Keywords: Forêts aléatoires, Régression, Classification, Importance des Variables, Sélection des Variables

Forêts aléatoires : remarques méthodologiques Résumé : On s'intéresse à la méthode des forêts aléatoires d'un point de vue méthodologique. Introduite par Leo Breiman en 2001, elle est désormais largement utilisée tant en classification qu'en régression avec un succès spectaculaire. On vise tout d'abord à confirmer les résultats expérimentaux, connus mais épars, quant au choix des paramètres de la méthode, tant pour les problèmes dits "standards" que pour ceux dits de "grande dimension" (pour lesquels le nombre de variables est très grand vis à vis du nombre d'observations). Mais la contribution principale de cet article est d'étudier le comportement du score d'importance des variables basé sur les forêts aléatoires et d'examiner deux problèmes classiques de sélection de variables. Le premier est de dégager les variables importantes à des fins d'interprétation tandis que le second, plus restrictif, vise à se restreindre à un sous-ensemble suffisant pour la prédiction. La stratégie générale procède en deux étapes : le classement des variables basé sur les scores d'importance suivi d'une procédure d'introduction ascendante séquentielle des variables.

Introduction

Random forests (RF henceforth) is a popular and very efficient algorithm, based on model aggregation ideas, for both classification and regression problems, introduced by [START_REF] Breiman | Random Forests[END_REF] [START_REF] Breiman | Random Forests[END_REF]. It belongs to the family of ensemble methods, appearing in machine learning at the end of nineties (see for example [START_REF] Dietterich | An experimental comparison of three methods for constructing ensembles of decision trees : Bagging, Boosting and randomization[END_REF] [START_REF] Dietterich | An experimental comparison of three methods for constructing ensembles of decision trees : Bagging, Boosting and randomization[END_REF] and (2000) [START_REF] Dietterich | Ensemble Methods in Machine Learning[END_REF]). Let us briefly recall the statistical framework by considering a learning set L = {(X 1 , Y 1 ), . . . , (X n , Y n )} made of n i.i.d. observations of a random vector (X, Y ). Vector X = (X1 , ..., X p ) contains predictors or explanatory variables, say X ∈ R p , and Y ∈ Y where Y is either a class label or a numerical response. For classification problems, a classifier t is a mapping t : R p → Y while for regression problems, we suppose that Y = s(X) + ε and s is the so-called regression function. For more background on statistical learning, see [START_REF] Hastie | The Elements of Statistical Learning[END_REF] [START_REF] Hastie | The Elements of Statistical Learning[END_REF]. Random forests is a model building strategy providing estimators of either the Bayes classifier or the regression function.

The principle of random forests is to combine many binary decision trees built using several bootstrap samples coming from the learning sample L and choosing randomly at each node a subset of explanatory variables X. More precisely, with respect to the well-known CART model building strategy (see [START_REF] Breiman | Classification And Regression Trees[END_REF] [START_REF] Breiman | Classification And Regression Trees[END_REF]) performing a growing step followed by a pruning one, two differences can be noted. First, at each node, a given number (denoted by mtry) of input variables are randomly chosen and the best split is calculated only within this subset. Second, no pruning step is performed so all the trees are maximal trees.

In addition to CART, another well-known related tree-based method must be mentioned: bagging (see [START_REF] Breiman | Bagging predictors[END_REF] [START_REF] Breiman | Bagging predictors[END_REF]). Indeed random forests with mtry = p reduce simply to unpruned bagging. The associated R 1 packages are respectively randomForest (intensively used in the sequel of the paper), rpart and ipred for CART and bagging respectively (cited here for the sake of completeness).

RF algorithm becomes more and more popular and appears to be very powerful in a lot of different applications (see for example D íaz-Uriarte and Alvarez de [START_REF] Íaz-Uriarte | Gene Selection and classification of microarray data using random forest[END_REF] [START_REF] Íaz-Uriarte | Gene Selection and classification of microarray data using random forest[END_REF] for gene expression data analysis) even if it is not clearly elucidated from a mathematical point of view (see the recent paper by [START_REF] Biau | Consistency of random forests and other averaging classifiers[END_REF] [START_REF] Biau | Consistency of random forests and other averaging classifiers[END_REF] and [START_REF] Bühlmann | Analyzing Bagging[END_REF] [START_REF] Bühlmann | Analyzing Bagging[END_REF] for bagging). Nevertheless, [START_REF] Breiman | Random Forests[END_REF] [START_REF] Breiman | Random Forests[END_REF] sketches an explanation of the good performance of random forests related to the good quality of each tree (at least from the bias point of view) together with the small correlation among the trees of the forest, where the correlation between trees is defined as the ordinary correlation of predictions on so-called out-of-bag (OOB henceforth) samples. The OOB sample which is the set of observations which are not used for building the current tree, is used to estimate the prediction error and then to evaluate variable importance.

Tuning method parameters

It is now classical to distinguish two typical situations depending on n the number of observations, and p the number of variables: standard (for n >> p) and high dimensional (when n << p). The first question when someone try to use practically random forests is to get information about sensible values for the two main parameters of the method. Essentially, the study carried out in the two papers [START_REF] Breiman | Random Forests[END_REF] and [START_REF] Íaz-Uriarte | Gene Selection and classification of microarray data using random forest[END_REF] give interesting insights but Breiman focuses on standard problems while D íaz-Uriarte and Alvarez de Andrés concentrate on high dimensional classification ones.

So the first objective of this paper is to give compact information about selected bench datasets and to examine again the choice of the method parameters addressing more closely the different situations.

RF variable importance

The quantification of the variable importance (VI henceforth) is an important issue in many applied problems complementing variable selection by interpretation issues. In the linear regression framework it is examined for example by [START_REF] Grömping | Estimators of Relative Importance in Linear Regression Based on Variance Decomposition[END_REF] [START_REF] Grömping | Estimators of Relative Importance in Linear Regression Based on Variance Decomposition[END_REF], making a distinction between various variance decomposition based indicators: "dispersion importance", "level importance" or "theoretical importance" quantifying explained variance or changes in the response for a given change of each regressor. Various ways to define and compute using R such indicators are available (see [START_REF] Grömping | Relative Importance for Linear Regression in R: The Package relaimpo[END_REF] [START_REF] Grömping | Relative Importance for Linear Regression in R: The Package relaimpo[END_REF]).

In the random forests framework, the most widely used score of importance of a given variable is the increasing in mean of the error of a tree (MSE for regression and misclassification rate for classification) in the forest when the observed values of this variable are randomly permuted in the OOB samples. Often, such random forests VI is called permutation importance indices in opposition to total decrease of node impurity measures already introduced in the seminal book about CART by [START_REF] Breiman | Classification And Regression Trees[END_REF] [START_REF] Breiman | Classification And Regression Trees[END_REF].

Even if only little investigation is available about RF variable importance, some interesting facts are collected for classification problems. This index can be based on the average loss of another criterion, like the Gini entropy used for growing classification trees. Let us cite two remarks. The first one is that the RF Gini importance is not fair in favor of predictor variables with many categories while the RF permutation importance is a more reliable indicator (see [START_REF] Strobl | Bias in random forest variable importance measures: illustrations, sources and a solution[END_REF] [START_REF] Strobl | Bias in random forest variable importance measures: illustrations, sources and a solution[END_REF]). So we restrict our attention to this last one. The second one is that it seems that permutation importance overestimates the variable importance of highly correlated variables and they propose a conditional variant (see [START_REF] Strobl | Conditional variable importance for Random Forests[END_REF] [START_REF] Strobl | Conditional variable importance for Random Forests[END_REF]). Let us mention that, in this paper, we do not notice such phenomenon. For classification problems, Ben [START_REF] Ishak | Sélection de variables en classification binaire : comparaisons et application aux données de biopuces[END_REF] [START_REF] Ishak | Sélection de variables en classification binaire : comparaisons et application aux données de biopuces[END_REF] and D íaz-Uriarte, Alvarez de Andrés (2006) [START_REF] Íaz-Uriarte | Gene Selection and classification of microarray data using random forest[END_REF] for example, use RF variable importance and note that it is stable for correlated predictors, scale invariant and stable with respect to small perturbations of the learning sample. But these preliminary remarks need to be extended and the recent paper by [START_REF] Archer | Empirical characterization of random forest variable importance measures[END_REF] [START_REF] Archer | Empirical characterization of random forest variable importance measures[END_REF], focusing more specifically on the VI topic, do not answer some crucial questions about the variable importance behavior: like the importance of a group of variables or its behavior in presence of highly correlated variables. This one is the second goal of this paper.

Variable selection

Many variable selection procedures are based on the cooperation of variable importance for ranking and model estimation to evaluate and compare a family of models. Three types of variable selection methods are distinguished (see [START_REF] Kohavi | Wrappers for Feature Subset Selection[END_REF] [START_REF] Kohavi | Wrappers for Feature Subset Selection[END_REF] and [START_REF] Guyon | An introduction to variable and feature selection[END_REF] [START_REF] Guyon | An introduction to variable and feature selection[END_REF]): "filter" for which the score of variable importance does not depend on a given model design method; "wrap-

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per" which include the prediction performance in the score calculation; and finally "embedded" which intricate more closely variable selection and model estimation.

