Pierre Bertrand

Michel Broniatowski

Jean-François Marcotorchino

Minimization with respect to divergences and applications

Keywords: Mathematical Relational Analysis, Logical Indetermination, Coupling Functions, Task Partitioning Problem, Guessing Problem

We apply divergences to project a prior guess discrete probability law on pq elements towards a subspace defined by fixed margins constraints µ and ν on p and q elements respectively. We justify why the Kullback-Leibler and the Chi-square divergences are two canonical choices based on a 1991 work of Imre Csiszár. Besides we interpret the so called indetermination resulting from the second divergence as a construction to reduce couple matchings. Eventually, we demonstrate how both resulting probabilities arise in two information theory applications: guessing problem and task partitioning where some optimization remains to minimize a divergence projection.

Introduction

The present document is interested in the poorly studied indetermination (also called indeterminacy) as a natural second canonical coupling, the first one being the usual independence. We extract both notions from a projection of a prior uniform guess U to a subset of the set of probabilities defined by fixed margins. The problem is closely linked with contingency table adjustment exposed by Maurice Fréchet in [START_REF] Fréchet | Sur les tableaux de corrélations dont les marges sont données[END_REF] or [START_REF] Fréchet | Sur les tableaux de corrélations dont les marges et les bornes sont données[END_REF] as well as by Edwards Deming and Frederick Stephan [START_REF] Deming | On the least squares adjustment of a sampled frequency table when the expected marginal totals are known[END_REF] or by Sir Alan G. Wilson [START_REF] Wilson | A statistical theory of spatial distribution models[END_REF] or [START_REF] Wilson | Entropy in urban and regional modelling[END_REF]. Although we will limit ourselves to the discrete case it can be transposed in a continuous domain (see for example [START_REF] Stummer | Optimal transport with some directed distances[END_REF] in the same proceeding as the present paper) leading to the optimal transport theory for which an overview and recent breakthroughs are presented in [START_REF] Villani | Optimal Transport, Old and New[END_REF].

When limited to the discrete case, a work of Csiszár ([7]) shows that two projection functions set apart from all the others as they are the only ones to convey "natural" properties. As we shall see, the first one leads to independence while the second one, since canonic, motivates our interest for the indetermination coupling it generates.

We gather here a unified construction for both couplings and show indetermination comes with a stunning property of couple matchings minimization that one could have expected to happen under independence. Eventually this property is applied to two concrete problems: guessing and task partitioning.

Projection using divergences

Consider two given discrete probabilities µ = µ 0 , . . . , µ p on p elements and ν = ν 0 , . . . , ν q on q elements with non zero values (it amounts to reduce p and q). We associate to them the subspace L µ,ν of probability laws π on pq elements where :

π u,• = q v=1 π u,v = µ u , ∀1 ≤ u ≤ p, π •,v = p u=1 π u,v = ν v , ∀1 ≤ v ≤ q,
namely, µ and ν are the margins of π. L µ,ν is thus defined by a set of linear constraints on the components of π where π is a probability law living inside the simplex S pq .

As spotted in [START_REF] Bertrand | Logical indetermination coupling:a method to minimize drawing matches and its applications[END_REF], we can interpret discrete transport problems related to contingency table adjustments, as projections of a prior guess, typically the uniform law U = 1 pq , . . . , 1 pq , to our just defined subspace L µ,ν . It happens that the projections of a prior guess that we fix equal to U to a subspace L of S pq defined by a set of linear constraints was deeply studied in [START_REF] Csiszár | Why least squares and maximum entropy? an axiomatic approach to inference for linear inverse problems[END_REF]. Formally, we are minimizing a divergence (see Definition 1) between U and any element of L as exposed in Problem 1.

Definition 1 (Divergence with functional parameter φ also called f-divergence). Given, a positive function φ : R + → R + , strictly convex at 1 with φ(1) = 0, and given two discrete probabilities m and n on pq elements, we define the divergence D φ as:

D φ (m|n) = p u=1 q v=1 n u,v φ m u,v n u,v = E n φ m u,v n u,v .
Additionally we set 0 * φ(x) = 0 for all x ∈ R + .

Problem 1 (Generic Projection Problem). Given a subspace L of S pq defined by a set of linear constraints, the projection problem of the uniform law U to L using a divergence D φ as defined in Definition 1, states as follows:

min π∈L D φ (π |U ) .

Remark 1.

As mentioned by an anonymous reviewer, since L is a m-affine subspace and D φ is a f -divergence, the unicity of the optimal is stated from information geometry theory (see [START_REF] Nielsen | What is an information projection[END_REF]).

A key result of the paper [START_REF] Csiszár | Why least squares and maximum entropy? an axiomatic approach to inference for linear inverse problems[END_REF] is that, provided we add some hypotheses on the cost function we can limit ourselves to two divergences. We do not gather here those hypotheses, but one can find them obviously in the quoted paper as well as summarized in [START_REF] Bertrand | Logical indetermination coupling:a method to minimize drawing matches and its applications[END_REF]. Eventually, we obtain the two eligible divergences by setting φ : x → x log(x) -x + 1 leading to the usual Kullback-Leibler divergence:

D KL (π |U ) = p u=1 q v=1 π u,v log(pqπ u,v ); (1)
and thus to the transport problem using entropy exposed in [START_REF] Wilson | A statistical theory of spatial distribution models[END_REF] or with φ : x → (x -1) 2 leading to the so-called Pearson Chi-square divergence function:

D 2 (π |U ) = pq p u=1 q v=1 π u,v - 1 pq 2 (2)
hence to the minimal transport problem exposed in [START_REF] Stemmelen | Tableaux d'échanges, description et prévision[END_REF]. Typical computations, using Karush-Kuhn-Tucker, provide for the two divergences, a closed-form expression of the projections that we summarize in the figure 1 below where π × is the independence coupling of the two margins:

π × (u, v) = µ u ν v ; (3)
and π + is the so-called indetermination (or indeterminacy) coupling:

π + (u, v) = µ u q + ν v p - 1 pq . (4)
At this point we must notice that π + is not always defined since it may contain negative values. We add a hypothesis on the margins µ and ν to get rid of this problem:

p min u µ u + q min v ν v ≥ 1.
(

) 5
Regarding Equation 5, a typical modification of any couple of margins to deduce eligible margins can be found in [START_REF] Bertrand | Logical indetermination coupling:a method to minimize drawing matches and its applications[END_REF] while the probability to draw adapted margins if µ and ν are uniformly drawn on S p and S q respectively is provided in [START_REF] Bertrand | Transport Optimal, Matrices de Monge et Pont relationnel[END_REF].

Using Mathematical Relational Analysis notations, one can show that π + leads to an equality between a "for" vs "against" notion which justifies the name indetermination historically coined in [START_REF] Marcotorchino | Utilisation des comparaisons par paires en statistique des contingences[END_REF]. Besides, the + notation we chose in [START_REF] Bertrand | Independence versus Indetermination: basis of two canonical clustering criteria[END_REF] comes from the Full-Monge property (see for instance the survey [START_REF] Burkard | Perspectives of Monge properties in optimization[END_REF]) that the associated contingency matrix respects. Eventually, we can also extend the notion of indeterminacy in a continuous space and estimate an associated copula as exposed in [START_REF] Bertrand | Continuous indetermination and average likelihood minimization[END_REF]. In the present paper we shall not detail further the properties π + conveys and refer the interested reader to the aforementioned articles.

Prior guess U Entropy DKL min π uv πu,v ln(πu,v) Square deviation D2 min π u,v π 2 u,v - 1 pq Constraints on margins L(µ, ν) v πu,v = µu, ∀1 ≤ u ≤ p u πu,v = νv, ∀1 ≤ v ≤ q 0 ≤ πu,v ≤ 1 Independence π × u,v = µuνv Indetermination π + u,v = µu q + νv p - 1 pq
Fig. 1. Summary of a symmetric construction

Mimisation of couple matchings

We insist on a property expressed at the very beginning of the indetermination construction. Let us say π is a probability law on pq elements. If we independently draw two occurrences of W = (U, V ) ∼ π, the probability of getting a matching for a couple, that is to say

U 1 = U 2 and V 1 = V 2 simultaneously, is given by π U1,V1 × π U2,V2 = π 2 U1,V1 . Besides, it turns out that D 2 (π |U ) = pq p u=1 q v=1 π u,v - 1 pq 2 (6) = pq p u=1 q v=1 (π u,v ) 2 -1 (7)
so that the shift introduced by U vanishes and minimizing D 2 (π |U ) amounts to minimize the probability of a couple matching.

This property is illustrated by a decomposition of π + whose proof is available in [START_REF] Bertrand | Logical indetermination coupling:a method to minimize drawing matches and its applications[END_REF] and which leads to a method for drawing under π + . We first set u 0 such that it corresponds to the smallest line meaning µ u0 is minimal. Then, quoting ∀u, δ u = µ u -µ u0 , we can show that the following drawing realizes indetermination:

1. We draw u through µ.

2. We roll a loaded dice with the Bernoulli's skew: I = Be δu µu . 3. If we get 0, we draw v under the distribution π + u0,1 , . . . , π + u0,q that we normalize to 1, else, we draw v uniformly hence under 1 q , . . . , 1 q .

The method this coupling uses to minimize couple matchings reads as follows. If U 1 is frequent then µ U1 is high, hence δ U1 is high, this means that I will often draw 1 and that v will often be uniform preventing V 1 = V 2 as much as possible. On the opposite, if U 1 is rare (typically U 1 = u 0 ) then U 1 = U 2 is already rare and I will be allowed to often draw 0 so that v will follow π + u0,v v . Best option of course would be to stay uniform every time, this is the prior guess U solution but it does not belong to L µ,ν . The trick with indetermination is that it hides the disequilibrium on v that is needed to respect margins inside rare lines.

A stunning result is that couple reduction is not ensured by independence but by indetermination; though the difference between both is small. One can show that in the discrete case, the expected L 2 -difference is 1 pq (see [START_REF] Bertrand | Independence versus Indetermination: basis of two canonical clustering criteria[END_REF]). Furthermore the probability of a couple matching with a variable under a second law π only depends on its margins as stated in Proposition 1.

Proposition 1 (Matchings with another variable).

Let us suppose that π is a second probability law on pq elements with margins µ and ν . We give ourselves a first couple W = (U 1 , U 2 ) drawn under π + and a second one W = (U 1 , U 2 ) independently drawn under π . Then the probability of couple matching only depends on the margins and is given by:

P π + ×π (W = W ) = p u=1 µ u µ u q + q v=1 ν v ν v p - 1 pq . ( 8
)
Proof. The result is due to the structure of π + . Formally:

P π + ×π (W = W ) = (u,v) π + u,v π u,v = (u,v) µ u q + ν v p - 1 pq π u,v = p u=1 µ u µ u q + q v=1 ν v ν v p - 1 pq
In particular, the last proposition 1 shows that any π ∈ L µ,ν has the same probability of a couple matching with π + than π + itself. The next section leverages on that ability to reduce couple matching in two applications.

