Teddy Furon

The illusion of group testing

Keywords: Group testing, hypothesis testing, identification, information theory Test par groupe, test d'hypothèse, identification, théorie de l'information

This report challenges the assumptions usually made in non-adaptive group testing. The test is usually modelled as a probabilistic mechanism prone to false positive and / or false negative errors. However, the models are still too optimistic because the performances of these non ideal tests are assumed to be independent of the size of the groups. Without this condition, the report shows that the promises of group test (a number of tests and a decoding complexity scaling as c log N ) do not hold.

Introduction

Group testing has recently received a surge of research works mainly due to its connection to binary compressed sensing [START_REF] Lam | Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms[END_REF][START_REF] Atia | Boolean compressed sensing and noisy group testing[END_REF][START_REF] Scarlett | Converse bounds for noisy group testing with arbitrary measurement matrices[END_REF][START_REF] Scarlett | Phase transitions in group testing[END_REF]or to traitor tracing [START_REF] Meerwald | Group testing meets traitor tracing[END_REF][START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF]. The usual setup is often described in terms of clinical screening as it was the first application of group testing [START_REF] Dorfman | The detection of defective members of large populations[END_REF]. Among a population of N individuals, there are c infected people, with c much smaller than N . Screening the whole population by individual blood test is too costly. However, it is possible to mix blood samples from several persons and to perform a test. Ideally, the test is negative if none of these persons are infected, and positive if at least one of them is infected. The application of group testing are nowadays DNA screening [START_REF] Ngo | A survey on combinatorial group testing algorithms with applications to DNA library screening[END_REF], signal processing [START_REF] Gilbert | Recovering simple signals[END_REF], machine learning [START_REF] Zhou | Parallel feature selection inspired by group testing[END_REF]. Indeed, group testing may be a solution to any 'needles in haystack' problem, i.e. aiming at identifying among a large collection the few 'items' sharing a peculiar property detectable by a test, provided that this test can be performed on groups of several items. In this paper, we use the terminology of items and defective items.

The dominant strategy nowadays is called non-adaptive group testing [START_REF] Atia | Boolean compressed sensing and noisy group testing[END_REF][START_REF] Lam | Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms[END_REF]. A first stage pools items into groups and performs the tests. A second stage, so-called decoding, analyses the result of these tests to identify the defective items. Tests and decoding are sequential. If the number of tests M is sufficiently big, the decoding stage has enough information to identify the defective items. In a nutshell, the groups are overlapping in the sense that one item is involved in several tests. Decoding amounts at finding the smallest subset of items which would trigger the observed positive tests.

The promises of group testing are extremely appealing. First, the theoretical number of tests M asymptotically scales as O(c log N ) as N goes to infinity [START_REF] Atia | Boolean compressed sensing and noisy group testing[END_REF][START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF][START_REF] Scarlett | Converse bounds for noisy group testing with arbitrary measurement matrices[END_REF]. This result holds even if c increases with N , but at a lower rate [START_REF] Scarlett | Converse bounds for noisy group testing with arbitrary measurement matrices[END_REF][START_REF] Scarlett | Phase transitions in group testing[END_REF]. Second, recent papers propose practical schemes non only achieving this efficiency (or almost, i.e. O(c log c log N )) but also within a decoding complexity of O(c log N ) (or almost, i.e. O(c log c log N )) [START_REF] Cai | Grotesque: Noisy group testing (quick and efficient)[END_REF][START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF].

This paper takes a complete opposite point of view. The number c of defective items is fixed, and we don't propose more efficient design. On contrary, we show that these promises hold only for some specific probabilistic models. These models are well known in the literature of group testing. They do take into account some imperfection in the test process, however, they are somehow optimistic. As group testing becomes popular, people applying this technique to their 'needles in haystack' problems might be disappointed. The promises of group testing (a number of tests in O(c log N ) together with a computational complexity of O(c log N )) fade away for applications not compliant with these models.

The goal of this paper is to investigate what is specific in these models and to better understand the conditions necessary for achieving the promises of group testing. This paper has the following structure. Section 2 describes the recent approaches achieving both the minimum asymptotic number of tests and the minimal decoding complexity. The usual models are introduced together with an information theoretic justification that these approaches are sound. Section 3 introduces some more general models and shows that the total number of tests no longer scales as O(c log N ) in most cases.

Previous works

A typical paper about non-adaptive group testing proposes a scheme, which is composed of a pooling design and a decoding algorithm. The pooling is the way M groups are composed from a collection of N items. The decoding receives the M binary test results (positive or negative) to infer the defective items. Under a definition of a successful decoding and some probabilistic models, the paper then shows how the necessary number of tests asymptotically scales. For instance, if the decoding aims at identifying all the defective items, the authors show how M should scale as N → ∞ to make the probability of success converge to one. The best asymptotical scaling has been proven to be in O(c log N ) in theoretical analysis [START_REF] Atia | Boolean compressed sensing and noisy group testing[END_REF][START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF].

Notations and models

The assumptions of the proof of a typical group testing paper concern the distribution of the defective items in the collection and the model of the test. Denote x a binary vector of dimension N encoding which items are defective: x i = 1 if the i-th item is defective, 0 otherwise. X is the random variable associated to this indicator vector. We assume that there are a fixed number c of defective s.t. P[X = x] = 1/ N c if |x| = c, 0 otherwise. As for the test, the models define the probabilistic behavior of its output. Suppose a group G i of n items, and let 0 ≤ K i ≤ max(n, c) be the random variable encoding the number of defectives in this group. Denote first by Z i a binary r.v. s.t.

Z i = 1 if K i > 0, 0 otherwise. Now denote by Y i ∈ {0, 1} the r.v.
Y i = Z i ⊕ N i with N i independent
and identically distributed as Bernouilli B( ) and ⊕ the XOR operator.

Dilution:

Y i = ∨ j∈Gi [X j ∧ W i,j ],
where ∧ and ∨ are the AND and OR operators and W i,j a binary r.v. modeling the detectability of the j-th item in the i-th group. These random variables are independent (both along i and j) and identically distributed:

W i,j ∼ B(1-υ).
For a given defective and test, the probability of being diluted (i.e. not detectable) is υ.

Threshold:

Y i = 0 if K i ≤ κ L and Y i = 1 if K i ≥ κ U .
There are plenty variants describing what happens for κ L < K i < κ U [START_REF] Cheraghchi | Improved constructions for non-adaptive threshold group testing[END_REF].

Note that some models can be 'concatenated': we can witness a dilution phenomenon of parameter υ followed by a noise channel of parameter . Another way to model a test is through the c + 1 parameters (θ 0 , • • • , θ c ) defined as the following probabilities:

θ k := P[Y i = 1|K i = k]. (1)
Parameter θ 0 is thus the probability of a false positive, whereas 1 -θ k for 0 < k ≤ c are the probabilities of false negative when k defectives pertain to the test group. For the models above mentioned, we have the equivalent formulation:

1. Noiseless test: θ 0 = 0 and θ k = 1 for 0 < k ≤ c.

2. Noisy test: θ 0 = and θ k = 1 -for 0 < k ≤ c.

Dilution:

θ k = 1 -υ k with the convention that x 0 = 1, ∀x ∈ R + . 4. Threshold: θ k = 0 if 0 ≤ k ≤ κ L , θ k = 1 if κ U ≤ k ≤ c.

Inria

Probabilistic group testing

The next step is to create the binary design matrix A ∈ {0, 1} M ×N . This matrix indicates which items belong to which groups: A i,j = 1 if item j is involved in test i, and 0 if not. There are constructions which are deterministic (up to a permutation over the N items) such as those relying on disjunct matrices [START_REF] Indyk | Efficiently Decodable Non-adaptive Group Testing[END_REF][START_REF] Ngo | A survey on combinatorial group testing algorithms with applications to DNA library screening[END_REF]. Another popular method is the probabilistic construction where A i,j is set to one depending on a coin flip: P[A i,j = 1] = p. These coin flips are independent w.r.t. indices i (groups) and j (items). The sequence (A 1,j , • • • , A M,j ) is often called the codeword of item j. We shall focus on this last construction.

Theoretical studies [START_REF] Scarlett | Phase transitions in group testing[END_REF][START_REF] Scarlett | Converse bounds for noisy group testing with arbitrary measurement matrices[END_REF][START_REF] Atia | Boolean compressed sensing and noisy group testing[END_REF] shows that there is a phase transition: it is possible to identify all defectives with an asymptotically vanishing probability of error (as

N → ∞) if M ≥ max ∈{1,••• ,c} log 2 N I(Y ; A G dif |A Geq ) (1 + η); (2)
whereas the error probability converges to one for any decoding scheme if

M ≤ max ∈{1,••• ,c} log 2 N I(Y ; A G dif |A Geq ) (1 -η). (3)
The sets of items (G dif , G eq ) compose a partition of the set of defective items such that |G dif | = and |G eq | = c -, and A G dif (resp. A Geq ) denote the codewords of the items in G dif (resp. A Geq ). These theoretical results are extremely powerful since they still hold when c is not fixed but slowly increasing with N . They are somehow weakly related to practical decoding schemes. For instance, equation (2) comes from a genie aided setup: a genie reveals to the decoder some defective items G eq , and the number of tests needed to identify the remaining ones, i.e. G dif , is evaluated. This is done for different sizes of G eq , from 0 to c -1. A decoder without any genie needs more than the supremum of all these quantities.

In the sequel, we consider simpler expressions of the total number of tests but related to practical (or almost) decoders.

Joint decoder

The joint decoder computes a score per tuple of c items. It spots the tuple of defective items (identifying all of them) with a probability at least 1 -α J ; and it incorrectly points a tuple of non defective items with probability β J . Denote γ J := log(α J )/ log(β J /N c ). T. Laarhoven [START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF] showed that a sufficient and necessary number of tests is at least:

M J = c log 2 N max p∈(0,1) I J (p) (1 + O( √ γ J )), (4)
where

I J (p) = I(Y i , (A i,j1 , • • • , A i,jc )|p)
is the mutual information between the output of the test and the codeword symbols of the defectives {j 1 , • • • , j c }. In other words, this corresponds to the case where the genie reveals no information: G eq = ∅ [START_REF] Scarlett | Phase transitions in group testing[END_REF]. Since lim N →∞ γ J = 0 for fixed (α J , β J ), this allows to state that M J scales as M J ≈ c log 2 N/I J (p J ) with p J = arg max p∈(0,1) I J (p). For the equivalent model (θ 0 , • • • , θ c ), this amounts to find the maximizer of the following function:

I J (p) := h (P (p)) - c k=0 π k h(θ k ), (5)
with

P (p) := P[Y i = 1|p] = c k=0 π k θ k , (6)
π k := c k p k (1 -p) c-k , ∀0 ≤ k ≤ c, (7)
and h(x) is the entropy in bits of a binary r.v. distribution as B(1, x). Laarhoven gives the expressions of p J for large c and for the usual models [START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF]. The maximizer and the maximum are functions of c and of the parameters of the test model (for example, or υ for the noisy or dilution model).

