Greqam

Ekaterina Melnik

Jean-Benoît Zimmermann
email: zimmermann@univmed.fr

AN ECONOMIC APPROACH TO VOLUNTARY ASSOCIATION

Keywords: Voluntary Association, Public Good, Volunteering. JEL classification: H41, L30, L31

We develop an economic model of association based on voluntary contributions. Different equilibria corresponding to the different modes of formation of associations are analyzed and the results are compared with existing empirical literature. The main contribution consists in formalizing the voluntary association as a means of providing collectiveconsumption goods or services. We introduce the concept of the subjective quality as a possible incentive for volunteering. The model stresses the importance of non-pecuniary rewards and of accepted differentiation for well-functioning of voluntary organizations.

Introduction

The voluntary sector is considered as an extra-governmental provider of collective consumption goods representing an alternative for unsatisfied demanders (Weis-brod, 1977(Weis-brod, , 1986.) It can be distinguished from the public and the private sectors by its means of providing goods and services. The public provision of goods or services is financed by compulsory taxes, while on the market individuals have to pay their consumption. With voluntary provision, on the other hand, individuals can choose to contribute to the provision of a good or service, whether they consume it or not [START_REF] Sugden | Reciprocity: The Supply of Public Goods Through Voluntary Contributions[END_REF] In this paper, we investigate the aspect of voluntary nonprofit organizations that involves individual commitment to the voluntary association.

Voluntary monetary contributions and voluntary work constitute important (but not the unique) resources for the functioning of associations. According to [START_REF] Havens | Charitable Giving: How Much, by Whom, to What, and How[END_REF], in the United States, about 90 percent of American households donate to nonprofits. In 2001, individual contributions accounted for 13.7 percent of the total resources of American nonprofits, while public support represented 9.9 percent (Tax Foundation, 2005.) Voluntary contributions are not necessarily monetary.

They can also be represented by contributions in effort or time. In 1993, for instance, the value of voluntary work in the American nonprofits was estimated at about 182 billion dollars.

The example of the voluntary sector illustrates the fact that although voluntary contributions alone are often insufficient to make possible the provision of public goods, they cannot be ignored by economists. According to standard public good theory, the amount of the good provided through voluntary contributions will be sub-optimal because of free-riding. In theory, when agents are considered as pure altruists (Andreoni, 1989), i.e. when they are interested only in the total outcome of the collective action, public support leads to the complete crowding-out of voluntary contributions. However, empirical studies show that voluntary contributions are not completely crowded-out by public financial support. Moreover, in experimental studies, complete free-riding is not observed. Experimental studies allow to distinguish between different determinants of voluntary contributions. Generally, these studies show that communication (for instance cheap-talk, negotiation or promises) and the absence of anonymity tend to increase voluntary contributions (Andreoni and Petrie, 2003.) Other social determinants, such as certain individual characteristics [START_REF] Glaeser | Measuring Trust[END_REF], cooperative values versus individualistic values (Offermann et al., 1996), social environment (Carpenter et al., 2004) also influence the provision of public goods. Clearly, the majority of experimental studies shed particular light on the influence of social factors in the provision of public goods.

Empirical studies in the voluntary sector focus largely on the factors influencing donors' behavior. These factors include state support of the provision of public goods, the total provision of public goods, the fund-rising expenditure of nonprofits and individual motivations. An interesting feature of the voluntary sector is that governmental support of the public provision of public goods does not crowd-out individual contributions. In some cases, one can even observe some crowding-in. [START_REF] Okten | Determinants of Donations in Private Nonprofit Markets[END_REF] find no significant parameters of crowding-out in the United States. Furthermore, they find that the area of higher education and research presents some crowding-in effects (with parameters of 0.06 and 0.1 respectively).

In a study of international relief and development organizations, [START_REF] Ribar | Altruistic and Joy-of-Giving Motivations in Charitable Behavior[END_REF] show that contributions to these organizations are only weakly affected by public support, but they are positively influenced by fund-raising expenditure. In experimental studies, the crowding-out is greater, but still incomplete. As Andreoni suggested (1993), this is due to the intentional elimination of social factors in experiments.

Another interesting feature is that voluntary organizations are recognized as being capable of revealing demands and practising price discrimination, or personalized prices [START_REF] Hansmann | The Role of Nonprofit Enterprise[END_REF][START_REF] Ben-Ner | Nonprofit Organizations: Why Do They Exist in Market Economies[END_REF] The concept of Lindahl equilibria using personalized prices shows that if such a price system is established, then efficiency can be attained in the provision of the public good. [START_REF] Ben-Ner | Nonprofit Organizations: Why Do They Exist in Market Economies[END_REF] pointed out that social relationships between the members, the constraint of non-distribution of benefits and the shared backgrounds of members all help nonprofits to establish personalized prices.

In this paper, we develop a model of voluntary association. Firstly, we consider a standard public good model where individuals can make voluntary monetary contributions. We derive some interesting properties concerning overcrowding, incentives, and the existence of different equilibria resulting from the different ways in which association can emerge.

Secondly, the concept of subjective quality is introduced into the basic model. The originality of the model is that we assume the public good to be characterized by at least two main components, namely the quantity and the quality. The quantity is considered here as a purely public component, insofar as all the members benefit equally from it. However, the quality of the public good is considered as a mixed (public and private) component. The agents can enjoy part of it in the same way, but there may exist certain characteristics of quality that are difficult or impossible to measure objectively. In a way, quality is always somewhat subjective, to the extent that perfect correspondence with the preferences of heterogeneous agents is unlikely to occur. In our model, the agents can contribute money or time and effort.

The latter, or volunteering, allows them to influence the quality of the provided good (or service) according to their own preferences. The aim of the model is to formalize a set of assumptions about the functioning of voluntary organizations.

The properties we derive from it help to understand better the recurring issues of nonprofit management and volunteering.

The paper is organized as follows. Section 2 presents the basic model and the conditions relating to overcrowding. Section 3 analyzes the individual and collective incentives to make voluntary contributions. Section 4 shows the existence of different symmetric equilibria. Section 5 shows the existence of non-symmetric equilibria both without and then with budget constraints. In section 6 we introduce volunteering and the concept of subjective quality. Finally, section 7 concludes.

The Basic Model

The model presented in this section is an extension of a standard public good model.

We consider a set of n agents i = 1 . . . N, with N > 1, who are members of an association. The number of members must be higher than one, by definition of an association. Each agent is endowed with an income w i , which can be used for private consumption and for the provision of a public good G. All the members have to pay a compulsory amount c, on top of which each member can add a voluntary pecuniary contribution d i in order to increase production of the public good. At this stage, the utility function of each agent can be written as follows:

U iG = w i -c -d i + θ N γ (Nc + N ∑ j=1 d j + X) α , (1) 0 < α < 1, 0 ≤ γ ≤ 1,
The variable X has a double economic meaning. A positive value is interpreted as a fixed amount of monetary resources coming from exogenous origins (public grants, for instance.) A negative X describes the net amount of fixed costs (fixed costs less exogenous resources).

The parameters α and θ describe the production technology of the public good contributing to the individual utility, in the non-linear and linear forms respectively.

The parameter α can be interpreted in terms of more or less important decreas-ing returns. The parameter θ is always positive, because contributing to the good is assumed to generate a positive externality. The introduction of this parameter allows to vary the attractiveness of the good. In fact, as has been noted in some experimental studies, the attractiveness of a public good can positively influence individual voluntary contributions to it [START_REF] Hichri | La contribution au bien public: étude théorique et évidence expérimentale[END_REF] The parameter γ denotes the publicness of the provided good by measuring its degree of rivalry [START_REF] Bergstrom | Private demands for public goods[END_REF][START_REF] Blecha | The Crowding Parameter and Samuelsonian Publicness[END_REF]. A γ tending towards zero corresponds to decreasing rivalry of the public good. At one extreme, γ equal to zero describes the case of a collective-consumption good in the sense of [START_REF] Samuelson | The Pure Theory of Public Expenditure[END_REF]. This good is completely non-rival and its consumption by one agent does not prevent others from consuming it. Think for instance of an environmental association campaigning against pollution, or an association fighting a disease. In these cases, each member benefits from the totality of the results of the collective action. On the contrary, a γ equal to 1 describes a completely rival good, equivalent to a collectively provided private good equally shared among the members.

As an illustration, this type of function may correspond to the running of housing cooperatives, fairly wide-spread nonprofit organizations in countries like Canada or Switzerland.

The Issue of Overcrowding

In our model, overcrowding may appear under some conditions. Actually, the share obtained by the agent i will depend on N and on the sharing rule characterized by γ. The overcrowding effects appear when the individual benefit for i from the good provided diminishes as the number of members grows.

The case of γ equal to zero implies the absence of overcrowding, insofar as it describes a pure public good [START_REF] Samuelson | The Pure Theory of Public Expenditure[END_REF]. To understand overcrowding effects when γ is strictly positive, we study the variations of the production function [START_REF] Rose-Ackerman | Ideals versus Dollars: Donors, Charity Managers, and Government Grants[END_REF] with the number of members N.

f (N) = 1 N γ (Nc + N ∑ j=1 d j + X) α , ( 2
)
Proposition 1 In the absence of fixed costs (namely when X ≥ 0) there is no overcrowding. More precisely:

• if γ ≥ α, association has no sense as the sharing rule cancels the marginal gain expected from any individual contribution.

• if γ < α, there exists a threshold N 1 beyond which any additional contribution increases individual utility.

Proposition 2 In the presence of fixed costs, (namely if X < 0) there can exist an overcrowding when γ > α. More precisely:

• if γ ≤ α, there exists a threshold N 0 beyond which any additional individual contribution increases individual utility;

• if γ > α an association makes sense only for the population size N ∈ [N 0 , N 1 ].

Beyond the threshold N 1 there is overcrowding (see proof in Appendix A).

Proofs are presented in Appendix A. We begin by assuming symmetrical voluntary contributions. Then we study the sign of the production function with respect to N in different configurations (fixed costs versus exogenous ressources). To summarize, the sign of the derivatives of the production function varies as follows:

Table 1 The size of the association and overcrowding

N N 0 = -X c N 1 = -γX (γ-α)c d f dN Unde f ined > 0 0 < 0 Association Overcrowding
At this stage, propositions 1 and 2 can be summarized in table 2. Then we extend the scope of the results by considering the internal dynamics of the association. [START_REF] Olson | Logic of Collective Action: Public Goods and the Theory of Groups[END_REF] has already stressed the role of the internal structure of associations, and in particular its impact on the group's capacity to organize the provision of public goods. These internal structures can bring some formal or informal coordination to the process of providing the public good or organizing a collective action. An informal coalition can be created by individuals who decide to provide the public good even if they have to support the total cost, or at least a higher cost than the others. According to Olson, such coalitions can be formed in relatively large groups. In this case, the public good may be provided even in non-structured groups, as some participants may be able to provide the public good to the entire group.

In a first extension, we demonstrate that the results hold when K first members, founders for instance, each make a fixed voluntary contribution while newcomers (i > K) only contribute the compulsory amount c.

Then we allow the possibility for i > K members to make a positive voluntary contribution, by letting voluntary contributions vary for any i ≤ K. We show that the general case results hold with these new extensions.