Applications

Guessing problem

In cryptography, a message u in a finite alphabet U of size p is typically sent from Alice to Bob while a spy whose name is Charlie tries to decode it. A common strategy for Alice to communicate efficiently and secretly with Bob consists in encoding the message using a couple of keys (public, private) for each character or a symmetric encryption which only requires one shared key between Alice and Bob. The literature concerned with the encryption method to choose according to the situation is diverse, the most-used standard is Advanced Encryption Standard described in various articles. Possibly, Charlie observes an encrypted message V in a second finite alphabet V of size q which is a function of the message u. We choose to gather in V any additional data Charlie may observe (the channel used, the sender's location, . . . ). In the end U and V are correlated but not always in a deterministic link. Related to the cryptography situation, the guessing problem quoted hereafter as Problem 2 was first introduced in the article [START_REF] Massey | Guessing and entropy[END_REF] and extended in [START_REF] Kumar | A unified framework for problems on guessing, source coding and task partitioning[END_REF] to take V into account. While in cryptography Charlie tries to decode a sequence of messages, the guessing problem focuses on decoding a unique message.

Let us formalize the situation: Charlie chooses its strategy (see Definition 2) according to the value taken by the observed second variable V : he typically adapts himself to the conveyed encryption. The probability law of the couple (U, V ) is quoted P U,V = π while its margins are P U = µ and P V = ν with no zero values.

Problem 2 (Original Guessing Problem or Spy Problem).

When Alice sends a message U = u to Bob, the spy Charlie observes a variable V = v and must find out the value u of the realization. He has access to a sequence of formatted questions for any guess ũ he may have: "Does u equal ũ?" for which the binary answer is limited to "yes/no".

Definition 2 (Strategy).

For any value v, a strategy σ v = S|v of Charlie is defined by an order on U representing the first try, the second and so on until number p. It can be deterministic or random: we quote P S|v its probability law. The global strategy quoted σ or S gathers all the marginal strategies S v and is a random variable dependent only on V (not U ) whose probability law is quoted P S .

Besides, for a given position i ∈ [1, p], and an order σ, σ[i] is the element in U corresponding to the i-th try.

The usual performance G(σ, u) of a spy consists in counting the number of questions required to eventually obtain a "yes" in Problem 2 when Charlie proposed the order S = σ and Alice generated the message U = u. For instance, with an alphabet U = {a, b, c, d}, if the message is u = c and the strategy σ of the spy consists in the order (b, c, a, d) (meaning he first proposes message b then c,. . . ) we have:

G(σ, u) = p i=1 i1 u=σ[i] = 1 • 1 u=b + 2 • 1 u=c + 3 • 1 u=a + 4 • 1 u=d = 2 • 1 u=c = 2
Eventually in [START_REF] Massey | Guessing and entropy[END_REF], they compute the ρ-moment of G as an estimation of the efficiency of a strategy S:

||G(S, U )|| ρ = E (S,U,V )∼P S,U,V (G(S, U ) ρ ) . (9)
We notice that G actually takes as granted that Charlie has up to p trials which is not adapted in case Alice sends a sequence of messages due to a lack of time. Let us move away from the literature and measure Charlie's performance by its probability to find out after one trial the message u Alice sent. It is a reasonable measure as, if a sequence of messages is sent, he may have to jump from one to the following after only one trial. What actually matters (as mentioned by an anonymous reviewer) is the number of attempts k Charlie can try between two messages from Alice. It could be captured with the one-shot performance M using a trick and supposing that Alice repeats each message k times.

Definition 3 (one-shot performance).

For a given strategy S (which can depend on V with a non deterministic link), we define the following performance measure as the probability to find out the value u after one trial, formally:

M (S, U, V ) = M (S, π) = P (S,U,V ) (S[1] = U ) .
We suppose as for the original optimal strategy that the spy has access to the distribution π = P U,V . We can imagine he previously observed the non-encrypted messages in a preliminary step.

Two strategies immediately stand out:

1. S max : systematically returns at v fixed (observed by hypothesis), the u associated with the maximal probability on the margin P U |V =v ; 2. S margin : returns at v fixed a random realization in U under the law P U |V =v .

Similarly as in see [START_REF] Massey | Guessing and entropy[END_REF] where they prove S max is the best strategy in case the performance measure is G, we can show it also maximizes the one-shot performance. Actually, M rewrites:

M (S, U, V ) = q v=1 ν v P (Sv,U )∼P Sv ⊗P U |V =v (S v [1] = U ) (10) = p u=1 q v=1 π u,v P Sv (S v [1] = u) . ( 11
)
Since P Sv only depends on v, M is maximal under S max , when P Sv = δ uv where u v = argmax u π u,v so that:

M (S, U, V ) ≥ M (S max , U, V ) = q v=1 π uv,v . (12)
Eventually, we quote u 1 = argmax u µ u and notice that q v=1 π uv,v ≥ µ u1 leading to the proposition 2.

Proposition 2 (Charlie's best performance).

We suppose that the margins µ and ν are fixed. Then, for any coupling probability π between messages U and ciphers V , the best one-shot performance Charlie can perform always happens under S max . Furthermore it admits a fixed lower-bound µ u1 independent on π; to summarize:

M (S, U, V ) ≥ M (S max , U, V ) = q v=1 π uv,v ≥ µ u1 . (13)
Let us suppose, commendable task if any, that Alice wants to minimize Charlie's best one-shot performance. We also suppose that the margins µ on U and ν on V are fixed. It is a common hypothesis: the alphabet U in which the messages are composed typically respects a distribution on letters; variable V on its own, if it represents frequencies for instance may have to satisfy occupation weights on each channel. Eventually, Alice can only leverage on the coupling between U and V .

Precisely, let us now compute the corresponding value for our two couplings, both are optimal (for Alice).

M × = M (S max , π × ) = µ u1 M + = M (S max , π + ) = µ u1
Regarding the second strategy S margin we know it is less efficient for Charlie in term of one-shot performance. Yet, it is by far harder to cope with for the sender who cannot easily prevent random conflicts. Consequently we come back to the reduction of couple matchings (here a success for Charlie), whose "indetermination coupling", we know, prevents us against. Let us unfold this remark hereafter.

Replacing P Sv by its value under the second strategy in Equation 11 allows us to estimate the one-shot performance of S margin which is given by:

M (S margin , U, V ) = u∈U v∈V ν v (π u|V =v ) 2 . ( 14
)
Concerning the strategy S margin we have the two bounds:

||π|| 2 2 min v∈V ν v ≥ M (S margin , U, V ) ≥ ||π|| 2 2 max v∈V ν v , (15)
with

||π|| 2 = u∈U v∈V (π u,v ) 2 .
Depending on ν the bounds are more or less tight. If ν v goes to the uniform, the situation becomes similar to S max : both couplings are equal and optimal.

In any case, Equation 15shows that studying the guessing problem brings us back to Problem 1 associated with cost D 2 whose solution is given by the indetermination coupling of the margins. It guarantees an efficient reduction of conflicts (see section 3) and eventually a weak one-shot performance under S margin as expressed in Equation ( 15) and an optimal one under S max .

Task Partitioning

Task partitioning problem is originally introduced in [START_REF] Bunte | Encoding tasks and Rényi entropy[END_REF] where the authors provide a lower bound on the moment of the number of tasks to perform. Let us follow the gathering work of [START_REF] Kumar | A unified framework for problems on guessing, source coding and task partitioning[END_REF] where they also coin a generalized task partitioning problem basically adapting it as a special case of the guessing problem.

Formally, we begin with the original problem of tasks partitioning: a finite set U of p tasks is given together with an integer q ≤ p. The problem consists in creating a partition A = (A 1 , . . . , A q ) of U in q classes to minimize the number of tasks to perform, knowing that if one needs to perform a task u ∈ A i , it is mandatory to launch simultaneously the whole subset of tasks included within A i .

Practically, a task U = u to perform is randomly drawn from U under a probability distribution P U = µ representing the tasks frequencies. As any task, the task u to perform is assigned to a unique class A i(u) of the arbitrary partition. Hence, A(u) = |A i(u) | counts the number of tasks to perform. Precisely, one plays on the partition knowledge to perform, in average, as few tasks as possible.

Similarly to the guessing problem, the performance of a partition A is estimated through the ρ-moment of the number of tasks A(U ) to perform:

E U ∼P U [A(U ) ρ ] . (16)
Inspired by the general guessing problem, paper [START_REF] Kumar | A unified framework for problems on guessing, source coding and task partitioning[END_REF] extends the task partitioning problem. Let us introduce here this generalized version, in which we are no longer interested in minimizing the number of tasks to perform but rather in reducing the number of tasks before a selected (or a chosen) task u.

Indeed, in the first version, as soon as u is drawn, an arbitrary rule imposes to perform the whole subset A i(u) leading to realize A(u) tasks. In the new version, tasks are performed sequentially in A i(u) according to a global strategy S that can be deterministic or random. Typically, tasks may consist in a flow of signatures which an administration requires while q would be the number of workers dedicated to perform those signatures on incoming documents. A worker can be given the entitlement to perform several signatures, assistants usually do. In that case, the partition encodes the assignments of tasks to workers. When a worker V = v is assigned a document, the depositor waits until the signature. At the same time the worker follows his own strategy S v to sign his assigned documents, meaning he can always follow the same order leading to a deterministic strategy or change every day leading to a random strategy.

With a global strategy S which gathers the workers' strategies S v , ∀1 ≤ v ≤ q and for a task u to perform, the performance of a partition A is measured using

N S,A (u) = j / S i(u) [j] = u,
which represents the number of tasks performed before the intended task u (u included) by the worker i[u] it is affected to according to partition A. A lower bound is provided in the paper [START_REF] Kumar | A unified framework for problems on guessing, source coding and task partitioning[END_REF].

Let us now suppose that the keys 1 ≤ v ≤ q are associated with offices that must perform a proportion ν v of the incoming tasks which still follow a distribution µ. It actually appears as a sensible problem where a manager would have to distribute in advance tasks among teams according to the usual observed distribution of tasks and a list of available teams with their capacities.

Besides, we suppose that each team uses the strategy S marge to perform tasks, meaning they randomly perform one according to their margin theoretical distribution; for a document signing, they randomly sign one.

From now on, we can rewrite our task partitioning problem under the form of a guessing problem:

-V = v, formerly corresponds to a worker, now it represents the information the spy has access to; -U = u, formerly represents a task to perform, now it represents a sent message; -S = σ, formerly represents the order in which tasks are performed, now it represents the order in which Charlie proposes his guesses.

Under this formalism, we are interested in measuring the probability M (S, U, V ) of executing u first as an extended application of the one-shot performance of Definition 3 and we have:

M (S, U, V ) ≥ ||π|| 2 2 max 1≤v≤q ν v ≥ ||π + || 2 2 max 1≤v≤q ν v . ( 17
)
This inequality provides a lower bound for any distribution of the tasks among the team, no distribution can generate a worst "one-shot probability" of satisfying the intended task. In task partitioning actually, each task u is uniquely associated to a worker v = i(u) so that the random variable representing the worker is deterministic conditionally to U : π u,v = δ v=i(u) .