The drawback is that the decoding is exhaustive: it scans the N c possible subsets of size c from a set of N items. Therefore its complexity is in O(N c ). This is called a joint decoder as it jointly considers a subset of c items. The joint decoder is mainly of theoretical interest since its complexity is hardly tractable. Some schemes propose approximations of a joint decoder with manageable complexity resorting to Markov Chain Monte Carlo [START_REF] Knill | Interpretation of pooling experiments using the Markov chain Monte Carlo method[END_REF], Belief Propagation [START_REF] Sejdinovic | Note on noisy group testing: asymptotic bounds and belief propagation reconstruction[END_REF] or iterative joint decoders [START_REF] Meerwald | Group testing meets traitor tracing[END_REF].

Single decoder

The single decoder analyses the likelihood that a single item is defective. It correctly identifies a defective item with probability 1 -α S while incorrectly suspecting a non defective item with probability less than β S . Denote γ S = log(β S )/ log(α S /N ). Laarhoven [START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF] showed that a sufficient and necessary number of tests is at least:

M S = log 2 N max p∈(0,1) I S (p) (1 + O(γ S )) (8)
where

I S (p) = I(Y i , A i,j1
|p) is the mutual information between the output of the test and the symbol of the codeword of one defective, say j 1 . Again, since lim N →∞ γ S = 0 for fixed (α S , β S ), this allows to state that M S scales as M S ≈ log 2 N/I S (p S ) with p S = arg max p∈(0,1) I S (p). For the equivalent model (θ 0 , • • • , θ c ), this amounts to find the maximizer of the following function:

I S (p) := h (P (p)) -ph(P 1 (p)) -(1 -p)h(P 0 (p)) (9)
with

P 1 (p) := P[Y i = 1|A i,j1 = 1, p] = c k=1 c -1 k -1 p k-1 (1 -p) c-k θ k (10)
P 0 (p) := P[Y i = 1|A i,j1 = 0, p] = c-1 k=0 c -1 k p k (1 -p) c-1-k θ k (11)
Laarhoven gives the expressions of p S for large c and for the usual models [START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF]. It always holds that I J (p) ≥ cI S (p), for any p ∈ [0, 1]. This yields M S inherently bigger than M J [START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF]: Both total numbers of tests scale as c log N , but with a bigger multiplicative constant for M S . The simple decoder computes a score for each item. Therefore its complexity is linear in O(N ).

Divide and Conquer

Papers [START_REF] Cai | Grotesque: Noisy group testing (quick and efficient)[END_REF][START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF] have recently proposed schemes meeting the promises of group testing as listed in the introduction: optimal scaling both in the total number of tests and decoding complexity.

Inria

Both of them are deploying a 'Divide and Conquer' approach. Identifying c defectives among a collection of N items is too complex. Their strategy splits this problem into S simpler problems. The collection is randomly split into S subsets. S is chosen such that any subset likely contains at most one defective. Indeed, their proof selects S big enough s.t., with high probability, each defective belongs at least to one subset where it is the only defective. Assume that it is possible to detect whether a subset has no, one or more defectives. Then, a group testing approach is applied on each subset containing a single defective (so called 'singleton' subset in [START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF]): The decoding identifies this defective thanks to the result of tests performed on groups composed of items of that subset. It turns out that identifying defectives in a collection is much simpler when knowing there is only one. In a non-adaptive framework, all group tests are performed in the first stage, but the decoding is only run on subsets deemed as 'singleton'.

We detail here our own view of this 'Divide and Conquer' approach. Papers [START_REF] Cai | Grotesque: Noisy group testing (quick and efficient)[END_REF][START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF] slightly differ in the way subsets are created. More formally, each subset S k ,1 ≤ k ≤ S, is composed independently by randomly picking N S items in the collection of N items. Denote by π the probability that an item belongs to a given subset: π = N S /N. Subset S k is not useful for identifying a given defective if:

• it doesn't belong to subset S k with probability 1 -π,

• else, if it is not the only defective in this subset with probability 1 -H(0; N, c, N S ), where H(k; N, c, N S ) is the hypergeometric distribution,

• else, if the decoding over this subset misses its identification with probability denoted by α.

Over all, this event happens with probability

g(N S ) := (1 -π) + π ((1 -H(0; N, c, N S ) + H(0; N, c, N S )α) (12)
which is minimized by selecting N S = N +1 /c+1 because g(N S ) -g(N S -1) ≤ 0 iff N S ≤ N +1 /c+1 (we assume that c + 1 divides N + 1). The probability that S k is useless for identifying a given defective simplifies in:

g(N S ) = 1 -N S (N -c -1)! N ! (N -N S )! (N -N S -c)! (1 -α), (13)
= 1 - c c + 1 c . 1 -α c + 1 .(1 + O(1/N )), (14)
where we use the fact that

Γ(N +a) /Γ(N+b) = N a-b (1 + (a + b -1) (a-b) /2N + O( 1 /N 2 )) [16].
Suppose that the goal of the decoding is to identify on expectation a fraction (1 -α S ) of the defectives. The probability of missing a given defective because none of the subset is useful for identifying it equals α S :

P[Not identifying a given defective] = g(N S ) S = α S . (15)
Since ( c /c+1) c ≥ 1 /e, it is safe to choose S = (c + 1)e(-log α S )/(1 -α) . Suppose now that the goal is to identify all the defectives with probability 1 -α J . The probability of identifying them all is given by:

P[identifying all of them] = 1 -g(N S ) S c = 1 -α J . ( 16
)
This can be achieved with S ≥ e(c + 1) log(c/α J )/(1 -α) .

RR n°9164

The point of this 'Divide and conquer' approach is that m = Θ(log 2 N S ) tests are needed for identifying a unique defective item in a subset of size N S and with a fixed probability of error α (see Sec. 2.4). Since the sizes of the subsets are all equal to N /c, the total number of tests scales as M DC = O(c log c log N /c) to identify all defectives with high probability, which is almost the optimal scaling. In [START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF], the authors show that the decoding can also exploit subsets containing two defectives (so-called 'doubleton') which reduces S = O(c) for identifying all the defectives.

To discover a fraction of the defectives the total number of tests scales as M DC = O(c log 2 N /c). This ends up in the optimal scaling achieved by GROTESQUE [START_REF] Cai | Grotesque: Noisy group testing (quick and efficient)[END_REF] and the 'singleton' only version of SAFFRON [START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF].

These schemes have also the following advantages:

• The decoding complexity scales like O(cm) = O(c log 2 N /c) if a deterministic construction is used as in [START_REF] Cai | Grotesque: Noisy group testing (quick and efficient)[END_REF] and [START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF]. We decode O(c) 'singleton' subsets in total. In the noiseless setup, decoding a 'singleton' amounts to read the outputs of the tests because it exactly corresponds to the codeword of the unique defective of that subset. If the setup is not noiseless, the outputs are a noisy version of this codeword. An error correcting code whose decoding is in O(m) gets rid of these wrong outputs. For instance, the authors of [START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF] uses a spatially-coupled LDPC error correcting code.

• The decoding complexity scales like O(cmN S ) = O(N log 2 N /c) if a probabilistic construction is used per subset. We decode O(c) 'singleton' subsets. Decoding a singleton amounts to compute the likelihood scores for N S items and identifying the defective as the items with the biggest score. The likelihood is a weighted sum of the m test outputs. The next section shows that finding the optimal parameter p of the probabilistic construction is also simple.

The main drawback of the 'Divide and Conquer' strategy is that it doesn't apply when θ 1 = θ 0 . This typically corresponds to the 'threshold' model where one unique defective is not enough to trigger the output of a test. Likewise, if θ 1 ≈ θ 0 , any efficient error correcting decoder will fail and the only option is the exhaustive maximum likelihood decoder. At that point, a probabilistic construction is preferable.

Identifying the unique defective in a 'singleton' subset

This 'Divide and conquer' approach greatly simplifies the model [START_REF] Atia | Boolean compressed sensing and noisy group testing[END_REF]. Since there is a single defective, we only need parameters θ 0 and θ 1 . By the same token, there is no need of joint decoding since the defective is unique. The mutual information in (8) takes a simple expression:

I DC (p) = H(Y i |p) -H(Y i |A i,j , p) = h(θ 0 + p(θ 1 -θ 0 )) -(1 -p)h(θ 0 ) -ph(θ 1 ), (17)
which is strictly positive on (0, 1) if θ 1 = θ 0 and whose maximisation is simpler than for the single and joint decoders. This concave function has a null derivative for:

p DC = 1 θ 1 -θ 0 • 1 2 h(θ 1 )-h(θ 0 ) θ 1 -θ 0 + 1 -θ 0 . ( 18
)
This gives the following application to the usual models:

1. Noiseless test: θ 0 = 1 -θ 1 = 0 so that p DC = 1 /2 and I DC (p ) = 1.

2. Noisy test: θ 0 = 1 -θ 1 = so that p DC = 1 /2 and I DC (p ) = 1 -h( ).

Inria 3. Dilution: θ 0 = 0 and θ 1 = 1 -υ so that

p DC (υ) = 1 1 -υ • 1 2 h(υ)/(1-υ) + 1 , (19)
I DC (p DC (υ)) = h ((1 -υ)p DC (υ)) -p DC (υ)h(υ). (20)
Denote f (υ) = 1 /p DC (υ). We have:

f (υ) = -1 -2 h(υ) 1-υ 1 + ln υ 1 -υ for υ ∈ [0, 1]. ( 21
) Since ln(υ) ≤ υ -1 -(1 -υ) 2 /2
, we have on one hand 2 h(υ)

1-υ ≥ 2 υ /(1 -υ)
and on the other hand

(1 + ln υ 1-υ )/(1 -υ) ≤ -1 /2. This shows that f (υ) ≥ 0. Function f is increasing, therefore p DC is a decreasing function of the dilution factor υ. As υ → 1, h(υ)/(1 -υ) = 1/ ln(2) -log 2 (1 -υ) + O(1 -υ), s.t. p DC → 1 /e ≈ 0.37.
The number of tests for a subset is given by ( 8) which multiplied by the number of subsets gives

M DC ≈ ce(-log α S ) I DC (p ) log 2 ( N /c) (22)
for identifying a fraction 1 -α S of defectives on expectation.