For non-parametric models, only a small number of methods are available, especially for the classification case. Let us briefly mention some of them, which are potentially competing tools. Of course we must firstly mention the wrapper methods based on VI coming from CART, see [START_REF] Breiman | Classification And Regression Trees[END_REF] [START_REF] Breiman | Classification And Regression Trees[END_REF] and of course, random forests, see [START_REF] Breiman | Random Forests[END_REF] [START_REF] Breiman | Random Forests[END_REF]. Then some examples of embedded methods: [START_REF] Poggi | Classification supervisée en grande dimension. Application à l'agrément de conduite automobile[END_REF] [START_REF] Poggi | Classification supervisée en grande dimension. Application à l'agrément de conduite automobile[END_REF] propose a method based on CART scores and using stepwise ascending procedure with elimination step; [START_REF] Guyon | Gene selection for cancer classification using support vector machines[END_REF] [START_REF] Guyon | Gene selection for cancer classification using support vector machines[END_REF] (and [START_REF] Rakotomamonjy | Variable selection using SVM-based criteria[END_REF] [START_REF] Rakotomamonjy | Variable selection using SVM-based criteria[END_REF]), propose SVM-RFE, a method based on SVM scores and using descending elimination. More recently, Ben [START_REF] Ishak | Sélection de variables en classification binaire : comparaisons et application aux données de biopuces[END_REF] [START_REF] Ishak | Sélection de variables en classification binaire : comparaisons et application aux données de biopuces[END_REF] propose a stepwise variant while [START_REF] Park | An L1 regularization-path algorithm for generalized linear models[END_REF] [START_REF] Park | An L1 regularization-path algorithm for generalized linear models[END_REF] propose a "LARS" type strategy (see [START_REF] Efron | Least angle regression[END_REF] [START_REF] Efron | Least angle regression[END_REF] for classification problems.

Let us recall that two distinct objectives about variable selection can be identified: (1) to find important variables highly related to the response variable for interpretation purpose; (2) to find a small number of variables sufficient for a good prediction of the response variable. The key tool for task 1 is thresholding variable importance while the crucial point for task 2 is to combine variable ranking and stepwise introduction of variables on a prediction model building. It could be ascending in order to avoid to select redundant variables or, for the case n << p, descending first to reach a classical situation n ∼ p, and then ascending using the first strategy, see [START_REF] Fan | Sure independence screening for ultra-high dimensional feature space[END_REF] [START_REF] Fan | Sure independence screening for ultra-high dimensional feature space[END_REF]. We propose in this paper, a two-steps procedure, the first one is common while the second one depends on the objective interpretation or prediction.

The paper is organized as follows. After this introduction, Section 2 focuses on random forests parameters. Section 3 proposes to study the behavior of the RF variable importance index. Section 4 investigates the two classical issues of variable selection using the random forests based score of importance. Section 5 finally opens discussion about future work.

2 Selecting method parameters 2.1 Experimental framework

RF procedure

The R package about random forests is based on the the seminal contribution of Breiman and Cutler [START_REF] Breiman | Random Forests[END_REF] and is described in [START_REF] Liaw | Classification and Regression by random-Forest[END_REF] [START_REF] Liaw | Classification and Regression by random-Forest[END_REF]. In this paper, we focus on the randomForest procedure. The two main parameters are mtry, the number of input variables randomly chosen at each split and ntree, the number of trees in the forest 2 .

A third parameter, denoted by nodesize, allows to specify the minimum number of observations in a node. We retain the default value (1 for classification and 5 for regression) of this parameter for all of our experimentations, since it is close to the maximal tree choice.

OOB error

In this section, we concentrate on the prediction performance of RF focusing on out-of-bag (OOB) error (see [START_REF] Breiman | Random Forests[END_REF]). We use this kind of prediction error estimate for three reasons: the main is that we are mainly interested in comparing results instead of assessing models, the second is that it gives fair estimation compared to the usual alternative test set error even if it is considered as a little bit optimistic and the last one, but not the least, is that it is a default output of the procedure. To avoid unsignificant sampling effects, each OOB errors is actually the mean of OOB error over 10 runs.

Datasets

We have collected information about the data sets considered in this paper: the name, the name of the corresponding data structure (when different), n, p, the number of classes c in the multiclass case, a reference, a website or a package. The two next tables contain synthetic information while details are postponed in the Appendix. We distinguish standard and high dimensional situations and, in addition, the three problems: regression, 2-class classification and multiclass classification.

Table 1 displays some information about standard problems datasets: for classification at the top and for regression at the bottom.

Name

Observations

Regression

About regression problems, even if it seems at first inspection that the seminal paper by Breiman [START_REF] Breiman | Random Forests[END_REF] closes the debate about good advice, it remains that the experimental results are about a variant which is not implemented in the universally used R package. Moreover, except this reference, at our knowledge, no such a general paper is available, so we develop again the Breiman's study both for real and simulated data corresponding to the case n >> p and we provide some additional study on data corresponding to the case n << p (such examples typically come from chemometrics). We observe that the default value of mtry proposed by the R package is not optimal, and that there is no improvement by using random forests with respect to unpruned bagging (obtained for mtry = p).

Standard problems

Let us briefly examine standard (n >> p) regression datasets. In Figure 1 for real ones and for simulated ones in Figure 2. Each plot gives for mtry = 1 to p the OOB error for three different values of ntree = 100, 500 and 1000. The vertical solid line indicates the value mtry = p/3, the default value proposed by the R package for regression problems, the vertical dashed line being the value mtry = √ p. Three remarks can be formulated. First, the OOB error is maximal for mtry = 1 and then decreases quickly (except for the ozone dataset, for reasons not clearly elucidated), then as soon as mtry > √ p, the error remains the same.

Second, the choice mtry = √ p gives always lower OOB error than mtry = p/3, and the gain can be important. So the default value proposed by the R package seems to be often not optimal, especially when ⌊p/3⌋ = 1. Lastly, the default value ntree = 500 is convenient, but a much smaller one ntree = 100 leads to comparable results. So, for standard (n >> p) regression problems, it seems that there is no improvement by using random forests with respect to unpruned bagging (obtained for mtry = p).

High dimensional problems

Let us start with a simulated data set for the high dimensional case n << p. This example is built by adding extra noisy variables (independent and uniformly distributed on [0, 1]) to the Friedman1 model defined by:

Y = 10 sin(πX 1 X 2 ) + 20(X 3 -0.5) 2 + 10X 4 + 5X 5 + ǫ INRIA
where X 1 , . . . , X 5 are independent and uniformly distributed on [0, 1] and ǫ ∼ N (0, 1). So we have 5 variables related to the response Y , the others being noise. We set n = 100 and let p vary. Let us give four comments. All curves have the same shape: the OOB error decreases while mtry increases. While p increases, both OOB errors of unpruned bagging (obtained with mtry = p) and random forests with default value of mtry increase, but unpruned bagging performs better than RF (about 25% of improvement). The choice mtry = √ p gives always worse results than those obtained for mtry = p/3. Finally, the default choice ntree = 500 is convenient, but a much smaller one ntree = 100 leads to comparable results.

Figure 4 and 5 show the results of the same study for the Friedman2 and Friedman3 models. The previous comments remain valid. Let us just note that the difference between unpruned bagging and random forests with mtry default value is even more pronounced for these two problems.

To end, let us now examine the high dimensional real data set PAC. Figure 6 gives for same ten values of mtry the OOB error for four different values of ntree = 100, 500, 1000 and 5000 (x-axis is in log scale). The general behavior is similar except for the shape: as soon as mtry > √ p, the error remains the same instead of still decreasing. The difference of the shape of the curves between simulated and real datasets can be explained by the fact that, in simulated datasets we considered, the number of true variables is very small compared to the total number of variables. One may expect that in real datasets, the proportion of true variables is larger.

So, for high dimensional (n << p) regression problems, unpruned bagging seems to perform better than random forests and the difference can be large.

Classification

About standard classification problems, we check that Breiman's conclusions remain valid for the considered variant and that the mtry default value proposed in the R package is good. However for high dimensional classification problems, we observe that larger values of mtry give sometimes much better results.

Standard problems

For classification problems for which n >> p, again the paper by Breiman is interesting and we just quickly check the conclusions.

Let us first examine in Figure 7 standard (n >> p) classification real data sets. Each plot gives for mtry = 1 to p the OOB error for three different values of ntree = 100, 500 and 1000. The vertical solid line indicates the value mtry = √ p, the default value proposed by the R package for classification. Three remarks can be formulated. The default value mtry = √ p is convenient for all the examples. The default value ntree = 500 is sufficient and a much smaller one ntree = 100 is not convenient and can leads to significantly larger errors. The general shape is the following: the errors for mtry = 1 and for mtry = p (corresponding to the unpruned bagging) are of the same "large" order of magnitude and the minimum is reached for the value √ p. The gain can be about 30 or 50%.

So, for these 9 examples, the default value proposed by the R package is quite optimal.

Let us now examine in Figure 8 standard (n >> p) classification simulated datasets. As it can be seen, ntree = 500 is sufficient and, except for the ringnorm already pointed out as a somewhat special dataset (see [START_REF] Cutler | Pert -Perfect random tree ensembles[END_REF] [START_REF] Cutler | Pert -Perfect random tree ensembles[END_REF]) the value mtry = √ p is good. Here, the general shape of the error curve is quite different compared to real datasets: the error increases with mtry. So for these four examples, the smaller mtry, the better.

High dimensional problems

Let us now consider the case n << p for which D íaz-Uriarte and Alvarez de Andrés (2006) [START_REF] Íaz-Uriarte | Gene Selection and classification of microarray data using random forest[END_REF] give numerous advice. We complete the study by trying Again the default value ntree = 500 is sufficient, and at the contrary the value ntree = 100 can leads to significantly larger errors. The general shape is the following: it decreases in general and the minimum value is obtained or is close to the one reached using mtry = p (corresponding to the unpruned bag- ging). The difference with standard problems is notable, the reason is that when p is large, mtry must be sufficiently large in order to have a high probability to capture important variables (that is variables highly related to the response) for defining the splits of the RF. In addition, let us mention that the default value mtry = √ p is still reasonable from the OOB error viewpoint but of course, since √ p is small with respect to p, it is a very attractive value from a computational perspective (notice that the trees are not too deep since n is not too large).