Maria Ruiz-Castell
email: maria.ruiz@lih.lu

Gwenaëlle Le Coroller
email: gwenaelle.lecoroller@lih.lu

Jean-Francois Landrier
email: jean-francois.landrier@univ-amu.fr

Djedgiga Kerkour
email: djedjiga.kerkour@outlook.com

Bernard Weber
email: bernard.weber@labo.lu

Guy Fagherazzi
email: guy.fagherazzi@lih.lu

Brice M R Appenzeller
email: brice.appenzeller@lih.lu

Michel Vaillant
email: michel.vaillant@lih.lu

Torsten Bohn
email: torsten.bohn@lih.lu

Micronutrients and Markers of Oxidative Stress and Inflammation Related to Cardiometabolic Health: Results from the EHES-LUX Study

Keywords: population based study, antioxidants, adipokines, cardio metabolic health, secondary plant compounds

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

Introduction

Metabolic syndrome (MetS) is defined as a constellation of different metabolic disorders (including abdominal obesity, insulin resistance, high cholesterol, and high blood pressure) that increases the risk of cardiovascular diseases (CVD), the leading cause of mortality worldwide. MetS is a global health problem, affecting nearly 25% of the world's adult population (with regional variabilities) [START_REF] Saklayen | The Global Epidemic of the Metabolic Syndrome[END_REF]. It is commonly associated with chronic inflammation and high levels of oxidative stress [START_REF] Rani | Oxidative stress and metabolic disorders: Pathogenesis and therapeutic strategies[END_REF]. Possible causes for MetS and associated risk factors include changes in lifestyle (e.g., sedentary lifestyle or unbalanced diet), chronic exposure to environmental pollutants, and genetic predisposition. Several studies have shown an association between a healthy diet rich in fruits and vegetables and a reduction in the risk of developing chronic diseases, including CVD, type 2 diabetes (T2D), and certain types of cancer [START_REF] Joshipura | The Effect of Fruit and Vegetable Intake on Risk for Coronary Heart Disease[END_REF][START_REF] Rees | Mediterranean' dietary pattern for the primary prevention of cardiovascular disease[END_REF]. There is also evidence of an inverse association between fruit and vegetable consumption and MetS [START_REF] Tian | Fruit and vegetable consumption and risk of the metabolic syndrome: A meta-analysis[END_REF][START_REF] Lee | Fruit and vegetable consumption and the metabolic syndrome: A systematic review and dose-response meta-analysis[END_REF]. The observed health benefits can be explained by the antioxidant and anti-inflammatory properties of the combination of certain micronutrients (e.g., vitamins and minerals), phytochemicals (e.g., polyphenols, carotenoids), and some types of fermentable fiber [START_REF] Alissa | Dietary Fruits and Vegetables and Cardiovascular Diseases Risk[END_REF], which typically relates well to the intake of fruits/vegetables from the diet [START_REF] Galan | Serum concentrations of β-carotene, vitamins C and E, zinc and selenium are influenced by sex, age, diet, smoking status, alcohol consumption and corpulence in a general French adult population[END_REF]. In parallel, studies have observed that individuals with MetS present a reduction in blood concentrations of antioxidant vitamins and phytochemicals [START_REF] Godala | The risk of plasma vitamin A, C, E and D deficiency in patients with metabolic syndrome: A case-control study[END_REF][START_REF] Ford | The Metabolic Syndrome and Antioxidant Concentrations: Findings from the Third National Health and Nutrition Examination Survey[END_REF]. This reduction could be due to the transfer of lipophilic phytochemicals, such as carotenoids and fat soluble vitamins (A, D, E, K), from the blood into fatty tissues [START_REF] Samouda | Relationship of oxidative stress to visceral adiposity in youth and role played by vitamin D[END_REF][START_REF] Landrier | Lipophilic Micronutrients and Adipose Tissue Biology[END_REF], causing a drop in the circulating proportion of these constituents in the bloodstream of individuals with more adipose tissue. Another possible explanation is the need for higher amounts of antioxidants in individuals with metabolic problems, which is also characterized by increased lipid peroxidation products (e.g., malondialdehyde (MDA) and oxidized low-density lipoprotein (ox-LDL)). These antioxidants may quench the increased levels of oxidative stress, such as reactive oxygen or reactive nitrogen species, causing a reduction in the concentration of circulating and/or stored antioxidants [START_REF] Andersen | Longitudinal associations between body mass index and serum carotenoids: The CARDIA study[END_REF]. There is also the possibility that individuals with metabolic disorders have a diet characterized by low intake of such antioxidants [START_REF] Wallström | Serum concentrations of β-carotene and α-tocopherol are associated with diet, smoking, and general and central adiposity[END_REF]. However, there is a clear relation between low dietary intake or circulating antioxidants and increased risk of chronic disease. For instance, epidemiological studies have shown positive correlations between the intake of carotenoids and lower risk of T2D [START_REF] Hamer | Intake of fruit, vegetables, and antioxidants and risk of type 2 diabetes: Systematic review and meta-analysis[END_REF], and negative correlations between carotenoid plasma levels and total mortality [START_REF] Buijsse | Plasma carotene and α-tocopherol in relation to 10-y all-cause and cause-specific mortality in European elderly: The Survey in Europe on Nutrition and the Elderly, a Concerted Action (SENECA)[END_REF] and total polyphenol intake and all-cause mortality [START_REF] Tresserra-Rimbau | Polyphenol intake and mortality risk: A re-analysis of the PREDIMED trial[END_REF]. This is supported by many in vitro and in vivo mechanistic studies showing that carotenoids can act on intracellular signaling cascades by inhibiting the formation of various downstream cytokines and/or enhancing the production of antioxidant enzymes [START_REF] Kaulmann | Carotenoids, inflammation, and oxidative stress-Implications of cellular signaling pathways and relation to chronic disease prevention[END_REF]. Results observed of the possible protective effects of antioxidant vitamins are inconclusive however, with studies showing no reduction in vitamin concentrations in individuals with MetS [START_REF] Beydoun | Carotenoids, vitamin A, and their association with the metabolic syndrome: A systematic review and meta-analysis[END_REF], sex differences [START_REF] Park | Effects of total vitamin A, vitamin C, and fruit intake on risk for metabolic syndrome in Korean women and men[END_REF], and contradictory health outcomes regarding supplementation [START_REF] Kaulmann | Carotenoids, inflammation, and oxidative stress-Implications of cellular signaling pathways and relation to chronic disease prevention[END_REF][START_REF] Lee | Vitamin E in the primary prevention of cardiovascular disease and cancer: The Women's Health Study: A randomized controlled trial[END_REF]. It should be further noted that there is a clear relation between oxidative stress and inflammation [START_REF] Khansari | Chronic Inflammation and Oxidative Stress as a Major Cause of Age-Related Diseases and Cancer. Recent Pat[END_REF][START_REF] Biswas | Does the Interdependence between Oxidative Stress and Inflammation Explain the Antioxidant Paradox? Oxidative Med[END_REF], with one potentially aggravating the other. The pro-inflammatory state in individuals with MetS is characterized by elevated concentrations of inflammatory markers, such as C-reactive protein (CRP), and alterations in the production of adipokines (proteins secreted by adipose tissue) [START_REF] Ouchi | Adipokines in inflammation and metabolic disease[END_REF]. This can be explained in part due to an excess in visceral fat characterized by dysfunctional adipocytes that secrete high concentrations of cytokines, such as leptin (pro-inflammatory role, though acting also as a hormone on satiety), and low concentrations of adiponectin (anti-inflammatory) [START_REF] Hajer | Adipose tissue dysfunction in obesity, diabetes, and vascular diseases[END_REF]. The present study aimed to assess possible associations between circulating concentrations of micronutrients/phytochemicals, inflammatory and oxidative stress markers, and MetS in a large subsample of the population of the European Health Examination Survey in Luxembourg.

Materials and Methods

Study Population and Design

We analyzed data from 606 randomly selected individuals (315 with MetS, 291 without MetS) from the 1529 participants of the European Health Examination Survey (EHES-LUX) [START_REF] Bocquet | Study design and characteristics of the Luxembourg European Health Examination Survey (EHES-LUX)[END_REF]. We selected the random sample stratified by MetS with the aim to produce a sample in which exposure varies. EHES-LUX is a cross-sectional population-based survey, representative of the general population, conducted between February 2013 and January 2015 in Luxembourg. The target sample of EHES-LUX was residents (excluding those who were living in institutions) of the Grand Duchy of Luxembourg, aged 25 to 64 years, who agreed to participate [START_REF] Bocquet | Study design and characteristics of the Luxembourg European Health Examination Survey (EHES-LUX)[END_REF]. For each participant, trained nurses performed a medical examination and collected health questionnaires and biological samples (hair, blood, and urine).

Examinations included blood pressure and anthropometric measurements (height, weight, waist circumference (WC), hip circumference, and thigh size), an electrocardiogram, visual acuity examination, and a spirometry. Blood samples were collected to analyze fasting blood glucose, plasma-triglycerides, plasma-cholesterol (total cholesterol, high-density lipoprotein cholesterol (HDL-C), low-density lipoprotein cholesterol (LDL-C)), and thyroid hormones. Moreover, we analyzed concentrations of selected micronutrients, markers of oxidative stress, and inflammation in the 606 selected individuals. Of these, we excluded 55 individuals with values below the limit of detection for any of the outcome measures above. We also excluded from the present analysis 16 outliers (one value of CRP > 90000 mg/mL, nine values of insulin > 2000 pmol/L, four values of F2-isoprostanes > 100.000 ng/mL, two values of folic acid > 90.8 nmol/L). Thus, 508 individuals had complete biological information. From those, 504 had complete information on socioeconomic and lifestyle characteristics (280 with MetS, i.e., cases, and 224 without MetS, i.e., controls) (Figure 1). EHES-LUX was approved by the National Ethics Committee of Luxembourg (CNER, N • 201205/07) and notified to the Commission Nationale pour la Protection des Données (CNPD). All individuals approved their participation by written informed consent.

vey, representative of the general population, conducted between February 2013 and Jan uary 2015 in Luxembourg. The target sample of EHES-LUX was residents (excluding those who were living in institutions) of the Grand Duchy of Luxembourg, aged 25 to 64 years who agreed to participate [START_REF] Bocquet | Study design and characteristics of the Luxembourg European Health Examination Survey (EHES-LUX)[END_REF]. For each participant, trained nurses performed a medica examination and collected health questionnaires and biological samples (hair, blood, and urine). Examinations included blood pressure and anthropometric measurements (height weight, waist circumference (WC), hip circumference, and thigh size), an electrocardio gram, visual acuity examination, and a spirometry. Blood samples were collected to ana lyze fasting blood glucose, plasma-triglycerides, plasma-cholesterol (total cholesterol high-density lipoprotein cholesterol (HDL-C), low-density lipoprotein cholesterol (LDL-C)) and thyroid hormones. Moreover, we analyzed concentrations of selected micronutrients markers of oxidative stress, and inflammation in the 606 selected individuals. Of these, we excluded 55 individuals with values below the limit of detection for any of the outcome measures above. We also excluded from the present analysis 16 outliers (one value of CRP > 90000 mg/mL, nine values of insulin > 2000 pmol/L, four values of F2-isoprostanes > 100.000 ng/mL, two values of folic acid > 90.8 nmol/L). Thus, 508 individuals had complete biologica information. From those, 504 had complete information on socioeconomic and lifestyle char acteristics (280 with MetS, i.e., cases, and 224 without MetS, i.e., controls) (Figure 1). EHES LUX was approved by the National Ethics Committee of Luxembourg (CNER, N° 201205/07 and notified to the Commission Nationale pour la Protection des Données (CNPD). All indi viduals approved their participation by written informed consent.