Less optimistic models

We would like to warn the reader that the promises of group testing are due to the simplicity of the models described in Sec. 2.1. These models are not naive since they do encompass the imperfection of the test over a group. The output of the test is modeled as a random variable. Yet the statistics of the test only depend on the number of defective items inside the group, but not on the size of the group. Consider the noisy setup with parameter modelling the imperfection of the test. The optimal setting is p DC = 1 /2 for the 'Divide and Conquer' scheme. This means that if N = 200, then the size of the groups is around 100, if N = 2 • 10 9 , the groups are composed of a billion of items and, still, the reliability of the test is not degraded. There are some chemical applications where tests can detect the presence of one single particular molecule among billions. But this is certainly not the case of all 'needles in haystack' problems.

Our proposed model

We believe there are many applications where the reliability of the test degrades as the size n of the group increases. Indeed, when the size of the group grows to infinity, the test might become purely random. For the noisy setup, should be denoted as n s.t. lim n→+∞ n = 1 /2 if the test gets asymptotically random. For the dilution model, υ should be denoted as υ n s.t. lim n→+∞ υ n = 1. This captures the fact that the defectives get completely diluted in groups whose size grows to infinity.

Instead of coping with the noisy or dilution setups, we prefer to consider the equivalent model where probabilities (θ 0,n , • • • , θ c,n ) now depend on the size of the group. We make the following assumptions:

• For all n, θ 0,n ≤ • • • ≤ θ c,n . Having more defective items in a group increases the probability that the test is positive.

• θ 0,n is a non decreasing function of n. Parameter θ 0,n is the probability of a false positive (the test is positive whereas there is no defective in the group). Increasing the size of the group will not help decreasing the probability of this kind of error.

• For 0 < k ≤ c, θ k,n is a non increasing function. Again, 1 -θ k,n is the probability of a false negative (the test is negative whereas there k defective items in the group). Increasing the size of the group will not help decreasing the probability of this kind of error.

• These probabilities are bounded monotonic functions, therefore they admit a limit as n → +∞, denoted as θk := lim n→+∞ θ k,n .

A test is deemed as asymptotically random if θ0 = • • • = θc , whatever the value of this common limit.

Application to group testing designs

We consider the three schemes above-mentioned: single, joint and 'Divide and conquer'. As described in Sec. 2.5, the 'Divide and conquer' design builds S subsets by randomly picking N S items out of N . The optimal size N S of a subset grows linearly with N . The three schemes then compose random groups from a population whose size N, be it N = N (single and joint schemes) or N = N S ('Divide and conquer' design), grows to infinity. We denote the size of the test groups by n(N) to investigate different choices as N goes to infinity. Note that n(N) ≤ N. When we pick up at random n(N) items the probability p that this group contains a given defective is p = n(N)/N.

We analyze two choices concerning the asymptotical size of the test groups: either lim N→∞ n(N) = +∞ or lim N→∞ n(N) < +∞. We derive the mutual information at stake for the three schemes ('Divide and Conquer', simple, and joint) and we deduce the asymptotical scaling of the total number of tests.

We introduce the non increasing functions δ k,n := θ k,n -θ 0,n ≥ 0. For the 'Divide and conquer' scheme, the decoding is performed on singleton subset. Therefore δ 1,n shows the speed at which the test gets closer to a random test. For the simple and joint decoder, there are up to c defective items in a group (we suppose that c < n) and δ c,n shows the speed at which the test gets closer to a random test.

The most important factor is the limit δk := lim n→∞ δ k,n . For the 'Divide and conquer' scheme, δ1 > 0 means that the two probabilities θ 0,n and θ 1,n have distinct limits, and Sect. 2.6 gives the optimum choice replacing θ 0 and θ 1 by their limits θ0 and θ1 . When there is a single defective in a subset, Eq. [START_REF] Knill | Interpretation of pooling experiments using the Markov chain Monte Carlo method[END_REF] shows that the number of tests to identify it is Θ(log 2 N S ), which in turn is Θ(log 2 N /c). Because the number of subsets S used by the 'Divide and Conquer' strategy in Sect. 2.5 asymptotically gets independent of N , the total number of tests scales as Θ(c log 2 N /c) (identification of a fraction of defective items) or Θ(c log c log 2 N ) (identification of all the items).

For the single and joint decoders, δc > 0 means that the test is not asymptotically random. Packing more and more items in groups always provides informative tests. The strategy of selecting n(N) s.t. lim N→∞ n(N)/N = p will deliver a non null mutual information. Parameter p is derived from Sec. 2 replacing (θ 0 , • • • , θ c ) by their limits ( θ0 , • • • , θc ). There might be other way giving a lower total number of tests, but at least this strategy delivers the promises of group testing with a total number of tests scaling as Θ(c log 2 N ).

The next sections investigate our main concern : the case where the test is asymptotically random.

Inria 4 First strategy: lim N→∞ n(N) = n

The first strategy makes the size of the test groups converging to the finite value n := lim N→∞ n(N) for which the test is not random: suppose θ 0,n < θ 1,n . On the other hand, the probability of an item being in a given test group vanishes as p = n/N.

4.1

The case where θ 0,n = 0

Assume first that θ 0,n = 0, Taylor series give the following asymptotics (See Appendices A.1, B.1, and C.1):

I J (p) ≈ c n N ∆h 0,1 , (23)
I S (p) ≈ n N ∆h 0,1 , (24)
I DC (p) ≈ n N S ∆h 0,1 , (25)
with ∆h 0,1 := [(θ 1,n -θ 0,n )h (θ 0,n ) + h(θ 0,n ) -h(θ 1,n )].
These three mutual informations only depend on the first two parameters of the model which is unusual for the joint and the single schemes. As the probability p vanishes, the tests are positive for a unique reason: there is a single defective in the groups. More formally, thanks to L'Hôspital's rule, the probability that there is a single defective knowing that there at least one converges to zero:

lim p→0 π 1 1 -π 0 = lim p→0 1 -(c -1) p 1 -p = 1 (26)
It is therefore quite normal that I DC (p) and I S (p) coincide (except that N S is replaced by N ). However, the 'Divide and Conquer' scheme runs a group testing procedure per subset s.t. it asymptotically needs e(-log α S ) more tests than the single decoder (comparison of ( 22) with ( 8)). I J (p) is exactly c times bigger than I S (p), which in the end offers the same scaling for the total number of tests: M J ≈ M S (comparison of ( 4) with ( 8)). This signifies that the joint decoding doesn't perform better than the single decoding. Indeed, the score computed for a tuple of c items by the joint decoder becomes asymptotically equal to the sum of the scores of the c items as computed by the single decoder.

The three schemes need a total number of tests scaling as O(N log N ). It is surprising that it doesn't depend on c, but the most important point is that this is much less appealing than the promise in O(c log N ).

4.2

The case where θ 0,n = 0 When θ 0,n = 0, the expressions above are no longer correct because lim x→∞ h (x) = ∞. New Taylor series give the following asymptotics (See Appendices A.1, B.1, and C.1):

I J (p) ≈ cθ 1,n n N log 2 N n , (27)
I S (p) ≈ θ 1,n n N log 2 N n , (28)
I DC (p) ≈ θ 1,n n N S log 2 N S n . ( 29
)
The same comments as above hold except that this time the schemes provide a better total number of tests scaling as O(N ). The explanation is that a test s.t. θ 0,n = 0 has the advantage of being positive if and only if there is at least one defective in the group. Indeed, there is a single defective in a positive group exactly in the 'Divide and Conquer' scheme and asymptotically for the joint and single decoders. This certainty eases a lot the decoding. These mutual informations show that the multiplicative factor of this scaling is 1/(nθ 1,n ) for the single and joint decoders. We thus need to select n s.t. nθ 1,n > 1 in order to be, asymptotically at least, preferable to an exhaustive search testing items separately. This raises an even more stringent condition for the 'Divide and Conquer' scheme because we need nθ 1,n > e(-log α S ).

5 Second strategy:

lim N→∞ n(N) = ∞
The second strategy makes the size of the test groups increasing as N → ∞. Therefore, the rate at which the test becomes random matters. This is reflected by the speed at which δ 1,n ('Divide and Conquer' ) or δ c,n (joint and single) converge to zero. Once again, we make the distinction between tests s.t. θ0 > 0 and those for which θ0 = 0.

The case where θ0 = 0

Assume first that θ0 = 0, Taylor series give the following asymptotics (See Appendices A.2.1, B.2.1 and C.2.1):

I J (p) ≈ - 1 2 h ( θ0 )Var[θ K,n ], (30)
I S (p) ≈ - 1 2 h ( θ0 ) 1 c 2 p(1 -p) Cov(K, θ K,n ) 2 , (31)
I DC (p) ≈ - 1 2 h ( θ0 )p(1 -p)δ 2 1,n , (32)
with

K ∼ B(c, p). Since Cov(K, θ K,n ) 2 ≤ Var[K]Var[θ K,n ]
and Var[K] = cp(1 -p), these series comply with the rule that I S (p) ≤ I J (p)/c. We can also check that if c = 1, these three series are equal. Another remark: If θ K,n = Kδ 1,n + θ 0,n ∀0 ≤ K ≤ c, then I DC (p) = I S (p) = I J (p)/c and the three schemes provide the same scaling of the total number of tests.

Note however that the probability p of belonging to a group equals n/N in the first two expressions, whereas it equals n/N S in the last expression.

Application to the noisy group testing: In this setup, θ 0,n = 1 -θ k,n = n → 1 /2. This simplifies the expressions above as follows:

I J (p) ≈ 2 ln 2 (1 -p) c (1 -(1 -p) c )δ 2 1,n (33)
I S (p) ≈ 1 ln 2 p(1 -p) 2c-1 δ 2 1,n (34)
I DC (p) ≈ 2 ln 2 p(1 -p)δ 2 1,n . (35)
If we suppose that δ 1,n = O(n -a ) with a > 0 and we increase the size of the group s.

t. n ∝ N b with 0 ≤ b ≤ 1 (or n ∝ N b
S for the 'Divide and Conquer' scheme), then the three schemes offer a mutual information of the same order:

Inria • If 0 < a ≤ 1 /2,
the best option is to choose b = 1, i.e. to fix the value of p, to achieve I = O(N -2a ), which ends up in a total number of test scaling as Ω(N 2a log N ).