Let us examine a simulated dataset for the case n << p, introduced by Weston et al. (2003) [START_REF] Weston | Use of the zero norm with linear models and kernel methods[END_REF], called "toys data" in the sequel. It is an equiprobable two-class problem, Y ∈ {-1, 1}, with 6 true variables, the others being some noise. This example is interesting since it constructs two near independent groups of 3 significant variables (highly, moderately and weakly correlated with response Y ) and an additional group of noise variables, uncorrelated with Y . A forward reference to the plots on the left side of Figure 11 allow to see the variable importance picture and to note that the importance of the variables 1 to 3 is much higher than the one of variables 4 to 6. More precisely, the model is defined through the conditional distribution of the X i for Y = y: for 70% of data, X i ∼ yN (i, 1) for i = 1, 2, 3 and X i ∼ yN (0, 1) for i = 4, 5, 6.

for the 30% left, X i ∼ yN (0, 1) for i = 1, 2, 3 and X i ∼ yN (i -3, 1) for i = 4, 5, 6.

the other variables are noise, X i ∼ N (0, 1) for i = 7, . . . , p.

After simulation, obtained variables are standardized. Let us fix n = 100.

The plots of Figure 10 are organized as previously, four values of p are considered: 100, 200, 500 and 1000 corresponding to increasing nuisance space For larger values of p (p ≥ 500), the shape of the curve is close to the one for high dimensional real data sets (the error decreases and the minimum is reached when mtry = p). Whence, the error reached by using random forests with default mtry is about 70% to 150% larger than the error reached by unpruned bagging which is close to 3% for all the considered values of p.

Finally, for high dimensional classification problems, our conclusion is that it may be worthwhile to choose mtry larger than the default value √ p. After this section focusing on the prediction performance, let us now focus on the second attractive feature of RF: the variable importance index.

Variable importance

The quantification of the variable importance (abbreviated VI) is a crucial issue not only for ranking the variables before a stepwise estimation model but also to interpret data and understand underlying phenomenons in many applied problems.

In this section, we examine the RF variable importance behavior according to three different issues. The first one deals with the sensitivity to the sample size n and the number of variables p. The second examines the sensitivity to method parameters mtry and ntree. The last one deals with the variable importance of a group of variables, highly correlated or poorly correlated together with the problem of correct identification of irrelevant variables.

As a result, a good choice of parameters of RF can help to better discriminate between important and useless variables. In addition, it can increase the stability of VI scores.

To illustrate this discussion, let us consider the toys data introduced in Section 2.3.2 and compute the variable importance. Recall that only the first 6 variables are of interest and the others are noise.

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Remark 3.1 Let us mention that variable importance is computed conditionally to a given realization even for simulated datasets. This choice which is criticizable if the objective is to reach a good estimation of an underlying constant, is consistent with the idea of staying as close as possible to the experimental situation dealing with a given dataset. In addition, the number of permutations of the observed values in the OOB sample, used to compute the score of importance is set to the default value 1. On each row, the first plot is the reference one for which we observe a convenient picture of the relative importance of the initial variables. Then, when p increases tremendously, we try to check if: [START_REF] Alon | Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays[END_REF] the situation between the two groups remains readable; (2) the situation within each group is stable; (3) the importance of the additional dummy variables is close to 0.

Sensitivity to n and p

The situation n = 500 (graphs at the top of the figure) corresponds to an "easy" case, where a lot of data are available and n = 100 (graphs at the bottom) to a harder one. For each value of n, three values of p are considered: 6, 200 and 500. When p = 6 only the 6 true variables are present. Then two very difficult situations are considered: p = 200 with a lot of noisy variables and p = 500 is even harder. Graphs are truncated after the 16th variable for readability (importance of noisy variables left are the same order of magnitude as the last plotted).

Let us comment on graphs on the first row (n = 500). When p = 6 we obtain concentrated boxplots and the order is clear, variables 2 and 6 having nearly the same importance. When p increases, the order of magnitude of importance decreases. The order within the two groups of variables (1, 2, 3 and 4, 5, 6) remains the same, while the overall order is modified (variable 6 is now less important than variable 2). In addition, variable importance is more unstable for huge values of p. But what is remarkable is that all noisy variables have a zero VI. So one can easily recover variables of interest.

In the second row(n = 100), we note a greater instability since the number of observations is only moderate, but the variable ranking remains quite the same. What differs is that in the difficult situations (p = 200, 500) importance of some noisy variables increases, and for example variable 4 cannot be highlighted from noise (even variable 5 in the bottom right graph). This is due to the decreasing behavior of VI with p growing, coming from the fact that when p = 500 the algorithm randomly choose only 22 variables at each split (with the mtry default value). The probability of choosing one of the 6 true variables is really small and the less a variable is chosen, the less it can be considered as important.

In addition, let us remark that the variability of VI is large for true variables with respect to useless ones. This remark can be used to build some kind of test for VI (see [START_REF] Strobl | Bias in random forest variable importance measures: illustrations, sources and a solution[END_REF] [START_REF] Strobl | Bias in random forest variable importance measures: illustrations, sources and a solution[END_REF]) but of course ranking is better suited for variable selection.

We now study how this VI index behaves when changing values of the main method parameters.

Sensitivity to mtry and ntree

The choice of mtry and ntree can be important for the VI computation. Let us fix n = 100 and p = 200. In Figure 12 we plot variable importance obtained using three values of mtry (14 the default, 100 and 200) and two values of ntree (500 the default, and 2000). The effect of taking a larger value for mtry is obvious. Indeed the magnitude of VI is more than doubled starting from mtry = 14 to mtry = 100, and it again increases whith mtry = 200. The effect of ntree is less visible, but taking INRIA ntree = 2000 leads to better stability. What is interesting in the bottom right graph is that we get the same order for all true variables in every run of the procedure. In top left situation the mean OOB error rate is about 5% and in the bottom right one it is 3%. The gain in error may not be considered as large, but what we get in VI is interesting.

Sensitivity to highly correlated predictors

Let us address an important issue: how does variable importance behave in presence of several highly correlated variables? We take as basic framework the previous context with n = 100, p = 200, ntree = 2000 and mtry = 100. Then we add to the dataset highly correlated replications of some of the 6 true variables. The replicates are inserted between the true variables and the useless ones. The first graph of Figure 13 is the reference one: the situation is the same as previously. Then for the three other cases, we simulate 1, 10 and 20 variables with a correlation of 0.9 with variable 3 (the most important one). These replications are plotted between the two vertical lines.

The magnitude of importance of the group 1, 2, 3 is steadily decreasing when adding more replications of variable 3. On the other hand, the importance of the group 4, 5, 6 is unchanged. Notice that the importance is not divided by the number of replications. Indeed in our example, even with 20 replications the maximum importance of the group containing variable 3 (that is variable 1, 2, 3 and all replications of variable 3) is only three times lower than the initial importance of variable 3. Finally, note that even if some variables in this group have low importance, they cannot be confused with noise.

Let us briefly comment on similar experiments (see Figure 14) but perturbing the basic situation not only by introducing highly correlated versions of the third variable but also of the sixth, leading to replicate the most important of each group. Again, the first graph is the reference one. Then we simulate 1, 5 and 10 variables of correlation about 0.9 with variable 3 and the same with variable 6. Replications of variable 3 are plotted between the first vertical line and the dashed line, and replications of variable 6 between the dashed line and the second vertical line.

The magnitude of importance of each group (1, 2, 3 and 4, 5, 6 respectively) is steadily decreasing when adding more replications. The relative importance between the two groups is preserved. And the relative importance between the two groups of replications is of the same order than the one between the two initial groups.

Prostate data variable importance

To end this section, we illustrate the behavior of variable importance on a high dimensional real dataset: the microarray data called Prostate. The global picture is the following: two variables hugely important, about twenty moderately important variables and the others of small importance. So, more precisely Let us comment on Figure 15. For the two most important variables (first row), the magnitude of importance obtained with ntree = 2000 and mtry = p/3 is much larger than to the one obtained with default values. In the second row, the increase of magnitude is still noticeable from the third to the 9th most important variables and from the 10th to the 20th most important variables, VI is quite the same for the two parameter choices. In the third row, we get VI closer to zero for the variables with ntree = 2000 and mtry = p/3 than with default values. In addition, note that for the less important variables, boxplots are larger for default values, especially for unimportant variables (from the 200th to the 250th). We distinguish two variable selection objectives:

1. to find important variables highly related to the response variable for interpretation purpose;

2. to find a small number of variables sufficient to a good prediction of the response variable.

The first is to magnify all the important variables, even with high redundancy, for interpretation purpose and the second is to find a sufficient parsimonious set of important variables for prediction. Two earlier works must be cited: D íaz-Uriarte, Alvarez de Andrés (2006) [START_REF] Íaz-Uriarte | Gene Selection and classification of microarray data using random forest[END_REF] and Ben [START_REF] Ishak | Sélection de variables en classification binaire : comparaisons et application aux données de biopuces[END_REF] [START_REF] Ishak | Sélection de variables en classification binaire : comparaisons et application aux données de biopuces[END_REF].

D íaz-Uriarte, Alvarez de Andrés propose a strategy based on recursive elimination of variables. More precisely, they first compute RF variable importance. Then, at each step, they eliminate the 20% of the variables having the smallest importance and build a new forest with the remaining variables. They finally select the set of variables leading to the smallest OOB error rate. The proportion of variables to eliminate is an arbitrary parameter of their method and does not depend on the data. Ben Ishak, Ghattas choose an ascendant strategy based on a sequential introduction of variables. First, they compute some SVM-based variable importance. Then, they build a sequence of SVM models invoking at the beginning the k most important variables, by step of 1. When k becomes too large, the additional variables are invoked by packets. They finally select the set of variables leading to the model of smallest error rate. The way to introduce variables is not data-driven since it is fixed before running the procedure. They also compare their procedure with a similar one using RF instead of SVM.

