André Lapidus
email: andre.lapidus@univ-paris1.fr

Stochastic Dominance of Any Type and Any Degree, and Expected Utility: A Unifying Approach

Keywords: stochastic dominance, expected utility, risk, decision. JEL classication: D81

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Introduction

Stochastic dominance as a way for partially ordering probability distributions was introduced in economics and decision theory through some pioneering articles published from the 1960's onwards 1 [START_REF] Quirk | Admissibility and Measurable Utility Functions[END_REF][START_REF] Hadar | Rules for Ordering Uncertain Prospect[END_REF][START_REF] Hanoch | Eciency Analysis of Choices Involving Risk[END_REF][START_REF] Rothschild | Increasing Risk: I. A Denition[END_REF][START_REF] Whitmore | Third-Degree Stochastic Dominance[END_REF]) and spawned later an important body of literature. From the very beginning, emphasis was laid on the equivalence between the attitude toward risk of a decision maker expressed by various types of stochastic dominance, and corresponding classes of utility functions, within an expected utility framework. Called congruence by [START_REF] Fishburn | Continua of Stochastic Dominance Relations for Bounded Probability[END_REF], this equivalence between a partial order, generated by stochastic dominance, and the intersection between complete orders, generated by a class of utility functions, led to the powerful conclusion that preference for a stochastically dominating distribution of such type might be represented by any utility function belonging to a specic class; and, conversely, that a decision maker endowed with any utility function from this class always prefers stochastically dominating distributions of this type. However, no general formulation was given till now.

The rst results [START_REF] Quirk | Admissibility and Measurable Utility Functions[END_REF][START_REF] Hadar | Rules for Ordering Uncertain Prospect[END_REF][START_REF] Hanoch | Eciency Analysis of Choices Involving Risk[END_REF][START_REF] Rothschild | Increasing Risk: I. A Denition[END_REF] related rst degree stochastic dominance (FSD) to the class of nondecreasing utility functions that is, to functions whose rst derivative was non-negative so that distributions 2 . In the last three papers [START_REF] Hadar | Rules for Ordering Uncertain Prospect[END_REF][START_REF] Hanoch | Eciency Analysis of Choices Involving Risk[END_REF][START_REF] Rothschild | Increasing Risk: I. A Denition[END_REF], it was also shown that second degree stochastic dominance (SSD) was linked to the class of non-decreasing concave utility functions whose rst and second derivatives are respectively positive and negative which characterize risk-aversion for expected utility decision makers. This result was extended to a more specic type of risk-aversion, taking skewness into account, by [START_REF] Whitmore | Third-Degree Stochastic Dominance[END_REF], who linked third degree stochastic dominance (TSD) to the class of non-decreasing concave utility functions with positive third derivative, provided the mean of the dominating distribution was at least equal to that of the dominated one. Later on, such attitude toward risk has been named prudence [START_REF] Kimball | Precautionary Savings in the Small and in the Large[END_REF], and unied accounts of the three rst degrees of stochastic dominance could be provided (see, for instance, Thorlund-Petersen 2001).

Again, the result on third degree stochastic dominance was generalized to n-th degree stochastic dominance, which was related to utility functions whose odd and even derivatives are respectively positive and negative, till the n-th degree [START_REF] Fishburn | Continua of Stochastic Dominance Relations for Bounded Probability[END_REF], therefore allowing a broader exploration of these renements of the attitude toward risk involved similarly in higher degree stochastic dominance and in the signing of higher degree derivatives of the utility function, like what was called temperance (4th degree, after [START_REF] Kimball | Precautionary Motives for Holding Assets[END_REF] or edginess (5th degree, Lajeri-Chaherli

2004).

Symmetrically, characterizations of risk-seeking appeared some years later, giving rise to what was to be called (in spite of an imperfectly xed vocabulary) inverse stochastic dominance 3 . [START_REF] Goovaerts | Insurance Premiums[END_REF] introduced second degree inverse stochastic dominance (SISD), which amounts to what Levy (2006, pp. 126-130) called risk-seeking stochastic dominance (RSSD), as related to the class of non-decreasing convex utility functions (with nonnegative rst and second derivatives). Like in the case of direct dominance, taking skewness into account led to third degree inverse stochastic dominance, which was divided by [START_REF] Zaras | Dominances Stochastiques pour Deux Classes de Fonctions d'Utilité[END_REF] between two types: third degree type 1 inverse stochastic dominance (TISD1), linked to the class of convex utility functions with non-positive third derivative [START_REF] Zaras | Dominances Stochastiques pour Deux Classes de Fonctions d'Utilité[END_REF]) therefore denoting imprudence; and third degree type 2 inverse stochastic dominance (TISD2), which also corresponds to convex utility functions, but with non-negative third derivative [START_REF] Goovaerts | Insurance Premiums[END_REF] which might be interpreted as prudence, like for TSD 4 .

Since the pioneering works of the 1960's and 70's, the concept of stochastic dominance has been subjected to several renements and extensions: quantile approach, introduction of riskless assets, extension to the measure of inequalities, multicriteria decision, almost dominance or fuzzy measures, consistency with non-expected utility theories like rank-dependent utility or cumulative prospect theory, etc. . . A review of this literature can be found in [START_REF] Guo | Stochastic Dominance and Its Applications in Portfolio Management[END_REF], and in the books by [START_REF] Sriboonchitta | Stochastic Dominance and Applications to Finance, Risk and Economics[END_REF] and by [START_REF] Levy | Stochastic Dominance Investment Decision Making under Uncertainty[END_REF]. But what seems to have been the most salient evolution concerns the way the orders generated by stochastic dominance 2 An intuitive knowledge of rst degree stochastic dominance in relation to decision seems to have been widely spread long before the contributions of 1969and 1970[START_REF] Pradier | De Usu Artis Conjectandi in Jure : Quid de Oeconomia (Politica) ?[END_REF] pointed out, Jacob Bernoulli, for instance, argued as early as the very beginning of 18th century, in the fourth part of his Ars Conjectandi, that what may be advantageous in one case, and can never harm, should be preferred [praeferendum est] to what is in no case benecial or harmful, and related this to popular wisdom expressed in a German saying, Hildt es nicht / so schadt es nicht (Bernoulli 1713, p. 320).

were considered. The original approach leads to view stochastic dominance of various types as generating possible partial preference preorders over a set of lotteries. This justies raising the question of the congruence with a class of utility functions within a context of expected utility.

However, once the idea of congruence has been accepted in principle, attention seems to have shifted toward the nature of the concept of stochastic dominance which allows the ecient selection of a lottery among a subset of lotteries, according to some utility functions belonging to the congruent class. Typical of this appraoach is Post and Kopa (2013) paper, which allows for comparing a given lottery with a discrete set of alternative lotteries, or with a set of linear combinations of these lotteries, on the basis of a generalization of the concept of convex stochastic dominance, which Fishburn introduced as early as 1974.

This paper goes a step back to the original approach, acknowledging that the question of congruence has been solved only for the few cases noted above. It follows on from Fishburn's solution, which concerned n-th degree direct stochastic dominance [START_REF] Fishburn | Continua of Stochastic Dominance Relations for Bounded Probability[END_REF]) but gives up this limitation to a specic type of stochastic dominance. Therefore, it rst provides a unifying treatment, now permitting any type of stochastic dominance, direct or inverse, and of any degree, in relation to classes of utility functions dened by the signing of their successive derivatives (section 2). This allows giving general conditions of congruence, that is conditions for the representation of each kind of stochastic dominance by a class of utility functions and reciprocally (section 3).

A formal framework

Denote X and Y two distinct random variables with support [a, b] ⊂ R. Assume that their respective distribution functions, F and G, are absolutely continuous, so that they can be represented by their density of probability functions f and g5 . Stochastic dominance of degree i (i = 1, ..., n) of f over g (X over Y ) amounts to the non-positivity of an index of dominance for each value of x on [a, b]. This index, which is said nal when i = n and intermediary otherwise, is the i-th integral of f (x) -g (x), each step of integration being either from a to x direct dominance or from x to b inverse dominance. The situation usually favored in the literature, where each step of integration from 1 to n is direct, will be termed hereafter complete direct dominance of degree n. Symmetrically, in an expected utility framework, preference given to f over g amounts to the non-negativity of the dierence between expected utilities, E u (f ) -E u (g), which depends on the properties of a utility function u assumed n-th dierentiable expressed in the signs of its successive derivatives, u 1 (x) , ...u n (x).

The formal framework within which the relation between stochastic dominance and expected utility is investigated results from the denitions of (1) the index and condition of stochastic dominance; (2) the classes of utility functions and of resulting preferences according to expected utility; and (3) a transformation procedure from the degrees of integration of the index of stochastic dominance to the degrees of derivation of the corresponding utility function.

Denition 1. Stochastic dominance

Let A ⊆ [n],
where [n] is the set of integers {1, ..., n}. Then:

(i) The stochastic dominance index of degree i ∈ [n], denoted H i A,n (x)
is dened by6 :

H i A,n (x) = 1 A (i) ˆx a H i-1 A,n (y) dy + (1 -1 A (i)) ˆb x H i-1 A,n (y) dy with H 0 A,n (x) = f (x) -g (x) .
This index is nal if i = n and intermediary if i < n.

(ii) Let f D A,n g denote the stochastic dominance of f over g at nal degree n, direct for all degrees in A and inverse for all degrees not in A. The condition of stochastic dominance of f over g is dened as:

f D A,n g ⇔ ∀x ∈ [a, b], H n A,n (x) ≤ 0
(since the two random variables, X and Y, are distinct, f = g, and strict inequality holds for at least one value of x).

Note.

1, l, m results from repeated integrations of f (x) -g (x)
, where the rst integration is from a to x 1 , the l-th from a to x l , the m-th from x m to b, and the n-th from a to x:

H n {1,...l,...n} (x) = ˆx a ... ˆb xm ... ˆxl a ... ˆx1 a (f (y) -g (y)) dy ...dx l-1 ...dx m-1 ...dx n-1 . If H n (x) ≤ 0 for each value of x on [a, b],
f is said to dominate stochastically g, directly at nal degree n and at intermediary degrees 1 and l, and inversely at intermediary degree m.