• If a > 1 /2, the best option is to set b = 0, i.e. to fix the size of the group, to achieve I = O(N -1 ), which ends up in a total number of tests scaling as Ω(N log N ). We rediscover here the results of Sect. 4.

These scalings are much bigger than the promised Θ(c log N ). Yet, if the test smoothly becomes random as n increases, i.e. when a < 1 /2, the situation is actually not that bad since the scale of the total number of tests is slower than Θ(N ), i.e. the scaling of the exhaustive screening (yet, we need a setup where the Ω becomes a Θ).

The appendices gives upper bounds of the mutual informations of the single and joint decoders in the general case:

I J (p) - 1 2 h ( θ0 )(1 -(1 -p) c )δ 2 c,n , (36)
I S (p) - 1 2 h ( θ0 ) p 1 -p δ 2 c,n (37)
These two upper bounds share the same decrease in

O(N -2a ) if δ 1,n = O(n -a ) with 0 < a ≤ 1 /2. Now to get M = O((log N ) d ) we need to have I = Ω((log N ) 1-d
) and therefore, for a fixed p, δ c,n (or δ 1,n for the 'Divide and Conquer' scheme) being Ω((log n) 1-d /2 ). The point of this chapter is to consider that δ c,n converges to zero, therefore d > 1. We are getting closer to the promise of group testing for tests becoming random at a very low speed.

The case where θ0 = 0

The appendices A.2.2, B.2.2 and C.2.2 show that:

I J (p) ≤ (-θ c,n log 2 θ c,n )(1 -π 0 -π c ) + θ c,n (-(1 -π 0 ) log 2 (1 -π 0 )) + o(θ c,n ) (38) I S (p) ≤ (-θ c,n log 2 θ c,n )(1 -π 0 -π c ) + θ c,n (-(1 -π 0 ) log 2 (1 -π 0 ) + (c -1)p c log 2 p) + o(θ c,n ) (39)
I DC (p) = θ 1,n (-p log 2 p) + o(θ 1,n ), (40)
with π 0 = (1 -p) c and π c = p c . For a fixed p, the mutual informations of the joint and single decoders are dominated by -θ c,n log 2 θ c,n . If θ c,n = O(n -a ), a > 0, the total number of tests scales as Ω(N a ). This does not hold for the 'Divide and Conquer' scheme: if θ 1,n = O(n -a ), the total number of tests scales as Ω(N a log N ).

If n ∝ N b , 0 ≤ b ≤ 1 so that p ∝ N b-1 , then the mutual informations of the joint and single decoders are O(N b(1-a)-1 log N ). If the test is slowly converging to a random test, i.e. a < 1, then we should set b = 1 and we are back to the option of freezing p. Otherwise, it is better to set b = 0 so that the total number of tests scales as Ω(N ), and we find back the first strategy fixing n. The same comment holds for the the 'Divide and Conquer' scheme.

Again, this case is preferable to the case θ0 = 0: M = Ω(N a log N ) and not Ω(N 2a log N ), and for a longer range 0 < a ≤ 1 (and not 0 < a ≤ 1 /2).

Last but not least, for the 'Divide and Conquer' scheme, to get M = O((log N ) d ) we need to have θ 1,n = Ω((log N ) 1-d ) with d > 1 to make θ 1,n vanishing as n = pN increases (p is fixed). With the same setup, M = O( (log N ) d /log log N ) for the single and joint decoders.

Conclusion

The point of this chapter is not to find the best choice concerning the asymptotical size of the test groups. We just show that whatever this choice, group testing fails delivering the promise of a total number of tests scaling as O(c log N ). The condition of utmost importance for such an appealing scaling is to have a test which doesn't become purely random as the size of the group grows to infinity. However, group testing almost keeps its promise, i.e. a total number of tests scaling as a power of log N , for setups where the test converges to randomness very slowly, i.e. at a rate in Ω( 1 /log g n) with g > 0.

For this kind of setups, it is better to fix p, which means that the size of the groups are proportional to N . However, if the test becomes random too rapidly, i.e. as fast as O(n -a ) with a ≥ 1 /2, it is useful to switch from a fixed p strategy to a fixed n strategy.

Setups where there is no false positive (θ 0,n = 0) or no false negative (θ k,n = 1 for k > 0) lead to better performances: the total number of tests is lower and the transition from fixed p to fixed n occurs at a higher rate, i.e. for a = 1.

A 'Divide and conquer'

A.1 First strategy: lim N→∞ n(N) = n

The first strategy makes the size of the test groups converging to the finite value n := lim N S →∞ n(N S ) for which the test is not random, i.e. θ 0,n < θ 1,n . On the other hand, the probability of an item being in a given test group vanishes as p = n/N S . Assume that θ 0,n = 0, a Taylor series of [START_REF] Zhou | Parallel feature selection inspired by group testing[END_REF] gives the following asymptotic:

I DC (p) = n N S [(θ 1,n -θ 0,n )h (θ 0,n ) + h(θ 0,n ) -h(θ 1,n )] + o(N -1 S ). (41)
If θ 0,n = 0, the result above does not hold because lim x→0 h (x) = +∞. A new Taylor series gives the following asymptotic:

I DC (p) = θ 1,n n N S log 2 N S n + o 1 N S log(N S ) . (42)
A.2 Second strategy:

lim N→∞ n(N) = ∞ A.2.1 When θ0 ∈]0, 1[
The assumption here is that θ1 = θ0 which lies in ]0, 1[. We denote η n := θ 0,n -θ0 and δ 1,n := θ 1,n -θ 0,n . With these notations, we have

I DC (p) = h( θ0 + η n + pδ 1,n ) -(1 -p)h( θ0 + η n ) -ph( θ0 + η n + δ 1,n ), (43)
Note that

η n ≤ η n + pδ 1,n ≤ η n + δ 1,n because 0 ≤ p ≤ 1, which implies that |η n + p n δ 1,n | ≤ max(|η n |, |η n + δ 1,n |). (44)
Both |η n | and |δ 1,n | converges to 0 so that, for > 0, there exist n 0 big enough s.t. ∀n ≥ n 0 , max(|η n |, |η n + δ 1,n |) ≤ . We then apply the following Taylor development for θ0 ∈]0, 1[:

h( θ0 + ) = h( θ0 ) + h ( θ0 ) + 2 h ( θ0 )/2 + o( 2 ), (45)
Inria on the three terms of (43) to simplify it to:

I DC (p) = - 1 2 δ 2 1,n h ( θ0 )p(1 -p) + o( 2 ). (46)
Since we assume in the text that θ 0,n is non decreasing, it has to converge to θ0 from below s.t. η n is non positive. In the same way, θ 1,n converges to θ0 from above s.t. η n +δ 1,n is non negative. This shows that ≤ δ 1,n ≤ 2 and therefore δ 1,n = Θ( ). This allows to replace o( 2 ) by o(δ 2 1,n ) in (46).

A.2.2 When θ0 ∈ {0, 1}

We detail the case for θ1 = θ0 = 0. With the same notations as in App. A.2.1, this case implies that η 0,n = 0 because θ 0,n is non decreasing and non negative. The mutual information in this context equals:

I DC (p) = h(pδ 1,n ) -ph(δ 1,n ). (47)
For > 0, there exist n big enough for which δ 1,n = and where h( ) = -log 2 ( ) + / ln 2 + o( ).

Applying this development, we obtain:

I DC (p) = δ 1,n (-p log 2 p) + o(δ 1,n ). (48)

B Joint decoder

We assume that the size of a group is always larger than c. Therefore, (c + 1) parameters define the test (θ 0,n , • • • , θ c,n ).

B.1 First strategy: lim N→∞ n(N) = n

The mutual information for the joint decoder (5) has the following Taylor series when p = n/N → 0, for θ 0,n > 0:

I J (p) = c n N [(θ 1,n -θ 0,n )h (θ 0,n ) + h(θ 0,n ) -h(θ 1,n )] + o(1/N ), (49)
and, for θ 0,n = 0 and θ 1,n > 0:

I J (p) = cθ 1,n n N log (N ) + o(N -1 log N ). (50)
We have supposed that n is chosen s.t. δ 1,n > 0. It is possible to relax this constraint and the first non nul parameter δ k,n will appear in the above equations. Yet, we also get a decay in N -k instead of N -1 , whence choosing n s.t. δ 1,n = 0 should be avoided if possible. This is a real issue for the threshold group testing model where θ 1,n = θ 0,n (see Sect. For 0 < θ0 < 1, we apply the development (45) to h(P (p)) and h(θ k,n ):

I J (p) = - 1 2 h ( θ0 ) c k=0 π k θ 2 k,n -P (p) 2 + o( 2 ) (51) = - 1 2 h ( θ0 )Var[θ K,n ] + o( 2 ). (52

C Single decoder

We assume that the size of a group is always larger than c. Therefore, (c + 1) parameters (θ 0,n , • • • , θ c,n ) define the test.

C.1 Asymptotical analysis for lim N S →∞ n(N S ) < ∞ Assume that θ 0,n = 0, a Taylor series of (9) gives the following asymptotic:

I S (p) = n N [(θ 1,n -θ 0,n )h (θ 0,n ) + h(θ 0,n ) -h(θ 1,n )] + o(1/N ). ( 56
)
If θ 0,n = 0, a Taylor series gives the following asymptotic: (60)

From ( 10) and [START_REF] Meerwald | Group testing meets traitor tracing[END_REF], it is clear that θ 0,n ≤ P 0 (p) and P 1 (p) ≤ θ c,n . We also have

p(1 -p)P (p) = E[(K -cp)θ K,n ] = E[Kθ K,n ] -E[K]E[θ K,n ] = Cov(K, θ K,n ). ( 61
)
with K ∼ B(c, p) because E[K] = cp. We introduce K ∼ B(c, p) independent of K. Then, on one hand:

Cov(K -K , θ K,n -θ K ,n ) = Cov(K, θ K,n ) + Cov(K , θ K ,n ) = 2Cov(K, θ K,n ), (62)
while, on the other hand,

Cov(K -K , θ K,n -θ K ,n ) = E[(K -K )(θ K,n -θ K ,n )] -E[K -K ]E[θ K,n -θ K ,n ] = 0≤k,k ≤c π k π k (k -k )(θ k,n -θ k ,n ). (63)

1 [

 1 2.1). B.2 Second strategy: lim N→∞ n(N) = ∞ Now suppose that the parameters of the model vary with n s.t. θ 0,n ≤ θ 1,n ≤ • • • ≤ θ c,n , and that δ c,n = θ c,n -θ 0,n vanishes to 0 as n increases. The analysis is made for a fixed 0 < p < 1. We remind that π k = c k p k (1 -p) c-k , ∀0 ≤ k ≤ c, the distribution of the binomial B(c, p). T. Furon B.2.1 When θ0 ∈]0, As in the previous section, η n = θ 0,n -θ0 and δ k,n = θ k,n -θ 0,n converge to zero. For any > 0, there exists n large enough for which max(|η n |, η n + δ c,n ) = . This implies that |η n + δ k,n |, ∀0 ≤ k ≤ c, and |η n + P (p) -θ 0,n | are smaller than .