We propose the following two-steps procedure, the first one is common while the second one depends on the objective:

1. Preliminary elimination and ranking:

Compute the RF scores of importance, cancel the variables of small importance;

Order the m remaining variables in decreasing order of importance.

Variable selection:

For interpretation: construct the nested collection of RF models involving the k first variables, for k = 1 to m and select the variables involved in the model leading to the smallest OOB error;

For prediction: starting from the ordered variables retained for interpretation, construct an ascending sequence of RF models, by invoking and testing the variables stepwise. The variables of the last model are selected.

Of course, this is a sketch of procedure and more details are needed to be effective. The next paragraph answer this point but we emphasize that we propose an heuristic strategy which is not supported by specific model hypotheses but based on data-driven thresholds to take decisions. Remark 4.1 Since we want to treat in an unified way all the situations, we will use for finding prediction variables the somewhat crude strategy previously defined. Nevertheless, starting from the set of variables selected for interpretation (say of size K), a better strategy could be to examine all, or at least a large part, of the 2 K possible models and to select the variables of the model minimizing the OOB error. But this strategy becomes quickly unrealistic for high dimensional problems so we prefer to experiment a strategy designed for small n and large K which is not conservative and even possibly leads to select fewer variables.

Starting example

To both illustrate and give more details about this procedure, we apply it on a simulated learning set of size n = 100 from the classification toys data model (see Section 2.3.2) with p = 200. The results are summarized in Figure 16. The true variables (1 to 6) are respectively represented by (£, △, •, ⋆, ¡, ). We compute, thanks to the learning set, 50 forests with ntree = 2000 and mtry = 100, which are values of the main parameters previously considered as well adapted for VI calculations.

Let us detail the main stages of the procedure together with the results obtained on toys data: The result is drawn on the top left graph for the 50 most important variables (the other noisy variables having an importance very close to zero too). Note that true variables are significantly more important than the noisy ones.

We keep this order in mind and plot the corresponding standard deviations of VI. We use this graph to estimate some threshold for importance, and we keep only the variables of importance exceeding this level. More precisely, we select the threshold as the minimum prediction value given by a CART model fitting this curve. This rule is, in general conservative and leads to retain more variables than necessary in order to make a careful choice later.

The standard deviations of VI can be found in the top right graph. We can see that true variables standard deviation is large compared to the noisy variables one, which is close to zero. The threshold leads to retain 33 variables.

Then, we compute OOB error rates of random forests (using default parameters) of the nested models starting from the one with only the most important variable, and ending with the one involving all important variables kept previously. The variables of the model leading to the smallest OOB error are selected.

Note that in the bottom left graph the error decreases quickly and reaches its minimum when the first 4 true variables are included in the model. Then it remains constant. We select the model containing 4 of the 6 true variables. More precisely, we select the variables involved in the model almost leading to the smallest OOB error, i.e. the first model almost leading to the minimum. The actual minimum is reached with 24 variables.

The expected behavior is non-decreasing as soon as all the "true" variables have been selected. It is then difficult to treat in a unified way nearly constant of or slightly increasing. In fact, we propose to use an heuristic rule similar to the 1 SE rule of [START_REF] Breiman | Classification And Regression Trees[END_REF] [START_REF] Breiman | Classification And Regression Trees[END_REF] used for selection in the cost-complexity pruning procedure.

We perform a sequential variable introduction with testing: a variable is added only if the error gain exceeds a threshold. The idea is that the error decrease must be significantly greater than the average variation obtained by adding noisy variables.

The bottom right graph shows the result of this step, the final model for prediction purpose involves only variables 3, 6 and 5. The threshold is set to the mean of the absolute values of the first order differentiated errors between the model with 5 variables (the first model after the one we selected for interpretation, see the bottom left graph) and the last one.

It should be noted that if one wants to estimate the prediction error, since ranking and selection are made on the same set of observations, of course an error evaluation on a test set or using a cross validation scheme should be preferred. It is taken into account in the next section when our results are compared to others.

To evaluate fairly the different prediction errors, we prefer here to simulate a test set of the same size than the learning set. The test error rate with all (200) variables is about 6% while the one with the 4 variables selected for interpretation is about 4.5%, a little bit smaller. The model with prediction variables 3, 6 and 5 reaches an error of 1%. Repeating the global procedure 10 times on the same data always gave the same interpretation set of variables and the same prediction set, in the same order.

Highly correlated variables

Let us now apply the procedure on toys data with replicated variables: a first group of variables highly correlated with variable 3 and a second one replicated from variable 6 (the most important variable of each group). The situations of interest are the same as those considered to produce Figure 14. Let us comment on Table 3, where the expression i j means that variable i is a replication of variable j.

Interpretation sets do not contain all variables of interest. Particularly we hardly keep replications of variable 6. The reason is that even before adding INRIA noisy variables to the model the error rate of nested models do increase (or remain constant): when several highly correlated variables are added, the bias remains the same while the variance increases. However the prediction sets are satisfactory: we always highlight variables 3 and 6 and at most one correlated variable with each of them.

Even if all the variables of interest do not appear in the interpretation set, they always appear in the first positions of our ranking according to importance. More precisely the 16 most important variables in the case of 5 replications are: (3 2 7 3 10 3 6 11 3 5 12 6 8 3 13 6 16 6 1 15 6 14 6 9 3 4), and the 26 most important variables in the case of 10 replications are: (3 14 3 8 3 2 15 3 6 5 10 3 13 3 20 6 21 6 11 3 12 3 18 6 1 24 6 7 3 26 6 23 6 16 3 25 6 22 6 17 6 19 6 4 9 3 ). Note that the order of the true variables (3 2 6 5 1 4) remains the same in all situations.

Classification

Prostate data

We apply the variable selection procedure on Prostate data. The graphs of Figure 17 are obtained as those of Figure 16, except that for the RF procedure, we use ntree = 2000, mtry = p/3 and for the bottom left graph, we only plot the 100 most important variables for visibility. The procedure leads to the same picture as previously, except for the OOB rate along the nested models which is less regular. The key point is that it selects 9 variables for interpretation, and 6 variables for prediction. The number of selected variables is then very much smaller than p = 6033. So, although the sets of variables are not identical for each run of the procedure, they are not completely different. And in addition the most important variables are included in all sets of variables.

High dimensional classification

We apply the global variable selection procedure on high dimensional real datasets studied in Section 2.3.2, and we want to get an estimation of prediction error rates. Since these datasets are of small size, we use a 5-fold cross-validation to estimate the error rate. So we split the sample in 5 stratified parts, each part is successively used as a test set, and the remaining of the data is used as a learning set. Note that the set of variables selected vary from one fold to another. So, we give in Table 4 the misclassification error rate, given by the 5-fold crossvalidation, for interpretation and prediction sets of variables respectively. The number into brackets is the average number of selected variables. In addition, one can find the original error which stands for the misclassification rate given by the 5-fold cross-validation achieved with random forests using all variables. This error is calculated using the same partition in 5 parts and again we use ntree = 2000 and mtry = p/3 for all datasets. The number of interpretation variables is hugely smaller than p, at most tens to be compared to thousands. The number of prediction variables is very small (always smaller than 12) and the reduction can be very important w.r.t the interpretation set size. The errors for the two variable selection procedures are of the same order of magnitude as the original error (but a little bit larger).

We compare these results with the results obtained by Ben Ishak and Ghattas (2008) (see tables 9 and 11 in [START_REF] Ishak | Sélection de variables en classification binaire : comparaisons et application aux données de biopuces[END_REF]) which have compared their method with 5 competitors (mentioned in the introduction) for classification problems on these four datasets. Error rates are comparable. With the prediction procedure, as already noted in the introductory remark, we always select fewer variables than their procedures (except for their method GLMpath which select less than 3 variables for all datasets).

Regression

A simulated dataset

We now apply the procedure to a simulated regression problem. We construct starting from the Friedman1 model and adding noisy variables as in Section The graphs are of the same kind as in classification problems. Note that variable 3 is confused with noise and is not selected by the procedure. This is explained by the fact that it is hardly correlated with the response variable. The interpretation procedure select the true variables except variable 3 and two noisy variables, and the prediction set of variables contains only the true variables (except variable 3). Again the whole procedure is stable in the sense that several runs give the same set of selected variables.

In addition, we simulate a test set of the same size than the learning set to estimate the prediction error. The test mean squared error with all variables is about 19.2, the one with the 6 variables selected for interpretation is 12.6 and the one with the 4 variables selected for prediction is 9.8. Three very sensible groups of variables appear from the most to the least important. First, the two temperatures (8 and 9), the inversion base temperature [START_REF] Cheze | Partial and Recombined Estimators for Nonlinear Additive Models[END_REF] known to be the best ozone predictors, and the month (1), which is an important predictor since ozone concentration exhibits an heavy seasonal component. A second group of clearly less important meteorological variables: pressure height (5), humidity [START_REF] Breiman | Bagging predictors[END_REF], inversion base height [START_REF] Breiman | Random Forests[END_REF], pressure gradient [START_REF] Bühlmann | Analyzing Bagging[END_REF] and visibility [START_REF] Cutler | Pert -Perfect random tree ensembles[END_REF]. Finally three unimportant variables: day of month (2), day of week (3) of course and more surprisingly wind speed [START_REF] Breiman | Classification And Regression Trees[END_REF]. This last fact is classical: wind enter in the model only when ozone pollution arises, otherwise wind and pollution are uncorrelated (see for example [START_REF] Cheze | Partial and Recombined Estimators for Nonlinear Additive Models[END_REF] [START_REF] Cheze | Partial and Recombined Estimators for Nonlinear Additive Models[END_REF] highlighting this phenomenon using partial estimators).

Ozone data

Let us now examine the results of the selection procedures. After the first elimination step, the 2 variables of negative importance are canceled, as expected.