Metabolic Syndrome

MetS was defined as having a WC ≥ 94 cm for men and ≥ 80 cm for women and at least the presence of two factors of the following: (1) total triglycerides ≥ 150 mg/dL or on related medication for both men and women; (2) HDL cholesterol < 40 mg/dL for men and < 50 mg/dL for women or being on related medication; (3) systolic blood pressure ≥ 130 or diastolic blood pressure ≥ 85 mm Hg or being on related medication for both men and women; and (4) fasting plasma glucose ≥ 100 mg/dL or previous diagnosis of diabetes for both men and women. We used the International Diabetes Federation definition [START_REF] Zimmet | A New International Diabetes Federation (IDF) Worldwide Definition of the Metabolic Syndrome: The Rationale and the Results[END_REF].

Micronutrients, Markers of Oxidative Stress, and Inflammation

We analyzed circulating concentrations of 14 micronutrients/phytochemicals and inflammatory/oxidative stress markers. Cortisol, folic acid, and vitamin D (as 25 hydroxyvitamin D) in serum aliquots [START_REF] Ford | The Metabolic Syndrome and Antioxidant Concentrations: Findings from the Third National Health and Nutrition Examination Survey[END_REF][START_REF] Hajer | Adipose tissue dysfunction in obesity, diabetes, and vascular diseases[END_REF], and 20 µL, respectively) were measured by a chemiluminescent enzymatic antibody-based method (Cobas e 602, Roche) by a local commercial laboratory with an accredited method (Laboratoires Réunis, Junglinster, Luxembourg). Vitamin A (retinol) and vitamin E (alpha-tocopherol) were also measured by this laboratory from serum aliquots (Agilent 1290 Infinity II, 30 µL) based on HPLC combined with UV-Vis detection, employing the Chromosystems 34,000 kit (Chromosystems, Munich, Germany). Total carotenoids were determined following the micro-extraction procedure with heptane, similar as described by Donaldson [START_REF] Donaldson | Development of a rapid, simple assay of plasma total carotenoids[END_REF]. In short, 100 µL of serum were mixed with 100 µL of methanol, including 1% butylated hydroxytoluene for protein precipitation, and then extracted with 120 and again with 70 µL of heptane. The combined fractions were measured at 450 and 470 nm spectrophotometrically (Spectra Max M2, Molecular Devices, San Jose, CA, USA) in a microcuvette (Micro Quartz Cuvette, Black, 0.7 mL, Spectrometer Cell; Science Outlet, Aliso Viejo, CA, USA). Quantification was carried out, assuming an average molecular extinction coefficient of 134.000 mol/(L * cm). Aliquots of the serum (50 µL) were measured for MDA based on the thiobarbituric acid (TBARS) assay, employing a commercial kit (10009055, Cayman Chemicals, Ann Arbout, MI, USA), based on fluorescence detection with an excitation wavelength of 525 nm and an emission wavelength of 565 nm (Spectra Max M2 spectrophotometer, Molecular Devices, San Jose, CA, USA). Total phenolics in serum aliquots were measured based on the Folin Ciocalteu assay, expressed as gallic acid equivalents. In short, 50 µL of serum and 80 µL of NaOH (2.5 M in 75% of methanol) were added to facilitate lipoprotein dissemblance. Following incubation at 37 • C for 30 min, 20 µL of 6.25 M metaphosphoric acid (MPA) was added and centrifuged (13,000× g, 5 min). The supernatant was removed and combined with a follow-up extraction of 250 µL of methanol (65%). Combined phases were quantified with 1:5 diluted Folin Ciocalteu reagent (Sigma Aldrich, St. Louis, MS, USA) at 750 nm in a wellplate reader (Polarstar, BMG Labtech, De Meern, The Netherlands); 8-iso-prostaglandin F2α in plasma was quantified by a direct enzyme immunoassay kit (ADI-901-091; Enzo Life Sciences AG, Villeurbanne, France). Ox-LDL in serum was measured by enzyme immunoassay (BI-20032; OLAB, Vienna, Austria). Leptin, adiponectin, CRP, and insulin in plasma (references DY398, DY1065, DY1707, DY8056, respectively) were measured using commercially available enzyme-linked immunosorbent assay kits purchased from Biotechne R&D systems (Lille, France). All quantifications were made according to the manufacturer's specifications, after preliminary dilution tests, and based on external calibration curves.

Covariates

We included age, sex (men/women), and country of birth (Luxembourg, Portugal, other EU countries, other non-EU countries) as sociodemographic variables. Socioeconomic status included the variables education level (categorized into no qualification, primary education, secondary education, and tertiary education) and job status (not working, working). Lifestyle characteristics included smoking (current smoking or quitted < 12 months, non-smokers or quitted > 12 months), alcohol consumption (non-alcohol consump-tion, ≤6 drinks/week, >6 drinks/week), and physical activity (aerobic physical activity ≥ 150 min, aerobic physical activity < 150 min per week).

Statistical Data Analysis

We used means (± standard deviation) and frequencies (%) to describe the general characteristics of the sample population. Associations between MetS and potential risk factors (e.g., age, sex, smoking history, socioeconomic status) were analyzed with Pearson's chi-squared test (χ2) or Student's t-test as appropriate. Associations between MetS and micronutrients and markers of oxidative stress and inflammation were analyzed by the Wilcoxon test for non-parametric values. All concentrations of micronutrients/phytochemicals and markers of oxidative stress and inflammation were tested for normal distribution using the Shapiro-Wilk test of normality. Concentrations were natural log-transformed to meet residual normality. We used the least absolute shrinkage and selection operator (LASSO) [START_REF] Tibshirani | Regression Shrinkage and Selection via the Lasso[END_REF] to select (from a total of twenty variables) the variables included in the final model by using the GLMNET package on R. The objective was to obtain the L1 penalization (λ) parameter to find the simplest model (i.e., the smallest number of coefficients) and a reduction of the bias and variance. We did a cross-validation and applied the rule of one standard error. In order to avoid over-fitting, a nonparametric bootstrapping with 1000 iterations was performed, and the selected variables were those that appeared more than 80% in all bootstrap samples. Following the selection of the variables, we calculated the odds ratios (ORs) and their 95% confidence intervals (CIs) to study the association between MetS and micronutrients, markers of oxidative stress, and inflammation. All analyses were stratified by sex. Analyses included participants with complete information for all variables. We performed a sensitivity analysis in which we replaced censored values of CRP, insulin, and leptin with the values of the detection limit divided by two. All tests were two-tailed. We considered a p-value of 0.05 as statistically significant. All analyses were performed using SAS 9.4 (SAS Institute Inc., Cary, NC, USA) and R (version 3.5.3).

Results

Participants' Characteristics

Men had higher values of weight, BMI, and waist circumference, and presented higher levels of cardiometabolic risk factors compared to women (e.g., higher systolic and diastolic blood pressure, concentration of triglycerides, total cholesterol and fasting glucose, lower concentrations of HDL-C). Nearly half of participants included in the present study had MetS (55.6% with MetS compared to 44.4% without MetS). In our study sample (N = 504), nearly 62% of men had MetS compared to 48% of women. Men also had more MetS components (e.g., hypertriglyceridemia, hyperglycemia, and low HDL-C) compared to women (Table 1). Younger participants with higher education, with a job, and who did aerobic physical activity ≥ 150 min per week were less likely to have MetS. The same results were observed when comparing men and women, with the exception of job status for women (Supplementary Table S1). N: number; SD: standard deviation; BMI: body mass index; HDL-C: high-density lipoprotein cholesterol; HbA1c: glycated hemoglobin A1c. 1 Pearson's chi-squared test for categorical variables, t-test for continuous variables. 2 Waist circumference ≥ 94 cm for men and ≥ 80 cm for women. 3 Total triglycerides ≥ 150 mg/dL or on related medication. 4 HDL cholesterol < 40 mg/dL for men and < 50 mg/dL for women or on related medication. 5 Systolic blood pressure ≥ 130 or diastolic blood pressure ≥ 85 mm Hg or on related medication. 6 Fasting plasma glucose ≥ 100 mg/dL or self-reported based on a physician diagnosis.

Association between MetS and Markers of Inflammation, Oxidative Stress, and Micronutrients

We observed differences in blood concentrations of micronutrients/phytochemicals and inflammatory/oxidative stress markers between participants with and without MetS (Table 2). Concentrations of vitamin E and A, leptin, CRP, insulin, and 8-iso-prostaglandin F2α were higher in participants with MetS. Concentrations of adiponectin and Ox-LDL were lower in participants with MetS. When stratified by sex (Figure 2, Supplementary Table S2), concentrations of vitamin E, leptin, CRP, and insulin were higher in men with MetS, whereas adiponectin and Ox-LDL concentrations were lower. Regarding women, concentrations of vitamins E and A, leptin, and CRP were higher for those with MetS, while concentrations of adiponectin and beta-carotene were lower.

In both men and women, we observed a positive correlation between CRP and leptin and insulin and vitamin E, and a negative correlation between CRP and adiponectin and vitamin D and beta-carotene (Figure 3). Leptin was positively correlated with insulin, CRP, and vitamin E, and negatively correlated with adiponectin and beta-carotene. We also observed a positive correlation between adiponectin and vitamin D and beta-carotene and cortisol, and a negative correlation between adiponectin and leptin and insulin and CRP in women.