)

  Note that Var[θ K,n ] ≤ E[(θ K,n -θ 0,n ) 2 ] ≤ δ 2 c,n (1 -(1 -p) c ) < 4 2 .On the other hand,Var[θ K,n ] ≥ π 0 (θ 0,n -P (p)) 2 + π c (θ c,n -P (p)) 2 ≥ π 0 π c π 0 + π c δ 2 c,n ≥ 2 . (53)This shows thatVar[θ K,n ] = Θ( 2 ) so that we can replace o( 2 ) by o(Var[θ K,n ]). B.2.2 When θ0 ∈ {0, 1}We start by applying the development h(x) = -x log 2 (x) + x /ln 2 + o(x) on h(P (p)) and h(θ k,n ):I J (p) = (-P (p) log 2 (P (p))) -c k=0 π k (-θ k,n log 2 θ θ k,n ) + o(θ c,n ).(54)The function x → -x log 2 (x) is increasing over [0, 1 /e) and P (p) ≤ θ c,n (1-π 0 ). On the other hand c k=0 π k (-θ k,n log 2 θ θ k,n ) ≥ π c (-θ c,n log 2 θ c,n ). This inequality follows from these arguments: I J (p) ≤ (-θ c,n log 2 θ c,n )(1 -π 0 -π c ) + θ c,n (-(1 -π 0 ) log 2 (1 -π 0 )) + o(θ c,n ). (55)

I S (p) = θ 1,n n N log 2 Nπ

 2 n + o (log(N )/N ) . (57) If θ 1,n = θ 0,n , the Taylor series wil show the role of the first non nul coefficient δ k,n but fraction n/N must be replaced by ( n/N ) k . This should be avoided as it Inria C.2 Second strategy: lim N→∞ n(N) = ∞We first present some relations between P (p), P 1 (p) and P 0 (p):P 1 (p) = P (p) + (1 -p)P (p)/c,(58)P 0 (p) = P (p) -pP (p)/c, k θ k,n (k -cp).

  Since θ k,n is increasing with k, the summands in the last equation are all non negative. We can also lower bound this sum by only keeping the terms |k -k | = c. This shows that0 ≤ δ c,n cp c-1 (1 -p) c-1 ≤ P (p) = 1 p(1 -p) Cov(K, θ K,n ). (64)An upper bound is given by noting that c k=0 π k θ k,n k ≤ cpθ c,n and c k=0 π k θ k,n ≥ θ 0,n , s.t. p(1 -p)P (p) ≤ δ c,n cp. This proves that P (p) = Θ(δ c,n ).Since P (p) ≥ 0, we haveθ 0,n ≤ P 0 (p) ≤ P (p) ≤ P 1 (p) ≤ θ c,n .(65)These five probabilities converge to θ0 as n → ∞.C.2.1 When θ0 ∈]0, 1[ For > 0, there exists n large enough s.t. max(|η n |, δ c,n + η n ) = .The Taylor series of (9) leads to:(1 -p) Cov(K, θ K,n ) 2 + o( 2 ).(67) Now, δ c,n = δ c,n + η n -η n s.t. ≤ δ c,n ≤ 2 and P (p) = Θ( ). This allows to replace o( 2 ) by o(P (p) 2 ) or o(Cov(K, θ K,n ) 2 ) in the equations above. The upper bound on P (p) is used to bound the mutual information: I S (p) ≤ -

  modeling the output of the test performed on this group. There are four well known models:1. Noiseless test: Y i = Z i . The test is positive if and only if there is at least one defective in a group,

2. Noisy test:

RR n°9164

Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399

T. Furon

C.2.2 When θ0 ∈ {0, 1}

We have:

Following the same rationale as in Sect. B.2.2, we get:


Pollyana Trigueiro

Silvia Pedetti

Baptiste Rigaud

Sebastien Balme

Jean-Marc Janot

Ieda M G Dos Santos

Régis Gougeon

Maria G Fonseca

Thomas Georgelin

Maguy Jaber
email: maguy.jaber@upmc.fr

Going through the wine fining: intimate dialogue between organics and clays

Keywords: adsorption, resveratrol, BSA, NMR, Fluorescence spectroscopy, wine, clay

Wine chemistry inspires and challenges with its complexity and intriguing composition. In this context, the composites based on the use of a model protein, a polyphenol of interest and montmorillonite in a model hydroalcoholic solution have been studied. A set of experimental characterization techniques highlighted the interactions between the organic and the inorganic parts in the the composite. The amount of the organic part was determined by ultraviolet-visible (UV-VIS) and thermal analysis. X-ray diffraction (XRD) and transmission electronic microscopy (TEM) informed about the stacking/exfoliation of the layers in the the composites.

Vibrational and nuclear magnetic resonance spectroscopies methods stressed on the formation of a complex between the protein and the polyphenol before adsorption on the clay mineral.

The mobility/rigidity of the organic parts were determined by fluorescence time resolved spectroscopy. Changes in the secondary structure of the protein occured upon complexation with polyphenol on clay mineral due to strong interactions. Although not representating faithfully enological conditions, these results highlight the range and nature of mechanisms possibly involved in wine fining.

Introduction

Proteins are present in wine in low concentrations, contributing weakly to their nutritive characteristics. Nevertheless, their presence in high proportion may affect the quality of the product, being responsible for the turbidity and time-instability of white wines. The proteins of wine are generally tolerant to the pH of the wine (pH = 3-3.6) [START_REF] Jaeckels | Influence of bentonite fining on protein composition in wine[END_REF][START_REF] Van Sluyter | Wine protein haze: Mechanisms of formation and advances in prevention[END_REF][START_REF] Moreno-Arribas | Analytical methods for the characterization of proteins and peptides in wines[END_REF]. Naturally occurring proteins of grapes, and in particular pathogenesis related (PR) proteins, have been shown to cause turbidity or haze formation in white wines, where the instability can be related to factors such as pH of the medium, ethanol content, ionic strength, temperature or concentration of the organic acids, tannins and polyphenolic compounds [START_REF] Dordoni | Effect of bentonite characteristics on wine proteins, polyphenols, and metals under conditions of different pH[END_REF][START_REF] Vincenzi | Study of combined effect of proteins and bentonite fining on the wine aroma loss[END_REF][START_REF] Esteruelas | Phenolic compounds present in natural haze protein of Sauvignon white wine[END_REF]. Therefore, it is necessary in the fining treatment to eliminate the risk of protein precipitation, which content can vary from 10 to more than 260 mg.L -1 [START_REF] Cilindre | It's time to pop a cork on champagne's proteome![END_REF][START_REF] Wigand | Analysis of Protein Composition of Red Wine in Comparison with Rosé and White Wines by Electrophoresis and High-Pressure Liquid Chromatography-Mass Spectrometry (HPLC-MS)[END_REF]. It must be noted though, that in Champagne, proteins can have a positive role on the foamability.

Polyphenols are responsible for the differences between white and red wine, especially in the determination of the color, body, flavor and structure of red wine [START_REF] Pérez-Magariño | Polyphenols and colour variability of red wines made from grapes harvested at different ripeness grade[END_REF]. Among the diversity of polyphenolic compounds, stilbenes and derivatives have been largely studied,in particular resveratrol (RESV), has antioxidant, bactericides, anti-inflamatory and vitamin properties, which are related to its protective effect against cardiovascular diseases [START_REF] Cao | Non-covalent interaction between dietary stilbenoids and human serum albumin: Structure-affinity relationship, and its influence on the stability, free radical scavenging activity and cell uptake of stilbenoids[END_REF].Even though it can also be found in food products and beverages such as peanut butter, mulberries and grape juice, Red wine is believed to be the main source of resveratrol in the human diet [START_REF] Lu | Resveratrol, a natural product derived from grape, exhibits antiestrogenic activity and inhibits the growth of human breast cancer cells[END_REF][START_REF] Wenzel | Metabolism and bioavailability of trans-resveratrol[END_REF]. The phenolic compounds found in wines may vary according to their structure as, tannins, non-flavonoids and flavonoids. RESV (Figure 1 SI, in Supplementary information) is a flavonoid, natural pigment, which can be present in high concentration in grape skin but not in grape flesh, so red wine contains significant amounts of resveratrol compared to white wine, with concentration ranging from 0.1 to 3.0 mg.L -1 , where its amount depends on the grape variety and the vinification process [START_REF] Minussi | Phenolic compounds and total antioxidant potential of commercial wines[END_REF][START_REF] Lucas-Abellán | Cyclodextrins as resveratrol carrier system[END_REF]. It has been reported in the literature that the complexation of resveratrol with macromolecules can modulate the expression of its bioviability and stability as well as its antioxidant effect [START_REF] Lucas-Abellán | Cyclodextrins as resveratrol carrier system[END_REF][START_REF] Richard | Recognition characters in peptide-polyphenol complex formation[END_REF]. Different fining agents in winemaking are reported in the literature, but clay minerals rich in montmorillonite such as bentonite, remain heavily used for white wines. Bentonite behaves as stabilizer by adsorbing proteins in wine wich typically have isoelectric point (IP) between 3 and 9 and a MW between 20-70 KDa [START_REF] Achaerandio | Protein Adsorption by Bentonite in a white wine model solution: Effect of Protein Molecualr Weight and Ethanol Concentration[END_REF]. Montmorillonite displays a layered structure of type 2:1, ideally formed by two sheets of tetrahedrally coordinated silicon linked through a sheet of octahedrally coordinated aluminum. Al 3+ and Mg 2+ generally occupy octahedral sites, whereas Hydrated exchangeable Na + or Ca 2+ cations are present in the interlayers to balance the negative layer charge [START_REF] Jaber | Synthesis, characterization and applications of 2:1 phyllosilicates and organophyllosilicates: Contribution of fluoride to study the octahedral sheet[END_REF][START_REF] Brigatti | Chapter 2 Structures and Mineralogy of Clay Minerals[END_REF][START_REF] Jaber | Mercaptopropyl Al-Mg phyllosilicate: synthesis and characterization by XRD, IR, and NMR[END_REF].