Therefore we keep 10 variables for interpretation step and then the model with 7 variables is then selected and it contains all the most important variables: (9 8 12 1 11 7 5).

INRIA

For the prediction procedure, the model is the same except one more variable is eliminated: humidity [START_REF] Breiman | Bagging predictors[END_REF] .

In addition, when different values for mtry are considered, the most important 4 variables (9 8 12 1) highlighted by the VI index, are selected and appear in the same order. The variable 5 also always appears but another one can appear after of before.

Discussion

Of course, one of the main open issue about random forests is to elucidate from a mathematical point of view its exceptionally attractive performance. In fact, only a small number of references deal with this very difficult challenge and, in addition to bagging theoretical examination by Bülmann and Yu (2002) [START_REF] Bühlmann | Analyzing Bagging[END_REF], only purely random trees, a simple version of random forests, is considered. Purely random trees have been introduced by Cutler and Zhao (2001) [START_REF] Cutler | Pert -Perfect random tree ensembles[END_REF] for classification problems and then studied by [START_REF] Breiman | Consistency for a simple model of Random Forests[END_REF] [START_REF] Breiman | Consistency for a simple model of Random Forests[END_REF], but the results are somewhat preliminary. More recently [START_REF] Biau | Consistency of random forests and other averaging classifiers[END_REF] [START_REF] Biau | Consistency of random forests and other averaging classifiers[END_REF] obtained the first well stated consistency type results.

From a practical perspective, surprisingly, this simplified and essentially not data-driven strategy seems to perform well, at least for prediction purpose (see Cutler and Zhao 2001 [13]) and, of course, can be handled theoretically in a easier way. Nevertheless, it should be interesting to check that the same conclusions hold for variable importance and variable selection tasks.

In addition, it could be interesting to examine some variants of random forests which, at the contrary, try to take into account more information. Let us give for example two ideas. The first is about pruning: why pruning is not used for individual trees? Of course, from the computational point of view the answer is obvious and for prediction performance, averaging eliminate the negative effects of individual overfitting. But from the two other previously mentioned statistical problems, prediction and variable selection, it remains unclear. The second remark is about the random feature selection step. The most widely used version of RF selects randomly mtry input variables according to the discrete uniform distribution. Two variants can be suggested: the first is to select random inputs according to a distribution coming from a preliminary ranking given by a pilot estimator; the second one is to adaptively update this distribution taking profit of the ranking based on the current forest which is then more and more accurate.

These different future directions, both theoretical and practical, will be addressed in the next step of the work.

Appendix

In the sequel, information about datasets retrieved from the R package mlbench can be found in the corresponding description file.

Standard problems, n >> p:

Binary classification

RR n 6729

Figure 1 :Figure 2 :

 12 Figure 1: Standard regression: 3 real data sets

Figure 3 :

 3 Figure 3: High dimensional regression simulated data set: Friedman1. The x-axis is in log scale Figure 3 contains four plots corresponding to 4 values of p (100, 200, 500 and 1000) increasing the nuisance space dimension. Each plot gives for ten values of mtry (1, √ p/2, √ p, 2 √ p, 4 √ p, p/4, p/3, p/2, 3p/4, p) the OOB error for three different values of ntree = 100, 500 and 1000. The x-axis is in log scale and the vertical solid line indicates mtry = p/3 the default value proposed by the R package for regression, the vertical dashed line being the value mtry = √ p.Let us give four comments. All curves have the same shape: the OOB error decreases while mtry increases. While p increases, both OOB errors of unpruned bagging (obtained with mtry = p) and random forests with default value of mtry increase, but unpruned bagging performs better than RF (about 25% of improvement). The choice mtry = √ p gives always worse results than those obtained for mtry = p/3. Finally, the default choice ntree = 500 is convenient, but a much smaller one ntree = 100 leads to comparable results.Figure4and 5 show the results of the same study for the Friedman2 and Friedman3 models. The previous comments remain valid. Let us just note that the difference between unpruned bagging and random forests with mtry default value is even more pronounced for these two problems.To end, let us now examine the high dimensional real data set PAC. Figure6gives for same ten values of mtry the OOB error for four different values of ntree = 100, 500, 1000 and 5000 (x-axis is in log scale). The general behavior is similar except for the shape: as soon as mtry > √ p, the error remains the same instead of still decreasing. The difference of the shape of the curves between simulated and real datasets can be explained by the fact that, in simulated datasets we considered, the number of true variables is very small compared

Figure 4 :Figure 5 :

 45 Figure 4: High dimensional regression simulated data set: Friedman2. The x-axis is in log scale

INRIA

Figure 6 :

 6 Figure 6: High dimensional regression: PAC data. The x-axis is in log scale

Figure 9 :

 9 Figure 9: High dimensional classification: 9 real data sets. The x-axis is in log scale

Figure 10 :

 10 Figure 10: High dimensional classification simulated data set: toys data for 4 values of p. The x-axis is in log scale

Figure 11

 11 Figure 11 illustrates the behavior of variable importance for several values of n and p. Parameters ntree and mtry are set to their default values. Boxplots are based on 50 runs of the RF algorithm and for visibility, we plot the variable importance only for a few variables.

Figure 11 :

 11 Figure 11: Variable importance sensitivity to n and p (toys data)

Figure 12 :

 12 Figure 12: Variable importance sensitivity to mtry and ntree (toys data)

Figure 14 :

 14 Figure 14: Variable importance of two groups of correlated variables (augmented toys data)

  Figure 15 compares VI obtained for parameters set to their default values (graphs of the left column) and those obtained for ntree = 2000 and mtry = p/3 (graphs of the right column).

INRIA

Figure 15 :

 15 Figure 15: Variable importance for Prostate data (using ntree = 2000 and mtry = p/3, on the right and using default values on the left)

Figure 16 :

 16 Figure 16: Variable selection procedures for interpretation and prediction for toys data

Figure 17 :

 17 Figure 17: Variable selection procedures for interpretation and prediction for Prostate dataIn addition, to examine the variability of the interpretation and prediction sets the global procedure is repeated five times on the entire Prostate dataset. The five prediction sets are very close to each other. The number of prediction variables fluctuates between 6 and 10, and 5 variables appear in all sets. Among the five interpretation sets, 2 are identical and made of 9 variables and the 3

INRIA

2. 2

 2 .2, a learning set of size n = 100 with p = 200 variables. Figure 18 displays the results of the procedure. The true variables of the model (1 to 5) are respectively represented by (£, △, •, ⋆, ¡).

Figure 18 :

 18 Figure 18: Variable selection procedures for interpretation and prediction for Friedman1 data

  Before ending the paper, let us apply the entire procedure to the ozone dataset. It consists of n = 366 observations of the daily maximum one-hour-average ozone together with p = 12 meteorologic explanatory variables. Let us first examine, in Figure 19 the VI obtained with RF procedure using mtry = p/3 = 4 and ntree = 2000. From the left to the right, the 12 explanatory variables are 1-Month, 2-Day of month, 3-Day of week, 5-Pressure height, 6-Wind speed, 7-Humidity, 8-Temperature (Sandburg), 9-Temperature (El Monte), 10-Inversion base height, 11-Pressure gradient, 12-Inversion base temperature, 13-Visibility.

Figure 19 :

 19 Figure 19: Variable importance for Ozone data

Figure 20 :

 20 Figure 20: Variable selection procedures for interpretation and prediction for Ozone data

Table 2 :

 2 High dimensional problems: data sets for classification at the top, and for regression at the bottom

			Variables Classes
	Ionosphere	351	34	2
	Diabetes	768	8	2
	Sonar	208	60	2
	Votes	435	16	2
	Ringnorm	200	20	2
	Threenorm	200	20	2
	Twonorm	200	20	2
	Glass	214	9	6
	Letters	20000	16	26
	Sat-images	6435	36	6
	Vehicle	846	18	4
	Vowel	990	10	11
	Waveform	200	21	3
	BostonHousing	506	13
	Ozone	366	12
	Servo	167	4
	Friedman1	300	10
	Friedman2	300	4
	Friedman3	300	4
	Table 1: Standard problems: data sets for classification at the top, and for
	regression at the bottom
	Table 2 displays high dimensional problems datasets: for classification at the
	top and for regression at the bottom.
				INRIA

Table 3 :

 3 Variable selection procedure in presence of highly correlated variables (augmented toys data)

Table 4 :

 4 Variable selection procedure for four high dimensional real datasets. CV-error rate and into brackets the average number of selected variables

	Dataset	interpretation prediction original
	Colon	0.16 (35)	0.20 (8)	0.14
	Leukemia	0 (1)	0 (1)	0.02
	Lymphoma	0.08 (77)	0.09 (12)	0.10
	Prostate	0.085 (33)	0.075 (8)	0.07

see http://www.r-project.org/ RR n 6729

In all the paper, mtry = m with m ∈ R stands for mtry = ⌊m⌋ RR n 6729

RR n 6729


Wassim El Housseini

Mathieu Etienne

Alain Walcarius

Dr Francois Lapicque
email: francois.lapicque@univ-lorraine.fr

Multiphase chemical engineering as a tool in modelling electromediated reactions-Example of Rh complex-mediated regeneration of NADH

Keywords: Beta-nicotinamide adenine dinucleotide hydrate, chloro(2,2′-bipyridyl) (pentamethylcyclopentadienyl)-rhodium (III), electromediated reactions, gas-liquid absorption, reaction kinetics

On the basis of the electromediated regeneration of beta-nicotinamide adenine dinucleotide hydrate (NADH) by chloro(2,2′-bipyridyl) (pentamethylcyclopentadienyl)-rhodium (III) chloride complex -Rh(III), the manuscript describes a modelling approach inspired by multiphase chemical engineering with gas absorption with chemical reaction. Here, the electrode surface is assimilated to the gas-liquid interface, and the reducing Rh(I) formed at the electrode corresponds to the dissolved gas. This method allows the kinetics of the liquid phase reaction between NAD + and Rh(I) to be estimated and the beneficial effect of NAD + concentration on the reduction current of Rh(III) to be successfully predicted. Moreover, because of the fast kinetics of the liquid phase reaction, the overall mediated regeneration of NADH by the redox Rh couple can be viewed as occurring as the direct reduction of NAD + to NADH. The approach could be applied to other electromediated processes, upon sufficient knowledge of the system physico-chemical specificities.