For the rest of the paper we now consider X ≥ 0 and γ < α.

θ α ≥ N γ (Nc + N-1 ∑ j=1 d j + X) 1-α . (3)
Thus, it diminishes with N.

The collective incentive assumes that:

θ α ≥ (Nc + X) 1-α N 1-γ . ( 4
)
Thus it increases with N until

N ≤ 1 -γ γ X c (5)
and then it diminishes.

For proof see Appendix B.

Example As a simple illustration, let us take the case where X = 0 and ∑ d i = 0. By substituting this in (3) and (4), we obtain the conditions for the individual incentives:

N ≤ (θ α) 1 1-α c ,
and for the collective incentives:

N ≥ ( c 1-α θ α ) 1 α
The figure below shows how the number of participants N varies with the amount of c due, for given parameters θ and α. is ceteris paribus higher when the production technology is less effective. Equiva-lently, the size above which the members have incentives to form a coalition inside the association is ceteris paribus larger when the technological parameter is low.

Corollary 3.1 To each size N of voluntary association corresponds an optimal level of fee c * (N, X, α, γ). In addition, this level increases with N, the number of members.

Proof is given in Appendix B.

Symmetric Equilibria

In this section, we consider the different equilibria corresponding to different modalities of formation of an association. Depending on these modes of emergence of association, the members can decide together to contribute an equal amount, or some members can form a coalition of volunteers. Some prior conditions may influence the process of formation of an association, for instance the heterogeneity in members' incomes. Sociologists [START_REF] Laville | Les fonctionnements associatifs[END_REF]) make a distinction between at least two main modes of formation of associations, depending on the position of the founders with regard to the action undertaken: -"for self" and "for others". In the first case, the members self-organize with no distinction between a dominant category of active members and beneficiaries. In the organization "for others", on the contrary, there is a differentiation a priori between so-called "weak"

and "strong" categories of agents (Laville and Sainsaulieu, 1997, p. 288.) These modes of emergence affect the internal functioning of the organization. In associations whose members are not the direct beneficiaries, it is common to observe some detachment of "weak" categories from the "strong" actors. In this kind of organization, the mobilization of passive beneficiaries becomes an important is-sue for management. In self-organized associations, on the contrary, the difficulty consists in reaching a consensus between the "strong actors" and pursuing the joint action in a sustained manner (Laville and Sainsaulieu, 1997, p. 290).

In the model proposed here, these situations are considered using the concepts of different equilibria. The N-symmetric equilibrium can be considered as the most socially desired result, in that it implies equal contributions from each member. It corresponds to the case of an association where all the participants belong to the same category of actors.

However, as [START_REF] Olson | Logic of Collective Action: Public Goods and the Theory of Groups[END_REF] noted, an internal coalition can be formed, whose members make additional voluntary contributions. This situation can correspond to the case of the association "for others", with a distinction between "strong" and "weak" categories of participants. This kind of association is studied using the concept of K-symmetric equilibrium. Finally, introducing a budget constraint allows us to analyze the partial K-symmetric equilibrium. In this case, the members belonging to the coalition of volunteers can each make an equal voluntary contribution, while members whose incomes do not allow them to contribute at the same level contribute less.

N-symmetric equilibrium

In the utility function without budget constraint, there is no particular reason for any asymmetry in the amount of voluntary contributions. In the case of N-symmetric equilibrium, the voluntary contribution

d i = d ∀i ∈ N is equivalent to a voluntary increase in dues, raising them from c to c = c + d.
Proposition 4 Without budget constraint, there exists a unique N-symmetric equilibrium of the voluntary contribution d * whose value is increasing in N, θ , and α and decreasing in c and X.

Proof is given in Appendix C.

Example Sign of d * . Let us not restrict the sign for the voluntary contribution. A negative contribution is therefore equivalent to a reduction in dues. A positive value is equivalent to an increase in dues. To illustrate this, take a case where θ = 1, γ = 0, and α = 0.5. Replacing these parameters in (C.3), we study the conditions making d * equal to zero:

d * = 1 4 N -c - X N = 0. ( 6
)
The solution of ( 6) gives a positive root N + and a negative root N -. d * is negative between the two roots, so ∀N ≤ N + and positive beyond.

Consequently, for

N ≤ 2(c + c 2 + X)
d ≤ 0, which is equivalent to a decrease in dues, and for

N ≥ 2(c + c 2 + X),
d ≥ 0, which is equivalent to an increase in dues.

The considerations above concern the choice of the size of the association. Members may have incentives to recruit new members rather than reducing their dues, which might also mean reducing the scope of the collective action. This may be the case for the organizations Olson wrote about, which almost always welcome newcomers (Olson, 1971, p. 59.) Depending on the level of dues and on the structure of distribution of agents' incomes, it may be more or less difficult to attract new members. Furthermore, when there are criteria for joining an association, this may limit its accessibility to potential members. [START_REF] Gordon | A Typology of Voluntary Associations[END_REF], who studied these membership criteria, identified the criteria of merit (as in the American So-ciological Society, for instance) and the criteria of attributes, such as gender (e.g. feminist associations), origins or culture (for instance an association of Ukrainians), or simply certain social links between members.

Some associations may, on the contrary, prefer to restrict entry, like the so-called "status organizations" [START_REF] Hansmann | A Theory of Status Organizations[END_REF], which grant their participants a certain social status [START_REF] Gordon | A Typology of Voluntary Associations[END_REF] In these organizations, the decision of an individual to join depends not only on the characteristics and the price of the collective good provided, but also on the characteristics of the members. This kind of association establishes a minimal status level (in social, economic or other terms) in addition to membership dues. When membership in an organization provides its members with social status, members may decide to limit the number of newcomers. This is the case described by [START_REF] Olson | Logic of Collective Action: Public Goods and the Theory of Groups[END_REF]:

"If the top "400" were to become the top "4000", the benefits to the entrants would be offset by the losses of old members, who would have traded an exalted social connection for one that might be only respectable" (Olson, 1971, p. 37.) In any case, there exists a trade-off between the size of the organization and the level of dues.

Coalitions and K-Symmetric Equilibrium

As we have already noted, one can consider a coordinated initiative of a group of K members deciding collectively to make a symmetric voluntary contribution. Generally, such coordination is difficult to achieve and the organization costs can be high. However, as [START_REF] Olson | Logic of Collective Action: Public Goods and the Theory of Groups[END_REF] noted, in some cases an existing group which has already met the organization costs may facilitate further collective action. More-over, he noted that the aptitude of a group to provide a collective good can be partly explained by its origins and the factors that sustain it.

The idea that pre-existing organizations can facilitate the collective action has also been suggested by [START_REF] Coleman | Social Capital in the Creation of Human Capital[END_REF] in the form of the concept of social capital.

Groups of individuals who desire to contribute more can have various different origins. Besides a pre-existing organization, it may be founded on kinship links or shared social and cultural characteristics.

Proposition 5 For any K ≤ N, there exists a K-symmetric unique equilibrium such

that d i = d K ∀i ∈ K, and d j = 0 ∀ j ∈ N -K. Moreover, d K is increasing in K, θ ,
and α and decreasing in N, X, and c.

The proof is given in Appendix C.

Example Sign of d K : Let us take an example of association characterized by the parameters θ = 1, γ = 0, and α = 0.5. Substituting them in (C.6) and resolving yields two solutions:

K + = 2 √ Nc + X and K -= -2 √ Nc + X. If K ≤ K + , then d ≤ 0, and if K > K + , then d > 0.
K + is interpreted here as a minimal size of K positive equilibrium. In other words, it is the minimal size of an internal coalition formed by members to make voluntary contributions within the limits of their budget constraint.

If, nevertheless K + > N, we return to a N-symmetric equilibrium with d * < 0 corresponding to a necessary reduction of compulsory payments.

The propositions 4 and 5 postulate, respectively, that at N-symmetric and K-symmetric equilibria, the voluntary contribution decreases with an increase in the external resources X. This result illustrates the effect of crowding out currently highlighted by the models of "pure altruism" [START_REF] Andreoni | Privately Provided Public Goods in a Large Economy: The Limits of Altruism[END_REF], where individuals are interested in total provision of the public good. As has been shown in a number of theoretical studies, under the "purely altruistic preferences" assumption, voluntary contributions are completely crowded out by public subsidies [START_REF] Andreoni | Privately Provided Public Goods in a Large Economy: The Limits of Altruism[END_REF][START_REF] Andreoni | Impure Altruism and Donations to Public Goods: a Theory of Warm-Glow Giving[END_REF][START_REF] Ribar | Altruistic and Joy-of-Giving Motivations in Charitable Behavior[END_REF]. However, this assumption has been challenged by empirical facts, notably in the nonprofit area [START_REF] Steinberg | Voluntary Donations and Public Expenditures in a Federalist System[END_REF][START_REF] Andreoni | Privately Provided Public Goods in a Large Economy: The Limits of Altruism[END_REF][START_REF] Andreoni | Impure Altruism and Donations to Public Goods: a Theory of Warm-Glow Giving[END_REF][START_REF] Ribar | Altruistic and Joy-of-Giving Motivations in Charitable Behavior[END_REF]. The assumption of "impurely altruistic preferences" [START_REF] Andreoni | Privately Provided Public Goods in a Large Economy: The Limits of Altruism[END_REF], according to which individuals derive utility not only from total provision of the public good, but also from the act of giving, accounts better for the persistence of voluntary contributions in the presence of public financial support. Under this assumption, the crowding-out of voluntary contributions is incomplete (see for instance [START_REF] Steinberg | Voluntary Donations and Public Expenditures in a Federalist System[END_REF] Moreover, Rose-Ackerman (1986) shows the possible crowding-in effect of government grants in situations where voluntary contributions and grants are not perfect substitutes. However, in her model, crowding-in effects are only possible under the condition that the government is able to impose its rules on the organizations it supports concerning the services they provide.

According to the proposition above, the K agents forming an internal coalition can contribute an equal amount, at the equilibrium, while others pay only the compulsory dues. The voluntary contribution decreasing in N sheds light on the character of the sharing between the members of the coalition and the rest of the group.

The larger the organization, the heavier the charge supported by the K volunteers.

Moreover, individuals may be averse to sharing a common good among a large number of people [START_REF] Alesina | Participation in Heterogeneous Communities[END_REF]La Ferrara, 2000, Bergstrom and[START_REF] Bergstrom | Private demands for public goods[END_REF] This aversion could result from the homophile preferences often used to explain the formation of homogeneous groups of individuals with common features [START_REF] Cohen | Source of Peer Group Homogeneity[END_REF] Finally, the proposition that voluntary contributions are increasing in θ describes the positive effects of the attractiveness of the public good on individual incentives to contribute. This theoretical finding is supported by a number of experimental studies, where an increase in the parameter of attractiveness leads to an increase in the mean of voluntary contributions (Hichri, 2004, p. 184.)

