode architectures. The assay can be easily extended to other small molecules and sandwich assays, representing a promising tool for detecting metabolites at the nanomolar level.

Introduction

Metabolites are low molecular-weight molecules, also referred to as small molecules, chemically transformed in biological processes. In clinical chemistry, these molecules represent useful markers for early diagnosis. [1][2][3] The development of fast and sensitive methods for their detection is an intensive area of research replacing the traditional methods based on chromatographic and spectroscopic techniques, which rely on extensive sample preparation and time-consuming procedures. Sensors represent an alternative, where a selective molecular recognition process occurs, generating a physicochemical change conveniently transduced to an electrical signal. Aptamers have played a key role in the development of sensors since they can be developed for a wide range of targets from metal ions to cells, [4] among them small molecules, where adenosine and cocaine have been extensively used as model systems [5][6][7][8][9] . Other features of the aptamers include: its synthesis does not involve animals; during the chemical synthesis, modifications can be introduced at defined positions; these biomolecules are way more stable than proteins and might be dried without loss of activity. In recent years, the sequence of several aptamers has been split into two parts, maintaining their affinity, [10][11][12][13][14][15][16] allowing the detection of a species in a similar way than the antibody sandwich assay. In a sandwich assay using electrochemical signaling, the capture antibody is immobilized on the working electrode surface and binds the analyte. A second labeled antibody subsequently binds to the analyte for the signal generation. Colloidal metal labels have been widely used in the last two decades in this type of assay. [17][18][19] A similar approach can be taken with the adenosine aptamer, whose original hairpin sequence can be cut in the loop domain (Scheme 1A, left). [20] The originated segments open a myriad of sandwich-like strategies. These fragments have been used grafted on gold nanoparticles (AuNPs) for colorimetric detection in solution, [7] or on surfaces as recognition elements in biosensors; in the last case, as examples of transduction methods we can mention, fluorescence, [21] surface plasmon resonance (SPR), [6,22] and electrochemical detection. [8,23] Electrochemical detection is a very convenient method since it can be used in turbid samples, a relevant issue in many clinical and biological studies, and due to its hardware simplicity, a low cost per assay can be easily achieved. The use of colloidal metal labels in sandwich assays allows a sensitive response; however, most of the examples presented in the literature involve the nanoparticle's chemical dissolution, involving the addition of reagents and introducing more steps to the detection process or complicated construction schemes. [17,19,23] In this work, we present an electrochemical sandwich assay able to generate an output current at 0.3 V vs. Ag/AgCl using squarewave voltammetry (SWV). The assay is based on the splitting of the adenosine aptamer into two fragments (Scheme 1A), one immobilized on the surface of the electrode (capture fragment, C) and the other one (signaling fragment, S) is immobilized on AuNPs. The AuNPs also contain a redox polyelectrolyte (polyallylamine derivatized with an osmium polypyridyl complex, OsPA) for the signal generation (AuNP/S/OsPA, Scheme 1B). The functionalized nanoparticles play a crucial role in the figures of merit of the assay since each nanoparticle presents several aptamers for adenosine recognition, improving the binding affinity, while the osmium redox centers distributed around all the nanoparticle have a multiplicative effect in the signal generation. In this way, each nanoparticle works as a multidentate ligand improving its binding to the surface, achieving state-of-the-art adenosine quantification with a limit of detection (LOD) of 3.1 nM.

Results and Discussion

Assay Concept

Scheme 1A (left) shows the DNA sequence used by Lin and Patel in their structural studies of the DNA-adenosine complex. This single-stranded DNA is composed of 27 nucleotides, which folds into a hairpin structure in the presence of adenosine. The nucleotides forming the loop are rich in guanine and are responsible for the adenosine recognition (depicted as pentagons in the scheme). At the left of the loop, 4 pairs of complementary nucleotides help to stabilize the complex. If this sequence is divided into two fragments by cutting it in the 14th nucleotide (indicated by the scissor in the scheme), it can reassemble in the presence of adenosine. [24] In order to adapt these two new sequences to a sandwich-type assay, some changes were introduced: [7] the top part was selected as capture sequence (C) to be immobilized on a gold electrode surface; a thiol group was added and the sequence extended with thymine nucleotides and a short 5 nucleotide sequence (AGAGA) (Scheme 1A, right). On the other hand, the bottom part of the sequence (S) was modified at the 5' terminus with a thiol group followed by 10 thymine nucleotides (Scheme 1A, right); while in the 3´ end, only four (in italics) of the eight nucleotides match with the complementary C sequence. The S sequence binds to the C sequence only in the presence of adenosine using these modifications. [6,22,25,26]