Remark 1. Let A (i) and A \(i) be identical subsets of [n], except that i belongs to A (i) and not to A \(i) . By construction, the following properties always hold:

H i A (i) (a) = H i A \(i) (b) = 0 (a) H i A (i) (b) = H i A \(i) (a) (b) H i A (i) (x) = H i A \(i) (a) -H i A \(i) (x) (c) H i A \(i) (x) = H i A (i) (b) -H i A (i) (x) (d) H i A (1) (x) = -H i A \(1) (x) (e)
Remark 2. Standard literature often favors situations of complete direct stochastic dominance, where the complementary of A, [n] A, is empty. For instance: rst (FSD), second (SSD), third (TSD) and n-th (NSD) degree stochastic dominance respectively correspond to congurations where the bipartitions of [n] are respectively such that A,

[1] A = ({1} , ∅), A, [2] A = ({1, 2} , ∅), A, [3] A = ({1, 2, 3} , ∅) and A, [n] A = ({1, ...n} , ∅).
The possibility that [n] A be not empty (that is, that f stochastically dominates g inversely at some intermediary or nal degree) is typically dealt with in order to explore risk-seeking, through either second degree inverse stochastic dominance (SISD or RSSD -risk-seeking stochastic dominance), where A, [2] A = ({1} , {2} ),

or third degree inverse stochastic dominance, either of type 1 (TSID1) when the third integration is from a to x so that A, [3] A = ({1, 3} , {2} ), or of type 2 (TISD2) when the third integration is from x to b so that A, [3] A = ({1} , {2, 3} ).

The construction of relevant classes of n-dierentiable utility functions u dened on [a, b] obeys the same principles as those which have governed the construction of an index of stochastic dominance. The starting point is the identication of the sets of degrees of derivation of a utility function, corresponding to either positive or negative derivatives.

Denition 2. Utility

Let B ⊆ [n].
Then:

(i) Dene a set U B,n of n-dierentiable utility functions as:

U B,n = u : [a, b] → R such that ∀i ∈ B, ∀x ∈ [a, b] , u i (x) ≥ 0 and ∀i ∈ [n] B, ∀x ∈ [a, b] , u i (x) ≤ 0}
(with for all i, strict inequalities for at least one x).

(ii) Following the expected utility approach, all decision makers whose utility function belongs to U B,n are said to prefer f to g when for each of them, the expected value of the utility of f is not smaller than that of g. Their common preference is denoted

f R B,n g: f R B,n g ⇔ ∀u ∈ U B,n , E u (f ) -E u (g) ≥ 0.
Remark 3. The signing of the derivatives of the utility function is usually linked to some typical attitudes toward risk through the ranking of random variables according to their expected utility.

For instance, it is well-known that increasing utility functions (1 ∈ B) characterize decision-makers with monotonous increasing preferences, whatever their attitude toward risk. When 2 ∈ [n] B, the decision-maker is risk-averse and, conversely, he or she is risk-seeking when 2 ∈ B. The intuitive meaning of 3 ∈ B is this of prudence [START_REF] Kimball | Precautionary Savings in the Small and in the Large[END_REF], so that 3 ∈ [n] B corresponds to imprudence. Higher degrees of derivation are not that easy to interpret. However, 4 ∈ [n] B and 4 ∈ B are currently viewed as, respectively, temperance [START_REF] Kimball | Precautionary Motives for Holding Assets[END_REF]) and intemperance.

Similarly, 5 ∈ B and 5 ∈ [n] B correspond to edginess [START_REF] Lajeri-Chaherli | Proper Prudence, Standard Prudence and Precautionary Vulnerability[END_REF] and to what might be called calmness. Typical utility functions like the logarithmic ones, which give rise to monotone increasing preferences, risk-aversion, prudence, temperance, edginess and so on, are characterized by alternatively positive and negative derivatives, and were sometimes called mixed risk averse utility functions [START_REF] Caballé | Mixed Risk Aversion[END_REF]: B = {1, 3, ...} and [n] B = {2, 4, ...}). Eeckhoudt and Schlessinger (2006) have shown the equivalence between such higher order risk attitudes and preferences over particular classes of lottery pairs, involving zero-mean independent noise random variables.

The relation between stochastic dominance and expected utility is taken up through a relation between two elements A and B of the powerset of [n] . Consider the Procedure AB hereafter, which allocates i (i = 1, ..., n) between B and [n] B as a degree of derivation, according to its belonging, as a degree of integration, to A or [n] A, and to the belonging of i -

1 to B or [n] B: Procedure AB • For all i ∈ A    if i -1 ∈ B, then i ∈ [n] B if i -1 ∈ [n] B or i = 1, then i ∈ B • For all i ∈ [n] A    if i -1 ∈ [n] B or i = 1, then i ∈ [n] B if i -1 ∈ B, then i ∈ B Denition 3. Transformation
The transformation of i = 1, ..., n, as degrees of integration belonging to either A or [n] A, into degrees of derivation belonging to either B or [n] B, is performed by

φ : 2 [n] → 2 [n] , A → φ (A) = B (1)
3 Congruence between stochastic dominance and expected utility

The issue is now that of the conditions of consistency between two partial orders: according to stochastic dominance, D A,n , and to expected utility, R B,n . Though presented dierently, such consistency was called congruence by P. Fishburn (1976, p. 303):

Denition 4. Congruence A stochastic dominance order D A,n and an expected utility order R B,n are congruent when, for each f and g, f D A,n g ⇔ f R B,n g.

It will be shown (Proposition 1) that, provided additional conditions on the bounds of the distributions are satised, congruence is achieved when B is the image of A through the transformation φ dened in (1). The structure of the relation between the orders generated by stochastic dominance and expected utility is further investigated in Proposition 2.

The following lemma shows that each possible allocation of the degrees of integration corresponds through φ to an allocation of the degrees of derivation, and reciprocally:

Lemma 1. ∀A ∈ 2 [n] , ∃B ∈ 2 [n] : B = φ (A)
and ∀B ∈ 2 [n] , ∃A ∈ 2 [n] : B = φ (A).

Proof. Check with Procedure AB that φ in (1) is a bijection of 2 [n] into itself.

A general condition for congruence between stochastic dominance and expected utility orders is presented in the following proposition.

Proposition 1. For all A, B ∈ 2 [n] such that B = φ (A), the two following propositions are equivalent for all f and g: 1. f R B,n g (expected utility). 2. f D A,n g (stochastic dominance)

and, if n ≥ 3, for all i = 2, ..., n -1:

H i A,n (γ i ) ≤ 0 (where γ i = 1 A (i) b + (1 -1 A (i)) a)
(conditions on upper or lower bounds).

Proof. See Appendix.

An intuitive interpretation of Proposition 1 is that provided conditions on bounds are satised, the partial order on random variables generated by any type of stochastic dominance, say D A,n , is identical to the intersection between all the complete preference preorders on random variables underlying all the possible utility functions belonging to U B,n , when B = φ (A). Proposition 1 may also be viewed as establishing a link between subjective assessments and objective properties. It means that on the one hand, what all possible decision makers, characterized by a utility function from the same class, have in common evidently rests on their respective subjective preferences; but, on the other hand, this common ranking of random variables may alternatively be viewed as model-free, that is, as depending only on some objective properties of the distribution functions expressed in stochastic dominance.

We know that in the case of complete direct n-th degree stochastic dominance, the conditions on upper bounds can be viewed as algebraic combinations of the dierences between the successive moments around zero of the distributions [START_REF] Jean | The Identication of Stochastic Dominance Ecient Sets by Moment Combination Orderings[END_REF]. A typical illustration, anticipated by [START_REF] Whitmore | Third-Degree Stochastic Dominance[END_REF], is that in order to have congruence between complete direct third degree stochastic dominance (A = {1, 2, 3}) and the class of increasing concave utility functions with a nonnegative third derivative (B = {1, 3}), the dierence between means (rst moments), E (f )-E (g),

had to be non-negative 7 . The conclusion of [START_REF] Jean | The Identication of Stochastic Dominance Ecient Sets by Moment Combination Orderings[END_REF] still holds in the case of Proposition 1 where dominance is not necessarily direct at each step since, by construction, the value of any index of dominance at the relevant bound is itself an algebraic sum of some complete direct dominance indices of equal and smaller degrees at the upper bound: any index of degree i at the relevant bound is therefore a linear combination of the dierences between the moments around zero j M f (0) -j M g (0) of all degrees j from 1 to i -1.

B 1 = φ ({1}) = {1}
that is with monotonous non-decreasing preferences, entailing universal preference for the dominating random variable, whatever the decision maker's attitude toward risk.

7

Recall that E (f ) -E (g) ≥ 0 is already a consequence of complete direct stochastic dominance at degrees 1 or SSD (second degree stochastic dominance: [START_REF] Hadar | Rules for Ordering Uncertain Prospect[END_REF][START_REF] Hanoch | Eciency Analysis of Choices Involving Risk[END_REF][START_REF] Rothschild | Increasing Risk: I. A Denition[END_REF] (mean preserving spread subcase MPS)) corresponds to

A 2 = {1, 2}, and was acknowledged congruent with

B 2 = φ ({1, 2}) = {1}
that is with monotonous non-decreasing preferences and risk-aversion.

TSD (third degree stochastic dominance: Whitmore 1970) corresponds to A 3 = {1, 2, 3},

and was acknowledged congruent with

B 3 = φ ({1, 2, 3}) = {1, 3}
that is with monotonous non-decreasing preferences, risk-aversion, positive skewness-seeking and negative skewness-aversion (typically, a decision maker interested in insuring his or her house, and in buying public lotteries tickets). This corresponds to what is usually called prudence since [START_REF] Kimball | Precautionary Savings in the Small and in the Large[END_REF]

B n = φ ({1, 2, 3, ...n}) = {1, 3, ...}
that is, with a class of utility functions whose odd and even derivatives are respectively non-negative and non-positive.

The same bringing together is a bit less immediate in the cases of stochastic dominance of degrees 2 and 3, related to monotonous non-decreasing preferences and risk-seeking, from the point of view of expected utility. These dierent cases were usually called inverse stochastic dominance (see, for example, Zaras 1989), and have in common that 2 / ∈ A n . A technical but simple problem of presentation arises from the fact that the corresponding indices of stochastic dominance generally relied on decumulative distribution functions, of type F (x) = ´b x f (y) dy, instead of cumulative functions F (x) = ´x a f (y) dy. The resulting successive integrals were therefore computed on the basis of H 1 ∅,1 (x), instead of H 1 {1},1 (x), although the concerned decision makers were endowed with non-decreasing preferences. Nonetheless, since H 1 ∅,1 (x) = -H 1 {1},1 (x), any higher degree of the intermediary or nal index is such that, assuming that 1 is an element

of A n , H i An\{1},n (x) = -H i An,n (
B 2 = φ ({1}) = {1, 2}
that is, with monotonous non-decreasing preferences and risk-seeking.