Budget constraint and partial K-symmetric equilibrium

What determines the emergence of homogeneous groups, besides homophily? As the sociological literature stresses, the size and composition of the population can also lead to the formation of homogeneous groups. For instance, [START_REF] Mcpherson | Homophily in Voluntary Organizations: Status Distance and the Composition of Face-to-Face Groups[END_REF] studied the influence of the size and diversity of voluntary associations on the formation of homophile links between members. They showed that homophily is not merely a natural tendency of individuals, but that the social structure can also favor or constrain individual choices. Therefore, among the factors influencing the internal groups in associations, we should distinguish between constrained and freely-chosen homophily.

We shall incorporate this idea into our model in the last section of this chapter, by introducing individual preferences for the subjective quality of the public good. Here, we introduce some heterogeneity by imposing the budget constraint that influences agents' choices. In the case with a budget constraint, and with Nsymmetric equilibrium, it is possible that

∃i ∈ N : w i -c -d * < 0.
Now, some members of the association cannot contribute at the optimal level d *

and their contributions cannot exceed the level of d = w i -c limited by their bud-get constraint.

Proposition 6 Given a distribution of revenues, there exists a unique value of K ≤ N and a unique partially K-symmetric equilibrium, such that

d i = d K ∀i ∈ K = {k|w k -c -d K ≥ 0} et d j = w j -c ∀ j ∈ (N -K).
The proof is given in Appendix C.

Non-symmetric equilibria

In this section, we study the relation between the level of voluntary contributions and incomes. This relation is founded on the non-constant character of the marginal contribution of a monetary unit to the individual utility. Taking the individual utility model based on division of the budget between private consumption and contributions to the public good, we can consider that the net income of the agent i, or w i -c -d i contributes to her utility in a concave way. Consequently, we can note:

U i = (w i -c -d i ) β + θ N γ (Nc + N ∑ j=1 d j + X) α . (7)

Equilibrium without budget constraint

Proposition 7 Without budget constraint, there exists a unique equilibrium, in which any agent i fixes her voluntary contribution in such a way that her net income is brought to the level w i -c -di = w 0 , where w 0 > 0 is a reference level common to all the agents.

The proof is given in Appendix D.

Example Let us consider a special case: if α = β , then

w 0 = Ω + X N + ( θ N γ ) 1 1-α . ( 8
)
The proposition 7 shows how the agents chose their level of voluntary contribution at a non-symmetric equilibrium. In fact, in this case, each agent pays a personalized price for the public good. The ability of voluntary organizations to establish personalized prices (or a system of price discrimination) has already been stressed in the economic literature [START_REF] Hansmann | Nonprofit Enterprise in the Performing Arts[END_REF][START_REF] Ben-Ner | Nonprofit Organizations: Why Do They Exist in Market Economies[END_REF] According to Ben-Ner, the non-market interactions between members, (kinship links, neighborhood, common background), as well as the non-profit distribution constraint, allow these organizations to reveal demands and to establish personalized prices.

Budget constraints

We now introduce budget constraints: ∀i ∈ N, d i ≥ 0, and d i ≤ w i -c. In a solution under constraint d i < 0 becomes d i = 0, while

d i > w i -c becomes d i = w i -c. Nevertheless, d i > w i -c ⇔ w i -c -w 0 > w i -c
, and thus ⇔ w 0 < 0, which contradicts the previous proposition.

Thus, the constraint d i ≥ 0 still remains.

Here, d i < 0 if and only if w i < w 0 + c, and the set of agents with incomes lower than w 0 + c do not bring voluntary contributions.

Proposition 8 There exists a unique equilibrium under constraint, in which the richest agents of a sub-group K ≤ N contribute at such a level that their net incomes are reduced to a common reference level w i -c -d i = w 0 > 0 for all the agents in K, while the agents from N -K, whose incomes are less than w 0 + c, do not make any voluntary contributions.

For proof see Appendix D.

Example As a special case, take α = β . Then

w 0 = Ω K + X + (N -K)c K + ( θ N γ ) 1 1-α . ( 9
)
Remark The result where all the agents are reduced to the same level w 0 of net income is somewhat simplistic. This is due to the fact that in this model, the share of the initial allocation dedicated to private consumption corresponds to an homogenous good with a unique technology of consumption. However, in real life, disposable income is rather devoted partly to differentiated baskets of goods characterized by diversified consumption technologies and partly to savings.

Volunteering and subjective quality

Now, let us assume that the agents can contribute to the public good not only money, but also their time and effort. One can differentiate three main types of theoretical framework for explaining volunteering: pure public good models, private consumption models and investment models [START_REF] Ziemek | Economic Analysis of Volunteers' Motivations-A Cross-Country Study[END_REF] Pure public good models consider that volunteers are motivated exclusively by total provision of the public good.

Private consumption models focus on personal rewards, like social status or simply a warm glow. Investment models may be considered as a special case of private consumption model, as volunteering provides certain benefits associated with the accumulation of human and social capital.

The model we describe below is situated between the "pure public goods" and "private consumption" approaches, as the act of volunteering is not independent of the characteristics of the public good to which it contributes. We consider that volunteering provides an additional utility to the one derived from total provision, as in [START_REF] Steinberg | Voluntary Donations and Public Expenditures in a Federalist System[END_REF], for instance. Steinberg's approach, considered as a "mixed public-private good approach", represents a case of an "impurely altruistic model" [START_REF] Andreoni | Impure Altruism and Donations to Public Goods: a Theory of Warm-Glow Giving[END_REF]. The individual utility function includes the public goods provided through the agent's individual voluntary contribution, public resources and the contributions of others. The individual contribution and the other agents' contributions are complementary when the individual is faced with social comparisons within her group of reference. When the individual is motivated only by total provision, her contributions and the contributions of others are perfect substitutes.

In our model, each member i can contribute to the public good by making a nonmonetary contribution of effort noted v i , which is subjective value. This effort allows agents to influence the quality of the public good according to their own preferences. The quality is subjective insofar as it cannot always be evaluated objectively. In other words, the quality of a good (or a service) is subjective when it is evaluated through individual perceptions rather than standardized measures.

One notable example of the use of subjective quality comes from the field of child day-care. Here, the measurement of quality is influenced by parents' preferences concerning certain service characteristics, making it more subjective than the standards elaborated by experts. For instance, it may be important to know whether the provider of the child-care service speaks the same language, shares the same values or the same religion. This implies that an individual's perception of quality may differ not only from others' perceptions, but also from the more objective evaluations made by experts using an index of quality [START_REF] Evers | Governing diversity -conceptual and practical challenges illustrated by the example of the German child-care system[END_REF][START_REF] Moss | Defining quality: Values, stakeholders and processes[END_REF] As [START_REF] Evers | Governing diversity -conceptual and practical challenges illustrated by the example of the German child-care system[END_REF] have shown in a German case study, the development of nonprofit child-care centers was mainly driven by the desire of some parents to create a service in keeping with certain pedagogical, ideological or religious principles.

Moreover, as [START_REF] Kushman | A Three-Sector Model of Day Care Center Services[END_REF] observed in the United States, nonprofit child-care centers tend to provide more labor-intensive services, sometimes including parent participation, than private ones.

We therefore suggest that the members of an association can volunteer in the hope of influencing the subjective quality of the good or service being provided. In the field of child care, for instance, the role of volunteers can consist in organizing activities according to their cultural background or pedagogical convictions. In the field of the arts, active members can influence programming choices according to their personal tastes. Finally, in a charity helping the poor, the more active volunteers can influence the criteria of eligibility or the order of priority (Rose-Ackerman, 1986.)

Two aspects must be taken into consideration here:

• the cost of the effort.

By convention we measure effort v i as a fraction of the individual's disposable time that is dedicated to the volunteering, or v i ∈ [0, 1]. For i, the effort v i represents an opportunity cost that is proportional to her income. We assume that each agent i works a length of time 1 to earn an income w i . Thus, the opportunity cost can be written as v i w i .

• the impact on the subjective value of the good.

The subjective value of the public good for the agent i depends both on the total amount of effort provided by all the volunteers and on the weight of her own effort as a proportion of the total. The combination of both can differ according to how "individualistic" the association is. So agents can attach more or less value to the public good itself, but they also value their own contributions more or less highly.

Thus, the subjective level of effort can be written as follows:

(

1 -δ ) N ∑ j=1 v j + δ v i = (1 -δ )v -i + v i . ( 10
)
where δ is the parameter of "individualism". Two extreme cases can be considered:

• δ = 0 leads to ∑ N j=1 v j and expresses a purely collective orientation. In other words, the effort of a volunteer is added to the sum of efforts provided by other agents;

• δ = 1 leads to v i and designates a purely individualistic orientation, where the personal contribution is the only one valued, in spite of the effort provided by others.

We consider that δ , the so-called parameter of individualism, characterizes the type of association. Under some conditions, it is possible to link this parameter with the typology of voluntary associations (see for instance [START_REF] Gordon | A Typology of Voluntary Associations[END_REF], defining them as "expressive" or "instrumental groups". The main objective of the first type of association is to provide common activities for their members. The members are direct beneficiaries of organized activities (for instance a countryclub.) A δ equal to one denotes a purely expressive association, inasmuch as participation in its activities provides a direct gratification for the members. On the contrary, the main function of so-called instrumental associations is directed outside the organization. This kind of association aims to create and maintain a normative condition or a commitment (Gordon and Babchuk,p. 25) and exercises a social influence. Examples include associations for the defence of the rights of minorities and the poor, or certain political and religious organizations. The extreme case of such an association is denoted by a value of δ equal to zero. In other words, it is assumed that the members are driven by ideological principles or common values, rather than the pleasure of participating in a shared activity. In real life, voluntary associations can accomplish both functions. For example, some associations may have an expressive function at the local level and an instrumental function at the national level. Intermediate associations may be situated between the two extreme cases with a δ more or less close to 0 or to 1.

In our model, we introduce effort into the production function of the public good by means of a Cobb-Douglas function. The utility of the agent i is now written as:

U i = w i -c -d i -w i v i + θ N γ (Nc + N ∑ j=1 d j + X) α ((1 -δ ) N ∑ j=1 v j + δ v i ) β (11)
with α and β ∈]0, 1[.

In a standard way, each agent i determines her pecuniary contribution d i and her effort v i so as to maximize her subjective utility U i , according to her rational expectations about the levels of voluntary contributions ∑ j =i d j = d -j and effort

∑ j =i v i = v -i
made by the other agents.

Proposition 9 At equilibrium, each agent fixes her levels of voluntary contribution and effort (d i , v i ) by attributing to the latter a relative weight depending on her income. The lower the income, the higher the relative weight of the effort. In other words, a poorer agent will compensate for her weak monetary contribution by a greater effort. Moreover, for all the agents, the level of direct effort is an increasing function of the degree of individualism δ .

For proof see Appendix E.

The model above shows that pecuniary contributions and volunteering can well be made jointly. This conforms to the stylized facts, whereby monetary donations and volunteering often come together. Generally, the explanation proposed for this phenomenon is that donors need to control the good use of their donations. In this paper, we put forward the idea that the trade-off between monetary contributions and volunteering can be based on the desire to influence the quality of the public output. Moreover, the valuation of the public good becomes socially-based, to the extent that it depends on the agents' characteristics (culture, education, ideological considerations, etc.)