Adsorption of proteins onto clay minerals has been widely reported in the literature [START_REF] Assifaoui | Structural Studies of Adsorbed Protein (Betalactoglobulin) on Natural Clay (Montmorillonite)[END_REF][START_REF] Yu | Adsorption of proteins and nucleic acids on clay minerals and their interactions: A review[END_REF][START_REF] Servagent-Noinville | Conformational Changes of Bovine Serum Albumin Induced by Adsorption on Different Clay Surfaces: FTIR Analysis[END_REF]. It may involve several types of physical and chemical interactions such as cation exchange, electrostatic forces or hydrogen bonding, on the interlayer surface of clay minerals, on the edges or both. The positive charge of the protein when the pH is below the IP allows the adsorption on the negatively charged surface of montmorillonite due to electrostatic forces. Various possible mechanisms can describe the interaction between proteins and clays: intercalation, exfoliation or both [START_REF] Assifaoui | Structural Studies of Adsorbed Protein (Betalactoglobulin) on Natural Clay (Montmorillonite)[END_REF]. Extensive penetration of protein chains into the interlayer space of clays can lead to exfoliation or delamination of the silicate layers [START_REF] Kumar | Effect of Type and Content of Modified Montmorillonite on the Structure and Properties of Bio-Nanocomposite Films Based on Soy Protein Isolate and Montmorillonite[END_REF][START_REF] Gopakumar | Influence of clay exfoliation on the physical properties of montmorillonite / polyethylene composites[END_REF][START_REF] Cai | Adsorption of DNA on clay minerals and various colloidal particles from an Alfisol[END_REF].

The use of clay minerals for the fine treatment of wine may induce not only adsorption of the protein but also of other molecules. As reported in the literature, there may be a decrease in the amount of resveratrol in the wine due to interactions with fining agents [START_REF] Donovan | Effects of Small-Scale Fining on the Phenolic Composition and Antioxidant Activity of Merlot Wine[END_REF][START_REF] Threlfall | Effects of fining agents on transresveratrol concentration in wine[END_REF] or an interaction of resveratrol with the protein due to electrostatic forces, hydrophobic interactions and/or hydrogen bond between these two compounds [START_REF] Liang | Interaction of b-lactoglobulin with resveratrol and its biological implications[END_REF][START_REF] N′ Soukpoé-Kossi | Resveratrol Binding to Human Serum Albumin[END_REF].

Bovine serum albumin (BSA, Figure 2 SI in Supplementary information) with MW of 66.5 KD a is used as model protein for this study. It is characterized by globular structure and is the most abundant soluble plasma protein with a typical circulating concentration of 0.6 mmol.L -1 . In addition, it is extensively used in biochemical studies due to its wide availability and high structural resemblance with human serum albumin [START_REF] Bourassa | Binding sites of resveratrol, genistein, and curcumin with milk α-And β-caseins[END_REF][START_REF] Carter | Structure of serum albumins[END_REF]. Resveratrol has a strong interaction with albumin. Indeed, several molecules can be bound to the hydrophobic pocket of albumin with an affinity constant around 2.10 4 M -1 or 5.10 4 M -1 . This binding occurs via both hydrophillic and hydrophobic interactions and/or hydrogen bonding [START_REF] Bourassa | Binding sites of resveratrol, genistein, and curcumin with milk α-And β-caseins[END_REF][START_REF] Nair | Biology Spectroscopic study on the interaction of resveratrol and pterostilbene with human serum albumin[END_REF][START_REF] Jiang | Study of the interaction between trans-resveratrol and BSA by the multi-spectroscopic method[END_REF].

The aim of the present work is to investigate the co-adsorption of a model protein, the BSA, and a polyphenol of interest, resveratrol, onto a synthetic montmorillonite (Mt)in an hydroalcoholic solution, in order to investigate, via a multi-technical approach, the presence and the nature of interactions between the clay mineral, the protein and the polyphenolic compound present in wine.

2.

Experimental part

Materials

Bovine Serum Albumin (>99%) and resveratrol (>99%) were purchased from Sigma-Aldrich.

Potassium phosphate monobasic (>99%) buffer 0.1M and ethanol (solvent system) were used to solubilize the protein and polyphenol. Anapproximation of amodel wine solution was made with the mixture of 25% distilled water, 50% phosphate buffer and 25% ethanol (ratio v/v) at pH = 4.5 (solvent system used in BSA solution and RESV solution). For the synthesis of the montmorillonite,the following reagents were used: Aerosil 130 (Evonik Industries) as source of silica, Bohemite (AlOOH) 74% Al2O3 (Sasol Germany), Magnesium acetate tetrahydrate (Sigma-Aldrich), Sodium acetate (Sigma-Aldrich) and hydrofluoric acid.

Synthesis of montmorillonite

For the synthesis of sodium montmorillonite, the reagents were mixed in the following order: deionized water, hydrofluoric acid and the sources of interlamellar cation: sodium acetate, magnesium acetate, alumina and silica. The hydrogels were aged under stirring at room temperature for 2h and then were autoclaved for reaction at 220 °C for 72h. The autoclaves were cooled to room temperature and the products were washed thoroughly with distilled water and centrifuged. The solids were then dried at 50 °C for 24 h [START_REF] Jaber | Selectivities in Adsorption and Peptidic Condensation in the (Arginine and Glutaminic Acid)/Montmorillonite Clay System[END_REF][START_REF] Tangaraj | Adsorption and photophysical properties of fluorescent dyes over montmorillonite and saponite modified by surfactant[END_REF].

Adsorption of proteins

Adsorption experiments were carried out by adding dropwise BSA solution, dissolved in the above described solvent system, in an aqueous suspension of the clay mineral (1.2mg.mL -1 ).

The initial concentration of BSA (1.0 mg.mL -1 ) was determined before addition of montmorillonite. The mixture was stirred at room temperature for 4 hours. After that, the suspension was centrifuged at 2700rpm for 8 minutes. Both supernatants and solids were collected and analyzed. Solids were washed with aqueous solution to remove weakly adsorbed BSA and dried at room temperature. The amount of protein remaining in the supernatant was determined by UV-Vis spectroscopy considering the absorption maximum wavelength at 281 nm. All measurements were performed in triplicate. The concentration of adsorbed protein on montmorillonite was determined by:

Г t = V ( C 0 -C t ) m ( 1
)
where V is the total volume of the protein solution, C0 is the initial concentration of protein, Ct is the concentration of protein in supernatant at time t, and m is the mass of the clay mineral.

With V (mL), C0 and Ct (mg.mL -1 ), the interfacial concentration Гt of protein is given in mg of protein per mg of clay mineral (mg.mgMt -1 ).

Adsorption of resveratrol onto sodium clay

The polyphenol adsorption experiments were performed as described for the protein. The initial concentration of RESV (0.5 mg.mL -1 ) was determined before addition of montmorillonite. The amount of resveratrol remaining in the supernatant was determined by UV-Vis spectroscopy with absorption at 306 nm.

Simultaneous adsorption of polyphenol and BSA onto sodium clay

BSA (1.0 mg.mL -1 ) was dissolved in solution of the above described solvent system. Separately, resveratrol with a concentration of (0.5 mg.mL -1 ) was dissolved in a solution of the above described solvent system. Then, the solution containing RESV was added dropwise to the BSA solution that was then kept under constant stirring for 2 hours. Resulting solution was added dropwise to the aqueous suspension of the clay mineral with a of concentration 1.2 mg.mL - 1 under constant stirring for 2 hours. The resulting mixture was stirred for 4 hours at room temperature. The resulting slurry was washed and centrifuged for removal of excess of protein and polyphenol. The resulting solid was separated by centrifugation and dried at room temperature.

UV-Visible

Ultraviolet-visible (UV Vis) absorption spectra were recorded using an Ocean View Optics spectrometer DH-2000-BAL. The light source was a Deuterium and Halogen lamp, equipped with 400μm diameters optic fibers, coupled with CUV 1cm cuvette holder. Spectra were acquired by Ocean View Spectroscope Software. The absorbance wavelength range was from 250 to 900 nm.

X-ray diffraction (XRD)

Powder X-ray diffractograms were recorded using D8 Advance Bruker-AXS Powder X-ray diffractometer with CuKα radiation (λ= 1.5405 Å). The diffractions patterns were measured between 5-70° (2θ) with a scan rate of 0.5 deg.min -1 . The active area of the detector was limited as much as possible in order to reduce the background scattering at low angle between 1-10° (2θ). Samples were kept 72 h before measurements under controlled humidity.

Transmission electron microscopy (TEM)

TEM study of the samples was performed on a JEOL 2010 microscope, 200kV LaB6 coupled Orius camera, from Gatan Company. Samples in the form of bulk powders were suspended in ethanol and then deposited on 400 mesh copper grids covered with an ultrathin carbon membrane of 2-3 nm thickness.

2.9.Attenuated Total Reflectance (ATR) Infrared spectroscopy

Fourier transform infrared spectroscopy (FTIR) of the solid samples was performed on Agilent Cary 630 FTIR spectrometer using Agilent diamond attenuated total reflectance ATR technique. Spectra were acquired on the 4000 and 650 cm -1 range and processed with the Microlab FTIR Software (Agilent Technologies).

Thermal analysis

Thermal analyses were carried out using a Q600-1694 SDT Q600 TA Instrument and TA Universal Analysis 2000, the heating rate was of 5 °C.min -1 from 25 °C to 1000 °C, with flow rate of 10 mL.min -1 , in air atmosphere.

Solid State Cross-Polarization Magic Angle Spinning Carbon 13 Nuclear

Magnetic Resonance ( 13 C CP-MAS Solid state nuclear magnetic resonance) [START_REF] Minussi | Phenolic compounds and total antioxidant potential of commercial wines[END_REF] CCP-MAS NMR spectra were obtained on a Bruker Advance 500 spectrometer operating at ΩL=500MHz ( 1 H ) and125 MHz ( 13 C ) with a 4 mm H-X MAS probe. Chemical shifts were calibrated using the CH2signal of adamantane (38.52 ppm) as an external standard. The CP spectra were acquired with a MAS rate of 14 kHz, an acquisition time of 40 ms, a ramp-CP contact time of 1ms, a 1 s recycle delay and with a 1 H spinal 64decoupling sequence. The number of scans to obtain the spectra depending on the S/N obtained for each sample. Spectra were processed with a zero filling factor of 2 and with an exponential decay corresponding to 25 Hz line broadening. Only spectra with the same line broadening were directly compared.