TISD1 (third degree type 1 inverse stochastic dominance: Zaras 1989) corresponds to A 3 = {1, 3} and was acknowledged congruent with

B 3 = φ ({1, 3}) = {1, 2}
that is, with monotonous non-decreasing preferences, risk-seeking, positive skewness-aversion and negative skewness-seeking (imprudence).

TISD2 (third degree type 2 inverse stochastic dominance: Goovaerts, De Vylder and Haezendonck 1984) corresponds to A 3 = {1} and was acknowledged congruent with

B 3 = φ ({1}) = {1, 2, 3}
that is, with monotonous non-decreasing preferences, risk-seeking, positive skewness-seeking and negative skewness-aversion (prudence).

However, the three following instances of stochastic dominance are still missing in current literature, and are worth being noted.

1. The rst instance is that of situations in which a decision maker would prefer the distribution whose probability to get at least this income is the smaller that is, the stochastically dominating random variable at degree 1 when A 1 = ∅:

FISD (stochastic dominance inverse for the degree 1). A 1 = ∅. This amounts to congruence with

B 1 = φ (∅) = ∅
that is with non-increasing utility functions. Of course, such situations might seem of little practical interest except in case the purpose were to consider the behavior toward risk of a decision maker whose objective would be to ruin himself or herself. However, it makes clear that, contrary to a hasty conclusion, though preference for distributions which are stochastically directly dominated at degree 1 by other distributions might seem a bit strange, it is by no way constitutively irrational.

2. But it is also obvious that, by contrast to TSD which has been extensively studied after Whitmore's 1970 paper, in relation to the more familiar idea of DARA (decreasing absolute risk aversion), a second type of third stochastic dominance TSD2, homologous to TISD2, doesn't seem to have aroused special interest:

TSD2 (third degree stochastic dominance, type 2) corresponds to A 3 = {1, 2}. According to Proposition 1, granted that third degree conditions on bounds are satised, TSD2 is congruent with

B 3 = φ ({1, 2}) = {1}
that is, with monotonous non-decreasing preferences, risk-aversion, positive skewness-aversion and negative skewness-seeking which amounts to imprudence. After all, we all know people who dislike risk, never buy public lotteries tickets, and nonetheless would reject the idea of insuring their house if they were not legally obliged to subscribe to an insurance contract.

3. Finally, whereas complete direct stochastic dominance of any degree n (NSD, where A n = {1, ..., n}) has been regularly taken up after [START_REF] Fishburn | Continua of Stochastic Dominance Relations for Bounded Probability[END_REF]paper (see, for instance, Levy 2006, pp. 131-132), such was not the case for stochastic dominance direct or indirect at any intermediary degree i or nal degree n, that is, when A n is any element of 2 [n] . Filling this gap was the main purpose of this article.

Remark 4. Stochastic dominance orders and classes of utility functions

According to a well-known sucient rule of stochastic dominance orders, if f stochastically dominates g at degree k, it also dominates g at degrees k + 1, k + 2, ... (see, for instance, Levy 2006, pp. 119-20, for the relations between complete direct dominances of degrees 1, 2 and 3). Typically, this means that risk-averse and risk-seeking decision makers with positive monotone preferences might disagree on the order between random variables not ordered by rst degree direct stochastic dominance, but that they agree on the order of random variables generated by the latter, which is consistent with both direct and inverse stochastic dominance of degree 2. This result can be easily generalized and related to preferences according to expected utility in the following proposition:

Proposition 2. Let m and n be two integers such that m

≤ n. If A m , B m ∈ 2 [m] and A n , B n ∈ 2 [n]
satisfy (a) and (b):

(a) B m = φ (A m ), and

B n = φ (A n ) (b) A m ⊆ A n , [m] A m ⊆ [n] A n or B m ⊆ B n , [m] B m ⊆ [n] B n then: 1. D Am,m ⊆ D An,n 2. U Bn,n ⊆ U Bm,m
(By abuse of notation, D Am,m and D An,n stand for the set-equivalents of the corresponding binary relations).

Proof. First note that, in reason of the denition of φ by (1), the two alternative conditions in (b) a, b] so that its integral, either from a to x, or from x to b, H m+1 (x) ≤ 0, and so on till H n (x). As a result, f D Am,m g ⇒ f D An,n g.

are equivalent: A m ⊆ A n , [m] A m ⊆ [n] A n ⇔ B m ⊆ B n , [m] B m ⊆ [n] B n 1. Obvious, since H m (x) ≤ 0 for each x over [

Observe that

, since B m ⊆ B n and [m] B m ⊆ [n] B n , u ∈ U Bn,n ⇒ u ∈ U Bm,m .
Bringing together the two parts of Proposition 2 shows the structure of the relation between the two partial orders generated either by stochastic dominance, or by expected utility. At the nal stage, Proposition 1 makes clear that when conditions on bounds are satised, they lead to the same result: D Am,m and R Bm,m order the same random variables (that is, D Am,m = R Bm,m ), just like D An,n and R Bn,n (that is, D An,n = R Bn,n ) and it can be concluded from Proposition 2

(1.) that their set is expanding from m to n. However, Proposition 2 (2.) shows that the same conclusion is obtained dierently from the expected utility point of view. Moving from m to n rst generates a contraction: for instance, such utility function, which was included in U Bm,m does not belong any more to U Bn,n , because of the signs of its derivatives from m + 1 to n. But since the resulting partial order R is the intersection between the complete orders underlying each utility functions belonging to U , R depends on less complete orders when moving from m to n. So that it is expanding from R Bm,m to R Bn,n , whereas U is contracting from U Bm,m to U Bn,n .

The aim of this paper was to set out a framework for analyzing the relations between subjective preferences between random variables founded on specic classes of utility functions, and the modelfree, objective proprieties, of the probability distributions of these random variables expressed in corresponding types of stochastic dominance. This framework rst consists in the construction of an index of stochastic dominance, which allows either direct or inverse dominance at any intermediate or nal degree. A proposition is demonstrated which establishes the conditions of congruence (in the sense [START_REF] Fishburn | Continua of Stochastic Dominance Relations for Bounded Probability[END_REF]) between the orderings generated by stochastic dominance and by classes of utility functions, extending the results of the pioneering works of [START_REF] Hadar | Rules for Ordering Uncertain Prospect[END_REF], Henoch and [START_REF] Hanoch | Eciency Analysis of Choices Involving Risk[END_REF], [START_REF] Whitmore | Third-Degree Stochastic Dominance[END_REF], [START_REF] Fishburn | Continua of Stochastic Dominance Relations for Bounded Probability[END_REF], [START_REF] Goovaerts | Insurance Premiums[END_REF] and [START_REF] Zaras | Dominances Stochastiques pour Deux Classes de Fonctions d'Utilité[END_REF]. Comparison with previous contributions also leads to the identication of neglected kinds of stochastic dominance, like a third degree type 2 stochastic dominance (TSD2) and the congruent class of utility functions.

Appendix: Congruence between stochastic dominance and expected utility (Proposition (1))

A Preliminaries concerning the construction of the expression of E u (f )-E u (g)

[At least till equation ( 2), this sub-section can be skipped by the readers familiar with the technique of demonstration already used in most pioneering papers since [START_REF] Hadar | Rules for Ordering Uncertain Prospect[END_REF] or [START_REF] Hanoch | Eciency Analysis of Choices Involving Risk[END_REF]]

The proof of Proposition 1 follows readily from the repeated integration by parts until degree n of E u (f ) -E u (g) = ´b a u (x) (f (x) -g (x)) dx = ´b a u (x) H 0 A,n (x) dx, the non-negativity of E u (f ) -E u (g) for all u ∈ U B,n being equivalent to f R B,n g. The result is quite classical, except that at each step of integration, it has to be given according to the belonging of i to either A or [n] A -which draws on Remark 1 (a)-(e). Starting from the rst degree of integration gives:

• Degree 1:

E u (f ) -E u (g) = ˆb a u (x) H 0 A (x) dx                            1 ∈ A : = u (x) H 1 A (x) b a -´b a u 1 (x) H 1 A (x) dx = -´b a u 1 (x) H 1 A (x) dx 1 ∈ [n] A : = -u (x) H 1 A (x) b a + ´b a u 1 (x) H 1 A (x) dx = ´b a u 1 (x) H 1 A (x) dx.
Note that the sign before the integrals in the right-hand side of the equations is negative when 1 belongs to A, and positive when it does not.

Example 2 .

 2 Usual and neglected types of stochastic dominance in relation to expected utility Such representation of congruence allows nding again the usual formulations of stochastic dominance from the end of the 1960's to the 1980's, and their correspondence with certain classes of utility functions. In return, it also allows nding the neglected or missing categories which can now be put to the fore. This bringing together is obvious in the case of complete direct dominance in the following usual cases, A n and B n denoting elements of the powerset 2 [n] : FSD (rst degree stochastic dominance: Quirk and Saposnik 1962 (discrete case); Hadar and Russell 1969; Hanoch and Levy 1969) corresponds to A 1 = {1}. It was acknowledged congruent with

  's paper, in relation to the positive sign of the third derivative of the utility function. NSD (n-th degree stochastic dominance: Fishburn 1976) corresponds to complete direct stochastic dominance with A n = [n] = {1, 2, 3...n} and was acknowledged congruent with

  x) (see Remark 1 on Denition 1). So that the non-positivity requirement for the index of stochastic dominance was usually replaced by a non-negativity condition insofar as it was applied to inverse dominance at intermediate or nal degree 2. Taking this into account allows establishing the following three correspondences: SISD (second degree inverse stochastic dominance: Goovaerts, De Vylder and Haezendonck 1984) or RSSD (risk-seeking stochastic dominance: Levy 2006) corresponds to A 2 = {1}, and was acknowledged congruent with

For instance, in a pioneering paper where they applied stochastic dominance to the question of of inequality measures,[START_REF] Muliere | A Note on Stochastic Dominance and Inequality Measures[END_REF] named inverse stochastic dominance the integration of dierences between the inverses of cumulative distribution functions, F -1 and G -1 , and not between decumulative functions.

For a systematic account of risk attitudes involved in the signing of the various derivatives of the utility function, see Eeckhout and Schlesinger 2006.