Here, the concept of service assumes particular importance. The outputs of nonprofit organizations are mostly represented by services [START_REF] Weisbrod | The Voluntary Nonprofit Sector. An Economic Analysis[END_REF][START_REF] Hansmann | The Role of Nonprofit Enterprise[END_REF]. It can be difficult to evaluate a service when it has both immediate and long-term impacts, or when the perceptions of its quality differ across individuals. The diversity of criteria by which a service can be appraised influences the possible trade-off between giving and volunteering. Pecuniary contributions and volunteering may well influence the characteristics of the service differently. Thus, agents can contribute both money and effort. Another interesting issue highlighted in the model is the level of effort provided by an agent as a function increasing in δ , the parameter of individualism characterizing the association. In other words, the model predicts that volunteering is likely to be more important in expressive associations [START_REF] Gordon | A Typology of Voluntary Associations[END_REF], i.e. those essentially oriented towards the interests of their members. One conclusion we can draw from this result is that to attract a greater effort of volunteering, an association should take into account the personal interests of its volunteers. This can be achieved through awards, such as public recognition and social status, but also through socialization, dialogue and communication (Laville and Sainsaulieu, 1997, p. 292).

This result is in keeping with some stylized facts about volunteering at the international level in general, and in the United States and France in particular.

At the international level, [START_REF] Salamon | Global Civil Society: An Overview[END_REF] highlight the important role of vol-unteers in the field of culture and arts, accounting for a quarter of total volunteering (p. 24). According to [START_REF] Schervish | Social participation and charitable giving: a multivariate analysis[END_REF], volunteers in the United States largely benefit from the activities they create. Finally, in France, more than half of the members of expressive associations (e.g. shared leisure activities, social clubs) participate at least once a week in the shared activities as organizers, while this fig-

ure is only 34 per cent in the instrumental associations, like those for the defence of minority interests. Moreover, nearly half of the time dedicated to volunteering is mobilized in the areas of sport, culture and leisure, which are the biggest consumers of this input [START_REF] Prouteau | Donner son temps: les bnvoles dans la vie associative[END_REF] The results of [START_REF] Prouteau | Donner son temps: les bnvoles dans la vie associative[END_REF] confirm the hypothesis that in France, at least, volunteering by adults is often reinforced by specific needs for services (children's education, for instance).

In an empirical study using U.S. data on married women's voluntary labor, Carlin (2001) finds that participation rates are positively related to the number of children.

As [START_REF] Ziemek | Economic Analysis of Volunteers' Motivations-A Cross-Country Study[END_REF] shows in a cross-country study, the presence of young children in a household positively influences the egoism and investment motivation of volunteers.

As regards the effect of income on volunteering, according to (E.3), the level of effort is a decreasing function of w. In other words, the effort of volunteering is likely to rise when the wage rate decreases. As empirical support to this finding, we can cite the inverse relation between voluntary labor and wage rates documented by Menchik and Weisbrod (1987). In addition, [START_REF] Andreoni | Charitable Contributions of Time and Money[END_REF] find a relatively substantial negative effect of the net wage on volunteer hours. However, the effects of wage rate on volunteering may be sensitive to the type of volunteers' motivations or other determinants.

Collective-consumption goods can be provided by voluntary associations. In this paper, we consider an important feature of these organizations, concerning the issue of personal commitment to a common action. We develop a model of voluntary association of individuals who are interested in obtaining advantages from the provision of a common good. At the first step, we consider a set of agents who benefit from a quantity of a good in a uniform manner. We show the existence of several symmetric and non-symmetric equilibria corresponding to the different modes of emergence of associations identified in the sociological literature [START_REF] Laville | Les fonctionnements associatifs[END_REF]. At the N-symmetric equilibrium corresponding to the case of a "for-self" association, all the agents are equal in the sense that they all make a fixed voluntary contribution. Voluntary contributions are crowded out by compulsory payments and by exogenous resources. At the K-symmetric equilibrium, where one can distinguish between two categories of agents, the "strong" and the "weak" ones, only K agents make voluntary contributions. This case corresponds to the case of an association created "for others". Here, the effects of crowding out are also present. Moreover, the level of voluntary contributions diminishes with the size of the organization. Both at the N-symmetric and the K-symmetric equilibria, the amount of voluntary contributions grows with the attractiveness of the public good.

Then, by introducing a budget constraint, we show the existence of a unique partial K-symmetric equilibrium, in which the K richest agents give an equal amount, while the others give the rest of their disposable income after dues. When it is assumed that the net income of the agent enters the utility function in a concave way, we find that there exists a non-symmetric equilibrium at which all the agents fix their voluntary contributions in such a way as to make their net disposable in-comes equal. Finally, in the same configuration and after re-introducing the budget constraint, we show the existence of a unique equilibrium at which only the richest members form an internal coalition to make additional voluntary contributions. The members of this group inside the association give an amount that makes their net incomes equal.

At the second step, we introduce the quality of the public good into the basic model.

Here, quality is considered simultaneously as a public and private component of the good provided, while quantity is generally considered as the only feature of the public good. An exception is provided by [START_REF] Ben-Ner | Nonprofit Organizations: Why Do They Exist in Market Economies[END_REF], who considers quality as a public component and quantity as a private component, in the sense that consumers only buy the quantity they demand. The model presented above is reinforced by a number of works in the empirical literature concerning the positive relationship between voluntary contributions and the attractiveness of the public good, incomplete crowding-out phenomena, the importance of the organization size and its effects on voluntary contributions, and the positive link between incomes and the level of voluntary contributions. Moreover, according to the stylized facts, the effort of volunteering is more important in the organizations oriented towards the direct interest of the members. Thus, the rewards of volunteers are not pecuniary. Rather, the model proposed above stresses how dialogue, social acknowledgment, and what some call "accepted differentiation" [START_REF] Laville | Les fonctionnements associatifs[END_REF] are important.

Even if most of the results of the model essentially correspond to the stylized facts, they can depend on the structure of the model. An important assumption made in the model is that individuals may desire not only to increase production of the public good, but also to influence its quality by volunteering. However, this does not exclude other possible incentives of volunteers, which are not taken into account in the model. Moreover, the incentives of volunteers to devote time and effort may differ, depending on the people and the social context. The degrees of motivation of individuals can also be heterogeneous. This heterogeneity of incentives could be an interesting issue for future works.

Finally, the main contribution of this paper consists in highlighting voluntary association as a group of individuals formed around a common intention. But this does not exclude that possibility that the members of an association may have their own aspirations and conceptions of the good or service they want to provide. This idea is formalized by introducing the concept of subjective quality, according to which the perceptions of the quality of a good (or service) may well differ across persons. Accepting these differences, what has been called "accepted differentiation", allows the actors to promote a collective action. As [START_REF] Laville | Les fonctionnements associatifs[END_REF] observed, associative activity is an exercise in social cohesion; it does not exclude the expression of differences that come together around a shared project. In other words, as the saying goes, "people can share the same bed without sharing the same dreams." (i.) When X ≥ 0, namely in the presence of public subsidies,

• γ ≥ α d f dN ≤ 0, ∀N. (A.5)
In this case, association does not make sense as the sharing rule described by γ cancels the gain of individual utility.

• γ < α

The variation of the sign of the derivative is given in the table below.

N N 1 = -γX (α-γ)c d f dN < 0 0 > 0
Beyond N 1 , each additional contribution leads to a gain in individual utility.

There is no overcrowding, but there exists a minimal size of association which is necessary for association to make sense.

(ii.) When X < 0, namely in the presence of fixed costs, two cases need to be studied:

• If γ ≤ α, the production function is increasing with the number of contributors if their number is higher than N 0 . In this case, there is no overcrowding.

Nc (α -γ) -γX > 0, ∀N. (A.6) Thus, d f dN ≥ 0, ∀N ≥ -X c = N 0 . (A.7)
Therefore we obtain the following table of variations, where two alternative cases depend on the relative position of K with regard to s.

N 0 s K N 1 d f dN < 0 > 0 < 0 < 0 > 0 K d f dN < 0 > 0 > 0
Thus, for a population that is large enough, there is no overcrowding.

• In the case where c(α -γ) -d ≤ 0 and X ≥ 0, the inequality (A.14) holds for any K ≥ 0 (here we are in the alternative case K ≤ N 1 ).

If X < 0, we find the two alternative cases above.

Finally, if Y < 0, we find the previous results.

A.2 Extension 2

If beyond the number K it is possible to make voluntary contributions, the situation of every member is improved. Overcrowding remains absent, but it can occur when Y < 0 and γ > α, beyond a threshold possibly moved forward.

A.3 Extension 3

If below K, the voluntary contributions vary, one can substitute Kd by D = ∑ K j=1 d j .

The result still holds.

B Proofs of proposition 3 and corollary 3.1

B.1 Proof of proposition 3

The utility for the member N is written as

U N = w N -c -d N + θ N γ (Y N + d N ) α , (B.1)
where

Y N = Nc + N-1 ∑ j=1 d j + X. (B.2)
N has an incentive to contribute when

∂U N ∂ d N ≥ 0. (B.3) ∂U N ∂ d N = -1 + θ α N γ (Y N + d N ) α-1 . (B.4)
Thus, we obtain the following condition:

θ α ≥ N γ (Y N + d N ) 1-α . (B.5)
Therefore, N will have incentives to contribute a positive amount if and only if

θ α ≥ N γ (Nc + N-1 ∑ j=1 d j + X) 1-α . (B.6)
The right-hand term is increasing in N.

Then, formation of the internal group is studied in the situation of symmetry. In other words, here we analyze the situation where a group of individuals in an association each decide to provide an equal amount. This is equivalent to the decision to raise the compulsory payment. Now, the initial individual utility is written as

U G 0 i = w i -c + θ N γ (Nc + X) α , (B.7)
We study the derivative with regard to c:

∂U ∂ c = -1 + θ αN 1-γ (Nc + X) α-1 . (B.8) Consequently, ∂U ∂ c ≥ 0 if and only if θ α ≥ (Nc + X) 1-α N 1-γ . (B.9)
We study the variations of the right-hand side term of (4):

f (N) = (Nc + X) 1-α N 1-γ (B.10) ∂ f ∂ N = (Nc + X) -α N -γ N 2(1-γ) (γNc -(1 -γ)X). (B.11) Consequently, sg ∂ f ∂ N = sg(γNc -(1 -γ)X), (B.12) and thus, ∂ f ∂ N ≥ 0 if and only if N ≥ 1 -γ γ X c . (B.13) 2

36

B.2 Proof of corollary 3.1

The level c * is obtained from (B.8) as solution of ∂U ∂ c = 0. It can be written as:

c * = (θ α) 1 1-α N 1-γ 1-α -X N . (B.14)
Now the variation of c * as a function of N can be studied:

∂ c * ∂ N = (θ α) 1 1-α N 1-γ 1-α α-γ 1-α + X N 2 . (B.15) Given the assumption X ≥ 0 et α > γ, ∂ c * ∂ N > 0.
2 C Proofs of propositions 4, 5, and 6

C.1 Proof of proposition 4

The utility function in N-symmetric equilibrium can be written as

U i = w i -c -d + θ N γ (N(c + d) + X) α , (C.1) ∂U i ∂ d = -1 + θ αN 1-γ (N(c + d) + X) α-1 . (C.2)
By assuming the right-hand term of (C.2) equal to zero, we obtain the optimal value of voluntary contribution in N-symmetric equilibrium without budget constraint1 :

d * = (θ α) 1 1-α N α-γ 1-α -c - X N . (C.3)
From equation (C.3) follows that d * is increasing in θ .