When stratified by sex (Figure 2, Supplementary Table S2), concentrations of vitamin E, leptin, CRP, and insulin were higher in men with MetS, whereas adiponectin and Ox-LDL concentrations were lower. Regarding women, concentrations of vitamins E and A, leptin, and CRP were higher for those with MetS, while concentrations of adiponectin and beta-carotene were lower. In both men and women, we observed a positive correlation between CRP and leptin and insulin and vitamin E, and a negative correlation between CRP and adiponectin and vitamin D and beta-carotene (Figure 3). Leptin was positively correlated with insulin, CRP, and vitamin E, and negatively correlated with adiponectin and beta-carotene. We also observed a positive correlation between adiponectin and vitamin D and beta-carotene and cortisol, and a negative correlation between adiponectin and leptin and insulin and CRP in women. Leptin, adiponectin, CRP, vitamin A, age, sex, and physical activity were selected by the LASSO logistic regression as the best predictors for MetS (Table 3). For both men and women combined, a one-unit increase in log CRP was associated with a 51% increase in the odds of having MetS (OR 1.51 (95% CI: 1.16, 1.98)), and a one-unit increase in log adiponectin was associated with a 76% decrease in the odds of having MetS (0.24 (0.13, 0.46)). Moreover, adults with a one-unit increase in log leptin were 3.10 times more likely to have MetS (3.10 (2.10, 4.72)), and a one-unit increase in vitamin A was associated with a 2% increase in the odds of having MetS (1.02 (1.01, 1.03)). We observed that, in men, a one-unit increase in log CRP was associated with a 71% increase in the odds of having MetS (1.71 (1.19, 2.51)), and men with a one-unit increase in log leptin were 3.34 times more likely to have MetS (3.34 (2.15, 5.42)). For women, a one-unit increase in log CRP was associated with a 38% increase in the odds of having MetS (1.38 (0.94, 2.06)), and a one-unit increase in vitamin A was associated with a 3% increase in the odds of having MetS (1.03 (1.01, 1.05)). Moreover, women with a one-unit increase in log leptin were 3.31 times more likely to have MetS (3.31 (1.63, 7.20)), and, in women, a one-unit increase in log adiponectin was associated with an 82% decrease in the odds of having MetS (0.18 (0.07, 0.46)). Leptin, adiponectin, CRP, vitamin A, age, sex, and physical activity were selected by the LASSO logistic regression as the best predictors for MetS (Table 3). For both men and women combined, a one-unit increase in log CRP was associated with a 51% increase in the odds of having MetS (OR 1.51 (95% CI: 1.16, 1.98)), and a one-unit increase in log adiponectin was associated with a 76% decrease in the odds of having MetS (0.24 (0.13, 0.46)). Moreover, adults with a one-unit increase in log leptin were 3.10 times more likely to have MetS (3.10 (2.10, 4.72)), and a one-unit increase in vitamin A was associated with a 2% increase in the odds of having MetS (1.02 (1.01, 1.03)). We observed that, in men, a oneunit increase in log CRP was associated with a 71% increase in the odds of having MetS (1.71 (1.19, 2.51)), and men with a one-unit increase in log leptin were 3.34 times more likely to have MetS (3.34 (2.15, 5.42)). For women, a one-unit increase in log CRP was associated with a 38% increase in the odds of having MetS (1.38 (0.94, 2.06)), and a oneunit increase in vitamin A was associated with a 3% increase in the odds of having MetS (1.03 (1.01, 1.05)). Moreover, women with a one-unit increase in log leptin were 3.31 times more likely to have MetS (3.31 (1.63, 7.20)), and, in women, a one-unit increase in log adiponectin was associated with an 82% decrease in the odds of having MetS (0.18 (0.07, 0.46)).

MetS Components and Markers of Inflammation

When examining MetS components separately (Table 4), we observed higher odds of abdominal obesity and hyperglycemia associated with leptin in both men (15.19 (7.04, 32.79)) and (15.14 (5.64, 40.62)), respectively) and women (2.08 (1.45, 2.98) and 1.97 (1.01, 3.82), respectively). Higher odds of high blood pressure were associated with leptin (1.57 (1.08, 2.27)), and higher odds of abdominal obesity were associated with CRP (2.14 (1.33, 3.45)) only in men. In women, higher odds of abdominal obesity, hypertriglyceridemia, low HDL, and high blood pressure were associated with lower concentrations of adiponectin (0.23 (0.07, 0.70), 0.21 (0.09, 0.53), 0.27 (0.11, 0.67), and 0.31 (0.13, 0.74), respectively). We also observed that, in women, vitamin A was associated with higher odds of hypertriglyc-

Discussion

In the present study, we have provided insights into the factors that may influence the development of MetS. In particular, our results reveal the necessity to target a variety of factors, such as dietary micronutrients, inflammation, and oxidative stress. Elevated circulating concentrations of CRP, leptin, insulin, and vitamin A were all positively associated with MetS and MetS components, whereas adiponectin was inversely associated. To our knowledge, this is the first study that uses the LASSO technique to identify specifically which circulating micronutrients/phytochemicals and inflammatory/oxidative stress markers are associated with MetS.

Regarding dietary micronutrients, in our study, we included serum vitamin E, polyphenols, and carotenoids based on their frequent presence in plant-based diets [START_REF] Rauma | Antioxidant status in long-term adherents to a strict uncooked vegan diet[END_REF], and vitamin A and D for their occurrence in animal-based diets [START_REF] Booth | Natural Food Sources of Vitamin A and Provitamin A[END_REF]. Contrary to observations made in other studies [START_REF] Godala | The risk of plasma vitamin A, C, E and D deficiency in patients with metabolic syndrome: A case-control study[END_REF][START_REF] Ford | The Metabolic Syndrome and Antioxidant Concentrations: Findings from the Third National Health and Nutrition Examination Survey[END_REF], our results showed higher concentrations of vitamins E and A in individuals with MetS. In the final model, only vitamin A remained statistically significant, with a positive association observed in women only. It should be noted that, compared to other studies where participants had vitamin deficiencies, in our sample, almost all participants showed concentrations of vitamins A/E within the normal values (>20 µg/dL and >500 µg/dL, respectively). The evidence regarding the association between vitamin A and MetS, however, remains inconclusive. In a recent meta-analysis, there was no association observed between vitamin A and MetS [START_REF] Beydoun | Carotenoids, vitamin A, and their association with the metabolic syndrome: A systematic review and meta-analysis[END_REF]. Ford et al. [START_REF] Ford | The Metabolic Syndrome and Antioxidant Concentrations: Findings from the Third National Health and Nutrition Examination Survey[END_REF] and Beydoun et al. [START_REF] Beydoun | Serum Antioxidant Status Is Associated with Metabolic Syndrome among U.S. Adults in Recent National Surveys[END_REF] observed similar results to the present study, although the positive association between vitamins A/E and MetS disappeared or changed after adjusting for cholesterol and triglycerides, a result that we did not observe. In this particular case, adjusting for cholesterol and triglycerides could be justified, since vitamin E is transported through LDL-C and HDL-C. This allows us to exclude variations caused by these transporters as the underlying reason. In Western societies, individuals generally obtain vitamin A from two main dietary sources: preformed vitamin A, i.e., retinol (ca. 75% of total vitamin A intake) or provitamin A in the form of carotenoids (e.g., beta-carotene, ca. 25%) [START_REF] Booth | Natural Food Sources of Vitamin A and Provitamin A[END_REF]. From our study, based on mainly healthy participants from the general population, vitamin A (mainly coming from animal sources, e.g., meat, dairy products, fish, and associated oils) could be an indicator of their type of diet. This would explain why participants with MetS showed higher concentrations of vitamin A, since their diet might be rich in animal sources and low in fruit-and vegetable-derived carotenoids. Further on this hypothesis and the results observed in other studies, we also observed lower serum levels of beta-carotene in individuals with MetS [START_REF] Beydoun | Carotenoids, vitamin A, and their association with the metabolic syndrome: A systematic review and meta-analysis[END_REF]. Studies have shown that a diet rich in carotenoids can have a protective function in the development of chronic diseases [START_REF] Ford | The Metabolic Syndrome and Antioxidant Concentrations: Findings from the Third National Health and Nutrition Examination Survey[END_REF][START_REF] Hamer | Intake of fruit, vegetables, and antioxidants and risk of type 2 diabetes: Systematic review and meta-analysis[END_REF][START_REF] Beydoun | Carotenoids, vitamin A, and their association with the metabolic syndrome: A systematic review and meta-analysis[END_REF]. This effect could be mediated by carotenoids acting as antioxidants, modulating transcription factors while also regulating inflammation [START_REF] Kaulmann | Carotenoids, inflammation, and oxidative stress-Implications of cellular signaling pathways and relation to chronic disease prevention[END_REF]. However, in our study, beta-carotene was not selected in our final model. As for carotenoids, studies have observed that higher intakes of polyphenols are associated with reduced cardiometabolic risk factors [START_REF] Del Bo | Systematic Review on Polyphenol Intake and Health Outcomes: Is there Sufficient Evidence to Define a Health-Promoting Polyphenol-Rich Dietary Pattern?[END_REF]. In our study, we did not observe this association, although it is possible that i) our analytical test was not specific enough for dietary polyphenols, since it also detected other reduced compounds, and ii) the low bioavailability of many polyphenols impeded clear findings [START_REF] Del Bo | Systematic Review on Polyphenol Intake and Health Outcomes: Is there Sufficient Evidence to Define a Health-Promoting Polyphenol-Rich Dietary Pattern?[END_REF].

Our results emphasize the inflammatory response related to MetS, since biomarkers of inflammation, such as CRP, adiponectin, and leptin, were selected in our final model. Levels of CRP are increased in response to inflammation and are considered an indicator of coronary risk and metabolic problems [START_REF] Pai | Inflammatory Markers and the Risk of Coronary Heart Disease in Men and Women[END_REF]. On the other hand, levels of adiponectin (secreted by adipose tissue) are reduced in response to chronic inflammation [START_REF] Ouchi | A Novel Role for Adiponectin in the Regulation of Inflammation[END_REF]. The role of adiponectin was mainly observed in women. Sex-based differences in the strength of the association between adiponectin and MetS could be explained by the fact that women had higher concentrations of circulating adiponectin compared to men. These sex differences seemed to be related to body composition and fat distribution, since women usually have less visceral adipose tissue and more subcutaneous adipose tissue than men [START_REF] Lew | Sex-Based Differences in Cardiometabolic Biomarkers[END_REF]. Both CRP and adiponectin are considered as key players in the development of MetS, characterized by an imbalance of pro-and anti-inflammatory response in favor of the first. Furthermore, we observed that leptin had the strongest association with MetS, although the main association was observed with abdominal obesity as expected, since concentrations of leptin are correlated with total body fat [START_REF] Kumari | An update on metabolic syndrome: Metabolic risk markers and adipokines in the development of metabolic syndrome[END_REF]. This association was observed in both men and women regardless of concentrations being higher in women than men. Leptin was also associated with high blood pressure in men and hyperglycemia in both men and women. Cross-sectional and prospective studies have observed an association between leptin concentrations and MetS independent of obesity [START_REF] Franks | Leptin Predicts a Worsening of the Features of the Metabolic Syndrome Independently of Obesity[END_REF][START_REF] Frühbeck | Involvement of the leptin-adiponectin axis in inflammation and oxidative stress in the metabolic syndrome[END_REF]. Leptin has different biological roles, including energy balance and endocrine function [START_REF] Charchour | Mutltifaceted biological roles of leptin[END_REF]. Studies have observed that a dysfunctional adipose tissue produces adipokines (e.g., adiponectin and leptin) in abnormal concentrations and is associated with cardiometabolic problems [START_REF] Kumari | An update on metabolic syndrome: Metabolic risk markers and adipokines in the development of metabolic syndrome[END_REF]. Leptin resistance has also been identified as a risk factor associated with MetS, similar to insulin resistance [START_REF] Könner | Selective Insulin and Leptin Resistance in Metabolic Disorders[END_REF]. Insulin concentrations were also higher in adults with MetS, though in the final model the association was not statistically significant. Although insulin resistance is a characteristic of individuals with MetS, MetS may appear without the presence of elevated insulin concentrations [START_REF] Han | A clinical perspective of obesity, metabolic syndrome and cardiovascular disease[END_REF]. This could explain why insulin was selected in our model, but its association was not statistically significant.