Usual regularities, like absolute continuity on a compact interval of R, are assumed throughout this paper for the sake of simplicity, in order to escape possible complications (requiring the use of Riemann-Stieltjes or Lebesgue-Stieltjes integrals) when Lebesgue's criterion for Riemann integrability does not hold for f and g. See the clarication and restatement of the rst results of[START_REF] Hanoch | Eciency Analysis of Choices Involving Risk[END_REF] and[START_REF] Hadar | Stochastic Dominance and Diversication[END_REF] by[START_REF] Tesfatsion | Stochastic Dominance and the Maximization of Expected Utility[END_REF].

Recall that the indicator function 1 A in Denition 1 is an application from[n] to {0, 1} which, for i ∈ [n], yields 1 if i belongs to A, and 0 if it does not.

• Degree 2:

To sum up, the sign before the integral of u 2 (x) H 2 A (x) in the right-hand side of each equation is positive when 2 is in A and when, at degree 1, the sign before the integral of u 1 (x) H 1 A (x) in the right-hand side is negative; or when 2 is not A and when, at degree 1, the sign before the integral of u 1 (x) H 1 A (x) in the right-hand side is positive. Conversely, it is negative when 2 is not A and when, at degree 1, the sign before the integral of u 1 (x) H 1 A (x) in the right-hand side is negative; or when 2 is in A and when, at degree 1, the sign before the integral of u 1 (x) H 1 A (x) in the right-hand side is positive. On the other hand, the sign before u 1 (γ 2 ) H 2 A (γ 2 ), -where γ 2 equals a or b -still in the right-hand side of each equation, is positive when at degree 1, the sign before the integral of u 1 (x) H 1 A (x) in the right-hand side is itself positive, and it is negative otherwise.

Carrying on till degree n yields:

• Degree n:

where for all i ∈ [n] , sgn (i) = 1 -21 C (i) .

The set C ∈ 2 [n] is the outcome of Procedure AC hereafter, which allocates each i to either C or [n] C and, therefore, determines the values of sgn (n) and sgn (i -1) in (2).

Procedure AC

• For all

In an expected utility framework, f R B,n g is equivalent to the non-negativity of E u (f ) -E u (g) for all u ∈ U B,n , as given in (2).

Note that Procedure AB and Procedure AC above are identical, C in AC standing for B in AB, so that B = C. An immediate consequence is that in (2), the signs of sgn (i -1) and u i-1 (x),

as well as those of sgn (n) and u n (x), are always opposite.

Suciency:

for all i = 2, ...n -1 :

By hypothesis, when n ≥ 3, for all i between 2 and n1, H i A,n (γ i ) ≤ 0 on the right-hand-side of (2). Still by hypothesis, H n A,n (x) ≤ 0 for all x on [a, b] -which also means that at the particular value γ n , H n A,n (γ n ) ≤ 0. Since the signs of sgn (i) and u i (x) are always opposite for all u ∈ U B,n ,

In the same way, sgn (n) ´b a u n (x) H n A,n (x) dx ≥ 0, on the right-hand-side of (2). And since the whole right-hand-side of (2) is non-negative, E u (f ) -E u (g) ≥ 0 for all u ∈ U B,n , which amounts to f R B,n g.

Necessity:

and, if n ≥ 3, for all i = 2, ...n -1 :

Assume there exists y such that H n A,n (y) > 0. As a result, since the signs of sgn (n) and u n (y) are always opposite, if n ∈ B (resp., n ∈ [n] (B)), a suciently high (resp., low) value of u n (y) would allow the right-hand-side of (2) to become negative. And, since E u (f ) -E u (g) would become negative, this would contradict the assumption that f R B,n g. One concludes that such value y does not exist, so that, for all x on [a, b], H n A,n (x) ≤ 0, which means that f D A,n g. Since this non-positivity of H n A,n (x) also holds for the the upper and lower bounds of x, and consequently for γ n , H n A,n (γ n ) ≤ 0 in (2). Let us turn, now, to the left-hand part of the right-hand-side of (2), i.e., n i=2 sgn (i -1) u i-1 (γ i ) H i A,n (γ i ). Note that this left-hand part only exists when n ≥ 2, and that we have just shown that it is non-negative for n = 2, since H n A,n (γ n ) ≤ 0. Focusing on the cases when n ≥ 3, imagine that there exists a value k of i (2 ≤ i ≤ n -1) for which γ k would lead to H k A,n (γ k ) > 0. Here again, a high enough absolute value of u k-1 (γ k )


Richard Fischl

Kresten Bertelsen

Fanny Gaillard

Susana Coelho

Markus Klinger

Catherine Boyen

Mirjam Czjzek
email: mirjam.czjzek@sb-roscoff.fr

Cécile Hervé
email: cecile.herve@sb-roscoff.fr

Gurvan Michel

The cell wall active mannuronan C5-epimerases in the model brown alga Ectocarpus: from gene context to recombinant protein

Keywords: alginate, brown algae, Ectocarpus, mannuronan C5-epimerase

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

Introduction

Alginate is a major cell wall constituent of brown algae (Phaeophyceae) also synthesized by some bacteria of the Azotobacter and Pseudomonas genera. This polysaccharide is of important commercial value with an annual worldwide production of about 45,000 tons, extracted from various types of brown seaweeds including species mostly from the orders of Laminariales and Fucales [START_REF] Smidsrod | Chemistry and physical properties of alginates[END_REF]. In many different applications, alginates act as stabilizing, thickening, and/or gelling agents. The main uses are food additives. Alginates are increasingly used in cosmetics, health care and pharmaceuticals [START_REF] Zhao | Alginate modified nanostructured calcium carbonate with enhanced delivery efficiency for gene and drug delivery[END_REF]) and lately received a renewed interest for their use in nanotechnology [START_REF] Kovalenko | A Major Constituent of Brown Algae for Use in High-Capacity Li-Ion Batteries[END_REF] and in fuel and ethanol productions [START_REF] Enquist-Newman | Efficient ethanol production from brown macroalgae sugars by a synthetic yeast platform[END_REF][START_REF] Wargacki | An engineered microbial platform for direct biofuel production from brown macroalgae[END_REF][START_REF] Wei | Marine macroalgae: an untapped resource for producing fuels and chemicals[END_REF].

Alginate is a linear copolymer of 1,4-linked uronic acid epimers differing only at C5: the β-D-mannuronic acid (M) and the α-L-guluronic acid (G). These residues are arranged in homopolymeric regions of MM-and GG-blocks, interspaced with random arrangements of both monomers (MG-blocks), each of which have different conformational preferences and, hence, different physico-chemical properties [START_REF] Donati | New hypothesis on the role of alternating sequences in calcium-alginate gels[END_REF][START_REF] Heyraud | NMR spectroscopy analysis of oligoguluronates and oligomannuronates prepared by acid or enzymatic hydrolysis of homopolymeric blocks of alginic acid. Application to the determination of the substrate specificity of Haliotis tuberculata alginate lyase[END_REF]. While the MM-block regions do not associate in the presence of divalent cations, GG-blocks regions will form 'egg-box' junctions with calcium [START_REF] Grant | Biological interactions between polysaccharides and divalent cations : the egg-box model[END_REF], bridging two antiparallel chains and leading to gel formation. The mechanical strength of an alginate gel thus correlates with the proportion of GG-blocks (Figure 1). Alternating structures take up rather disordered conformations, yielding more flexible gels.

G-rich alginates with a M/G ratio ≤ 1.0 constitute around 25% of the total alginate market, with a yearly growth of about 10%, mainly in the food market. This type of alginate is also widely used in the pharmaceutical area [START_REF] Draget | Chemical, physical and biological properties of alginates and their biomedical implications[END_REF]. As most seaweed species have a M/G ratio of 1.2-1.6 in alginate [START_REF] Haug | Uronic acid sequence in alginate from different sources[END_REF][START_REF] Smidsrod | Chemistry and physical properties of alginates[END_REF], brown seaweeds with a naturally low M/G ratio of 0.7-0.5 are in high demand. Alginate with an M/G ratio of 0.4 is exclusively found in stipes of Laminaria hyperborea and Lessonia trabeculata [START_REF] Smidsrod | Chemistry and physical properties of alginates[END_REF]. The M/G ratio of alginate from the whole L. hyperborea plant is 1.0 [START_REF] Draget | Alginates from Algae[END_REF]) and separation of fronds from stipes is laborious.

Thus, novel ways to ensure an accessible and efficient production of alginate with low M/G ratios are desirable. Although chemical epimerisation can be performed on the polysaccharide [START_REF] Pawar | Alginate derivatization: a review of chemistry, properties and applications[END_REF], the high temperature required for this process leads to undesirable depolymerisation. At lower temperature actual processes use chemical catalysts that are difficult to remove from polyanionic material with strong cationic complexing properties, such as alginates.

The alginate biosynthesis pathway is yet not fully resolved in brown algae but genes homologous to their bacterial counterparts have been identified [START_REF] Michel | The cell wall polysaccharide metabolism of the brown alga Ectocarpus siliculosus. Insights into the evolution of extracellular matrix polysaccharides in Eukaryotes[END_REF]. The final steps were proposed to be the production of polymannuronate and the subsequent epimerization on the polymer of the M residues into G residues; the latter being catalysed by putative mannuronan C5-epimerases (ManC5-Es) resembling bacterial ManC5-Es [START_REF] Nyvall | Characterization of mannuronan C-5-epimerase genes from the brown alga Laminaria digitata[END_REF]. The biochemical characterisation of native and hybrid ManC5-E of bacterial sources has shown their abilities to generate distinct alginate structures [START_REF] Wolfram | Catalytic mechanism and mode of action of the periplasmic alginate epimerase AlgG[END_REF], Bjerkan et al. 2004[START_REF] Hartmann | Mode of action of recombinant Azotobacter vinelandii mannuronan C-5 epimerases AlgE2 and AlgE4[END_REF]. The presence of a large number of ManC5-E genes in brown algae also suggests that their gene products might yield polymers with different patterns of G and M distributions. Given that alginates in brown algae affect the texture of the cell wall in a similar manner as do pectins in the cell walls of land plants, this multigene situation is reminiscent of pectin methyl-esterases (PMEs) [START_REF] Micheli | Pectin methylesterases: cell wall enzymes with important roles in plant physiology[END_REF]. By analogy to PMEs, which can yield partially methylesterified pectins with either random or blockwise distribution of the methyl groups, the different isoforms of ManC5-E might yield alginates with different patterns of GG-block distribution. Moreover, the diversity of enzymes present in brown algae to remodel the algal cell wall polysaccharide, is much larger than that found in bacterial alginate synthetizing systems [START_REF] Bjerkan | Construction and analyses of hybrid Azotobacter vinelandii mannuronan C-5 epimerases with new epimerization pattern characteristics[END_REF]. It can thus be assumed that these algal enzymes could be used to produce alternative "tailor-made alginates" with novel physico-chemical properties Heterologous expression of algal genes is notoriously difficult and has presented a bottleneck in projects aiming at characterising enzymes of macroalgal origins (Groisillier et al. 2010, Kurland and[START_REF] Kurland | Errors of heterologous protein expression[END_REF]. Recent access to the first genome sequence of a brown alga [START_REF] Cock | The Ectocarpus genome and the independent evolution of multicellularity in brown algae[END_REF]) allowed the identification of all the genes shaping the ManC5-E family in Ectocarpus. By providing multiple targets, it opened a new door toward the expression of recombinant ManC5-Es. We took advantage of a medium throughput strategy we recently developed for protein expression [START_REF] Groisillier | MARINE-EXPRESS: taking advantage of high throughput cloning and expression strategies for the post-genomic analysis of marine organisms[END_REF], to heterologously express the ManC5-Es from Ectocarpus. Our work has generated the first recombinant and active ManC5-E from a brown alga. An additional comprehensive analysis of the gene context and transcript expression during the life cycle of Ectocarpus has provided new insights into the possible roles of ManC5-E genes in the brown algae.