Let us note

d * = (λ α) 1 1-α 1 N -c - X N , with λ = θ N 1-γ . (C.4)
We study the function f (α) = (λ α) 1 1-α avec α < 1.

d f dα = λ 1 -α (λ α) α 1-α .
d f dα > 0, d * is therefore increasing in α.

We now study the variation of d * with N:

∂ d * ∂ N = (θ α) 1 1-α α -γ 1 -α N α-γ 1-α -1 + X N 2 > 0.
On the contrary, it follows immediately from (C.3) that d * is decreasing in c and in X. 2

C.2 Proof of proposition 5

At the K-symmetric equilibrium, the utility function is written as follows: Now, the individual utility is written for any i ∈ K as follows:

U K i = w i -c -d K + θ N γ (Nc + Kd K + D N-K + X) α , (C.7)
where

D N-K = ∑ j>K w j -c.
or

U K i = w i -c -d K + θ N γ (K(c + d K ) + Ω N-K + X) α , (C.8)
where

Ω N-K = ∑ j∈(N-K) w j .
For any i ∈ K,

∂U i ∂ d K = -1 + θ αK N γ (K(c + d K ) + Ω N-K + X) α-1 . (C.9)
By assuming ∂U i ∂ d K = 0, we obtain an optimal value of contributions for any i ∈ K: Now, the first order condition is written as

d K = [( θ αK N γ ) 1 1-α -Ω N-K -X] 1 K -c.
∂U i ∂ d i = - β (w i -c -d i ) 1-β + θ N γ α (Nc + ∑ N j=1 d j + X) 1-α .
(D.1)

The second order condition becomes

∂ 2 U i ∂ d 2 i = - β (1 -β ) (w i -c -d i ) 2-β - θ N γ α(1 -α) (Nc + ∑ N j=1 d j + X) 2-α < 0. (D.2)
The non-constrained equilibrium is obtained by assuming

D.2 Proof of proposition 8

As in [START_REF] Foray | Open Software: Knowledge Openness and Cooperation in Cyberspace[END_REF], if the N agents are ranged in a decreasing order of incomes, there exists a K ≤ N satisfying the following conditions:

The individual utility is written ∀i ∈ K

U i = (w i -c -d i ) β + θ N γ (Nc + K ∑ j=1 d j + X) α . (D.7)
In the same way as in the proposition 7, ∀i ∈ K d i = w i -w 0 -c. To simplify the expression of the utility function, let us note w = w i , m = d i , M =

However, this time

Nc + X + d -i , f = v i , F = (1 -δ )v -i , z = θ N γ etx = w i -c -d i .
The latter represents the share of the budget of i available for her private consumption. Thus, the agent i is solving the following:

MaxU(x, m, f ) = x -w f + z(M + m) α (F + f ) β (E.1)
under a budget constraint

x + m + c -w = 0 (E.2)

Using the Kuhn and Tuker theorem and noting the multiplicator corresponding to the constraint (E.2) as λ , we write the marginal utilities and the respective first order conditions (the marginal utilities are proportional to the prices):

∂U ∂ x = 1 = λ p x = λ (i) ∂U ∂ m = αz (F + f ) β (M + m) 1-α = λ p m = λ (ii), and ∂U ∂ f = β z (M + m) α (F + f ) 1-β -w = λ p f = 0. (iii)
From (i) it follows that λ = 1.

Consequently, the first order conditions (ii) and (iii) can be written as αz(F + f ) β = (M + m) 1-α (ii ) and

β z(M + m) α = w(F + f ) 1-β (iii )
By making a cross product of the terms of these equations, we obtain:

αw(F + f ) = β (M + m) ⇔ f = β αw (M + m) -F. (E.3)

Fig. 1 .

 1 Fig. 1. Illustration: N as a function of c

2 C. 3

 23 figure (C.1).

Fig

  Fig. C.1. The distribution of voluntary contributions

2 D

 2 and Zimmermann (2007), the value of K is adjusted in such a way, that w k -c ≥ d K and w K+1 -c < d K . Proofs of propositions 7 and 8 D.1 Proof of proposition 7

  ∂U i ∂ d i = 0 ∀i ∈ N under condition w i -c -d i > 0. (D.3)We obtain thereforeβ (w i -c -d i ) 1-β = θ N γ α (Nc + ∑ N j=1 d j + X) 1-α . (D.4) It follows consequently that ∀i = j β (w i -c -d i ) 1-β = β (w j -c -d j ) 1-β or that ∀i ∈ N, w i -c -d i = w 0 ,and thus d i = w i -w 0 -c. (D.5) Thus, after voluntary contribution, all the agents are brought to the same level of net income w 0 N(c + w 0 ) = Ω -N(c + w 0 ), from which, by substituting in (D.4) we obtain X -Nw 0 ) 1-α (D.6) It follows that w 0 solution of the implicit equation (D.6) is strictly positive, which satisfies the condition (D.3). 2

2 E

 2 K(c + w 0 ) = Ω K -K(c + w 0 ), + X -Kw 0 + (N -K)c) 1-α . (D.8) Proof of proposition 9

Table 2

 2

	Overcrowding
	Cases X > 0	X < 0
	γ < α No overcrowding	No overcrowding
	γ > α Association makes no sense Overcrowding

Table A .

 A 1 Variations of the production function with N in the presence of external subsidies

The second order condition holds if X ≥ 0 and α < 1.

Acknowledgments

The authors gratefully acknowledge Walid Hichri, Alain Rallet, and Sylvie Thoron for helpful comments, discussions, and suggestions.

A Proofs of propositions 1 and 2

Let us assume first symmetric voluntary contributions, in other words d i = d for each i, and note c = c + d. Now, the equation ( 2) can be re-written as

To make f (N) defined and continuous, let us assume that

When X < 0 (namely, in the presence of fixed costs), this condition means that the number of participants must be high enough for the sum of individual contributions c to compensate the fixed cost X.

When X ≥ 0, this condition holds for any value of N ≥ 0.

The derivative of the production function of the public good ( 2) can be written as follows:

2), the sign of the derivative will be the one of the numerator (A.3), or

for a number of participants N equal to

the table of variations of the derivative will be the following:

Variations of the production function with N in the presence of fixed costs

Beyond N 1 , the production function generates overcrowding. In other words, when the number of participants is higher than N 1 , the amount of good produced diminishes with the number of members.

Note that N 1 > N 0 . The number of members N 0 represents a minimal size of association. Beyond this size and up to N 1 , effects of overcrowding are absent, namely the production of the public good increases with the number of members.

2

A.1 Extension 1

Now let us suppose that K first members, founders for instance, have fixed the level of the voluntary contribution, while the newcomers contribute only the compulsory amount c. For each N ∈ [N 0 , K], the previous results hold.

Beyond this threshold, ∀N > K the production function is written as follows:

This situation is equivalent to the previous case with c = c and

Thus, the previous results are found with

and

An important issue here is to situate N 1 with regard to K.

If Y ≥ 0 and γ < α,

if and only if γY ≥ (α -γ)cK.


BS Seyeon Kim

PhD Sara Baldassari

Suk Nam

MD, PhD Sim

MD, PhD Mathilde Chipaux

MD Georg Dorfmüller

; Dong

MD, PhD Seok Kim

; Won

MD, PhD Seok Chang

PhD Valérie Taly

MD, PhD Jeong Ho Lee

Ph.D Stéphanie Baulac
email: stephanie.baulac@icm-institute.org

Detection of brain somatic mutations in CSF from refractory epilepsy patients Running head: Detect somatic variants in epilepsy patients CSF

Keywords: Brain mosaicism, refractory epilepsy, cortical malformations, CSF, somatic variants, cell-free DNA

Brain mosaic mutations are a major cause of refractory focal epilepsies with cortical malformations such as focal cortical dysplasia, hemimegalencephaly, malformation of cortical development with oligodendroglial hyperplasia in epilepsy, or ganglioglioma. Here, we collected cerebrospinal fluid (CSF) during epilepsy surgery to search for somatic variants in cell-free DNA (cfDNA) using targeted droplet digital PCR. In 3/12 epileptic patients with known somatic mutations previously identified in brain tissue, we here provide evidence that brain mosaicism can be detected in the CSF-derived cfDNA. These findings suggest future opportunities for detecting the mutant allele driving epilepsy in CSF.

INTRODUCTION

Brain somatic variants have been shown to be an important etiology of structural intractable epilepsy. [START_REF] Poduri | Somatic Mutation, Genomic Variation, and Neurological Disease[END_REF][START_REF] Marsan | Mechanistic target of rapamycin (mTOR) pathway, focal cortical dysplasia and epilepsy[END_REF][START_REF] Koh | Brain somatic mutations in epileptic disorders[END_REF][START_REF] Gama | Somatic mosaicism and neurodevelopmental disease[END_REF] We recently reported that brain mosaic variants give rise to early-onset epilepsies associated with various focal cortical malformations, including focal cortical dysplasia type 2 (FCD2), hemimegalencephaly (HME), malformation of cortical development with oligodendroglial hyperplasia in epilepsy (MOGHE), and ganglioglioma (GG), a low-grade neuronal-glial tumor. [START_REF] Marsan | Mechanistic target of rapamycin (mTOR) pathway, focal cortical dysplasia and epilepsy[END_REF][START_REF] Koh | Brain somatic mutations in epileptic disorders[END_REF][START_REF] Gama | Somatic mosaicism and neurodevelopmental disease[END_REF][START_REF] Lim | Brain somatic mutations in MTOR cause focal cortical dysplasia type II leading to intractable epilepsy[END_REF][START_REF] Lim | Somatic Mutations in TSC1 and TSC2 Cause Focal Cortical Dysplasia[END_REF][START_REF] Koh | BRAF somatic mutation contributes to intrinsic epileptogenicity in pediatric brain tumors[END_REF][START_REF] Sim | Brain somatic mutations in SLC35A2 cause intractable epilepsy with aberrant N-glycosylation[END_REF][START_REF] Baldassari | Dissecting the genetic basis of focal cortical dysplasia: a large cohort study[END_REF][START_REF] Suk | Precise detection of low -level somatic mutation in resected epilepsy brain tissue[END_REF][START_REF] Thomas | Frequent SLC35A2 brain mosaicism in mild malformation of cortical development with oligodendroglial hyperplasia in epilepsy (MOGHE)[END_REF] Epilepsy associated with these aforementioned brain lesions is commonly pharmacoresistant to antiepileptics, and surgical resection of the epileptogenic zone is often the only treatment option effective at controlling seizures. While the genetic etiology of these disorders is now mostly known, molecular diagnosis requires genomic DNA derived from resected brain tissues, thus limiting a complete genetic assessment in cases not eligible for surgery.