1-Introduction

Beta-nicotinamide adenine dinucleotide hydrate (NAD+/NADH) cofactor and its phosphorylated derivative (NADP + /NADPH) are well-known electron mediators in biological systems, making it possible a number of enzymatic catalysed oxidation and reduction reactions [START_REF] Steckhan | Electroenzymatic Synthesis[END_REF][START_REF] Kohlmann | Electroenzymatic Synthesis[END_REF]. Because of their high costs, these cofactors must be regenerated in order to enable syntheses with oxidoreductases or bio-fuelcells applications economically viable [START_REF] Kara | Recent trends and novel concepts in cofactor-dependent biotransformations[END_REF][START_REF] Weckbecker | Regeneration of nicotinamide coenzymes: principles and applications for the synthesis of chiral compounds[END_REF]. In particular, various techniques can be used to regenerate the reducing forms NAD(P)H [START_REF] Zhang | Functional Electrodes for Enzymatic and Microbial Bioelectrochemical Systems[END_REF] but electrochemical methods are often employed [START_REF] Chenault | Regeneration of Nicotinamide Cofactors for use in Organic Synthesis[END_REF][START_REF] Chenault | Regeneration of Nicotinamide Cofactors for use in Organic Synthesis[END_REF][START_REF] Cheikhou | Electrochemical microreactor for chiral syntheses using the cofactor NADH[END_REF][START_REF] Immanuel | Recent progress and perspectives on electrochemical regeneration of reduced nicotinamine adenine dinucleotide[END_REF]) because of their high total turnover number (TTN) allowed and no need of additional sacrificial electron donors. However, great care should be taken to produce only the bioactive 1,4-NADH and avoiding the enzymatically inactive products (such as 1,6-NADH, 1,2-NADH or NAD2 dimers) [START_REF] Saba | NADH regeneration: a case study of Pt-catalyzed NAD + reduction with H 2[END_REF]. The direct regeneration of NADH by reduction of NAD + at an electrode surface occurs at very cathodic potentials, thus enters in competition with hydrogen evolution and leads to the formation of the inactive dimer of NAD + [START_REF] Jaegfeldt | A study of the products formed in the electrochemical reduction of nicotinamide-adenine-dinucleotide[END_REF]. For this reason, redox mediators are usually employed for this purpose [START_REF] Immanuel | Recent progress and perspectives on electrochemical regeneration of reduced nicotinamine adenine dinucleotide[END_REF][START_REF] Yuan | Efficient NADH regeneration by a redox polymer-immobilized enzymatic system[END_REF]. These mediators can be chemical species -such as complexes of transition or precious metals cations, or organic molecules e.g. methyl viologen -biological molecules e.g. flavine adenine dinucleotide or ferredoxin NADP + reductase (FNR). All of them allow the electrochemical reduction to be conducted at less cathodic potentials and have a very high selectivity towards the regeneration of the active form of the NADH cofactor (1,4-NADH)a crucial aspect, considering the high cycling rate of NAD + /NADH in enzymatic processes.

The complex chloro(2,2′-bipyridyl) (pentamethylcyclopentadienyl)-rhodium (III) chloride was shown to be the best non-enzymatic mediator for the regiospecific regeneration of NADH [START_REF] Ruppert | Efficient indirect electrochemical in-situ regeneration of nadh:electrochemically driven enzymatic reduction of pyruvate catalyzed by d-ldh[END_REF][START_REF] Lo | Regioselective reduction of NAD + models with [Cp*Rh(bpy)H] + : structure-activity relationships and mechanistic aspects in the formation of the 1,4-NADH derivatives[END_REF]Holmann et al., 2003;[START_REF] Zhang | Functional Electrodes for Enzymatic and Microbial Bioelectrochemical Systems[END_REF] when electrochemically reduced into the monovalent rhodium (I) complex form. Using Rh(III) and Rh(I) names to relate to the two mediator forms, NADH regeneration occurs after the reaction below, with consumption of one proton per NAD + ion (which is an accepted simplified view of a multistep process, [START_REF] Höfer | Polymer-modified electrodes with pendant [Rh III (C 5 Me 5 )(L)Cl] + -complexes formed by γ-irradiation cross-linking[END_REF]:

Rh(III) + 2 e → Rh(I)
(1)

Rh(I) + H + + NAD + → Rh(III) + NADH (2)
The principle is shown in Figure 1.

The rhodium complex can be employed as dissolved in the solution for liquid-phase reaction

(2) as reported by [START_REF] Vuorilehto | Indirect electrochemical reduction of nicotinamide coenzymes[END_REF]. or in a companion paper (El Housseini et al. 2021), or grafted at the electrode surface [START_REF] Tan | Regeneration of the NADH Cofactor by a Rhodium Complex Immobilized on Multi-Walled Carbon Nanotubes[END_REF][START_REF] Zhang | Functional Electrodes for Enzymatic and Microbial Bioelectrochemical Systems[END_REF], for possible recycling of this expensive mediator in real bioelectrochemical processes. In both cases, NAD + reduction leads to active NADH as shown by as shown by 1 H NMR spectroscopy or by integration of an enzymatic reaction e.g. lactate dehydrogenase-catalysed reduction of pyruvate. Experimental measurements related to the mediated regenerations show that: (i) voltammetric curves of Rh(III) reduction exhibit far larger currents (in absolute value) in the presence of NAD + in the solution than without. (ii) batch conversion of NAD + in the presence of the Rh complex under various concentrations of the two reactants does not simply depend on one of the two concentrations. In addition, for modelling and scale-up of mediated electrochemical processes, the rate constant, k, of reaction (2) needs to be estimated.

Because of its high interest in biochemistry, the regeneration of NADH using bi-pyridines pentamethylcyclopentadienyl (cp)-rhodium complexes has been largely investigated, depending on the nature of the pyridyl functionalized groups (Steckhan et al., Lo et al., Ganesan et al.), and the nature of the Rh(III) reduction process. For this concern, the incorporation of one proton after the two electron transfers from the cathode, has often been In case that the redox couple is attached to the cathode surface by chemical grafting or adsorption, experimental datain the form of voltammetric curves in motionless conditions, or under forced convectionare treated using electrochemical models developed for estimation of electrochemical and chemical kinetic parameters [START_REF] Ikeda | Kinetics of electron-transfer reactions of metalcomplexes at impermeable redox active polymeric films on electrode surfaces and charge transport within the polymer film[END_REF], Lyons et al., 1996;[START_REF] Lyons | The mechanism of mediated electron transfer at redox active surfaces[END_REF]. When the mediating redox couple is dissolved, other interpretation methods were developed [START_REF] Wienkamp | Indirect Electrochemical Regeneration of NADH by a Bipyridinerhodium(I) Complex as Electron-Transfer Agent[END_REF][START_REF] Shono | Electroorganic chemistry in organic synthesis[END_REF]. Although not corresponding to mediated electrosynthesis, the case of an electrochemical process followed a chemical reaction occurring in the liquid bulk has been considered in the past [START_REF] Labrune | Modelling of an indirect electrosynthesis process-1 Theoretical study of the effect of dismutation of the mediator[END_REF]. Modelling of the whole system was shown to successfully benefit from previous works on gas-liquid (G-L) absorption processes with chemical reaction [START_REF] Danckwerts | Gas-Liquid reactions[END_REF], to calculate the enhancement of gas absorption rate by occurrence of a fast chemical reaction, as it is for the absorption of acidic CO 2 and SO 2 gases in alkaline media.

The aim of this work was to use a comparable method to model the performance of an electrochemical cell for mediated electrosynthesis -in terms of current density and conversion rate of the reactant, on the example of the above Rh/NAD + system (El Housseini et al.). In both cases, the approach relies on the existence of a diffusion layer of thickness  L near the (G-L or electrode-electrolyte) interface, in which transfer phenomena only occur upon diffusion, i.e. with a negligible liquid velocity [START_REF] Danckwerts | Gas-Liquid reactions[END_REF]). For the above example of electromediated reaction, Rh(III) is named R3, Rh(I) complex, R1, whereas NAD + is referred as NAD. In the model (text and equations) for evident chemical meaning, Rh(I), Rh(III) and NAD + will still be used as such. The first step of the process is the dissolution of the gas in the liquid or the reduction of Rh(III) species on the cathode surface: in both cases, this interfacial step is assumed to be very fast. R1 formed diffuses from the cathode or reacts with NAD after

(2) in the diffusion layer as shown in Figure 2.

On the basis of the above analogy, a model has been developed here for calculation of concentrations profiles in the vicinity of the electrode surface depending on the operating conditions and the rate constant k of the reaction. The limiting current density of Rh(III) reduction in the absence of NAD + allowed mass transfer coefficient of this species to be estimated. Moreover, from the experimental values for the current density at various NAD + concentrations, rate constant k could be estimated using the model. Then, progress of NAD + electromediated conversion in discontinuous tests could be simply predicted assuming a fast chemical reaction, with successful comparison to previous data for identified operating conditions.

2-Materials and methods

The electrochemical reactor depicted in Figure 3 was formed by two 4 x 4 cm 2 compartments.