Results

Ectocarpus contains a multigenic family of putative ManC5-Es

The functional annotation of the genome of Ectocarpus previously indicated the occurrence of 28 genes encoding putative mannuronan C5-epimerases (ManC5-Es) [START_REF] Michel | The cell wall polysaccharide metabolism of the brown alga Ectocarpus siliculosus. Insights into the evolution of extracellular matrix polysaccharides in Eukaryotes[END_REF]. A more detailed analysis led to the identification of additional genes that shape a final multigenic family of 31 members that accounts for 0.19% of the total predicted gene number in Ectocarpus. The gene names and the corresponding acronyms (which use a 5-characters code) are listed in Supplementary Table S1. For phylogenetic analyses, 10 genes were discarded due to doubtful predicted gene structure. The phylogenetic tree indicated that at least four groups of ManC5-Es can be identified in Ectocarpus (Figure 2A). Among them one group contains modular proteins that are predicted to contain additional modules appended to the catalytic ManC5-E domain: a schematic diagram showing the protein structure of the eight candidates is shown in Figure 2B and Supplementary Figure S1. Five of the eight encoded proteins are predicted to contain a signal peptide targeting the protein to the cell surface. In addition to the catalytic ManC5-E domain, three main supplementary domains are found: the Wall Sensing Component (WSC), the AGP-protein core domain and a module of unknown function. The WSC domains are known to be involved in cell-wall sensing in yeast. They also shape a large multigenic family in the phylogenetic-distant brown algae where they are expected to play similar functions [START_REF] Cock | The Ectocarpus genome and the independent evolution of multicellularity in brown algae[END_REF][START_REF] Hervé | Arabinogalactan-proteins have deep roots in Eukaryotes: identification of genes and epitopes in brown algae and their role in Fucus serratus embryo development[END_REF]. Arabinogalactanproteins (AGPs) are highly glycosylated, hydroxyproline-rich proteins found at the cell surface of plants, but they have also been recently investigated in brown algae [START_REF] Hervé | Arabinogalactan-proteins have deep roots in Eukaryotes: identification of genes and epitopes in brown algae and their role in Fucus serratus embryo development[END_REF]. In both systems they play key roles in developmental processes. The module of unknown function (MUF) was identified manually and up to now we could not detect any significant homology to other proteins or domains deposited in public databases. The MUF sequence is highly conserved (Supplementary Figure S2) and is often found as repetitive motifs in the ManC5-E sequences under concern (i.e. MEP3, MEP1, MEP4) (Figure 2B). The protein motif consists of a sequence of 56 amino acids in average, and displays a particular amino acid composition (cysteine-, glycine-and serine-rich) (Supplementary Figure S2). Such particular amino acid patterns are often unstructured, i.e. linkers, or display propensity to selfassemble and we thus speculate that these domains might play a role in structuring these proteins within the cell wall.

The ManC5-E family includes genes that are differentially regulated during the lifecycle of Ectocarpus

Thirty of the 31 predicted ManC5-E gene family members are transcribed during the life cycle of Ectocarpus (http://bioinformatics.psb.ugent.be/orcae/overview/EctsiV2). The remaining gene that exhibits no expression in any of the conditions tested (MEP30), contains a large insertion of the Ectocarpus virus (EsV1-163) which may alter gene expression. The life cycle of Ectocarpus involves an alternation between two morphologically distinct generations, the sporophyte and the gametophyte [START_REF] Cock | Emergence of Ectocarpus as a model system to study the evolution of complex multicellularity in the brown algae[END_REF]. Microarray expression data [START_REF] Coelho | OUROBOROS is a master regulator of the gametophyte to sporophyte life cycle transition in the brown alga Ectocarpus[END_REF]) obtained for a subset of the ManC5-E genes revealed an interesting expression pattern during the life cycle generation stages. While the majority of these twelve genes were indistinctively expressed in both generations of the life cycle, four ManC5-E genes (MEP6, MEP26, MEP27, MEP28) were preferentially expressed in the gametophyte generation (Figure 3, t-test P<0.05), suggesting a role for these genes in the development of the gametophyte. These four genes belong to different ManC5-E phylogenetic clusters. Therefore, within a common cluster of ManC5-E genes, some can be either mostly expressed in the gametophytes, or showing no difference in their expression pattern. We conclude that the expression of some of the ManC-Es is differentially regulated during the life cycle of Ectocarpus and that the ManC5-Es do not cluster in the phylogenetic tree in relation to their expression patterns.

Gene structure analyses

Gene exon-intron structures analyses of the ManC5-E family were conducted. Figure 2C shows the mosaic structure of all the ManC5-E genes included in the tree. The average gene length of ManC5-E genes is larger than the total average gene length in Ectocarpus (Table 1A). In addition most ManC5-E genes possess more than 9 introns, with the average intron number being nearly twice that of the other Ectocarpus genes. The most common system for getting recombinant proteins used bacterial strains derived from Escherichia coli. The ManC5-E genes from Ectocarpus were examined regarding the codon usage preference: three of the 5 rare codons for E. coli show critical relative adaptiveness values (Table 1B). All these characteristics are likely to render heterologous expression of the ManC5-E genes difficult and we payed special attention during the process (described below) to overcome this problem.

Expression, purification and refolding of the mannuronan C5-epimerase MEP13

Although different strategies were used (e.g. E. coli strains, mammalian cells, in vitro assays), our previous attempts during the past fifteen years to express two putative ManC5-Es from Laminaria digitata [START_REF] Nyvall | Characterization of mannuronan C-5-epimerase genes from the brown alga Laminaria digitata[END_REF], were proved unsuccessful. The genome sequence of Ectocarpus offers the opportunity to have access to a higher number of genes, and potentially proteins. In order to increase our chances of getting soluble expression of active and recombinant ManC5-Es, a medium throughput strategy was applied [START_REF] Groisillier | MARINE-EXPRESS: taking advantage of high throughput cloning and expression strategies for the post-genomic analysis of marine organisms[END_REF].

The 31 putative ManC5-E genes were analysed and the corresponding encoded catalytic domains tentatively expressed. The medium throughput strategy rapidly showed that obtaining soluble heterologous protein expression of this eukaryotic protein family in bacterial expression systems was challenging, if not impossible. Attempts to express some ManC5-E genes in soluble form in other organisms than E. coli (i.e. the bacterium Lactococcus lactis, the yeasts Pichia pastoris and Hansenula polymorpha, and insect cells)

were not successful. The targeted catalytic module of MEP13 (MEP13-C5) was insolubly but highly expressed in E. coli (Supplementary Figure S3A), and our efforts in getting an active recombinant ManC5-E were subsequently focused on this specific target. In order to overcome a codon usage preference in E. coli strains, we used a codon-optimized sequence of MEP13-C5, the latter one being designated MEP13-C5 syn (Table 1B andSupplementary Figure S4). None of the different media or strains used improved the soluble expression of MEP13-C5, and a high level of insoluble protein was obtained in all cases (Supplementary Figure S3A,C). Nevertheless, usage of different complex media such as TB resulted in a higher cell density and consequently in a higher amount of inclusion bodies. In order to increase our chances of getting recombinant proteins, we included two additional E. coli strains developed for the expression of toxic protein or rare codon containing genes. Usage of E. coli Rosetta-gami2(DE3)pLysS produced larger and more significant inclusion bodies than E. coli BL21CodonPlus(DE3)RIPL. This discrepancy might be due to the different growth rates of these strains. Rosetta-gami2(DE3)pLysS grows more slowly than BL21CodonPlus(DE3)RIPL and incubation overnight seems to be more suitable to obtain the targeted protein expression with this strain.

The heterologous expression of MEP13-C5 in E. coli Rosetta-gami2(DE3)pLysS yielded about 15 mg/L of insoluble enzyme. The His-tag containing protein was then successfully refolded using an on-column refolding procedure. The analysis of the SDS-PAGE gels and corresponding western-blots (Supplementary Figure S3A,B) showed that after eluting the refolded protein, the samples were essentially pure, also indicating that only large aggregates, eluted in the flow-through, or refolded protein were present, yielding a final quantity of 6.5 mg of active MEP13-C5 enzyme. The identity of the protein was confirmed by peptide mass fingerprinting (Supplementary Figure S5). The DLS spectra of the protein solutions showed high monodispersity of the refolded samples, as compared to the unfolded protein samples (Supplementary Figure S6).

The refolded MEP13-C5 shows a mannuronan C5-epimerase activity

The activity of the refolded MEP13-C5 was monitored by measuring the change of M/G ratio in two different alginate solutions (having an initial M/G ratio of 1.5 and 1.7, respectively) using 1 H-NMR. In both cases a decrease of the M/G ratio indicated the action of MEP13-C5, epimerizing some mannuronate units into guluronate (Figure 4A). The change of blockiness could be analysed by integrating and measuring the relative surface of the NMR-peaks that appear in the range of 4.4.40 to 5.15 ppm (Figure 4B,C, Supplementary Figure S7), following the method described previously (Grasdalen et al. 1981[START_REF] Grasdalen | High-field, 1 H-N.M.R. spectroscopy of alginate: sequential structure and linkage conformations[END_REF][START_REF] Hartmann | Mode of action of recombinant Azotobacter vinelandii mannuronan C-5 epimerases AlgE2 and AlgE4[END_REF]. The increase of G-block structure (Figure 4B) is observed with both alginates but is only significant for the sample with a M/G-ratio of 1.5. Figure 4C illustrates that for both alginate samples, the repeated MM structures disappear while GG and GGG structures significantly increase. However, these data do not yet allow clear and full identification of which patterns are preferentially recognized and which are produced by the enzyme.