Liquid biopsy techniques based on the sequencing of cell-free DNA (cfDNA) are emerging as noninvasive methods for tumor diagnosis and progression. [START_REF] Schwarzenbach | Cell-free nucleic acids as biomarkers in cancer patients[END_REF] Interestingly, cerebrospinal fluid (CSF) has been shown to be a source of circulating tumor DNA released upon tumor cell death and a potentially powerful biomarker for the diagnosis and characterization of tumors of the central nervous system (CNS), such as gliomas. [START_REF] Pan | Brain tumor mutations detected in cerebral spinal fluid[END_REF][START_REF] Wang | Detection of tumor-derived DNA in cerebrospinal fluid of patients with primary tumors of the brain and spinal cord[END_REF][START_REF] Miller | Tracking tumour evolution in glioma through liquid biopsies of cerebrospinal fluid[END_REF] Moreover, aberrant neuronal death has also been observed in epilepsy patients and experimental animal models. [START_REF] Ekstrand | Lithium pilocarpine-induced status epilepticus in postnatal day 20 rats results in greater neuronal injury in ventral versus dorsal hippocampus[END_REF][START_REF] Scholl | Neuronal degeneration is observed in multiple regions outside the hippocampus after lithium pilocarpine-induced status epilepticus in the immature rat[END_REF][START_REF] Mao | Redox-related neuronal death and crosstalk as drug targets: Focus on epilepsy[END_REF] To date, however, the clinical and diagnostic utility of liquid biopsies from patients with intractable epilepsy and brain malformations has not yet been addressed. Here, we provide evidence in epileptic patients with previously identified mutations in the brain resected tissues, that somatic mutations can be detected in the CSF-derived cfDNA collected during epilepsy surgery.

METHODS

Patient cohort

We collected brain tissues, blood and cerebrospinal fluid (CSF) samples from a cohort of 12 patients (mean age 6.

Cerebrospinal fluid sampling and DNA extraction

All CSF samples were collected after dura opening before surgery (resection or hemispherotomy).

For all KR samples and FR-4 sample (8/12), CSF was collected from the subarachnoid space adjacent to the epileptogenic area; for FR-1, FR-2 and FR-3 cases, CSF was collected from the former resection cavity; for FR-5, it was collected from the inter-hemispheric fissure. The volume of collected CSF ranged from 1 to 6mL depending on the surgical site (See supplementary table for details). To remove any genomic DNA contaminant from cells, all CSF samples (except case FR-4) were centrifuged at 400g for 5 min at 4°C (FR cases) or 2000g for 5 min at 4°C followed by 10000g for 5 min at 4°C (KR cases). The supernatant was divided into 1mL aliquots and stored at -80°C. Cell-free DNA (cfDNA) samples were extracted from 1mL of CSF (except 0.5mL of CSF for KR-3 sample) with the Maxwell RSC cfDNA kit (Promega; French cohort) or the QIAamp circulating nucleic acid kit (Qiagen; South Korean cohort) and quantified using the high sensitivity dsDNA Qubit assay (Thermo Fisher Scientific). Using the same procedure described above, we also included cfDNA from 1mL of CSF, collected from the subarachnoid space after dura opening, of three other refractory epileptic patients with FCD1, FCD2a, and TSC, to be used as mutation-negative controls in the droplet digital PCR assays.

Droplet digital PCR assays and statistical analysis

The droplet digital PCR (ddPCR) QX200 system (Bio-Rad) was used for variant specific detection.

All cfDNA samples were first subjected to a mutation-site targeted 13 cycles preamplification (for each variant to test) with the SsoAdvanced™ PreAmp Supermix (Bio-Rad), optimized for unbiased target-specific preamplification of a limited amount of DNA, following the standard protocol for 20X TaqMan Assays. To avoid possible amplification errors, at least 2 independent preamplifications were done per sample. Diluted (1ng/µL) bulk brain DNAs were also preamplified to control for ddPCR accuracy. All ddPCR reactions were done in replicates (at least 2 replicates for controls, ≥3 for patients) using the ddPCR Supermix for probes according to the manufacturer's protocol. ddPCR mutation detection assays FAM+HEX (Bio-Rad) were purchased to detect the following variants: MTOR p.Ser2215Phe, MTOR p.Ala1459Asp, AKT3 p.Glu17Lys, PIK3CA p.Glu545Lys, SLC35A2 p.Ser212Leufs*9, SLC35A2 p.Glu254*, SLC35A2 p.Gln168*, and BRAF p.Val600Glu. In each assay, 3 to 5 mutation-negative cfDNAs controls (either the mutated patients used as crossed negative controls or the 3 mutation-negative controls) were preamplified and assessed by ddPCR. In all reactions, 5U of HindIII digestion enzyme was added. Amplification products were run on a QX200 droplet reader and data analyzed with the Quantasoft Analysis Pro software (version 1.0.596). To ensure sufficient sensitivity, only assays with > 100 copies/µL of WT (HEX+) DNA were kept for analysis. The VAF was calculated for each sample by the Quantasoft Analysis Pro software as the fractional abundance of mutant (FAM+) to total (wild type (HEX+) + mutant (FAM+)) DNA copies. Two-tailed t test (unequal variance) was used to assess if the average VAFs were statistically different between the mutated patient and the mutation-negative controls.

RESULTS

We assembled a collection of CSF samples from 12 neurosurgery cases of FCD2, HME, MOGHE, and GG with known mosaic pathogenic variant identified in the genomic DNA from the resected frozen brain specimen (absent from blood samples indicating their brain-specificity): brain variants in genes of the mTOR pathway (PIK3CA, AKT3, MTOR) in FCD2 and HME subjects (n=6); 9,10 brain somatic variants in the galactose transporter encoding gene SLC35A2 in MOGHE cases (n=3); [START_REF] Sim | Brain somatic mutations in SLC35A2 cause intractable epilepsy with aberrant N-glycosylation[END_REF][START_REF] Baldassari | Dissecting the genetic basis of focal cortical dysplasia: a large cohort study[END_REF][START_REF] Thomas | Frequent SLC35A2 brain mosaicism in mild malformation of cortical development with oligodendroglial hyperplasia in epilepsy (MOGHE)[END_REF] and the recurrent brain somatic BRAF p.Val600Glu variant in GG patients (n=3) (Table 1, Fig. 1). [START_REF] Koh | Brain somatic mutations in epileptic disorders[END_REF][START_REF] Suk | Precise detection of low -level somatic mutation in resected epilepsy brain tissue[END_REF] We asked whether brain somatic variants can be detected in cfDNA from the patients' CSF collected from the subarachnoid or ventricle space during epilepsy surgery. Since cfDNA concentrations from CSF were generally low (mean 0.38ng/µL, range 0.05-2.7ng/µL, total amount of collected cfDNA ranged from 0.27ng to 28ng) (Supplementary Table 1), we performed a targeted preamplification for all cfDNA samples. We set up a droplet digital PCR (ddPCR) mutation detection assay to detect the presence of a given variant in the matched cfDNA sample. To exclude potential amplification bias or errors, we compared the mean VAF of patients versus mutation-negative controls. We obtained significant p-values in 3 out of 12 patients: one HME (FR-4) with a CSF cfDNA VAF of 1.358% for PIK3CA p.Glu545Lys; one GG (KR-7) with a CSF cfDNA VAF of 0.214% for BRAF p.Val600Glu, and one MOGHE (KR-4) with a CSF cfDNA VAF of 0.145% for SLC35A2 p.Gln168*. Variant allele frequencies in the brain samples ranged from 1 to 24% in the whole cohort. There was no obvious correlation between VAFs in the brain tissue and cfDNA samples.

DISCUSSION

Currently, identification of brain somatic mutations relies on direct access to brain tissue from epilepsy surgery for both cortical malformations and GG lesions. Achieving a genetic diagnosis before surgery (or when surgery is not possible) may help in understanding mutation-related disease prognoses and adopting targeted therapies. We hypothesized that mutated DNA could be detected in CSF-derived cfDNA from patients with non-tumoral cortical malformations, as described for other CNS malignant tumors. By using a targeted ddPCR-based approach, we were able to detect somatic brain mutations in CSF-derived cfDNA in 3/12 (25%) surgical cases with a mean VAF of 0.57%. A recent study with a sensitivity threshold to detect down to 0.25% mutant rates, reported the detection of somatic mutations from the cfDNA of 3 cases with focal epilepsies by ddPCR, with a mean mutational burden of 6.8%: in one patient with subcortical band heterotopia (widespread mosaic variant in LIS1 p.Lys64*, VAF at 9.4% in CSF from lumbar puncture and 13% in the blood); one patient with FCD2b (TSC1 p.Phe581His*6, VAF at 7.8% in CSF from dural puncture and 2.8% in the resected tissue); and one with GG (BRAF p.Val600Glu, VAF at 3.2% in CSF from dural puncture and 20.4% in the tumor). [START_REF] Ye | Cerebrospinal fluid liquid biopsy for detecting somatic mosaicism in brain[END_REF] Overall, these results raise several questions. First, we were not able to detect known somatic mutations in CSF cfDNA from a substantial proportion of patients, indicating that the detection of low VAF variants (<0.1%) still needs technical improvement before a clinical use of CSF for genetic testing in epileptic patients. Moreover, in current clinical practice the described approach can only be applied when the mutation is known or suspected (as for the recurrent BRAF p.Val600Glu mutation), using variant-specific approaches. Second, all CSF samples in this study were collected during brain surgery before resection (mostly from the subarachnoid space). Genetic diagnosis based on CSFderived cfDNA would require the proof-of-concept that brain-specific variants can be detected in CSF obtained from lumbar puncture (thus in a pre-surgical assessment) and may require multiplexing ddPCR assays targeting recurrent variants. Third, it remains unknown whether mutated cells found in FCD2/HME (e.g. dysmorphic neurons and balloon cells) and MOGHE samples undergo cell death and therefore release their DNA.

Recent advances have shown that tumor-derived cfDNA in CSF samples can be used as a biomarker for monitoring tumor progression and response to therapy. [START_REF] Miller | Tracking tumour evolution in glioma through liquid biopsies of cerebrospinal fluid[END_REF] Brain surgery is not always effective in treating seizures since only 30-50% of patients are seizure-free after the first surgery and multiple resections are needed in some cases. [START_REF] Lamberink | Seizure outcome and use of antiepileptic drugs after epilepsy surgery according to histopathological diagnosis: a retrospective multicentre cohort study[END_REF] Identification of mutated brain DNA in CSF may be of use in predicting possible seizure relapse and personalized therapy. Thus, this work opens new avenues for improving the diagnostic workflow of patients with various brain malformations and research on the pathological mechanisms and progression of refractory focal epilepsies. FCD2: Focal cortical dysplasia type 2, HME: Hemimegalencephaly, MOGHE: Malformation of cortical development with oligodendroglial hyperplasia in epilepsy, GG: Ganglioglioma, VAF: variant allele frequency, ǂ : p-values referring to significant positive result, NA: Not applicable, ¶ : FR-4 CSF sample was not centrifuged prior freezing, therefore a cellular genomic DNA contamination cannot be excluded, $ : the significant p-value obtained in KR-6 highlights a significant lower VAF in the patient compared to the controls, thus confirming the absence of mutation detection. None of the variants was detected in the matched blood samples. Patients (except FR-4) were previously reported: KR-3 and KR-4 [START_REF] Sim | Brain somatic mutations in SLC35A2 cause intractable epilepsy with aberrant N-glycosylation[END_REF][START_REF] Thomas | Frequent SLC35A2 brain mosaicism in mild malformation of cortical development with oligodendroglial hyperplasia in epilepsy (MOGHE)[END_REF] ; FR-1, FR-2 and FR-3 9 ; FR-5 [START_REF] Baldassari | Dissecting the genetic basis of focal cortical dysplasia: a large cohort study[END_REF][START_REF] Thomas | Frequent SLC35A2 brain mosaicism in mild malformation of cortical development with oligodendroglial hyperplasia in epilepsy (MOGHE)[END_REF] ; KR-1, KR-2, KR-5 and KR-7 10 ; KR-6 [START_REF] Koh | BRAF somatic mutation contributes to intrinsic epileptogenicity in pediatric brain tumors[END_REF][START_REF] Suk | Precise detection of low -level somatic mutation in resected epilepsy brain tissue[END_REF] .