Gold nanoparticle functionalization

Layer by layer assembly is a powerful technique for the construction of complex nanoarchitectures in small areas through a simple method mainly based on electrostatic interactions. [27,28] This technique can be easily applied to charged macromolecules such as polyelectrolytes, [29,30] proteins, [31,32] and DNA. [33] Considering that the aptamer and the redox polyelectrolyte have opposite charges, we decided to combine in the same nanoparticle the recognition element (S sequence) and the amperometric transduction element (OsPA) by layer-bylayer self-assembling. AuNPs were exposed to different concentrations of S representing different S:AuNP molar ratios. In Table 1, the different tested combinations are presented. The nanoparticles were incubated in the presence of different concentrations of S for 1 hour. They were centrifuged at 15000 g (Scheme 2) and resuspended in 10 mM HEPES buffer, pH 7.4, remaining stable in solution due to the negative charges provided by the oligonucleotide phosphate moieties. The generated combinations (AuNP/Sxx, Table 1) were exposed to the same concentration of OsPA, incubated for 1 hour, centrifuged and resuspended in HEPES buffer. Stable suspensions are obtained in all cases in agreement with previous results. [34][35][36][37] . Scheme 2. AuNP modification steps, first with S and then with OsPA. [a] Ratio between final concentrations of S and AuNP. For AuNP, the final concentration is 6.4 nM in all cases.

The morphology and the size of the AuNPs were determined by TEM and UV-Vis spectroscopy (Figure 1), from these images, we can observe that AuNPs have 21±3 nm diameter (n=22). In Figure 1A, typical aptamer modified AuNPs can be observed, while Figure 1B shows the same nanoparticles after modification with the redox polyelectrolyte. The presence of the osmium atoms is evidenced by a different contrast surrounding the AuNP.

The UV-Vis spectra obtained before and after functionalization with S present a 2 nm shift in the maximum wavelength absorbance; while after OsPA functionalization, this shift increases in 4 nm, in agreement with the increase in the AuNPs diameter [6,22] (Figure 1C). Also, from the UV-Vis analysis of the initial solution and the supernatant recovered after AuNP/S/OsPA centrifugation, it was possible to determine the number of osmium probes assembled in each nanoparticle yielding a AuNPs:Os molar ratio of 1:3.1x10 4 . The modified nanoparticles (AuNP/S/OsPA) were left to dry onto an electrode surface and rinsed with water. Then, the adsorbed NPs were tested to determine the presence of the osmium probes in the nanoparticle (Figure 2). It can be observed that the amperometric response increases as the amount of S used in the modification of the AuNP increases. This behavior indicates that the adsorption of OsPA is maximized on the nanoparticle surface when the highest S concentration is used. Then, the following experiments have been carried with AuNP/S10/OsPA.

Surface modification

Gold electrodes were modified with the thiolated capture aptamer (C) to form a self-assembled monolayer (SAM) (Scheme 1B). Thiolated DNA probes bind to gold through the specific Au-S interaction; however, nonspecific adsorption of DNA occurs on gold due to its interaction with the nitrogen atoms present in each nucleotide. The formation of mixed SAMs with an inert mercapto-derivative improves the aptamer's orientation toward the solution and reduces the nonspecific binding. [38] Among the most used thiolated species, we can find mercaptohexanol [39,40] and polyethyleneglycol; [6,41] the last one has shown a higher resistance to the nonspecific adsorption of DNA and proteins. The gold electrode modification with C (AuE/C) was complemented with 5000 Da mercaptopolyethyleneglycol (HS-PEG) at two different concentrations, 10 and 150 µM. The anti-fouling properties of both surfaces were tested, exposing them to AuNP/S10/OsPA for 4 hours. Even though in previous works, the use of 10 µM was enough to guarantee a low nonspecific binding, in our assays, a higher concentration of HS-PEG was needed to guarantee good coverage of the surface. In this condition, 150 µM, a drastic reduction of nonspecific adsorption was observed (Figure S2, Supporting Information).