The anode for hydrogen oxidation into H + was a commercial gas diffusion electrode (GDE) (Ion Power) supported by a gas diffusion layer and loaded at 0.2 mg Pt/cm 2 was deposited on a Nafion N-212 membrane. This membrane separated the gas/solid part of the cell from the cathode chamber for NAD + mediated reduction. The mediated reduction of NAD + was carried out on the surface of a 4 x 4 x 0.4 m 3 graphite felt (GFD4 6EA, SGL, Germany). The material formed from polyacrylonitrile precursor was highly porous (81%) with pores larger than 100 µm. Whereas the anode was fed with a stream of fresh hydrogen (20 Ncm 3 .min -1 ), the liquid phase, with a volume of 40 cm 3 was recirculated in the cell at 20 cm 3 .min -1 in batch NADH runs carried out at a fixed voltage and at ambient temperature. Because of the very low change in potential of the anode GDE operated with current density in the order of 1 mA.cm -2 , the cathode potential was referred to the hydrogen anode, noted ref.

(H 2 ) in the manuscript.

The two electrode reactions are written as:

Rh(III) + 2 e -→ Rh(I) (1)
with Rh(I)

+ H + + NAD + → Rh(III) + NADH (2) H 2 → 2 H + + 2 e - (3)
From the above reactions, formation of one NADH molecule is accompanied by one proton gained by the solution. To avoid any visible change in the pH which has to be kept near 7, the solution contained phosphate buffer at 0.1 M.

3-Presentation of the model

The model was developed by analogy with G/L absorption with chemical reaction, first to predict the concentration profiles and the enhancement of the limiting current density for Rh(III) reduction by the presence of NAD + . In this model, the system was assumed at steady state, on the basis of the initial concentrations of the two reactants. In contrast, time dependence was accounted for in modelling NAD + conversion in the discontinuous cell.

Concentration profiles near the electrode surface

For the system presented above, one-dimensional mass balances were written at steady state in the diffusion layer, thus neglecting the convective terms;

(4) where x is the coordinate from the electrode surface; D R3 and D NAD are the diffusion coefficients of Rh(III) and NAD + in the solution, and k the rate constant of the 2 nd order kinetic law. The assumptions of the model are listed below:

 Rh(I) species was assumed to be present only in the solution even though Rh(I) was

shown to be partially adsorbed on the electrode surface (Vuorilehto et al.), due to the effective solution recirculation. NAD + reduction is thus assumed to occur only in the liquid phase which justifies the expression for the kinetic term in equation ( 4).

 As expressed above, the electrode reduction of Rh(III) is carried out under diffusion control, so its surface concentration is zero. The current density is then at its limiting value, i L .

 Rh(III) and Rh(I) complexes have identical diffusion coefficients. Therefore, since Rh(III) is converted to Rh(I) with a perfect selectivity, it can be shown that:

anywhere in the solution.

(

) 5
where subscript b refers to the solution bulk. corresponds to the concentration of Rh complex introduced.

 Consumption of Rh(I) complex by the chemical reaction is assumed to be fast thus occurring only in the diffusion layer . Therefore, two boundary conditions were written:

x = 0, and (6a)

x =  L , (6b)
 The kinetic law was written regardless of proton concentration. The presence of phosphate buffer and the pH fixed at 7.2 justify this assumption; k value holds for the pH considered.

Since NAD + is not formed or consumed at the electrode surface, its gradient at x=0 is zero; moreover, its concentration at  L is assumed to be that in the bulk:

x = 0, (7a) x =  L , (7b)
Without occurrence of chemical reaction (no NAD + added, or k=0), the profile of Rh(III) concentration is linear, so S 0 is equal to -1, and relation (13

) reduces to E = -S 0 , without (15)
E is easily deduced from the voltammetric curves by dividing the current at the diffusion plateau for the considered concentration of NAD + , to the corresponding recorded current without NAD + . For a postulated value of k, integration of (11) has been carried out by a finite element scheme, considering the boundary conditions (12).

4-Results and discussion

Experimental results

Voltammetric curves with Rh have first been recorded at 5 mV/s between 0.2 and -0.6 V/H 2 , in the cell containing various concentrations of Rh(III) in the 0.1 M PBS, NAD + -free solution at pH 7.2. This scan rate was formerly used with this system (Zhang et al., El Housseini et al.); in other works (Steckhan et al), the scan rate was larger or equal to 9 mV/s. Although

showing the increasing effect of Rh(III) concentration, together with the synergetic action of NAD + presence in the reduction current observed, the possible control of mass transfer could not be observed by a well-defined current plateau in all cases. As exemplified by Figure 4a, evidence of mass transfer control was observed near -0.25 V vs ref (H 2 ), either by a current hump peak or on the contrary by a shouldering from which the limiting current density could not be estimated with a sufficient accuracy. It was therefore preferred to proceed to four activation cycles at 5 mV/s in the above potential range, before performing a linear scan at 1 mV/s between 0 V and -0.6 V vs. H 2 . The conversion of NAD + to NADH was estimated by the charge passed to be in the order or below 5%. Because of the above consumption of NAD + in the voltammetric experiments, each couple of Rh(III) and NAD + concentrations was investigated with a freshly prepared solution. As shown in Figures 4b andc, better-defined plateaus of Rh(III) cathodic reduction could be observed at 1 mV/s, with or without NAD + present in the solution. Replicate experiments led to deviations in the current plateau lower than 10%. Mass transfer rates of Rh(III) species could be estimated from the curves after graphical subtraction of the residual current. The limiting current with an uncertainty estimated at 10%, was shown to obey a linear function of Rh(III) concentration (Figure 5), however with a constant term. This non-zero current was attributed to a residual current I res not subtracted by the above procedure. The data obtained for tests with six different Rh concentrations are comprised between two linear variations with a slope respectively equal to the average slope with plus or minus 10%. The slope of the average variation led to mass transfer coefficient k L , on the basis of two electrons exchanged and the area of the electrode taking as its geometrical area i.e. 16 cm 2 (see Appendix 1). Moreover, diffusivities D R3 and D NAD have been estimated as explained in Appendix 2.

k L = 4.6 10 -5 m/s ± 10%, D R3 = 3.48 10 -10 m 2 /s, D NAD = 2.4 10 -10 m 2 /s (16)

Concentration profiles in the diffusion layer

Calculations have been carried out for a Rhodium complex concentration of 50 µM, with 2 mM NAD + , for rate constant k varying between 20 to 100 m 3 .mol -1 , corresponding to Ha values ranging from 2.9 to 6.47.

For the lowest k value, Rh(I) concentration profile in the film deviates noticeably from the linear profile expected without NAD + . The concentration gradient at X=0 was found near -2.83. However R1 is still near 0.06 for X=1 (Figure 6), expressing the fact that the boundary conditions do not hold perfectly: as a matter of fact, the chemical reaction although relatively fast, does not only occur in the diffusion layer. This phenomenon is less visible for k = 40 m 3 mol -1 s -1 for which gradient S 0 is more negative and R1 is slightly below 0.02 at X = 1: calculated concentration profiles are more reliable. For k = 100 m 3 mol -1 s -1 , R1 is below 0.005 at the edge of the film (Figure 6), and the chemical reaction can be considered as "fast", i.e. occurring only in the film.

Because of its large excess in comparison to Rh complex, NAD + is little depleted in the diffusion film (Figure 6). Moreover, increasing the rate constant resultsas expected -in more significant depletion of NAD + near the electrode surface.

These calculations carried out for various k values led to the corresponding values for the concentration gradients at the wall and enhancement factor E, as shown in Figure 7a below.

The value for Ha number deduced from k affects strongly the enhancement factor, as shown in Figure 7b. In both figures, the curve in dotted line corresponds to Ha number below 3, i.e.

for which boundary conditions related to Rh(I) conversion in the film are not fulfilled.

Estimation of the rate constant from electrochemical data

Residual current density i res evidenced in Figure 5 was assumed to be constant at 0.3 mA/cm 2 in all voltammetric curves, i.e. for all NAD + concentrations. In Rel (13) giving access to enhancement factor E, both limiting current densities and had to be corrected for current density i res .

The experimental value for E factor was used for estimation of rate constant k: equations ( 11) subject to (12) were integrated: the trial and error procedure relied on the convergence of gradient S 0 , which has to be equal to the opposite of the E value (rel. 15).

Sensitivity analysis

Because of the uncertainty on the limiting current in the order of 10%, sensitivity analysis of the method for k estimation was achieved on the example of 125 µM Rh complex and 2 mM NAD + . In addition to the E value determined at 3.83, we considered its lower and higher estimates, within 10%. For this experiment example, the estimate for k from the "central" E value was found at 46 m 3 .mol -1 , and at 35 and 58 m 3 .mol -1 .s -1 from the lower and higher E values respectively. The above values show the strong sensitivity of the electrochemical technique, for which a 10% uncertainty in the limiting current recorded induces a change in k value in the order of 25%. In the three cases, R1 at the edge of the film was lower than 0.015, validating in an acceptable manner the assumption of fast reaction with complete consumption of Rh(I) in the film. The dimensionless profiles of R1 and NAD are shown in Figure 8.

Determination of rate constant k

The experimental values for the enhancement factor E estimated from CV curves for the various concentrations of NAD + and Rh complex, have been treated following the above procedure. Only experiments leading Ha number larger than 3 were shown here. Because of the uncertainties commented above, constant k was evaluated within its confidence interval:

k = 47 ± 20 m 3 .mol -1 .s -1 (pH = 7.2) (17)
Comparison between theory and practice has been made on the basis of the enhancement factor. For each experiment treated, the predicted enhancement factor could be estimated from the profile of R1 in the film. As shown in Figure 9, the predicted E values are in accordance with their experimental values within 20%.

The expression of the Hatta number (rel. 10) clearly shows that the chemical rate constant k is subject to the value for mass transfer coefficient k L . Because k L was estimated on the basis the geometrical area of the cathode, i.e. S=16 cm 2 as explained in the Appendix, the estimate for k is also subject to the area assumption. More precisely, k value given in rel. ( 17) does not have to be considered per se, but in comparison with k L . On the contrary, dimensionless criterion Ha related to the kinetics of the chemical process, is determinant to examine whether this process can be considered as fast, i.e. occurring exclusively in the diffusion film or not: the various experiments carried out, clearly showed that the reaction could be constant as fast.