  3 years at surgery) with lesional refractory epilepsy from France at the Fondation Rothschild (n=5) and South Korea at the Severance children's hospital (n=7). Neuropathological diagnoses were established by expert pathologists according to the ILAE classification for cortical malformations and the WHO classification for tumors. The cohort comprised FCD2 (n=1), HME (n=5), MOGHE (n=3), and GG (n=3) cases. Informed consent for the use of biological samples was obtained from all patients. The study protocol received approval by the ethical committees of CPP Ile de France II (N° ID-RCB/EUDRACT-2015-A00671-48) and Severance Hospital and the KAIST Institutional Review Board and Committee on Human Research.

Figure 1 .

 1 Figure 1. Representative ddPCR 2D scatter plot for the targeted detection of mutant alleles. 2D fluorescence amplitude plot shows wild-type-only droplets (HEX+ green circles), mutant-only droplets

Table 1 . Genetic findings in cfDNA and brain tissue samples

 1

	Patient ID	Pathological Diagnosis	Age at CSF collection	Gene HGVSp	Brain VAF (%)	Mean cfDNA VAF (%)	Mean cfDNA VAF from negative controls (%)	p-value
	FR-4 ¶	HME	12.4y	PIK3CA p.Glu545Lys	23	1.358	0.066	< 0.0001 ǂ
	KR-7	GG	17.3y	BRAF p.Val600Glu	4	0.214	0.073	0.0015 ǂ
	KR-4	MOGHE	4.2y	SLC35A2 p.Gln168*	18	0.145	0.053	0.0014 ǂ
	FR-5	MOGHE	7.8y	SLC35A2 p.Ser212Leufs*9	23	0.000	0.000	NA
	KR-2	HME	0.7y	AKT3 p.Glu17Lys	4	0.019	0.018	0.9205
	KR-5	GG	7.1y	BRAF p.Val600Glu	24	0.064	0.073	0.7443
	FR-1	FCD2	11.3y	MTOR p.Ser2215Phe	1	0.031	0.039	0.4398
	KR-3	MOGHE	5.2y	SLC35A2 p.Glu254*	16	0.000	0.003	0.3388
	KR-1	HME	0.1y	MTOR p.Ser2215Phe	23	0.059	0.039	0.3312
	FR-3	HME	3.6y	AKT3 p.Glu17Lys	12	0.033	0.018	0.1398
	FR-2	HME	1.4y	MTOR p.Ala1459Asp	9	0.019	0.008	0.0560
	KR-6	GG	7.1y	BRAF p.Val600Glu	13	0.021	0.073	0.0014 $

ACKNOWLEDGEMENTS

We thank the families that took part in this study, and the neuropediatricians who referred the patients.

We also thank the sequencing ICM core facility (iGenSeq). This work was funded by the European Research Council (N°682345 to SB), the "Investissements d'avenir" ANR-10-IAIHU-06 and ANR-18-RHUS-0005 program, the Suh Kyungbae Foundation (to J.H.L), the National Research Foundation of Korea (NRF) grant funded by the Korean government, Ministry of Science and ICT (No. 2019R1A3B2066619 to J.H.L), and Korean Health Technology R&D Project, Ministry of Health & Welfare, Republic of Korea (H15C3143).

Author Contributions

JHL and SB contributed to the conception and design of the study; SYK, SBal, NSS, MC, GD, DSK, WSC, and VT contributed to the acquisition and analysis of data; JHL, SB, SYK, SBal and NSS contributed to drafting the text and preparing the figures.

Potential Conflicts of Interest

Nothing to report.


Sylvain Prigent

Cesar Augusto Valades-Cruz

Ludovic Leconte

Leo Maury

Jean Salamero
email: jean.salamero@curie.fr

Charles Kervrann
email: charles.kervrann@inria.fr

BioImageIT: Open-source framework for integration of image data-management with analysis

Open science and FAIR principles have become major topics in the field of bioimaging. This is due to both new data acquisition technologies that generate large datasets, and new analysis approaches that automate data mining with high accuracy. Nevertheless, data are rarely shared and rigorously annotated because it requires a lot of manual and tedious management tasks and software packaging. We present BioImageIT, an open-source framework for integrating data management according to FAIR principles with data processing.

Main text

Advances in biological imaging over the past 20 years have been accompanied by a multitude of computerized approaches for indexation, reconstruction, processing, analysis, and exploitation of image data. As they address a variety of biological models at diverse scales of life, they are often dedicated to target case-studies (single cell scale, tissue-scale, embryos…). Consequently, imagining and developing a unified solution able to tackle any scenario in biological imaging very challenging. For more than ten years, several architectures and tools have been proposed to surmount many technological difficulties. The existing software platforms (e.g., Fiji 1 , Icy 2 , Knime 3 ), developed in various programming languages (e.g., java, c++, python) so far, are unfortunately not interoperable. Additional efforts in scripting and codes development are generally required to gather some heterogeneous image processing components in ad-hoc workflows.

Meanwhile, data is hosted in storage servers associated with a database system of their own (e.g., OMERO 4 ), without much possible interaction with processing and analysis tools. A more integrative approach of organizing, viewing, and analyzing data, is more and more required in data science to avoid manual management or script writing which can be tedious for non-expert scientists. Also, the management of often-massive datasets requires dedicated expertise in computer science to scale up the storage and computing resources. Finally, with the emergence artificial intelligence (AI) such as deep learning in bioimaging (e.g. DeepImageJ 5 , ImJoy 6 ), the automation of processes and the implementation of processing pipelines must take all stages of the data life cycle and new human-machine interactions into account. As such, data management and processing/analysis must meet quality criteria ensuring identification, accessibility, and interoperability of the data with their processing, storage, and analysis, and finally, anticipation of future reuse. These "FAIR" principles (findable, accessible, interoperable, and reusable) 7 impose new procedures and ethical issues to scientists whose research relies heavily on biological imaging, leading to a change in paradigm for image-based production of knowledge.

To address all these requirements, we developed BioImageIT, a unique open-source system integrating image data management and analysis, and an operational solution for handling large image data sets to the

requirements of open science (Fig. 1a). In BioImageIT, data is automatically annotated and processed in a single framework, and once processed. Unlike previous solutions, BioImageIT integrates any existing data management and image processing software. For instance, data can be hosted in OMERO 4 database using Bio-Formats 8 and processed by applying machine learning algorithms embedded in the TensorFlow 9 framework. b, Due to the geometry of LLS scanning, 3D raw images are skewed. c, g, First, realignment (deskew) of raw stacks was performed using Pycudadecon (https://github.com/tlambert03/pycudadecon). d, h, Richardson Lucy deconvolution 12 is performed by using Pycudadecon. e, CD-M6PR-eGFP vesicles are tracked by using Trackmate 13 . f, i, Deconvolved stacks and tracks are rendered by using Napari 14 . Full workflow, including Pycudadecon, Trackmate 13 and Napari 14 , was gathered in BioImageIT.

Unlike previous frameworks (e.g., Galaxy Project 15 , Snakemake 16 , BIAFLOWS 17 ), BioImageIT enables to address the following end-user's issues:

BioImageIT reconciles data management and data analysis in a common and interactive framework. Most open-source bio-image software are developed separately and specialized in either data management or data analysis. Therefore, users need to write ad-hoc scripts or to apply manual operations to process the data. BioImageIT allows importing and tagging data for each experiment. Each operation automatically generates metadata that allows to keep track of any processing or analysis and then complies with the recommended FAIR principles 7 in data science.

BioImageIT is interoperable and reusable. The processing tools are packaged in Docker 18 BioImageIT is user-friendly (Supplementary Note 2). BioImageIT consists of 3 layers (backend plugins, python API, Graphical User Interface (GUI)). Users can choose the most appropriate level of interaction with BioImageIT. Biologists may prefer to be assisted step-by-step; GUI is then appropriate for data management and analysis. Data analysts are more familiar with scripts and Jupyter 10 notebooks using the python API. Finally, data scientists may adopt the packing backend system to provide a standalone demonstration of the new processing tool.

In summary, BioImageIT is a generic framework for data management, analysis, and traceability. Unlike previous software platforms, it meets the needs of end-users and is a very flexible solution to link annotated data and processing tools. Furthermore, it facilitates interactions between experimental and data scientists. As BioImageIT is built upon existing technologies, it may be considered as a computer overlay allowing a user-friendly use of existing software by end users, without hindering the addition of new analytical methods by more experienced developers. Finally, the ability of BioImageIT to plug in data interaction tools allows for the deployment of highly specialized instances of BioImageIT for a specific application domain, as illustrated in Figure 1 depicting Lattice Light Sheet Microscopy 11 Data processing (Fig. 1b-i), reconstruction of Multi-Angle TIRF Azimuthal 20 (Fig. 2), as well as advanced image processing using conventional techniques (e.g. ND-SAFIR 21 ) and deep learning (e.g. N2V [START_REF] Krull | IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)[END_REF] ) (Supplementary Fig. 1 et Supplementary Video 4). These examples show the ability of BioImageIT to exploit advanced imaging analysis tools and workflows applied to datasets from various advanced microscopies. BioImageIT is being deployed on several imaging platforms, members of the France BioImaging infrastructure. On this basis, our next step is to create a BioImageIT community. Documentation, source code and video tutorials are available at https://bioimageit.github.io, and help is available via GitHub https://github.com/bioimageit. passage number were used and grown in DMEM-F12 medium without phenol red (Life Technologies, Carlsbad, CA) supplemented with 10% (vol/vol) fetal bovine serum (FBS) at 37°C in humidified atmosphere with 5% CO2. RPE1 cells were transiently transfected with plasmid coding for TfnR-pHluorin 24 , using Lipofectamin 3000, following basically the manufacturer recommendations (Invitrogen, Waltham, MA). Briefly, we used 2 μg of DNAs, completed with 125 μL with OptiMEM (Invitrogen, Waltham, MA) and 4µl of Reagent 3000 and mixed with 3.75 μL Lipofectamin 3000 completed with 125 μL with OptiMEM. The mixed solution was then incubated for 15-20 min at room temperature and finally added to cells grown, in 2ml culture medium on glass bottom petri dish at 80% confluency, for a further overnight incubation at 37 °C. When specified, RPE1 cells stably expressing mCherry-Lifeact 25 were imaged with spinning disk microscope. They were cultured as described above for wild type RPE1 and were plated 5 to 6 hours before acquisition at 200-250.103 per glass bottom 35 mm diameter Petri dishes (Ibidi GmBH, Gräfelfing, Germany).