Assay performance

Once we obtained the most appropriate electrode surface and modified AuNPs, we optimized the incubation time in adenosine presence. In previous works, using split aptamers and electrochemical detection, short incubation times were used (few minutes); [8,23] however, the detected concentrations ranged from micro-to millimolar. As the analyte and the aptamer concentrations are relevant in the kinetics of the recognition process, we decided to set longer incubation time to achieve the detection at nanomolar levels. We found a signal saturation at 10 nM concentration if we incubate the sample for 4 hours. Therefore, to balance a useful dynamic range with a competitive limit of detection, we reduced the incubation time to two hours, this time is similar to the one used by Wang et al. for low adenosine concentrations. [42] Figure 3 shows the calibration curve obtained for samples ranging from 5 to 75 nM of adenosine per duplicate, with a sensitivity of 82 nA nM -1 cm -2 . Each determination was carried out in a freshly prepared modified electrode and subtracting an average background current of two electrodes produced in the same batch. It can be observed that the dynamic range covers all the tested concentrations, and the calculated limit of detection is 3.1 nM of adenosine (see Supporting Information). Also, good reproducibility of the construction method is achieved since the calibration curve involves 10 independent electrodes (10 samples at 5 different concentrations, each measured in a new disposable electrode). From these series of measurements, only one result was discarded as an outlier. The major challenges to the assay selective are the presence of nucleobases and their derivatives. These molecules generally generate a small signal, around 2-10% of the adenosine signal, but always remarkably similar among them. [5,[43][44][45][46] This fact can be structurally understood considering the NMR solved aptamer structure. Adenosine is a nucleoside consisting of adenine, a nucleobase derivative of purine, attached to a ribose. Lin and Patel [24] established that the aptamer´s two recognition domains can locate two adenosine molecules through their adenine bases inside the loop structure. On the other hand, Huizenga and Szostak [20] have established that any modification on the adenine structure difficult the aptamer recognition. In this sense, the guanine base, another purine derivative, and its nucleosides are the most similar molecules to be used as a negative control. In this work, 2'-deoxyguanosine was used to test the selective; Figure 4 shows the results obtained for this molecule against a similar adenosine concentration. The difference between the current obtained in the presence of 75 nM 2´-deoxyguanosine is lower than the current obtained for 5 nM adenosine. This response is similar to those observed in previous assays, [5,[43][44][45][46] indicating that some random interactions between the aptamer and these nucleosides exist, a plausible behavior since all of them are molecules rich in amino and hydroxyl groups that form hydrogen bonds. One exciting feature of the results presented here is the assay's ability to detect a concentration of adenosine as low as 5 nM, considering the binding constant of the original aptamer (6 µM). [8] This behavior can be explained if we picture each AuNP as a molecule with more than one site to bind the complementary DNA fragments present on the electrode surface.

Bearing in mind the size and composition of one nanoparticle, and the area that covers when is bound to the electrode surface, we can check if this represents a reasonable model. We carried out quartz crystal microbalance experiments to evaluate the amount of C fragment adsorbed on gold yielding a surface coverage of 7.7 pmol cm -2 . On the other hand, considering the average AuNPs diameter (21 nm), the projected area of the whole particle onto the surface is 7.7x10 -12 cm -2 . If we consider that the C sequences are homogeneously distributed on the gold surface, there are around 16 C molecules below the projected area to interact with a single AuNP (outlined in Scheme 3). The number of DNA strands per nanoparticle can be estimated using the results obtained by Mirkin's group; [47] for AuNPs with a 21 nm diameter, an average of 100 strands per nanoparticle can be considered. Given this number, more than one strand can be found at the bottom part of the AuNP. Therefore, we can argue that the modified nanoparticle is working as a multisite binding nanostructure improving the formation constant of the adenosine aptamer complex. Regarding the signal generation, the presence of more than 30000 osmium probes per each nanoparticle generates a multiplicative effect in the response, improving its sensitivity. We can consider that each nanoparticle works as a multidentate ligand improving its binding to the surface, achieving a state-ofthe-art adenosine quantification with a limit of detection (LOD) of 3.1 nM, as it can be observed in Table 2, where our system is compared to other works published in the last decade. It can be observed in the table that those works presenting an expanded dynamic range have the limitation to scale in a logarithmic way, and the concentration can be determined in a semiquantitative way (orders of magnitude). In particular, the work informing a LOD of 0.1 nM is the result corresponding to the lowest concentration determined, which in turn presents a significant misfit with the determined regression curve (see Figure 4 in the cited reference). [43] It is worth mentioning that in all the cases reviewed for this work the old definition for LOD of S/N = 3 is used, yielding a more favorable result, while in our system the statistical analysis was carried out according to current IUPAC guidelines (see Supporting Information). [52]

Conclusion

This work describes an electrochemical based sandwich-type assay. AuNPs are modified with a split aptamer and a redox polyelectrolyte. The first DNA fragment is used as a recognition element, while the other for the electrochemical signal generation. The split-aptamer used here can detect a small molecule, represented by adenosine; besides, the increasing development of oligonucleotide engineering allows replication of this strategy to other small molecules of toxicological or metabolic relevance. [10][11][12][13][14][15][16] The assay achieves state-of-the-art figures of merit as it can be observed in Table 2.