An additional activity assay, using a GG-specific alginate lyase (AlyA1; [START_REF] Thomas | Comparative characterization of two marine alginate lyases from Zobellia galactanivorans reveals distinct modes of action and exquisite adaptation to their natural substrate[END_REF], was set up that allowed indirect evidence of ManC5-E activity. Indeed, under identical conditions, the MEP13-C5 pre-treated alginate samples produced up to 1.5-fold higher abundance of unsaturated sugar units upon incubation with the GG-specific alginate lyase than the initial untreated alginate samples (Figure 5).

Discussion

Functional validation of a putative mannuronan C5-epimerase gene from brown alga

To date, fully biochemically characterized mannuronan C5-epimerases have only been reported for enzymes of bacterial origin [START_REF] Buchinger | Structural and functional characterization of the R-modules in alginate C-5 epimerases AlgE4 and AlgE6 from Azotobacter vinelandii[END_REF][START_REF] Campa | Biochemical analysis of the processive mechanism for epimerization of alginate by mannuronan C-5 epimerase AlgE4[END_REF][START_REF] Hartmann | Mode of action of recombinant Azotobacter vinelandii mannuronan C-5 epimerases AlgE2 and AlgE4[END_REF][START_REF] Rozeboom | Structural and mutational characterization of the catalytic A-module of the mannuronan C-5-epimerase AlgE4 from Azotobacter vinelandii[END_REF][START_REF] Wolfram | Catalytic mechanism and mode of action of the periplasmic alginate epimerase AlgG[END_REF]. [START_REF] Nyvall | Characterization of mannuronan C-5-epimerase genes from the brown alga Laminaria digitata[END_REF] were the first to isolate full length cDNAs from a brown alga (Laminaria digitata) displaying similarity with bacterial mannuronan C5-epimerase genes. They proposed that these genes encoded functional ManC5-Es involved in the terminal step of alginate biosynthesis in brown algae.

Analysis of the genomic sequence of Ectocarpus provided candidate genes for the whole alginate biosynthetic pathway, including the putative ManC5-E genes [START_REF] Michel | The cell wall polysaccharide metabolism of the brown alga Ectocarpus siliculosus. Insights into the evolution of extracellular matrix polysaccharides in Eukaryotes[END_REF].

Nonetheless, the predicted function of these brown algal genes has never been biochemically proven until now. Our study reports, for the first time, the heterologous production, refolding and purification of an active ManC5-E, namely MEP13-C5, of brown algal origin. At least for the here reported gene, this result thus validates the hypothesis formulated by [START_REF] Nyvall | Characterization of mannuronan C-5-epimerase genes from the brown alga Laminaria digitata[END_REF] twelve years ago. To our knowledge this is also the first characterized enzyme specific of the biosynthesis of a cell wall polysaccharide from brown algae.

Fine-tuning of alginate structure

In the context of industrial applications, based on various alginates with different physicochemical properties, the use of specific ManC5-Es that could modulate rheological properties in a defined manner would be of precious biotechnological advantage. Moreover, since significant variations impact the hydrocolloid properties of alginates as a function of seasonal factors, tissue or species, having at hand specific ManC5-Es that could adjust the rheological properties after or during the industrial processing and extraction is also of large interest. In the past, several ManC5-Es of bacterial origin have been biochemically characterized [START_REF] Ertesvåg | Alginate-modifying enzymes: biological roles and biotechnological uses[END_REF][START_REF] Wolfram | Catalytic mechanism and mode of action of the periplasmic alginate epimerase AlgG[END_REF][START_REF] Campa | Biochemical analysis of the processive mechanism for epimerization of alginate by mannuronan C-5 epimerase AlgE4[END_REF][START_REF] Hartmann | Mode of action of recombinant Azotobacter vinelandii mannuronan C-5 epimerases AlgE2 and AlgE4[END_REF] and even engineered to produce tailored alginates [START_REF] Tøndervik | Mannuronan C-5 epimerases suited for tailoring of specific alginate structures obtained by high-throughput screening of an epimerase mutant library[END_REF], Bjerkan et al. 2004). To be useful in existing processes of commercial alginate production, the alginate ManC5-Es need to meet a number of requirements. For example, it is important that the enzymes are selected to work on polymeric substrate, not only on shorter oligosaccharides. This is the case for MEP13-C5, similarly to those of bacterial origin [START_REF] Campa | Biochemical analysis of the processive mechanism for epimerization of alginate by mannuronan C-5 epimerase AlgE4[END_REF][START_REF] Hartmann | Mode of action of recombinant Azotobacter vinelandii mannuronan C-5 epimerases AlgE2 and AlgE4[END_REF].

In our study, low but significant epimerization activity of the algal MEP13-C5 could be measured by 1 H-NMR and by the indirect measure of the activity of a GG-specific alginate lyase. However at this stage, it remains difficult to compare these preliminary analyses of activity to the mode of action of the bacterial enzymes. All alginate-producing bacteria expressed a periplasmic epimerase, AlgG, and the putative mannuronan C5-epimerases from Ectocarpus show highest sequence identity to these bacterial epimerases, with around 32% identity of MEP13-C5, for example, with AlgG from P. aeruginosa (Figure 6). Nevertheless, while all characterized bacterial enzymes of the AlgG-type have been reported to be rather calcium-independent [START_REF] Ertesvåg | Alginate-modifying enzymes: biological roles and biotechnological uses[END_REF][START_REF] Wolfram | Catalytic mechanism and mode of action of the periplasmic alginate epimerase AlgG[END_REF], the activity of MEP13-C5 could only be measured in presence of Ca 2+ . This situation is reminiscent of the AlgE-type enzymes produced by bacteria of the Azotobacter genera, which are calcium-dependent mannuronan C5-epimerases [START_REF] Wolfram | Catalytic mechanism and mode of action of the periplasmic alginate epimerase AlgG[END_REF][START_REF] Hartmann | Mode of action of recombinant Azotobacter vinelandii mannuronan C-5 epimerases AlgE2 and AlgE4[END_REF]. However the sequences of the algal mannuronan C5-epimerases are much more distantly related to these enzymes, with identities often less than 17%. Despite the lack of sequence similarity and different characteristics, the enzymes of bacterial origins that have been biochemically characterized all display a processive mode of action and common active site residues [START_REF] Wolfram | Catalytic mechanism and mode of action of the periplasmic alginate epimerase AlgG[END_REF][START_REF] Campa | Biochemical analysis of the processive mechanism for epimerization of alginate by mannuronan C-5 epimerase AlgE4[END_REF][START_REF] Hartmann | Mode of action of recombinant Azotobacter vinelandii mannuronan C-5 epimerases AlgE2 and AlgE4[END_REF]), but differences appear with respect to the preference of formation of MG or GG-blocks. Our algal enzyme seems to display a somewhat different specificity than the bacterial enzymes of the AlgG-type, in that it appears to recognize the presence of GG-blocks to preferentially increase the GG block sizes. This observation might be deduced from the NMR study, where the enzyme increased GG-block structures in a substrate already rich in GG-block motifs, but this preference is subtle and not confirmed by the alginate lyase-based assays. A more exhaustive library of standards should therefore be used in future biochemical studies to determine the alginate patterns, which are more susceptible to be epimerized. If verified, this subtle difference in mode of action is not surprising in the view of the differences that can be identified at the primary sequence level, as illustrated in Figure 6, although the characteristic active site signature 'DPHD' and the regular features specific of the β-helix fold align well. In the structural sequence alignment one can notice that several residues directly adjacent to the active site, and identified as being involved in substrate binding and processing in AlgE4 and AlgG [START_REF] Rozeboom | Structural and mutational characterization of the catalytic A-module of the mannuronan C-5-epimerase AlgE4 from Azotobacter vinelandii[END_REF][START_REF] Wolfram | Catalytic mechanism and mode of action of the periplasmic alginate epimerase AlgG[END_REF], have a different nature in MEP13-C5 (blue triangles in Figure 6). Moreover, single residue insertions or deletions (green triangles in Figure 6) or large inserted or deleted regions within the catalytic domain (blue dashed lines in Figure 6) may influence the positioning of important residues and influence substrate specificity, which may differ among the different algal ManC-5Es.

Studies of the bacterial multigenic C5-epimerase family from Azetobacter vinelandii (Hartman et al. 2002[START_REF] Buchinger | Structural and functional characterization of the R-modules in alginate C-5 epimerases AlgE4 and AlgE6 from Azotobacter vinelandii[END_REF]) have highlighted the subtle differences in substrate specificities of different AlgE-type enzymes. In the biological context of brown algal cell wall, the presence of a large multigenic ManC5-E family thus points towards the fact that variability in substrate specificity is to be expected. Interestingly, besides the rather well conserved catalytic domain of about 400 amino acids, the modular organisation is highly variable among ManC5-Es from Ectocarpus. The functional role of additional R-modules in bacterial C5-epimerases has only recently been analysed [START_REF] Buchinger | Structural and functional characterization of the R-modules in alginate C-5 epimerases AlgE4 and AlgE6 from Azotobacter vinelandii[END_REF]) and showed that they do influence alginate-pattern recognition and mode of action. Our study provides a first experimental indication that given their different modular organisation and fine structure at the primary sequence level, the mannuronan C5-Epimerases in brown algae are likely to recognise and/or generate distinct motifs, which are important for the physiological functions and the regulation of alginate synthesis and remodelling in the cell wall. Future structural and biochemical analyses will be necessary to elucidate in detail the potential of fine-tuning alginate structures by these algal ManC5-Es.

The roles of ManC5-Es in cell-wall remodelling in brown algae

By acting on a major component of the brown algal cell walls, the ManC5-Es are likely to be of importance in many physiological responses, and various studies are supporting the idea.