Lattice light-sheet microscopy. LLSM was done on a commercialized version of a previously described setup 11 from 3i (Denver, USA). Cells were scanned incrementally through a 20 μm long light sheet in 600 nm steps using a fast piezoelectric flexure stage equivalent to ∼325 nm with respect to the detection objective and were imaged using a sCMOS camera (Orca-Flash 4.0; Hamamatsu, Bridgewater, NJ). Excitation was achieved with 488-or 642-nm diode lasers (MPB Communications, Canada) at 10 % acousto-optic tunable filter transmittance with 300 mW and 100 mW respectively (initial box power) through an excitation objective (Special Optics 28.6× 0.7 NA 3.74-mm water-dipping lens) and detected via a Nikon CFI Apo LWD 25× 1.1 NA water-dipping objective with a 2.5× tube lens with a final pixel size of 104 nm. Lattice light-sheet imaging was performed using an excitation pattern of outer NA equal to 0.55 and inner NA equal to 0.493. Composite volumetric datasets were obtained using ~5-ms exposure/slice/channel at a time resolution of 0.833 s per total cell volume (about 40 slices). One hundred time points were acquired within 1 to 2 minutes. Acquired data were deskewed, a necessary step to realigned image frames, using pycudadecon, a python wrapper for cudaDecon, which is a CUDA/C++ implementation of an accelerated Richardson Lucy Deconvolution algorithm 12 (copyright to T. Lambert, Harvard Medical School, Boston, USA; https://github.com/tlambert03/pycudadecon). Then deskewed images were deconvolved also using pycudadecon. Later, tracking was performed using Trackmate 13 . Napari 14 , a multi-dimensional image viewer for Python (https://github.com/napari/napari), was used for 3D rendering. Full workflow, including Pycudadecon, Trackmate 13 and Napari 14 , was implemented in BioImageIT. Rab5 stably expressing Hela cell was scanned incrementally through a 20 μm long light sheet in 500 nm steps using a fast piezoelectric flexure stage equivalent to ∼270 nm with respect to the detection objective and were imaged using a sCMOS camera (Orca-Flash 4.0; Hamamatsu, Bridgewater, NJ). Excitation was achieved with a 488-diode laser (Coherent, Santa Clara, CA) at 15 % acousto-optic tunable filter transmittance with 300 mW. Volumetric datasets were obtained using ~10-ms exposure/slice at a time resolution of 2s per total cell volume (about 50 slices). One single volume was used for the analysis. Deskew was performed using pycudadecon prior denoising. LLSM Deskewed 3D noisy data could also be directly treated using Noise to Void (N2V) [START_REF] Krull | IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)[END_REF] as a training-free denoising methods embedded in BioImageIT and as such could be applied as another preprocessing step in a workflow when other methods cannot.

Multiangle Azimuthal Averaging TIRFM. Acquisition and high-resolution three-dimensional reconstruction of Multiangle Stacks were carried out as previously published and using the same setup 20 , except that the custom C++ code based on the CImg library (https://github.com/dtschump/CImg ) used for reconstruction, was integrated in BioImageIT. Our current setup is based on a Nikon Ti Eclipse equipped with the Ring-TIRF ILas2 module (GATACA systems, Massy, FR), with a PLAN APO TIRF 100X oil objective (1.49 NA). The acquisition of multiangle TIRF image stacks is performed from the iLas2 software interacting with the Metamorph 8.6 software (Molecular Devices, San Jose, CA). This system is interfaced with a hardware and software implementations allowing the full control of the illumination incident angle as a function of penetration depth for multiple wavelengths. TfR-Phluorine was imaged with excitation at a wavelength of 488nm. Nominal laser powers were adjusted at 10% for the 488nm laser (100 mW). The time-gated fluorescence detection was done an Evolve 512 (Teledyne Photometrics, Tucson, AZ) EMCCD camera. Readout time was down to 30 ms per frame.

Fast confocal microscopy.

It was performed with a spinning disk confocal based on a Ti-Eclipse inverted microscope (Nikon, Tokyo, JP) equipped with X1 confocal head (Yokogawa, Tokyo, JP), Prime95B, an sCMOS camera (Teledyne Photometrics, Tucson, AZ) and interfaced with a super resolution Live-SR module (GATACA systems, Massy, FR) based on optically demodulated structured illumination technique. Cells were scanned incrementally through ~10 μm Z-stacks in 300 nm step at 30ms/frame, using a 100X, Plan Apo NA 1.4 oil objective (Nikon, Tokyo, JP). Excitation was achieved at 561-nm (50 mW), diode laser (GATACA systems, Massy, FR), at low illumination of the sample adjusting the acousto-optic tunable filter transmittance at 20 % in the Live-SR mode, allowing relatively long-term imaging without photobleaching and phototoxicity at high resolution. The acquisition and reconstruction of the Live-SR module is performed with the Metamorph 8.6 software (Molecular Devices, San Jose, CA). Reconstruction of resulting low signal to noise ratio (SNR) super resolution images was performed directly using the setup-software before denoising using the ND-SAFIR 4D software 21 implemented in BioImageIT.

BioImageIT: An open-source framework for integration of image data management with analysis

Figure 1 .

 1 Figure 1. BioImageIT overview. a, Schematic view of BioImageIT architecture. BioImageIT core is composed of data management and data processing functionalities. Users can access plugins by a script editor, Jupyter 10 or BioImageIT graphical interface. Data management functionalities exploit local files, remote files or OMERO database. Data processing can perform computations in remote job, container, or local runner. Image analysis is provided by plugins, potentially written in different programming languages. Developers can implement their own plugins in BioImageIT and design their Graphical Interface. b-i, Example of Lattice Light Sheet 11 workflow of 3D reconstruction and tracking of live Genome-edited Hela cell (CD-M6PR-eGFP) stained with Tubulin TrackerTM Deep Red for Microtubules (see Methods).b, Due to the geometry of LLS scanning, 3D raw images are skewed. c, g, First, realignment (deskew) of raw stacks was performed using Pycudadecon (https://github.com/tlambert03/pycudadecon). d, h, Richardson Lucy deconvolution12 is performed by using Pycudadecon. e, CD-M6PR-eGFP vesicles are tracked by using Trackmate13 . f, i, Deconvolved stacks and tracks are rendered by using Napari14 . Full workflow, including Pycudadecon, Trackmate13 and Napari14 , was gathered in BioImageIT.

Figure 2 .

 2 Figure 2. BioImageIT a method for advanced reconstruction TIRF Azimuthal Multi-Angle 20 . Live cells RPE1 cells were transfected for Transferrin Receptor-Phluorine as indicated in Methods before acquisition with Multi-Angle Ring TIRFM. a, Maximum intensity projection of an image stack constituted of ten angles were acquired at 30ms exposure time per angle. Scale Bar 5µm. b, Thumbnails shows selected times of three appearing exocytosis spots in the areas selected and represented in squares of 3 different colors green, red, and yellow in (a). c, The graphs represent the lifetime of the three same spots by measuring intensity. Note the behavioral differences of the red curve compared to the green and yellow ones. d, MATIRF image reconstruction 20 was implemented in BioImageIT. MATIRF reconstruction parameters: depth: 300 nm, number of planes: 20, regularization: 0.001, numbers of iterations: 100. e, Representation of the first spot (Green square in (a)) reconstructed in 3D with three time points, pixels size in x, y, z (0.16 µm, 0.16 µm, 0.05 µm), using Napari for visualization. Scale bar: 5µm.

  Figure, Tables, Notes Sylvain Prigent 1,2,& , Cesar Augusto Valades-Cruz 1,2,& , Ludovic Leconte 1,2,& , Leo Maury 1,2 , Jean Salamero 1,2,* , Charles Kervrann 1,2,* 1 SERPICO Project Team, Inria Centre Rennes-Bretagne Atlantique, F-35042, Rennes, France 2 SERPICO Project Team, UMR144 CNRS Institut Curie, PSL Research University, F-75005, Paris, France & These authors contributed equally to the work * Correspondence should be addressed to: jean.salamero@curie.fr, charles.kervrann@inria.fr

Acknowledgements

This work was supported by the France-BioImaging Infrastructure (French National Research Agency -ANR-10-INBS-04-07, "Investments for the future"). We acknowledge the Cell and Tissue Imaging (PICT IBiSA, Institut Curie) platform, also member of the national infrastructure France-BioImaging (ANR-10-INBS-04-01) for maintaining the spinning-disk confocal microscope.

Data availability

Original datasets are available for download at https://github.com/bioimageit/manuscript .

Code availability

All source code used in this publication is open-source and published under the BSD 4-Clause. The latest stable releases used in this publication and the current versions that include bugfixes and updates can be downloaded from GitHub (https://github.com/bioimageit). Details on how to use the software are described in website of the project https://bioimageit.github.io/.

Author contributions

S.P. conceived the idea in discussions with J.S., C.K., C.A.V-C. and L.L.; S.P. developed the algorithms and implemented the software; C.A.V-C. and L.L. performed all experiments and reconstructions; J.S., C.A.V-C. and L.L. designed and implemented the analysis workflows; L.M. implemented algorithm wrappers and the software packaging; J.S. and C.K. supported and supervised the project; and S.P., J.S., C.K. and C.A.V-C. wrote the manuscript with input from the co-authors.

Competing interests

The authors declare no competing interests.

Additional information

Supplementary information is available for this paper at

Reprints and permissions information is available at

Correspondence and requests for materials should be addressed to J.S. or C.K.

Methods

Cell culture and fluorescence labelling. Hela cells stably expressing a CD-Mannose-6 Phosphate Receptor eGFP tagged construct (eGFP-M6PR small Ca++ dependent receptor) kindly provided by Dr B. Hoflack, (TU-Dresden, Germany) and Hela cells stably expressing eGFP-Rab5 were grown in DMEM-Glutamax medium supplemented with 10% FCS, as previously described [START_REF] Nielsen | [END_REF] . For LLSM (Lattice Light Sheet Microscopy 11 ) imaging, they were seeded 6 to 7 hours before image acquisition at 150.103 per well of a 6 well plate, containing 3 to 4 coverslips with a 5mm diameter, treated as previously described 11 . Coverslips were directly transferred to the lattice light-sheet microscope (LLSM) sample holder and inserted into the imaging chamber thermoregulated at 37°C containing 6 ml of LLSM medium (DMEM, without phenol red, with 1% BSA, 0.01% penicillin-streptomycin, 1mM pyruvate, and 20mM HEPES) containing the Tubulin TrackerTM DR (Tub- This file includes the developer manual. This developer manual was generated using Sphinx 4 . Code source can be found in https://github.com/bioimageit/bioimageit_core/tree/master/docs Supplementary Note 2:

SUPPLEMENTARY FIGURE

Description of Additional Supplementary Files

This file includes the user manual. This user manual was generated using Sphinx 4 . Code source can be found in https://github.com/bioimageit/bioimageit_gui/tree/master/docs