Zheng Tong
email: zheng.tong@hds.utc.fr

Philippe Xu
email: philippe.xu@hds.utc.fr

Thierry Denoeux
email: thierry.denoeux@utc.fr

ConvNet and Dempster-Shafer Theory for Object Recognition

Keywords: Pattern Recognition, Belief Function, Convolutional Neural Network, Supervised Learning, Evidence Theory

We propose a novel classifier based on convolutional neural network (ConvNet) and Dempster-Shafer theory for object recognition allowing for ambiguous pattern rejection, called the ConvNet-BF classifier. In this classifier, a ConvNet with nonlinear convolutional layers and a global pooling layer extracts high-dimensional features from input data. The features are then imported into a belief function classifier, in which they are converted into mass functions and aggregated by Dempster's rule. Evidence-theoretic rules are finally used for pattern classification and rejection based on the aggregated mass functions. We propose an end-to-end learning strategy for adjusting the parameters in the ConvNet and the belief function classifier simultaneously and determining the rejection loss for evidence-theoretic rules. Experiments with the CIFAR-10, CIFAR-100, and MNIST datasets show that hybridizing belief function classifiers with ConvNets makes it possible to reduce error rates by rejecting patterns that would otherwise be misclassified.

Introduction

Dempster-Shafer (DS) theory of belief functions [START_REF] Dempster | Upper and Lower Probabilities Induced by a Multivalued Mapping[END_REF][START_REF] Shafer | A mathematical theory of evidence[END_REF] has been widely used for reasoning and making decisions with uncertainty [START_REF] Yager | Classic works of the Dempster-Shafer theory of belief functions[END_REF]. DS theory is based on representing independent pieces of evidence by completely monotone capacities and aggregating them using Dempster's rule. In the past decades, DS theory has been applied to pattern recognition and supervised classification in three main directions. The first one is classifier fusion, in which classifier outputs are converted into mass functions and fused by Dempster's rule (e.g., [START_REF] Liu | Classifier fusion with contextual reliability evaluation[END_REF][START_REF] Bi | The impact of diversity on the accuracy of evidential classifier ensembles[END_REF]). Another direction is evidential calibration: the decisions of classifiers are transformed into mass functions (e.g., [START_REF] Xu | Evidential calibration of binary svm classifiers[END_REF][START_REF] Minary | Face pixel detection using evidential calibration and fusion[END_REF]). The last approach is to design evidential classifiers (e.g., [START_REF] Denoeux | A neural network classifier based on Dempster-Shafer theory[END_REF]), which represent the evidence of each feature as elementary mass functions and combine them by Dempster's rule. The combined mass functions are then used for decision making [START_REF] Denoeux | Analysis of evidence-theoretic decision rules for pattern classification[END_REF]. Compared with conventional classifiers, evidential classifiers can provide more informative outputs, which can be exploited for uncertainty quantification and novelty detection. Several principles have been proposed to design evidential classifiers, mainly including the evidential k-nearest neighbor rule [START_REF] Denoeux | A k-nearest neighbor classification rule based on Dempster-Shafer theory[END_REF][START_REF] Denoeux | A new evidential k-nearest neighbor rule based on contextual discounting with partially supervised learning[END_REF], and evidential neural network classifiers [START_REF] Denoeux | A neural network classifier based on Dempster-Shafer theory[END_REF]. In practice, the performance of evidential classifiers heavily depends on two factors: the training set size and the reliability of object representation. With the development of the "Big Data" age, the number of examples in benchmark datasets for supervised algorithms has increased from 10 2 to 10 5 [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF] and even 10 9 [START_REF] Sakaguchi | Efficient elicitation of annotations for human evaluation of machine translation[END_REF]. However, little has been done to combine recent techniques for object representation with DS theory.

Thanks to the explosive development of deep learning [START_REF] Lecun | Deep learning[END_REF] and its applications [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF][START_REF] Tong | Recognition, location, measurement, and 3d reconstruction of concealed cracks using convolutional neural networks[END_REF], several approaches for object representation have been developed, such as restricted Boltzmann machines [START_REF] Bengio | Learning deep architectures for AI[END_REF], deep autoencoders [START_REF] Vincent | Extracting and composing robust features with denoising autoencoders[END_REF][START_REF] Vincent | Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion[END_REF], deep belief networks [START_REF] Salakhutdinov | Deep boltzmann machines[END_REF][START_REF] Salakhutdinov | Learning with hierarchical-deep models[END_REF], and convolutional neural networks (ConvNets) [START_REF] Kim | Convolutional neural networks for sentence classification[END_REF][START_REF] Leng | 3D object understanding with 3D convolutional neural networks[END_REF]. ConvNet, which is maybe the most promising model and the main focus of this paper, mainly consists of convolutional layers, pooling layers, and fully connected layers. It has been proved that ConvNets have the ability to extract local features and compute global features, such as from edges to corners and contours to object parts. In general, robustness and automation are two desirable properties of ConvNets for object representation. Robustness means strong tolerance to translation and distortion in deep representation, while automation implies that object representation is data-driven with no human assistance.