4.4. Simulating NAD + mediated conversion in the cell with H 2 anode For these tests conducted at a fixed cathode potential corresponding to diffusion control, the solution with a volume V=40 cm 3 was circulated continuously in the cell. Former works dealing with G/L absorption (van Swaaij and Versteeg, 1992) often mention criterion R, which is the ratio of the maximum flow rate of A 1 consumption in the whole solution volume over the maximum mole flow rate of A 1 generated at the interface. R can be then expressed as:

(18

)
where S is the geometrical area of the cathode (16 cm 2 ). Considering k = 47 m 3 mol -1 s -1 and = 2 mM, leads to R near 2.6 10 4 , a value expressing that the overall mediated regeneration of NADH is controlled by the rate of Rh(III) reduction. Therefore, modelling of NADH regeneration can be viewed as carried out in a batch (discontinuous) electrochemical reactor with a rate calculated regardless of the occurrence of the chemical reaction, i.e. as if NAD + reacts directly at the cathode surface. It can be observed that the nature of the ratecontrolling process was the same for the median k values and for its lower and upper estimates.

The mass balance in the simplified discontinuous process carried out at the limiting current is:

Coefficient k L , B in the above relation is the mass transfer coefficient for species B: because  L is assumed independent of the species nature in the film model, k L,B derives from k L (related to A 1 ) after:

(20)

Integration of rel. ( 19) yields the usual exponential expression of the concentration expected from the above first order kinetics, from which conversion X NAD+ of NAD + can be deduced: confirming the reaction mechanism. However, for ratio C NAD /C R3 at 80, NADH regeneration appears sluggisher (Figure 10) with a quite slower reaction from 20 minutes; for this ratio value, the conversion could not exceed 62% even after 120 min (data not shown). Obviously, the above simple model does not hold for such case, presumably because the mechanism of mediated electrosynthesis deviates from the simple mechanism considered here for too different concentrations of Rh complex and NAD + , with possible effect of the surface reduction of NAD + by adsorbed Rh(I), a phenomenon not accounted for in the present model.

5-Conclusion

The present work was aimed at modelling the effectiveness of mediated electrochemical reactions conducted at fixed potential in terms of current density and conversion rates, using the approach developed in gas-liquid absorption accompanied by chemical reaction of the dissolved gas. The method relying here upon diffusion-controlled electrochemical process and applied to NADH regeneration mediated by a Rh complex, led to estimates of the rate constant of the chemical reactions between Rh(I) and NADH + , in comparison to the mass transfer coefficient to the electrode surface. Moreover, because the chemical process was shown to be fast enough, i.e. occurring nearly only in the diffusion film, the electrochemical conversion of NAD + to NADH at a fixed potential can be modelled as if NAD + was directly reduced at the cathode surface. The method could be successfully applied to other electrochemical processes involving indirect reduction or oxidation, a situation often encountered in electro-organic synthesis or bioelectrochemistry.

The validity of the simple approach is nevertheless dependent on the validity of the assumptions made and related to, first diffusion-controlled electrochemical process, then fast chemical reaction between substrate B and generated A 1 species. In case that the reaction is not fast enough (Ha < 3), the model for the batch electrochemical process has to be modified by inserting the chemical term in the related mass balance. Moreover, care has to be taken to check the kinetic mechanism of the chemical process in the operating conditions investigated.

Figure captions

Figure 1: Principle of the electromediated system investigated here, with occurrence of chemical reaction in the liquid phase. Figure 5: Variations of the limiting current density estimated from voltammetric curves at 1 mV/s with the Rh(III) solutions. Data are given with a 10% error bar, the liner regression is in solid lines and the two dotted lines are linear variations with a slope 10% higher or lower than the regression slope.

Figure 6: Dimensionless concentration profiles of species A 1 (Rh(I) complex) and B (NAD + ) in the diffusion film. Concentration of Rh complex and NAD + at 50 µM and 2 mM respectively. Effect of the rate constant k in m 3 .mol -1 .s -1 . Figure 9: Enhancement factor calculated for k=50 m 3 .mol -1 .s -1 versus their experimental values; dotted lines correspond to the diagonal ± 20%.

  diffusion layer film model, mass transfer coefficient k L of Rh(III) to the electrode is defined as the ratio (9) Finally, Ha number was finally defined as: , the assumption related to a fast reaction i.e. occurring only in the diffusion layer is only valid for Ha ≥ 3, for sufficient value of rate constant k in comparison to mass transfer coefficient k L , and sufficient concentration in "scavenging" NAD + species. Also introduced in G/L absorption, the enhancement factor E, defined as the ratio of the flux of Rh(III) consumption at the interface in the presence of chemical reaction to the flux without chemical reaction. In the present case, this flux is proportional to the current density, so that for mediated electrosynthesis, E is expressed as:(13) If S 0 is the concentration gradient of Rh(III) complex at the electrode:

  observed that according to this simple model, the electromediated conversion of NAD + depends neither on Rh complex concentration, nor on NAD + initial concentration, thus it might be supposed that the model holds for any value of the concentration ratio C NAD,0 /C R3,0 . Practically, it is often considered to use concentration of Rh complex at least at a few 10 -5 M for possible observation of a diffusion plateau in voltammetric curves, and NAD + concentration not larger than 10 mM. Data reported in (El Housseini et al.) obtained with various concentrations of Rh complex and NAD + with the same initial C NAD /C R3 concentration ratio at 40 and at E = -0.5 V vs. ref.(H 2 ), could be fairly well modelled by rel. (21) using the above value for mass transfer coefficient k L , without further adjustment (Figure10). Moreover, it can be observed that the various data in the Figure correspond to a turnover frequency of Rh complex near 370 h -1 for NADH regeneration. Because of the sufficiently fast chemical reaction between Rh(I) and NAD + , regeneration of NADH can be modelled as though this species is directly reduced at the electrode surface. Additional data with C NAD /C R3 ratio at 10 only, follow the above trend,

Figure 2 :

 2 Figure 2: Schematic view of the overall NADH regeneration mediated by Rh complex:Electrode reduction of R3 forming R1, which chemically reacts with dissolved NAD species.

Figure 3 :

 3 Figure 3: Electrochemical cell for NADH generation with recirculation of the Rh complex -NAD + containing solution Figure 4a) Cyclic voltammograms recorded at 5 mV.s -1 (Scan 3) with the graphite felt electrode in the buffer solution with 125 µM [Cp*Rh(bpy)Cl] + : two replicates.

Figure

  Figure4b and c: Voltammetric curves recorded at 1 mV/s after 4 successive cycles at 5 mV/s for Rh complex solution with increasing amounts of NAD+: b) with 50 µM Rh(III), c) with 100 µM Rh(III) complexes. Experiments were carried out in 0.1 M phosphate buffer solution at pH 7.2 under nitrogen atmosphere with a solution and hydrogen flow rates of 20 mL.min -1 each.

Figure 7 :

 7 Figure 7: Enhancement factor E corresponding to occurrence of the chemical reaction. Concentrations of Rh complex and NAD + at 50 µM and 2 mM respectively. Fig. 7a): variation of E with rate constant k; Fig. 7b: variation of E with Hatta number Ha.

Figure 8 :

 8 Figure 8: Dimensionless concentration profiles in the diffusion film for the three estimates of the enhancement factor (± 10%). 125 µM Rh complex and 2 mM NAD + ; E = 3.83. k values are in m 3 .mol -1 .s -1 .

Figure 10 :

 10 Figure 10: NAD + conversion versus time for various NAD + -over Rh complex concentration ratio. Rh complex concentrations varied from 12.5 -62.5 µM. Experimental data (some of them from El Housseini et al.) compared to the model predictions (rel. 21).

Acknowledgements: Thanks are due to CNRS for the inter-department PhD grant allocated to W. El Housseini and to contribution to the research costs.

Appendices

1-Experimental estimation of mass transfer coefficient

This coefficient has been estimated from the limiting current for Rh(III) reduction, I L , was deduced after subtraction of the residual current recorded at the same potential without Rh complex. Mass transfer coefficient was estimated considering the porous graphite felt as a flat, the very large pores of the materials allowing internal diffusion to be not controlling:

with S= 16 cm 2 . For replicate tests for Rh complex concentration in the range 50 -250 µM, k L was estimated at 4.6 10 -5 ± 10% m.s -1 .

2-Estimation of diffusion coefficients

Most relationships for estimation of the diffusivity of a soluble species in the solvent rely upon the molar volume of this species, V m , in addition to the temperature and the molecular weight of the solvent. Molecular volumes of the two solutes have been estimated using Le Bas' incremental method [START_REF] Reid | The properties of gases and liquids, 3 rd Edition[END_REF] at 337.8 cm 3 mol -1 for the Rh complex, and 588.8 cm 3 .mol -1 for NAD + . For estimation of diffusivities in the aqueous solutions, the reference case of dissolved oxygen was used. Use of Wilke and Chang's relation (1955) led to estimates for diffusivities of oxygen and NAD + at least two times larger than the expected values, respectively at 1.96 10 -9 and 2.40 10 -10 m 2 .s -1 at 293 K [START_REF] Aizawa | Electrochemical regeneration of nicotinamide adenine dinucleotide[END_REF]. It was then preferred to consider the diffusivity in a solid as in the form:

For the molecular volume of oxygen, 25.6 cm 3 . mol -1 , the two diffusivity values led to m near -0.67, slightly different from Wilke and Chang's value at -0.6. Finally, the diffusivity of the Rh complex was estimated from either oxygen or NDA + , the two methods yielding the same value at 3.48 10 -10 m 2 .s -1 .