While inflammation was related to MetS, no marker of oxidative stress (including OxLDL, MDA, or 8-iso-prostaglandin F2α) was associated with MetS or its components. This contrasts with previous findings that have associated these markers with MetS [START_REF] Hurtado-Roca | Oxidized LDL is associated with metabolic syndrome traits independently of central obesity and insulin resistance[END_REF][START_REF] Van't Erve | Classifying oxidative stress by F(2)-isoprostane levels across human diseases: A meta-analysis[END_REF]. It is possible that the limited differences in dietary status and the overall relatively healthy population prevented an imbalance in oxidative stress homeostasis.

Limitations of the present study include the cross-sectional design of our study, which does not allow drawing conclusions about causality. Moreover, we only analyzed data from participants with complete measurements, and therefore lost information by excluding participants with incomplete data. Strengths include the individual and objective measurement of a large number of risk factors. In addition, the LASSO selection technique allowed us to select the variables that best explained the association.

Conclusions

The present study contributes to a better understanding of the key determinants (selected from a large set of possible factors of MetS). Oxidative stress, inflammation, and nutritional status are among several factors that may influence the onset of MetS, and this study emphasizes that chronic inflammation appears to best characterize individuals with MetS.

Supplementary Materials:

The following are available online at https://www.mdpi.com/2072-6 643/13/1/5/s1, Table S1: Participants' sociodemographic, economic, and lifestyle characteristics by metabolic syndrome, stratified by sex (N = 504); Table S2: Concentration of micronutrients and markers of inflammation and oxidative stress by metabolic syndrome, stratified by sex (N = 504).

Figure 1 .

 1 Figure 1. Flowchart sample selection. EHES-LUX: European Health Examination in Luxembourg; SES: socioeconomic status.

Figure 1 .

 1 Figure 1. Flowchart sample selection. EHES-LUX: European Health Examination in Luxembourg; SES: socioeconomic status.

Figure 2 .

 2 Figure 2. Concentrations of markers of dietary nutrient intake, inflammation, and oxidative stress by metabolic syndrome, stratified by sex. All concentrations were natural log-transformed with the exception of vitamin A. Original units prior to log-transformation were vitamin E (μg/dL), adiponectin (μg/dL), ox-LDL (U/dL), leptin (µ g/dL), CRP (µ g/dL), insulin (nmol/dL), and beta-carotene (nmol/dL). Ns: p-value > 0.05; * p-value ≤ 0.05; ** p-value ≤ 0.01; *** p-value ≤ 0.001; **** pvalue ≤ 0.0001. Ox-LDL: oxidized low-density lipoprotein; CRP: c-reactive protein.

Figure 2 .

 2 Figure 2. Concentrations of markers of dietary nutrient intake, inflammation, and oxidative stress by metabolic syndrome, stratified by sex. All concentrations were natural log-transformed with the exception of vitamin A. Original units prior to log-transformation were vitamin E (µg/dL), adiponectin (µg/dL), ox-LDL (U/dL), leptin (µg/dL), CRP (µg/dL), insulin (nmol/dL), and beta-carotene (nmol/dL). Ns: p-value > 0.05; * p-value ≤ 0.05; ** p-value ≤ 0.01; *** p-value ≤ 0.001; **** p-value ≤ 0.0001. Ox-LDL: oxidized low-density lipoprotein; CRP: c-reactive protein.

Figure 3 .

 3 Figure 3. Correlations between micronutrients and markers of inflammation and oxidative stress. * p-value < 0.05; ** pvalue < 0.01; *** p-value < 0.001. T.phenolics: total phenolics; MDA: malondialdehyde; Ox-LDL: oxidized low-density lipoprotein; CRP: c-reactive protein; 8-iso-PGF: 8-iso-prostaglandin-F2α. All concentrations were natural log-transformed with the exception of vitamin A, MDA, and cortisol.

Figure 3 .

 3 Figure 3. Correlations between micronutrients and markers of inflammation and oxidative stress. * p-value < 0.05; ** p-value < 0.01; *** p-value < 0.001. T.phenolics: total phenolics; MDA: malondialdehyde; Ox-LDL: oxidized low-density lipoprotein; CRP: c-reactive protein; 8-iso-PGF: 8-iso-prostaglandin-F2α. All concentrations were natural log-transformed with the exception of vitamin A, MDA, and cortisol.

Table 1 .

 1 Cardiometabolic characteristics of the sample stratified by sex (N = 504).

		Total (N = 504)	Men (N = 272)	Women (N = 232)	p-Value 1
		Mean ± SD, N (%)	Mean ± SD, N (%)	Mean ± SD, N (%)
	Anthropometric characteristics
	Weight, kg	81.61 ± 18.47	88.37 ± 17.10	73.67 ± 16.83	<0.0001
	BMI, kg/m 2	28.34 ± 5.670	28.71 ± 4.830	27.90 ± 6.510	0.12
	Waist, cm	96.24 ± 15.19	101.0 ± 13.32	90.62 ± 15.36	<0.0001
	Hip size, cm	104.3 ± 11.06	102.8 ± 9.130	106.0 ± 12.78	<0.01
	Cardiometabolic risk factors
	Systolic blood pressure, mm Hg	127.1 ± 16.63	132.3 ± 14.41	121.0 ± 17.02	<0.0001
	Diastolic blood pressure, mm Hg	81.89 ± 11.14	84.10 ± 10.93	79.28 ± 10.83	<0.0001
	Triglycerides, mg/dL	134.8 ± 81.65	160.9 ± 91.56	104.1 ± 54.18	<0.0001
	HDL-C, mg/dL	48.22 ± 13.53	42.81 ± 10.58	54.56 ± 13.86	<0.0001
	Total cholesterol, mg/dL	203.6 ± 38.53	204.6 ± 40.43	202.3 ± 36.24	0.50
	Glucose, mg/dL	101.5 ± 21.28	104.8 ± 25.73	97.53 ± 13.45	<0.0001

Table 1 .

 1 Cont.

		Total (N = 504)	Men (N = 272)	Women (N = 232)	p-Value 1
		Mean ± SD, N (%)	Mean ± SD, N (%)	Mean ± SD, N (%)
	HbA1c, mol/mol	38.51 ± 6.840	38.72 ± 7.960	38.27 ± 5.250	0.45
	Metabolic syndrome	280 (55.56)	169 (62.13)	111 (47.84)	<0.01
	Metabolic syndrome components
	Abdominal obesity 2	350 (69.44)	185 (52.86)	165 (47.14)	0.45
	Hypertriglyceridemia 3	230 (45.63)	153 (66.52)	77 (33.48)	<0.0001
	Low HDL-C 4	219 (43.35)	148 (67.58)	71 (32.42)	<0.0001
	High blood pressure 5	265 (52.68)	145 (54.72)	120 (45.28)	0.76
	Hyperglycemia 6	289 (57.46)	182 (62.98)	107 (37.02)	<0.0001

Table 2 .

 2 Concentration of micronutrients and circulating markers of inflammation and oxidative stress by metabolic syndrome (N = 504).

		Metabolic Syndrome (N = 280)		Non Metabolic Syndrome (N = 224)
		Median	Q25	Q75	Min	Max	Median	Q25	Q75	Min	Max	p-Value 1
	Markers of dietary intake
	Vitamin D, µg/dL	0.213	0.147	0.273	0.046	0.710	0.224	0.156	0.282	0.062	0.536	0.32
	Vitamin E, µg/dL	1274	1087	1538	657.0	3100	1154	993.0	1327	587.0	2487	<0.0001
	Vitamin A, µg/dL	71.50	59.50	82.00	26.00	142.0	65.00	53.00	75.00	27.00	125.0	<0.0001
	Folic acid, nmol/dL	122.0	89.50	178.0	29.00	546.0	132.0	93.50	180.0	26.00	686.0	0.36
	Beta-carotene, nmol/dL	1078	751.0	1510	28.00	5459	1220	851.0	1822	284.0	4268	<0.01
	Total phenolics, mmol/dL	1072	1006	1144	785.3	1384	1065	1014	1125	781.2	1351	0.66
	Markers of oxidative stress
	MDA, µM	1.101	0.847	1.389	0.171	2.605	1.102	0.807	1.343	0.132	2.520	0.49
	Ox-LDL, U/dL	0.157	0.067	0.374	0.014	1.368	0.231	0.100	0.647	0.012	1.453	<0.001
	8-Iso-prostaglandin-F2α, µg/dL	0.019	0.009	0.086	0.002	0.412	0.017	0.009	0.067	0.001	0.317	0.34
	Marker of stress
	Cortisol, nmol/dL	4153	3243	5150	112.0	12980	4152	3346	5202	1025	16240	0.95
	Markers of inflammation
	CRP, µg/dL	0.016	0.008	0.028	0.001	0.138	0.007	0.003	0.014	0.000	0.070	<0.0001
	Adiponectin, µg/dL	0.027	0.020	0.036	0.007	0.117	0.034	0.025	0.047	0.008	0.132	<0.001
	Markers of energy metabolism and hormones
	Leptin, µg/dL	0.176	0.112	0.268	0.010	1.365	0.096	0.045	0.192	0.004	0.667	<0.0001
	Insulin, nmol/dL	1.148	0.760	1.761	0.180	17.97	0.919	0.540	1.228	0.157	18.00	<0.0001

N: number; Q: quartile; MDA: malondialdehyde; Ox-LDL: oxidized low-density lipoprotein; CRP: c-reactive protein; 1 Wilcoxon test.

Table 3 .

 3 Variables selected by the LASSO logistic regression, measuring the association between metabolic syndrome and micronutrients and markers of inflammation and oxidative stress (N = 504).

	Metabolic Syndrome
	Men	Women	Total
	OR (95% CI)	OR (95% CI)	OR (95% CI)

04, 1.12) 1.13 (1.09, 1.18) 1.10 (1.08,

1.14)

Table 3 .

 3 Variables

		Metabolic Syndrome
	Men	Women	Total
	OR (95% CI)	OR (95% CI)	OR (95% CI)

selected by the LASSO logistic regression, measuring the association between metabolic syndrome and micronutrients and markers of inflammation and oxidative stress (N = 504).