For instance, in Laminaria digitata, some ManC5-E genes are differentially expressed during stress responses such as elicitation and protoplast generation [START_REF] Tonon | Expression profiling of the mannuronan C5epimerase multigenic family in the brown alga Laminaria digitata (Phaeophyceae) under biotic stress conditions[END_REF], and upon seasonal variations [START_REF] Nyvall | Characterization of mannuronan C-5-epimerase genes from the brown alga Laminaria digitata[END_REF]. Probably one of the most important (but the less explored) field of action of ManC5-Es, is during developmental processes. Variations of the alginate structure upon tissues are known in Laminariales species, and our analysis of microarray expression data highlighted the possible role for ManC5-E during the development of gametophyte versus sporophytes. Interestingly, almost all of the ManC5-E genes are constitutively expressed, strongly suggesting that ManC5-Es are major actors during Ectocarpus development.

In land plants the homogalacturonan motifs of pectins are secreted into the wall as highly methylesterified forms, subsequently modified by the multigene PME family. The PMEs can either act randomly, promoting the action of cell wall hydrolases and contributing to cell wall loosening, or linearly, giving rise to blocks of demethylesterified motifs, which contribute to cell-wall stiffening through calcium chelation [START_REF] Micheli | Pectin methylesterases: cell wall enzymes with important roles in plant physiology[END_REF]. In brown algae, the functional analogy of ManC5-Es to PMEs is tempting but challenges our view of their modes of action. In contrast to PMEs in plants, our present knowledge on their epimerase activity indicates a 'dead-end' reaction, leading to cell wall strengthening only. The Ectocarpus genome does not encode any known alginate lyases, but the presence of enzymes possibly acting on unsaturated oligosaccharides (i.e. GH88; [START_REF] Michel | The cell wall polysaccharide metabolism of the brown alga Ectocarpus siliculosus. Insights into the evolution of extracellular matrix polysaccharides in Eukaryotes[END_REF] suggests that new families of polysaccharide lyases are present, or alternatively that some ManC5-Es have the dual activity, although this is not the case of MEP13-C5 (i.e. no lyase activity; data not shown). Also, the analysis of the ManC5-E gene structure in Ectocarpus indicates additional modules such as the WSC domains and the AGP-core motifs, which might putatively influence the ManC5-E activity [START_REF] Hervé | Arabinogalactan-proteins have deep roots in Eukaryotes: identification of genes and epitopes in brown algae and their role in Fucus serratus embryo development[END_REF]. The knowledge of the mechanisms involved in cell wall remodelling in brown algae is still in its infancy, partly because expressing algal enzymes into heterologous systems remains challenging. Nevertheless, our success on this first heterologous and refolded algal C5-epimerase, as well as the access to recent genome sequences [START_REF] Cock | The Ectocarpus genome and the independent evolution of multicellularity in brown algae[END_REF][START_REF] Ye | Saccharina genomes provide novel insight into kelp biology[END_REF] should help future analysis on deciphering cell wall metabolism in brown algae.

Material and Methods

Identification and bioinformatical analyses of the ManC5-E sequences

Ectocarpus contains 31 genes of ManC5-Es, available at the genome website of Ectocarpus siliculosus at https://bioinformatics.psb.ugent.be/gdb/ectocarpus and listed in Supplementary Table S1. For phylogenetic analyses, 10 genes were discarded due to doubtful predicted gene structure. The remaining deduced amino acid sequences encoding the ManC5-E catalytic domains were aligned using the MAFFT program at http://bioinformatics.uams.edu/mafft/ (Supplementary Fig. S8). Unrooted maximum likelihood phylogenetic trees were conducted using MEGA6 [START_REF] Tamura | MEGA4: Molecular Evolutionary Genetics Analysis (MEGA) software version 4.0[END_REF]). The best-fit model among 48 different amino acid substitution models was the LG model. The tree with the highest log likelihood (-8620.9504) is shown in Figure 1A. Initial trees for the heuristic search were obtained automatically by applying Neighbor-Joining and BioNJ algorithms to a matrix of pairwise distances estimated using a JTT model, and then selecting the topology with superior log likelihood value. A discrete Gamma distribution was used to model evolutionary rate differences among sites (5 categories (+G, parameter = 1.1283)). All positions containing gaps and missing data were eliminated. Bacterial sequences were used as an outgroup [START_REF] Michel | The cell wall polysaccharide metabolism of the brown alga Ectocarpus siliculosus. Insights into the evolution of extracellular matrix polysaccharides in Eukaryotes[END_REF]). There were a total of 24 sequences and 253 positions in the final data set. The robustness of the branches was evaluated using 100 bootstrap replicates. The modularity of each deduced amino acid sequence was further examined using Blast queries against the UniProt database and domain searches against the InterPro server. Additional refinements were performed manually by examination of the individual HCA plots. Signal peptides and transmembrane domains were predicted using the SignalP and TMHMM servers, respectively. Predictions of chimeric AGPmotifs were performed as described [START_REF] Hervé | Arabinogalactan-proteins have deep roots in Eukaryotes: identification of genes and epitopes in brown algae and their role in Fucus serratus embryo development[END_REF]. The cluster of ManC5-Es containing modular proteins is described in more details in Figure 1B. For each ManC5-E included in the phylogenetic tree, the gene structure of the full-length sequence is indicated (Figure 1C).

Microarray analysis of the ManC5-E mRNA abundances

We explored the microarray dataset that was obtained in [START_REF] Coelho | OUROBOROS is a master regulator of the gametophyte to sporophyte life cycle transition in the brown alga Ectocarpus[END_REF] to carry out the analysis of changes in mRNA abundances of ManC5-E genes in the gametophyte and sporophyte generations (ArrayExpress accession no. E-MTAB-485).

Medium throughput screening for cloning and protein expression of the ManC5-Es

The ManC5-E genes were tentatively cloned and heterologously expressed using a medium throughput strategy in a microtiter-plate format, as previously described [START_REF] Groisillier | MARINE-EXPRESS: taking advantage of high throughput cloning and expression strategies for the post-genomic analysis of marine organisms[END_REF]. In short, the open reading frames, trimmed to the catalytic domains only, were amplified by PCR and using primers incorporating specific restriction sites compatible with our ligation strategies (Supplementary Table S2). From 31 starting genes, 12 were discarded due to either doubtful predicted gene structure or incompatibility with our cloning strategy.

The amplification products were cloned into the pFO4 vector (a vector modified from pET15b from Novagen, to be compatible with the BamHI/EcoRI ligation strategy) and the pGEX (GE Healthcare Life Science) vectors, encoding an N-terminal His 6 -tag and a GST-tag, respectively. Recombinant plasmids were used to transform E. coli strains DH5α (Promega).

The validated plasmids were used to transform appropriate E. coli strains BL21(DE3) and BL21 (Novagen), respectively. Screening of protein expression was performed at a smallscale range using a 24-well plate format and the auto-inducible ZYP5052 media as described [START_REF] Groisillier | MARINE-EXPRESS: taking advantage of high throughput cloning and expression strategies for the post-genomic analysis of marine organisms[END_REF]). The plates were centrifuged at 1,200 g for 20 min at 4°C and the pelleted bacteria were resuspended in a lysis buffer containing 50 mM Tris pH 8, 300 mM NaCl, 1 mg/ml lysozyme, 0.1 mg/ml DNAse and an anti-proteases cocktail. The cell lysates were further centrifuged at 12,000 g for 20 min. The supernatants contained the soluble expressed fractions. The remaining cell pellets, which contained the insoluble expressed fractions, were extracted with 6 M urea. All fractions were analysed by SDS-PAGE. All further steps were performed on the single insolubly but highly expressed target module of MEP13.

Codon optimisation and scale-up expression and purification of MEP13-C5

Codon-optimisation of the encoded catalytic domain of MEP13 (named MEP13-C5) was performed by GeneArt within the framework PolyModE (www.polymode.eu). The sequence was extended with terminal restriction sites for BamHI and EcoRI to be compatible with our cloning strategy (Supplementary Table S2). The codon-optimised sequence was delivered as being cloned into the shuttle vector pMA. To differentiate between the cDNA and the codonoptimized sequence, the latter one is designated MEP13-C5 syn , where syn indicates the synthetic and codon-optimised sequence. MEP13-C5 syn was excised from the pMA vector using the BamHI and EcoRI restriction enzymes, purified, and further cloned into the pFO4 vector. In order to increase our chances of getting recombinant proteins, the codon-optimised sequence was also cloned into the pQE-80L vector (Qiagen). In addition to encode a His 6 -tag, this vector contains a PT5 promoter and two lac-operators to allow tight regulation of basal level expression, which makes this vector suitable to express toxic genes. The codonoptimised sequence was amplified by PCR from the pMA vector using specific primers (Supplementary Table S2), purified and digested by BamHI and SacI before ligation into the pQE-80L vector. The recombinant plasmids were used to transform E. coli strains DH5α. The plasmid DNA of the candidate clones was purified using Wizard® Plus SV Minipreps DNA Purification Systems (Promega) and the correctness of cloning was confirmed by DNA sequencing.

In order to increase our chances of getting recombinant proteins, we decided to use diverse expression strains and included two additional E. coli strains developed for the expression of toxic or rare codon containing genes. We therefore transformed E. coli BL21(DE3), E. coli Rosetta-gami2(DE3)pLysS and E. coli BL21CodonPlus(DE3)RIPL cells (all from Novagen) as expression strains. The transformed strains were cultivated in different complex media, including LB and TB (Sambrook and Russel, 2001;[START_REF] Korf | Large-scale protein expression for proteome research[END_REF]. The culture volume ranged from 200 mL to 1 L. To trigger formation of insoluble proteins, induction (1 mM IPTG final) has been carried out permanently at 37 °C overnight at 180 rpm.

After 24 hrs, the cells were analysed by microscopy for the presence of inclusion bodies, which contain the insoluble proteins. Cells were harvested by centrifugation at 6,000 g for 30 min and mechanically lysed using a French press.