The decomposition of the spectra was performed using Dmfit-2015 software. Decay analysis was performed using a Levenberg-Marquardt algorithm. For the analysis, the fluorescence decay law at the magic angle IM(t) was assumed as a sum of exponentials. We assumed a Poisson distribution of counts in the calculation of the χ2 criterion; residual profiles and autocorrelation function as well as Durbin-Watson and skewness factor were used in order to estimate the quality of the adjustment. The number of exponentials used for the fit was increased until all the statistical criterions were improved. All details about calculation of both lifetime are given elsewhere [START_REF] Balme | Highly efficient fluorescent label unquenched by protein interaction to probe the avidin rotational motion[END_REF].

2.12.Fluorescence spectroscopy

Steady

Results and Discussions

BSA and RESV adsorption onto sodium clay

The UV-Vis spectra of the BSA and RESV before and after adsorption on montmorillonite are

shown in Figure 1. The intensity of the band maxima of the BSA in Figure 1(a) at 281 nm decreases after adsorption on montmorillonite. The RESV showed band maxima at 306 nm as we can see in Figure 1(b) and follows the same trend as BSA. The influence of the contact duration was also investigated. The adsorption kinetics (C0 = 1.0 mg.mL -1 ) curves are reported in Figure 3 SI in Supplementary information and similar behavior was observed for both protein and resveratrol. In the present study, the Langmuir isotherm model adequately describes the adsorption data, that are as widely reported for protein adsorption on clay minerals [START_REF] Bajpai | Study on the adsorption of hemoglobin onto bentonite clay surfaces[END_REF]. The Langmuir isotherm of pseudo-second-order model can be linearized in agreement to the equation:

𝑡𝑡 Γ 𝑡𝑡 = 1 𝑘𝑘 2 Γ 𝑒𝑒 2 + 1 Γ 𝑒𝑒 𝑡𝑡 (2)
where Γ 𝑡𝑡 and Γ 𝑒𝑒 is the interfacial concentration (mg.mgMt) at time t and at the equilibrium respectively, k2 is the equilibrium rate constant of pseudo-second-order adsorption (mg mgMt - 1 .min -1 ).

Pseudo-second order equation were used to research the adsorption kinetic behavior for protein and polyphenol on montmorillonite, reported in Figure 4 SI in Supplementary files. The slope and intercept of the plot of t/Γt versus t were used to calculate the K2 rate constant. Values of the Γ 𝑒𝑒 (interfacial concentration maximum) and K2 (adsorption kinetic constant) are reported in Table I. K2 values indicates affinity of the protein and polyphenol for the inorganic support and in the present case the value of K2=0.16 obtained demonstrates the high affinity of the BSA with the surface of the montmorillonite [START_REF] Assifaoui | Structural Studies of Adsorbed Protein (Betalactoglobulin) on Natural Clay (Montmorillonite)[END_REF][START_REF] Lepoitevin | BSA and lysozyme adsorption on homoionic montmorillonite: Influence of the interlayer cation[END_REF].

Table I. Kinect parameters obtained by Langmuir isotherm for BSA and resveratrol adsorption on montmorillonite.

Resveratrol BSA Γ 𝑒𝑒 (mg.mgMt -1 ) 0.32 0.87 K2 (mg mgMt -1 .min -1 )

1.90 0.16

Localization of the organic molecules in the clay mineral

XRD patterns of the montmorillonite and all hybrids composites are reported in Figure 2. The X-Ray pattern of the raw montmorillonite shows typical reflections characteristics of the dioctahedral smectite (d060 = 0.149 nm) [START_REF] Lepoitevin | BSA and lysozyme adsorption on homoionic montmorillonite: Influence of the interlayer cation[END_REF][START_REF] Reinholdt | Synthesis and characterization of montmorillonite-type phyllosilicates in a fluoride medium[END_REF]. The d001 value is about 1.26 nm before adsorption corresponding to the thickness of the layer and the presence of hydrated sodium in the interlayer space. Upon adsorption of the protein, the d001 peak disappears while the other characteristic reflection peaks of montmorillonite remained intact, hence the hypothesis of a possible partial exfoliation or delamination can be considered [START_REF] Assifaoui | Structural Studies of Adsorbed Protein (Betalactoglobulin) on Natural Clay (Montmorillonite)[END_REF]. After adsorption of resveratrol, the interbasal spacing of the clay mineral cannot be detected, making difficult to evaluate the degree of intercalation of the polyphenol which may possibly be non-uniformly distributed between the montmorillonite layers. After adsorption of the BSA and RESV together, it is not possible to measure the inter-basal spacing d001, which may suggest partial or total exfoliation of the montmorillonite layers upon inclusion of the protein and the polyphenol.

These studies will be complemented with TEM analysis. Presence of the crystalline phase of resveratrol is common, characteristic peaks of the drug at 2θ=16.

Transmission electron microscopy (TEM)

Transmission electron microscopy (TEM) micrographs show layered structures with alternate dark and bright fringes with a repeat length of 1.26 nm for the raw clay, synthetic clay without modification.

In Figure 3 BSA-RESV-Mt (e) and (f). Experimental conditions: BSA (1.0 mg.mL -1 ), RESV (0.5 mg.mL -1 ), Mt (1.2 mg.mL -1 ) in a buffer phosphate solution at pH 4.5 and ethanolic solution in ratio of 25 %.

Spectroscopic characterization

Study of BSA and RESV conformation before and after adsorption onto clay mineral was performed using ATR-Infrared spectroscopy (Figure 4). The presence of the physisorbed water of montmorillonite is observed by the bending vibration at 1634 cm -1 [START_REF] Georgelin | Inorganic phosphate and nucleotides on silica surface: Condensation, dismutation, and phosphorylation[END_REF] while the characteristic peak at 1015 cm -1 is due to stretching vibrations of the Si-O groups [START_REF] Madejová | FTIR techniques in clay mineral studies[END_REF].

According to the literature [START_REF] Bourassa | Resveratrol, genistein, and curcumin bind bovine serum albumin[END_REF][START_REF] Roy | Spectroscopic and docking studies of the binding of two stereoisomeric antioxidant catechins to serum albumins[END_REF][START_REF] Liu | Molecular Modeling and Spectroscopic Studies on the Interaction of Transresveratrol with Bovine Serum Albumin[END_REF] the spectral ranges for the most intense peaks for proteins occur in the region 1700-1600 cm -1 and 1600-1500 cm -1 that are assigned to amide I and amide II vibrations, respectively. Infrared absorption of resveratrol showed three characteristics intense bands (Figure 4b) at 1606 cm -1 corresponding to C-C aromatic double band stretching, 1585 cm -1 band assigned to C-C olefinic stretching and the band observed at 1381 cm -1 corresponding to a ring C-C stretching [START_REF] Popova | Preparation of resveratrol-loaded nanoporous silica materials with different structures[END_REF][START_REF] Billes | Vibrational spectroscopy of resveratrol[END_REF].

Conformational changes of the structure of the BSA upon confinement onto montmorillonite were pointed. A shift in amide I and amide II bands from 1640 cm -1 to 1643 cm -1 and from 1521 cm -1 to 1525cm -1 in Figure 4(a) were observed. This is consistent with the change in the secondary structure of the protein [START_REF] Servagent-Noinville | Conformational Changes of Bovine Serum Albumin Induced by Adsorption on Different Clay Surfaces: FTIR Analysis[END_REF][START_REF] Della Porta | RSC Advances Conformational analysis of bovine serum albumin adsorbed on halloysite nanotubes and kaolinite : a Fourier transform infrared spectroscopy study[END_REF]. Upon adsorption of RESV onto Mt, we observed a spectral shift of characteristics bands of the RESV in Figure4(b) that move from 1606 to 1608cm -1 ,from 1585 to 1589cm -1 and from 1381 to 1386 cm -1 . The significant shift of this last band corresponding to the ring C-C stretching suggests the interaction of RESV with the Mt surface [START_REF] Popova | Preparation of resveratrol-loaded nanoporous silica materials with different structures[END_REF].

The absorption bands characteristic of the BSA-RESV-Mt composite are showed in Figure 4(c).

Shifts from 1640 to 1648 cm -1 and from 1521to 1524cm -1 as well as increase in the intensity of the amide I and amide II bands of protein, respectively were observed. Changes from 1585 to 1588cm -1 and from 1381 to 1388 cm -1 of the characteristics bands of resveratrol after interaction with BSA and montmorillonite were observed. This could be attributed to the presence of interactions between the protein and the polyphenol with montmorillonite or it may suggest the co-adsorption of resveratrol in the protein before interaction with the clay mineral. This information can be complemented with the fluorescence analysis.

3.5.Thermogravimetric Analyses

TG curve and their corresponding derivative of the montmorillonite are reported in Figure 5 SI in Supplementary files. The curve relative to montmorillonite alone is characterized by three steps of mass loss associated to endothermal events. The first one, at 82 o C with weight loss of 8.6 %, corresponds to the release of water physically adsorbed on the surface, the weight loss of 1.4 % at 184 o C is due to the departure of the interlayer water molecules. Finally, two peaks at 428 o C and 645 o C are due to dehydroxylation with weight loss of about 3.5 % [START_REF] Xie | Thermal characterization of organically modified montmorillonite[END_REF].

Thermogravimetric analyses of BSA and RESV alone and of BSA-RESV-Mt composite were studied and reported in Figure 6 SI and Figure 7 SI in Supplementary information. DTG curve of free BSA exhibits a broad mass loss at 61 o C related to dehydratation while two peaks observed at 275 o C and 316 o C are likely due to the polypeptide chain thermal decomposition of proteins that corresponds to 55% mass loss. In addition, a second thermal decomposition step of the organic matter is observed between 450-650 o C (mass loss around of 40%) with a maximum at 493 o C and 617 o C, includes both the decomposition of the hard residues of the proteins [START_REF] Duce | Loading of halloysite nanotubes with BSA, α-Lac and β-Lg: A Fourier transform infrared spectroscopic and thermogravimetric study[END_REF]. The free RESV is thermally stable under air flow up to about260°C and its degradation shows two maximum at 310 o C and 549°C, with a total mass loss around 90%.

After adsorption of the protein on Mt, an increase of the decomposition temperature for the BSA was observed, which occurs at 308°C and 537°C (mass loss 22%). This is likely due to the changes in the conformational structure of the proteins upon interactions with the clay mineral. After adsorption of RESV on Mt, an increase in thermal stability of the polyphenol was observed with a maximum decomposition temperature of polyphenol at 428 o C, with a mass loss of 14%. This may suggest intercalation or interactions of resveratrol in the external and/or internal surface of clay. The peak observed at 632°C with a mass loss of 2.6 % is attributed to matrix dehydroxylation.