Nutrients 2021, 13, 5. https://doi.org/10.3390/nu13010005 https://www.mdpi.com/journal/nutrients

Acknowledgments:

We are grateful to the population of Luxembourg and to all of the EHES-LUX team who have contributed to this study. We would like to thank Couffignal, S., Kuemmerle, A., Dincau, M., Mormont, D., Barre, J., Chioti, A., Delagardelle, C., Michel, G., Schlesser, M., Schmit, J.C., Gantenbein, M., Lieunard, C., Columeau, A., Kiemen, M., Weis, J., Ambroset, G., Billy, A., Larcelet, M., Marcic, D., Gauthier, C., Leners, B., Viau-Courville, M., Couturier, C. and Astier, J. for their valuable contributions. This research was funded by the National Research Fund of Luxembourg ([C17/BM/11653863/iMPACT.lu to MRC)], the Directorate and Ministry of Health, and the Ministry of Higher Education and Research (MESR).

Funding: This research was funded by the National Research Fund of Luxembourg (C17/BM/11653863/ iMPACT.lu to MRC), the Directorate and Ministry of Health, and the Ministry of Higher Education and Research (MESR).

eridemia (1.04 (1.02, 1.06)), high blood pressure (1.03 (1.01, 1.05)), and hyperglycemia (1.02 (1.00, 1.04)).

Conflicts of Interest:

The authors declare no conflict of interest.


Edouard Claparède

Clap & Lach Johannes Lachmann

John R Dolan
email: dolan@imev--mer.fr

On

Keywords: History of Protistology, Ciliates, Rhizaria, Dinoflagellates, Haeckel, Darwin

Accounts are given of the lives and careers of Edouard Claparède (1832--1871) and Johannes Lachmann (1832--1860), the authors of the landmark work of 19th century protistology "Etudes sur les Infusoires et les Rhizopodes", published in 3 parts in 1859, 1860 and 1861. Accounts are also given on the origin of the monograph, the relationship of Claparède and Lachmann with Ernst Haeckel, and Claparède's role as a promoter of Darwin's theories.

Introduction

Have you run across a species name noted as "Clap & Lach" and wondered who they were? They were Edouard Claparède (1832Claparède ( --1871) ) and Johannes Lachmann (1832--1860). They authored a massive tome, "Etudes sur les Infusoires et les Rhizopodes" of some 773 pages and 37 plates, containing about 200 protist species descriptions, most described as new. It was published over 4 years (1858--1861) in 3 separate volumes of a Swiss journal, Mémoires de l'Institut National Genevois, and also issued in a single large volume in 1868. Because of this major work, Claparède and Lachmann have been given some attention in histories of protistology (e.g., Saville Kent 1880--1881[START_REF] Cole | The History of Protozoology[END_REF][START_REF] Corliss | A salute to fifty--four great microscopists of the past: a pictorial footnote to the history of protozoology. Part 1[END_REF]). However, these accounts included few details of their lives and careers and neglect the origins of their landmark work on protists. Here though an attempt is made to shine a light on Claparède and Lachmann (Fig. 1) and their "Etudes". First, biographical sketches of both men will be given. These will include little known information on their relationships with their contemporary, Ernst Haeckel and Claparède's role in defending Darwin's views. A short description of the contents and the confusing publication history of the "Etudes" will then be provided along with evidence of the confusion in the protistology literature as to how to cite it. A supplementary file is provided which lists the species currently credited to Claparède and Lachmann's "Etudes", along with the names as given by them, and page locations in "Etudes" (see supplementary file).

Edouard Claparède and Johannes Lachmann

There is a good deal on information available on Edouard Claparède. Several articles of various lengths were published shortly after his death (Anon 1871[START_REF] De Saussure | Notice sur Edouard Claparède[END_REF][START_REF] Lankester | Edouard René Claparède[END_REF][START_REF] Monnier | Edourad Claparède[END_REF], Strickler 1871[START_REF] Vaucher | Edouard Claparède[END_REF] or in later years [START_REF] Vogt | Edouard Claparède[END_REF][START_REF] Yung | Un membre Illustre de l'Institut Edouard Claparède[END_REF][START_REF] Roussy De Sales | Claparède protisologue[END_REF]. In contrast, very little is available on Johannes Lachmann. The only information we have is from a short 'memorium' by Claparède in the "Etudes" (Claparède 1861a), two brief death notices (Anon 1860a,b), and a short entry in a dictionary of German Botanists (Bohmon 1977). While admittedly fragmentary, and sometimes containing obvious errors, the articles cited above form the basis of the following biographical accounts. Some additional information was obtained from 'Lettres de René--Edouard Claparède' (Morsier 1971). Jean--Louis--René--Antoine--Edouard Claparède was born on April 24, 1832 into a protestant family in Chancey, a Swiss village near the French border, not far from Geneva. His ancestors had fled the south of France during a period of religious persecution. His father was a minister. After attending 'Gymnase' in Geneva he took classes at the Academie in Geneva from 1850 to 1852. He graduated with a Bachelor's degree and then in 1853 went to the University of Berlin ostensibly to study to be a medical doctor. Once in Berlin, Claparède joined the group of medical students studying under the renowned physiologist and comparative anatomist Johannes Müller and among them was Johannes Lachmann. The two were the same age and became close friends. With regard to Lachmann, we know that he was from Braunsweig (Brunswick; now in Germany), born August 1, 1832, and that his father was a medical doctor. We do not when Lachmann joined Müller's group, only that he was awarded his medical degree in 1855, two years before Claparède. When Claparède arrived in Berlin, Lachmann was already an established member of Müller's group. He had even traveled to Trieste with Müller on a collecting expedition in 1852. In Müller's group, studies of natural history, rather than strictly medical studies appeared to be the rule. Lachmann's thesis work was on the structure of root nodules in legumes and was published in Müller's journal (Archiv für Anatomie und Physiologie) in 1857 (Lachmann 1857a). Claparède's thesis was on a freshwater bivalve, also published in Müller's journal (Claparède 1857). However, however the first publications of both Claparède and Lachmann concerned protists. Claparède's was on the freshwater rhizopod Actinophrys (Claparède 1854). Lachmann's first paper was on food processing in vorticellid ciliates [START_REF] Lachmann | Ueber die Organisation der Infusorien, besonders der Vorticellen[END_REF]). Lachmann's paper received international attention. A version in English was published in the Annals and Magazine of Natural History (Lachmann 1857b). Their bents toward natural history, rather than experimental physiology, is perhaps what motivated Müller to invite both Claparède and Lachmann to accompany him on a trip to Norway in the summer of 1855. According to Claparède's letters (Morseir 1971), the expedition was originally planned to sample in Messina, Italy and the destination was changed to Bergen, Norway due to an outbreak of cholera in Italy. They arrived in Bergen in early August 1855. In early September Müller left, as he did not find the echinoderm larvae he was seeking. They nearly left with Müller but he urged them to stay as they were having success in 'fishing for the infinitely small'. Lucky it was that they did not leave with Müller as his return trip ended dramatically in a shipwreck that Müller survived but the one of the students traveling with him did not. Claparède and Lachmann spent the months of September and October sampling protists, mostly of the marine plankton, traveling around to various localities on the coast, happy to be working on their own rather than under watchful eyes of Müller. They stayed in Norway until late October. The lasting importance of their work in the Bergen area over the three months spent there becomes evident in examining the species described in their "Etudes". The majority of forms described as new were found and observed during their stay in Norway (66 of 111 'new' species and 42 still considered today as valid first descriptions). One cannot help but wonder what they would have found had they gone to Messina rather than Bergen. Claparède and Lachmann returned to Berlin in early November. It was soon after their return to Berlin that they finished work on their first protist monograph. It was created in response to a tempting challenge from the Academie des Science (Paris). On January 30th of 1854 the Academie announced the subject of the competition for the Grand Prix des Sciences Physiques for the year 1856 [START_REF] Anon | Sciences Physiques. Grand Prix des Sciences Physiques, proposé en 1854 pour 1856[END_REF]. The challenge was to submit a rigorous and methodical study of life-history stages and reproduction of infusoria, i.e. Ehrenberg's "Polygastrica." The questions to be addressed included determining the natural affinities of the major groups of infusoria as some seemed to be more plant--like and others clearly animal--like. The facts established by the study were to be illustrated using figures as much as possible. The prize was a gold medal valued at 3,000 francs, about 10,000 Euros in today's currency. The documents were to be submitted to the Academie by January 1st 1856. Claparède and Lachmann submitted their tome "Sur La Reproduction des Infusoires" in December of 1855. It consisted of 302 pages and 11 plates containing 192 figures. Given that the prize subject was announced in early 1854, it is likely they had begun work on the monograph well before their stay in Norway that ended in early November of 1855. However, many observations recorded during their stay in Norway appear in the monograph. For example, their studies of dinoflagellates were based largely on observations made on marine species while in Bergen. Their observations led them to state that many forms appeared to contain ingested food items (Fig. 2), suggesting that dinoflagellates were more animal--like than plant--like.

the topics the prize candidates were to address.

While the prize was to be awarded in early 1856, just like today, the review process delayed a final decision until the following year. On February 8th, the announcement was made that 3 studies had been submitted by the deadline and evaluated by a committee (which differed from the composition of the committee originally announced). The Grand Prix was to be shared by the authors of two of the studies, whose conclusions largely agreed with one another: the one by Claparède and Lachmann and another one by Nicolas Lieberkühn (Anon. 1858a). The Grand Prix was an important award and news of the identity of the prizewinners was distributed widely. It was reported on by newspapers in France (e.g. Anon 1858b,c) with one reprinting entirely the reviewer's report (Anon. 1858d), in Germany (Anon. 1858e) and in Austria (Anon 1858f). However, nowhere was there any mention of how the prize, a gold medal, was to be split in two! The monograph they finished in 1855 would not be published until 1861, a year after the death of Johannes Lachmann. It was published as the third and last part of their "Etudes". They were given permission to publish it by the Academy but with the stipulation that it be published as submitted in 1855. The other prize-winning work, that of Lieberkühn, who was a colleague of Claparède and Lachmann in Müller's group, was entirely in Latin and apparently was never published. He went on to become an expert on sponges (e.g. [START_REF] Lieberkühn | On the motory phenomena of the sponges[END_REF][START_REF] Lieberkühn | Ueber das contractile Gewebe der Spongien[END_REF]. It bears noting that Claparède and Lachmann were quite young when they won (albeit shared) this prestigious prize. Both were but 26 years old, while Lieberkühn was 36. By the time the prize was awarded, Johannes Lachmann had taken a faculty position, at the Royal Institute of Agriculture, Poppelsdorf, in 1857. This was the first steady job he was offered, and was spurred by his recent marriage according to Claparéde's memorium on Lachmann (Claparéde 1861a and see also the following section). Meanwhile Edouard Claparède had returned to Geneva and married in 1860. His former teacher obtained for him a teaching post at the Academy in Geneva. Lachmann died on July 7, 1860 having not quite reached his 28th birthday. He died, according to Haeckel's letters, of some sort of blood infection following a burst boil. He left behind a widow and two small children. Claparède turned his attention to a variety of invertebrate groups following the 1861 publication of the last part of the "Etudes". Claparède was of renowned perpetual ill health and died in 1871, shortly after his 39th birthday, likely of tuberculosis. Like Lachmann, he left behind a widow and two small children. Thus, both Lachmann and Claparède died remarkably young but left behind a landmark work of protistology, the "Etudes". Before leaving historical considerations to focus on the "Etudes", a few words on two neglected topics are warranted: the close ties of Claparède and Lachmann with Ernst Haeckel and the Claparède's role as a defender of Darwin.