The cell lysate was clarified by ultracentrifugation at 100,000 g during 1 hour. The insoluble protein present in the cell pellet was purified and refolded on-column in a single step. For this purpose the pellet was solubilised using a tissue homogeniser and a refolding buffer containing 25 mM Tris-HCl pH 8.0, 200 mM NaCl, 10 mM 2-mercaptoethanol, 10% glycerol, 8 M urea, 40 mM imidazole and 4 mM CaCl 2 . The solubilised inclusion bodysolution was centrifuged at 100,000 g for 30 min and the supernatant loaded onto a 5 mL HisTrap HP column charged with 100 mM NiSO 4 . The refolding on-column was carried out by a slow gradient over two hours against the refolding buffer and using a buffer containing 25 mM Tris-HCl pH 7.8, 200 mM NaCl, 3 mM reduced glutathione, 0.3 mM oxidised glutathione, 10% glycerol and 4 mM CaCl 2 . The bound proteins were finally eluted in a gradient with the elution buffer containing 25 mM Tris-HCl pH 7.8, 200 mM NaCl, 3 mM reduced glutathione, 0.3 mM oxidised glutathione, 10% glycerol, 500 mM imidazole and 4 mM CaCl 2 . Protein purification was carried out on an Äkta Avant Chromatographic System from GE Healthcare. Columns and resins used were from GE Healthcare. All fractions collected were analysed by SDS-PAGE. The targeted eluates containing the refolded MEP13-C5 were pooled and the volume was reduced by ultrafiltration to approximately 3.5 mL of a final protein concentration of 1.94 g/L. In the aim to separate the putative different folding intermediates, a second purification step was performed by size exclusion chromatography using a Superdex 200 column from GE Healthcare. Equilibration and running buffers were those that were used in the final step of the refolding procedure.

Western-blotting

The sequences of the catalytic domains of all the brown algal ManC5-Es were aligned with the sequence of the catalytic domain of AlgE4 from Azetobacter vinelandii in order to identify specific sequence stretches that, when reported onto the structural coordinates of AlgE4, appeared to be located at the surface of the protein. Two stretches (CDGRAKNEMGECRMDIINS and NHGIIASKRCNNVKIF) were identified and polyclonal antibodies raised in mice against these peptides were purchased from Eurogentech. The antibodies were tested by ELISA and further used to verify heterologous expression of algal ManC5-Es (Supplementary Figure S3).

Analysis of protein identity by peptide mass fingerprinting

The refolded and purified MEP13-C5 was analysis on a SDS-PAGE gel. Bands at 50 kDa, which correspond to the expected size of the protein, were excised and analysed by peptide mass fingerprinting (PMF). Coomassie blue-stained protein slices were cut into small pieces, washed with distilled water and de-stained using acetonitrile. The cysteine residues were reduced by 100 µl of 10 mM DTT at 56°C and alkylated by 150 µl of 55 mM iodoacetamide at room temperature. Iodoacetamide solution was replaced by 100 µl of 100 mM NH 4 HCO 3 and gel dehydration was achieved with acetonitrile. After evaporation in Speed Vac (Thermo), proteins were digested overnight at 37°C by a solution containing 0.9 µg of a modified bovine trypsin (Promega, Madison, WI, USA) prepared in 25 mM NH 4 HCO 3 .

Finally, a double extraction was performed, first with 1% (v/v) trifluoro acetic acid solution, and subsequently with 100% (v/v) ACN. The resulting peptide mixture was extracted and dried under vacuum and re-suspended in 50% ACN, 0.1% TFA solution. PMF by MALDI-TOF MS was performed using a Voyager DE-STR MALDI-TOF mass spectrometer (Applied Biosystems). Each pooled tryptic digest (1 µL) was mixed with an equal volume of CHCA matrix (Sigma; 10 mg/mL in 50% ACN, 0.1% TFA) and spotted onto the MALDI target. Spectra were acquired in positive ion reflector mode under 20 kV accelerating voltage and a mass range of 450-5000 Da. Internal calibration was performed using trypsin autolysis fragments at m/z 1433.70, 2163.05, and 2289.1. The samples resulting from attempts to produce soluble proteins mostly contained proteins from the E. coli host strains, while the samples containing insoluble proteins released eighteen peptides specific of MEP13-C5 syn , covering 52% of the protein sequence (Supplementary Figure S5), and there was no contamination with E. coli proteins. This result demonstrated that MEP13-C5 was indeed highly expressed in an insoluble form, whereas each attempt to produce the enzyme in its soluble form failed.

Dynamic light scattering (DLS)

DLS was performed to control the mono-dispersity of the refolded protein. 100 µL of sample proteins (unfolded; refolded fractions) were centrifuged at 20,817 rcf on an Eppendorf Centrifuge 5417R for 10 minutes. 60 µL of supernatant was transferred into a quartz cuvette and placed into a Zetasizer Nano S from Malvern. Measurement followed at 20°C during 20 minutes.

Enzyme activity assays

All alginate samples were provided by DuPont. 100 µg of sample proteins were incubated in presence of 0.5% sodium alginate (M/G ratio of 1.5 and 1.7) in a buffer containing 20 mM MOPS pH7.0, 4 mM CaCl 2 , 100 mM NaCl, using a final volume of 1 mL. The reaction was incubated overnight at 30°C. To avoid signal-interference in subsequent NMR measurements, it was necessary to remove the CaCl 2 after the enzymatic reaction. Removal of CaCl 2 started with acidification of the samples by mixing them with equal volume (1 mL) of acidic water (300 µL of concentrated H 2 SO 4 in 100 mL of water, pH 1.4) and incubation for one hour at room temperature on a shaker. The samples were centrifuged at 141 g for 5 minutes and the top phase was removed. The samples were washed with an equal volume of water (1 mL) and incubated in the same way as before. Each washing step with acidic water and normal water was done three times. In the final step, the top phase (water) was removed and the samples were neutralised with Na 2 CO 3 (0.2 g in 10 g water) to a pH of 7.0. MEP13-C5 untreated alginates were submitted to the same procedure as the MEP13-C5 treated samples. The samples were lyophilized prior to measurements. Immediately before performing NMR measurements the samples were solubilized with D 2 O. NMR-spectra were measured without depolymerisation of the samples.

Non-lyophilized samples, without calcium removal, after treatment with refolded MEP13-C5 were also analysed in a second, indirect, activity assay using the action of a Gspecific alginate lyase known as AlyA1 from Zobellia galactanivorans [START_REF] Thomas | Comparative characterization of two marine alginate lyases from Zobellia galactanivorans reveals distinct modes of action and exquisite adaptation to their natural substrate[END_REF].

The MEP13-C5 treated alginate samples were incubated at 30°C with 100 µg of AlyA1 in a solution containing 100 mM Tris-HCl buffer at pH 7.0. Alginate lyase activity was assayed by measuring the increase in absorbance at 235 nm (A 235 ) of the reaction products (unsaturated uronates) for 5 min in a 1 cm quartz cuvette containing 0.5 ml of reaction mixture in a thermostated spectrophotometer. The difference in the AlyA1 activity between the MEP13-C5-treated and -untreated alginates was assigned to the MEP13-C5 activity.

NMR analysis

The untreated and enzyme-treated alginate samples were prepared as described above. All NMR experiments were performed on a Bruker Avance III spectrometer (Karlsruhe, Germany) equipped with a double-tuned direct-detection broadband 5 mm probe (and deuterium lock) and a 14.1 T magnet corresponding to a 1 H resonance frequencies of 600 MHz. Each spectrum was run with a pulse angle of 30º, 2.7 second acquisition and one second delay between each scan with a total of 64 scans. The data was zero filled to 65536 and a Gaussian apodiaztion function was added prior to the Fourier transform yielding a spectrum with a 20 ppm spectral width. Further processing such as deconvolution, was done using python scripting in combination with the nmrglue python library [START_REF] Helmus | Nmrglue: an open source Python package for the analysis of multidimensional NMR data[END_REF].

The chemical shifts are expressed in parts per million (ppm) downfield from the signal for 3-(trimethylsilyl)propanesulfonate.

For the alginate substrate with a M/G ratio of 1.7, four control samples were measured, giving an average M/G ratio of 1.68 and a standard deviation (sd) of 0.02. When the enzyme was added (4 replicates) the average M/G ratio decreased to 1.51 with an sd of 0.01. For the alginate substrate with a M/G ratio of 1.5, four control samples were measured, giving an average M/G ratio of 1.46 and a standard deviation (sd) of 0.01. When the enzyme was added (4 replicates) the average M/G ratio decreased to 1.33 with a sd of 0.02. The measure of block distribution in terms of "GG", "MM", "GM = MG ", "GGG", "MGM", "GGM = MGG", was undertaken by deconvolution or integration of the peaks in the region between 4.40 and 5.15 ppm following standard protocols (Grasdalen et al. 1981[START_REF] Grasdalen | High-field, 1 H-N.M.R. spectroscopy of alginate: sequential structure and linkage conformations[END_REF][START_REF] Hartmann | Mode of action of recombinant Azotobacter vinelandii mannuronan C-5 epimerases AlgE2 and AlgE4[END_REF]. Fig. 5. Assessment of mannuronan C5-epimerase activity of MEP13-C5 by analysis of the degradation pattern using a GG-specific bacterial alginate lyase. The production of unsaturated sugar units produced by the action of AlyA1 (a strictly GG-specific alginate lyase from Z. galactanivorans) was followed during 5 minutes by photospectrometry at a wavelength of 235 nm on the two alginate samples (M/G ratios of 1.5 and 1.7). In each case, data are means of three independent replicates ± SD.

Legends to figures

Fig. 1 .

 1 Fig. 1. Schematic representation of the effect of ManC5-Es on alginate structures. Alginate in

Fig. 2 .

 2 Fig. 2. Bioinformatical analyses of the ManC5-E family from Ectocarpus. (A) Phylogenetic

Fig. 3 .

 3 Fig. 3. Microarray analysis of the abundance of ManC5-E transcripts in Ectocarpus

Fig. 4 .

 4 Fig. 4. Assessment of mannuronan C5-epimerase activity of MEP13-C5 by 1 H-NMR. (A) The

Fig. 6 .

 6 Fig. 6. Extract of a structural sequence alignment of selected mannuronan C5-epimerases

Table 1 .

 1 Statistics for several features of the ManC5-E genes, including MEP13-C5. (A) Structure of genes compared to the Ectocarpus genome, (B) relative adaptiveness values (%) for 5 rare codons for E. coli, compared to the nuclear codon usage of E. coli

	Tables
	A				B
		genome (average)	ManC5-Es (average)	MEP13		AGA AGG ATA CTA CCC
	Gene length (bp) 6,974	9,712	5,494	E. coli (genome)	43 23 45 16 43
	CDS length (bp) 1,607	2,083	1,899	ManC5-Es (average)	40 60 9 11 79
	Number of introns per gene	7.1	13.7	10.0	MEP13-C5 MEP13-C5 syn	33 100 4 0 0 0	0 91 0 0

Acknowledgments

The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. 222628 (www.polymode.eu). This work was also supported by the Region Bretagne and the Centre National de la Recherche Scientifique (CNRS).

Abbreviations