In our assay, the redox polyelectrolyte surrounded the nanoparticle provides a label that immediately can generate a signal, in contrast to previous electrochemical sandwich immunoassay involving metallonanoparticles, where the signal is generated after several steps involving first a chemical dissolution step, with the introduction of new reagents, followed by an electrochemical step to complete the determination. [17][18][19]23] Therefore, the combination of two dynamics areas of materials science (nanotechnology and soft matter) can improve the detection limit of a recognition assay by generating the equivalent to a macromolecule constituted by a scaffold with multiple binding sites and multiple signal-generating moieties. This concept is represented here by the AuNP (scaffold) containing several aptamers (multiple binding sites) and the redox moieties (signal generation). These features also lead to other advantages. For example, the analysis can be performed using a low volume of liquid (25 µL); eventually, it can be carried out in turbid media, and it involves low-cost instrumentation, represented here by a disposable three-electrode system (Figure S3), that it can be immediately adopted to field-portable devices. In particular, the materials involved in this work are screen-printed electrodes, produced by a well-established technology used in the fabrication of glucose strips [53] and gold nanoparticles, widely used in chromatographic immunoassay. [START_REF] Burtis | Tietz Textbook of Clinical Chemistry and Molecular Diagnostics[END_REF] The results presented here regarding linearity and limit of detection, where all the samples were tested on new freshly modified electrodes indicate that the system is reproducible and, therefore, scalable, representing a promising tool in bioassays that can be easily adopted to other analytes, providing a sandwich type recognition system.

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Integrating advantages. Modification of AuNP with a split aptamer and a redox polyelectrolyte produces a sensitive electrochemical sandwich assay for adenosine; the redox polyelectrolyte surrounded the nanoparticle provides a label that immediately can generate a signal using square wave voltammetry on a modified screen-printed electrode. The method's sensitivity opens new alternatives for detecting metabolites through a simple amperometric assay using well-established components of the dignositics industry.

Scheme 1 .

 1 Scheme 1. General principle of the split aptamer-based assays. (A) The original sequence of the adenosine aptamer (left) and the sequences used in this work (C and S, right). Pentagons in blue represent adenosine molecules. (B) Recognition assay: the modified electrode (AuE/C) is exposed to adenosine and AuNP/S/OsPA. After incubation, the amount of adenosine-aptamer complex is determined by square wave voltammetry (SWV).

Figure 1 .

 1 Figure 1. AuNPs characterization. TEM image of A) AuNP/S and B) AuNP/S/OsPA. C) UV-Vis spectra for AuNP (black), AuNP/S (blue) and AuNP/S/OsPA (red). The numbers indicate the maximum wavelength absorbance in each case.

Figure 2 .

 2 Figure 2. Square wave voltammograms for the different AuNP/Sxx/OsPA: AuNP/S10/OsPA (red), AuNP/S05/OsPA (green), AuNP/S02/OsPA (blue), AuNP/S01/OsPA (black).

Figure 3 .

 3 Figure 3. (A) Square wave voltammograms at different concentrations and (B) Amperometric response as a function of the adenosine concentration. The dotted line corresponds to the linear fitting.

Figure 4 .

 4 Figure 4. Square wave voltammograms for 75 nM adenosine (black), 75 nM 2'-deoxyguanosine (gray) and 75 nM adenosine plus 75 nM 2'deoxyguanosine (green).

Scheme 3 .

 3 Scheme 3. Possible contact points between AuNP and the electrode surface.

Table 1 .

 1 Modification of AuNP with different S concentrations.

		AuNP/S10 AuNP/S05	AuNP/S02	AuNP/S01
	12.8 nM AuNP /µL	50	50	50	50
	10 µM S/ µL	10	5	2	1
	10 mM HEPES/ µL	40	45	48	49
	final [S]/nM	1000	500	200	100
	S:AuNP [a]	156	78	31	16

Table 2 .

 2 Analytical performances of recently published methods for adenosine detection

	Method	Dynamic	Response	LOD / nM	ref
		range /nM
	Colorimetry	5-1000	Logarithmic	0.13	[48]
	Fluorometry	Up to 320	Linear	2.4	[49]
	Electrochemical	0.1-10 6	Logarithmic	0.1	[43]
	Electrochemical	5-200	Linear	5	[44]
	Electrochemical	10-1000	Logarithmic	3	[45]
	Electrochemical	10-40	Linear	2.5	[46]
	Fluorometry	Up to	Linear	130	[50]
		6000
	Photoelectrochemical	0.01-1000	Logarithmic	3.3x10 -3	[51]
	Electrochemical	5-75	Linear	3.1	This
					work

Acknowledgements

The authors wish to acknowledge the support of the NanoAndes IRN program (funded by the CNRS) and the ARCANE Labex (CBH-EUR-GS/ANR-17-EURE-0003),UBACYT 20020170100341BA, and BID PICT 2015-0801.

Supporting information Summary

Experimental section and statistical analysis are presented in this part.

Conflict of Interest

The authors declare no conflict of interest.