Lachmann and Haeckel and Claparède

The group studying under the tutelage of Johannes Müller in Berlin included not only Lachmann and Claparède, but among many others, Ernst Haeckel from 1854 to 1857. From Haeckel's letters (available at https://haeckel--briefwechsel-projekt.uni--jena.de) one learns that he first met Lachmann and other students during a visit to Berlin in 1853 before joining Müller's group. Both Lachmann and Claparède became life--long friends of Haeckel and his family. Lachmann especially appears to have been close to Haeckel. They apparently vied for the affections of the same young woman in 1856, Luiza Passow, one of the daughters of another Professor at the University. Lachmann won out, with Haeckel graciously conceding defeat (June 8, 1856 letter). Haeckel made his first sampling trip to the Mediterranean in the fall of 1856 and in preparation asked Lachmann for advice. Lachmann had collected samples from Trieste, and spent two months sampling in Norway with Claparède. Haeckel asked what equipment he should take with him, which microscope, and how to sample (July 30, 1856 letter). According to Claparède's letters, on his way to Nice, Haeckel stopped and visited Claparède and his parents and brother in Geneva (Morsier 1971). Haeckel's subsequent success in sampling in Villefranche was likely a key occurrence in his life. It was during his visit to Nice and Villefranche that he first saw radiolaria alive and, on the advice of Müller, decided to study them [START_REF] Dolan | Ernst Haeckel's Radiolarians and Medusa: The influence of his visits to Villefranche on his science and his art[END_REF]. When Haeckel defended his thesis in 1857, Christian Gottfried Ehrenberg presided and Haeckel's opponents were his friends Johannes Lachmann, Edouard Claparède and Ernst Weiss (March 7, 1917 letter). Two years later Haeckel heeded Müller's advice and traveled to Messina to study radiolaria [START_REF] Richards | The Tragic Sense of Life, Ernst Haeckel and the Struggle over Evolutionary Thought[END_REF]. In 1864, while on a hiking trip in the Alps, Haeckel injured his foot and then spent several days recuperating at Claparède's home in Geneva (13 September 1864 letter). Many years later Haeckel wrote to Darwin boasting of his friend Claparède's ardent defense of the "The Origin of Species" (Darwin Correspondence Project, "Letter no. 4646"). Haeckel, his parents and his brother remained in contact with both Lachmann and Claparède over the years as well with their widows after each passed away. It is in the realm of speculation to posit that Haeckel's work on protists, especially radiolaria was due in some small part to the influences of Lachmann and Claparède. It is clear, however, that they were protistologists in Müller's group when Haeckel joined them and that they became close friends of his.

Claparède and Darwin

Edouard Claparède is credited as the first French--speaking proponent of Darwin's theories as presented in his "The Origin of Species" [START_REF] Monnier | Edourad Claparède[END_REF]. He wrote a detailed distillation of Darwin's theories published in two parts in 1861 (Claparède 1861b,c) "There is a favourable & long Review in Revue Germanique, which is important from coming from so good observer, as Claparede." Claparède is also credited with the Darwinian quote "Il vaut mieux être un singe perfectionée qu'un Adam dégénéré" with the version in English as "It is better to be a perfect ape than a degenerate Adam" [START_REF] Wood | Dictionary of quotations from ancient and modern, English and foreign sources: including phrases, mottoes, maxims, proverbs, definitions, aphorisms, and sayings of the wise men, in their bearing on life, literature, speculation, science, art, religion, and morals, especially in the modern aspects of them, selected and[END_REF]. If in England during Darwin's time, Thomas Henry Huxley could be thought of as "Darwin's Bulldog", perhaps Edouard Claparède could be considered "Darwin's St. Bernard", rescuing him from the attacks of both the religious and the followers of Lamarck in Switzerland and France.

Etudes sur les Infusoires et les Rhizopodes

The "Etudes sur les Infusoires et les Rhizopodes" is one of the fundamental works of 19th century protistology. It followed Dujardin's 1841 "Histoire naturelle des zoophytes: Infusoires, comprenant la physiologie et la classification de ces animaux, et la manière de les étudier à l'aide du microscope" and its predecessor Ehrenberg's 1838 "Die Infusionsthierchen als vollkommene Organismen. Ein Blick in das tiefere organische Leben der Natur". While Dujardin's monograph was critical of Ehrenberg's work, Claparéde and Lachmann's Etudes was critical of both Ehrenberg and Dujardin. The enduring importance of the Etudes likely lies in the contributions it made to knowledge of marine forms which had been previously neglected (Saville Kent 1880--1882), as well as detailed accounts of protist life histories [START_REF] Cole | The History of Protozoology[END_REF]. Of the many species described as new in the Etudes, most of the species currently accepted as having been first described in the Etudes are marine (see supplemental file). In the preface to the first part of the "Etudes", dated January 1858, Claparéde states (without explanation) that, sadly, Lachmann had no part in writing the first two parts of the 3--part work and participated only in writing the third part which, as we know, dated back to 1855. Nonetheless, in his text, Claparède repeatedly cites Lachmann's observations and sketches, from both their time in Norway and sampling in the environs of Berlin, as the basis of many species descriptions given in the first two parts. For example, in the first part, Tintinnus ventricosa is said to have been found in the North Sea by Lachmann, and in the second part Loxophyllum armatum is said to have been found by Lachmann from sampling peat near Berlin. Such contributions likely justified a co--authorship. The first two parts are actually one single work published in two separate volumes, the first in 1858 and the second in 1859, of the Mémoires de l'Institut National Genevois. As the two do not stand alone, publication in two volumes was quite possibly due to a page limit of a volume. The parts one and two total 482 pages plus the plates. The preceding 1857 volume of the Mémoires de l'Institut National Genevois consisted of about 200 pages. While the two first parts of 1858 and 1859 do not stand alone, they could, and likely should, be cited separately depending on the material cited. For example, the first part published in 1858 contains all the species descriptions of tintinnid ciliates, so reference to any tintinnid species description concerns only material in the 1858 publication. Likewise, any reference to a dinoflagellate species description should be 1859 as only it contains dinoflagellate species descriptions. In contrast, with regard to the possibility of phagotrophy in dinoflagellates, the subject is treated in both the 1859 volume as well as in the 1861 volume. The 1868 reprinting of all 3 parts should not be cited in any case as 'first published in...' as it consists of previously published material. The many covers of the "Etudes" are shown in Fig. 3, along with bullet descriptions of the contents. Given the publication history of the "Etudes", stretched out over 4 years, it is not surprising that it has been cited in various forms. However, it is a bit surprising that it has been incorrectly cited in recent years in our own journals of protistology. Table 1 shows a variety of incorrect citations in Acta Protozoologica, European Journal Protistology, Journal of Eukaryotic Microbiology and Protist in articles published from 2000 to 2021. In deference to authors who likely made inadvertent mistakes, reviewers and editors who cannot be expected to verify references, and copy editors who are not protistologists, specific articles are not given, only the year published and journal are indicated. The incorrect literature references include oddities ('vol 5 mem. 3'), mismatches of volume number with year published, omitting volume number and page range, etc. As the "Etudes" is available and searchable online (e.g. Biodiversity Heritage Library, Google Books, etc.) it is easy to verify that the reference you want to cite is correct. Edouard Claparède and Johannes Lachmann lived short lives but made a major contribution to protistology. Today, Johannes Lachmann is better known to botanists than protistologists. His thesis work on the root nodules of legumes belatedly earned him recognition as an early proponent of the importance of the nodules and their bacterial communities in plant nutrition (Bohmon 1977). Edouard Claparède worked on a wide variety taxa and among them were worms, and insects. In his day, and in more recent times, he is recognized as invertebrate zoologist more than as a protistologist (e.g., [START_REF] Lankester | Edouard René Claparède[END_REF], Petit & Théoridèse 1972). I hope that telling the stories of their lives, and of their monograph on protists, will lead to an appreciation of them as both remarkable people and pioneer protistologists.

Fig. 1 .

 1 Fig. 1. Edouard Claparède at about age 34 and Johannes Lachmann at about 27. Images from Wikimedia.

Fig. 3 .

 3 Fig. 3. The many covers of the "Etudes sur les Infusoires et les Rhizopodes". Bullet descriptions and page ranges of the contents were added.

  , a substantial review totaling 67 pages. He sent copies to Darwin who replied thusly in April of 1862 (Darwin Correspondence Project, "Letter no. 4371") Dear Sir I thank you sincerely for sending me your excellent Review & for the generous & most kind manner in which you have spoken of my views. I have read a great many critiques & abstracts of my Book; but I cannot remember one written with so much vigour & clearness & with so full an appreciation of the bearing of all the leading points. It is admirably done; & could have been so done only by one who entered on the subject with spirit, as well as with knowledge. I have been particularly struck by the eloquence of the latter portion of your article. You will have helped much in spreading in France sound views (as I hope & believe they are) on the descent and modification of species; & I have been told that my views were more unpopular in France even than in England; in Germany naturalists are beginning with some rapidity to adopt them. Pray accept my sincere thanks & respects & believe me Dear Sir | Yours very faithfully About a year later, Darwin wrote to George Bentham on the many reviews his book had received (Darwin Correspondence Project, "Letter no. 4100") saying that few reviews were worth noting but singling out the Claparède review, stating

Table 1 .

 1 The variety of incorrect citations to material in the Etudes sur les Infusoires et les Rhizopodes in Acta Protozoologica (AP), European Journal of Protistology (EJOP), Journal of Eukaryotic Microbiology (JEM) and Protist. The "Etudes" were published in 3 volumes of the Mémoires de l'Institut National

	Genevois: in 1858 vol. 5:1--260, in 1859 vol. 6:261--482, and in 1861 vol. 7:1--291.
	Year	Journal	Cited Year	Cited Vol	Cited pp
	2000	AP	1858	none	none
	2002	EJOP	1859	1--2	none
	2004	EJOP	1858	none	none
	2007	JEM	1858--1861	1--2	1--481
	2010	JEM	1858	5: mem. 3	203--204
	2011	AP	1859	5	1--260
	2015	JEM	1858	5: mem. 3	203--204
	2015	EJOP	1868	none	none
	2017	EJOP	1859	5	1--260
	2018	EJOP	1859	5	1--260
	2019	Protist	1859	none	261--482
	Conclusion

Acknowledgements

The comments and suggestions of the reviewers on a previous version led to significant improvements in the manuscript. Their thoughtful work is appreciated. Nonetheless, I retain full responsibility for all errors of fact, interpretation, or omission

Supplementary File Information

The supplementary file is a spreadsheet file composed of two sheets: one listing the protist species described as new in the "Etudes" that are currently listed as valid first descriptions and attributed to the "Etudes" by the World Register of Marine Species (http://www.marinespecies.org/index.php). The second sheet gives the full references (titles are not abbreviated) cited to facilitate location of the references.