Thermal analysis BSA-RESV-Mt composite showed mass loss peaks shifted with respect to single Mt composites, observed in Figure 7 SI in Supplementary files. The differential peaks at 46°C with mass loss of 18 % and 219°C with mass loss of 3 % correspond to the removal of physisorbed water and to the dehydratation of interlayer water, respectively. The progressive decomposition of the organic matter, with mass loss 37%, is indicated by the peaks maximum at 316°Cand 489°C. The significant enhancement in thermal stability of the composites can be attributed to the exfoliation of silicate layers with intercalation of the protein and polyphenol complex preventing the fast decomposition of the products. The results obtained are similar to those observed in the adsorption tests. onto Mt, peaks are shifted to 178.5; 175.8 and 173.0 ppm. For the BSA-RESV-Mt composite the shift to 178.6; 175.7 and 173.7 ppm was observed. It must be noted also that resveratrol is significantly co-adsorbed with BSA, as illustrated by the corresponding intense peaks for the spectrum of the BSA-RESV-Mt composite. Together, these results indicate that an interaction between the inorganic surface and the two organic molecules probably takes place leading to a modification of the secondary structure of the protein.

3.7.Fluorescence spectroscopy

The fluorescence is an interesting method to characterize the interaction between resveratrol and protein [START_REF] N′ Soukpoé-Kossi | Resveratrol Binding to Human Serum Albumin[END_REF][START_REF] Nair | Biology Spectroscopic study on the interaction of resveratrol and pterostilbene with human serum albumin[END_REF][START_REF] Jiang | Study of the interaction between trans-resveratrol and BSA by the multi-spectroscopic method[END_REF][START_REF] Balme | Structure, orientation and stability of lysozyme confined in layered materials[END_REF]. The BSA contains 2 Trp residues. Trp212 is located within a hydrophobic binding pocket of the protein and Trp134 is located on the surface of the molecule [START_REF] Bourassa | Resveratrol, genistein, and curcumin bind bovine serum albumin[END_REF][START_REF] Poklar | Interactions of different polyphenols with bovine serum albumin using fluorescence quenching and molecular docking[END_REF]. BSA and resveratrol thus exhibit different photophysical properties. The resveratrol absorbance spectrum covers the emission of the BSA. This induces a fluorescence quenching of the BSA in presence of resveratrol due to a non-radiation energy transfer [START_REF] Xiao | Probing the interaction of trans-resveratrol with bovine serum albumin: A fluorescence quenching study with tachiya model[END_REF][START_REF] Cao | Interaction between trans-resveratrol and serum albumin in aqueous solution[END_REF]. The adsorption mechanism is not totally elucidated since two assumptions have been formulated: (i) The BSA and resveratrol are adsorbed independently or (ii) the BSA binds the resveratrol. The fluorescence emission spectra of BSA-RESV-Mt composite exhibits a similar emission spectra that RESV-Mt under excitation at 290 nm. At the maximum of emission wavelength its intrinsic fluorescence and also confirm that a complex between resveratrol and BSA occurred [START_REF] Liu | Molecular Modeling and Spectroscopic Studies on the Interaction of Transresveratrol with Bovine Serum Albumin[END_REF]. Thus the most likely scenario is a binding of resveratrol by BSA before the adsorption onto montmorillonite.

Conclusion

The use of montmorillonite as a fining agent in winemaking adsorbed protein thus decreasing

  -state and time-resolved fluorescence spectra were obtained by the time-correlated single-photon counting technique. The excitation wavelength was achieved using a SuperK EXTREME laser (NKT Photonics, model EXR-15) as a continuum pulsed source combined with SuperK EXTEND-UV super continuum (NKT Photonics, model DUV); the wavelength was selected by coupling to a monochromator (Jobin-Yvon H10). The repetition rate was set to 19.4 MHz; the excitation pulse duration on this device is around 6 ps (full-width-at-halfmaximum, FWHM). The emission of fluorescence is detected, after passing through a polarizer oriented at the magic angle (54.73°) to the polarization of the excitation, through a double monochromator Jobin-Yvon DH10 on a hybrid PMT detector HPM-100-40 (Becker & Hickl).The instrumental response function of the equipment was measured by using a dilute suspension of polystyrene nanospheres in water (70 nm of diameter) as a scattering solution; it was typically about 130-160 ps FWHM. Decays were collected at a maximum counting rate of 17 kHz into 4096 channels using an acquisition card SPC-730 (Becker & Hickl). This limiting count rate was achieved by dilution in water of the sample and after sedimentation of the suspension in order to minimize as much as possible the scattering of the particles. The time per channel was set around 6 ps ch-1 in order to fit a full decay in the experimental time window.

Figure 1 .

 1 Figure 1. UV-Vis absorption spectra of supernatant solution during adsorption reaction of BSA (a) and RESV (b) on montmorillonite. Experimental conditions: BSA (1.0 mg.mL -1 ), RESV (1.0 mg.mL -1 ), Mt (1.0 mg.mL -1 ) in a buffer phosphate solution at pH 4.5 and ethanolic solution in ratio of 25%.

  5, 22.6, 23.8 and 30.1° were observed in RESV-Mt sample and at 16.4, 22.5 and 28.5° (2θ) in the BSA-RESV-Mt. Shifts in the d(001) reflexion in the RESV-Mt sample is probably due to a different hydration state of the interlayer space. In the pattern of BSA-Mt sample, the (001) reflexion is not visible due probably to an heterogeneity in the layer stacking or a delamination of the layers[START_REF] Bertacche | Host-guest interaction study of resveratrol with natural and modified cyclodextrins[END_REF][START_REF] Popova | Preparation of resveratrol-loaded nanoporous silica materials with different structures[END_REF]. These results will be confirmed by TEM.

Figure 2 .

 2 Figure 2. XRD patterns diffractograms and insert at low angle 1-10 o (2θ) of the montmorillonite (black), BSA absorbed onto clay (violet), RESV absorbed onto clay (pink) and BSA-RESV-Mt composite (green).

  (a) and 3(b), after BSA adsorption on Mt, the measured d001 value reaches3.2 nm indicating the intercalation and partial delamination of the protein in the interlayer space of the Mt. After resveratrol adsorption, the d001value varies from 1.5 to 1.85 nm, attesting of the heterogeneous incorporation of resveratrol, in Figure 3(c) and 3(d), in agreement with the XRD results. The sample containing both resveratrol and BSA presented different populations of layers: some are exfoliated and others are intercalated with a d spacing of BSA-RESV-Mt composite varying between 2.96-3.9 nm. From these results the incorporation of the BSA and RESV together is supposed to cause exfoliation of the clay mineral shown in Figure 3(e) and 3(f).

Figure 3 .

 3 Figure 3. Transmission electron micrographs (TEM) of the BSA-Mt (a) and (b); RESV-Mt (c) and (d);

Figure 4 .

 4 Figure 4. ATR-IR spectra of (a) free BSA and upon adsorption on Mt; (b) free RESV and upon adsorption onto Mt and (c) free BSA, free RESV and BSA-RESV-Mt composite.

3. 6 . 13 CFigure 5

 6135 Figure 5(a) exhibits spectra of the protein, the polyphenol and the composites prepared with the

Figure 5 (

 5 Figure 5(a). 13 C CP-MAS NMR spectra with a MAS frequency of 14kHz. Free RESV and free BSA and BSA-Mt, RESV-Mt and BSA-RESV-Mt composites.

Figure 5 (

 5 Figure 5(b). Deconvoluted spectra for BSA alone and upon adsorption on montmorillonite and BSA-RESV-Mt hybrid as illustrated by components attributed to carbonyl peaks.

Figure 5 (

 5 Figure 5(b) exhibits the decomposition of the spectra for the BSA, BSA-Mt and BSA-RESV-

Figure 6 (

 6 Figure 6(a). Fluorescence emission spectra of free BSA(black) and RESV-Mt (green), BSA-Mt(red) and BSA-RESV-Mt (blue) hybrids.

Figure 6

 6 Figure 6(b). Fluorescent decay of BSA-Mt (red), RESV-Mt (green), BSA-RESV-Mt (blue and violet) hybrids.

(

  390 nm), the average fluorescence lifetime 2.254 ns is longer than the one of RESV-Mt (under excitation 340 nm). The emission band of BSA has disappeared meaning that the Trp fluorescence is quenched by the polyphenol. In addition, we can observe a decrease of fluorescence lifetime to 1.302 ns measured at 340 nm. The fluorescence quenching of BSA is explained by a non-radiation energy transfer. The lifetime values of BSA-RESV-Mt τ1=2.587 ns, τ2=0.680 ns, τ3=0.03 ns are different from those of BSA-Mt, suggesting that the adsorbed BSA conformation is different in presence of RESV. Both fluorescence quenching and the different fluorescence lifetime of BSA indicates that RESV can interact with protein and quench

  turbidity or haze formation and can modulate the polyphenol concentration of wine. The results suggest a non denaturated intercalation of the protein in the interlayer space of montmorillonite by electrostatic forces and hydrogen bonding causing delamination and/or partial exfoliation of the layers of clay mineral observed in XRD and TEM. The direct interaction of resveratrol with montmorillonite can also be observed. It can be located on the surface and the edges of the clay mineral. Extensive penetration of the protein and resveratrol together causes exfoliation of the lamellar structure of the montmorillonite. Time resolved fluorescence experiments highlight the strong interaction between resveratrol and BSA and changes in the environment of amino acid residues due to energy transfer from Trp to resveratrol. This is the first example of coadsorption of a polyphenol and protein in presence of montmorillonite. Current studies are under progress to transpose this work on a real wine solution.

Table II .

 II Time resolved fluorescence results for free BSA and BSA-Mt, RESV-Mt and BSA-RESV-Mt hybrids.

		λEx(nm) λEm (nm) τ1	(Y1 %)	τ2 (ns) (Y2 %) τ3 (ns) (Y3 %) τav(ns) χ 2
			(ns)
	BSA	290	340 6.287 (88.8)	2.359	(10.7) 0.189	(0.6) 5.833 1.01
	BSA-Mt	290	340 4.556 (71.8)	1.553	(24.6) 0.427	(3.7) 3.667 1.18
	RESV-Mt	340	390 1.517 (33.4)	0.808	(42.4) 0.123	(24.2) 1.193 .93

Acknowledgements

This is work was supported by Capes/Cofecub Project (N 835/15) as well as with participation of the Archeology Molecular and Structural Laboratory and National School of Chemistry of Montpellier, in France and Fuel and Material Laboratory and Fast Solidification Laboratory, in Brazil.

Supplementary information :