Motivated by recent advances in DS theory and deep learning, we propose to combine ConvNet and DS theory for object recognition allowing for ambiguous pattern rejection. In this approach, a ConvNet with nonlinear convolutional layers and a global pooling layer is used to extract high-order features from input data. Then, the features are imported into a belief function classifier, in which they are converted into Dempster-Shafer mass functions and aggregated by Dempster's rule. Finally, evidence-theoretic rules are used for pattern recognition and rejection based on the aggregated mass functions. The performances of this classifier on the CIFAR-10, CIFAR-100, and MNIST datasets are demonstrated and discussed.

The organization of the rest of this paper is as follows. Background knowledge on DS theory and ConvNet is recalled in Section 2. The new combination between DS theory and ConvNet is then established in Section 3, and numerical experiments are reported in Section 4. Finally, we conclude the paper in Section 5.

Background

In this section, we first recall some necessary definitions regarding the DS theory and belief function classifier (Section 2.1). We then provide a description of the architecture of a ConvNet that will be combined with a belief function classifier later in the paper (Section 2.2).

Dempster-Shafer Theory

Evidence Theory The main concepts regarding DS theory are briefly presented in this section, and some basic notations are introduced. Detailed information can be found in Shafer's original work [START_REF] Shafer | A mathematical theory of evidence[END_REF] and some up-to-date studies [START_REF] Denoeux | Representations of uncertainty in artificial intelligence: Beyond probability and possibility[END_REF].

Given a finite set

Ω = {ω 1 , • • • , ω k }, called the frame of discernment, a mass function is a function m from 2 Ω to [0,1] verifying m(∅) = 0 and A⊆Ω m(A) = 1.
(

) 1
For any A ⊆ Ω, given a certain piece of evidence, m(A) can be regarded as the belief that one is willing to commit to A. Set A is called a focal element of m when m(A) > 0. For all A ⊆ Ω, a credibility function bel and a plausibility function pl, associated with m, are defined as bel

(A) = B⊆A m(B) (2)
pl(A) = A∩B =∅ m(B). (3)
The quantity bel(A) is interpreted as a global measure of one's belief that hypothesis A is true, while pl(A) is the amount of belief that could potentially be placed in A.

Two mass functions m 1 and m 2 representing independent items of evidence can be combined by Dempster's rule ⊕ [START_REF] Dempster | Upper and Lower Probabilities Induced by a Multivalued Mapping[END_REF][START_REF] Shafer | A mathematical theory of evidence[END_REF] as

(m 1 ⊕ m 2 ) (A) = B∩C=A m 1 (B) m 2 (C) B∩C =∅ m 1 (B) m 2 (C) (4)
for all A = ∅ and (m 1 ⊕ m 2 )(∅) = 0. Mass functions m 1 and m 2 can be combined if and only if the denominator on the right-hand side of ( 4) is strictly positive. The operator ⊕ is commutative and associative.

Belief Function Classifier Based on DS theory, an adaptive pattern classifier, called belief function classifier, was proposed by Denoeux [START_REF] Denoeux | A neural network classifier based on Dempster-Shafer theory[END_REF]. The classifier uses reference patterns as items of evidence regarding the class membership. The evidence is represented by mass functions and combined using Dempster's rule. In this section, we describe the architecture of a belief function classifier. For a more complete introduction, readers are invited to refer to Denoeux's original work [START_REF] Denoeux | A neural network classifier based on Dempster-Shafer theory[END_REF]. Step 1: The distance between x and each prototype p i is computed as

d i = x -p i i = 1, • • • , n, (5)
and the activation of the corresponding neuron is defined by introducing new pa-

rameters η i (η i ∈ R) as s i = α i exp(-η i d i 2 )
, where α i ∈ (0, 1) is a parameter associated to the prototype p i .

Step 2: The mass function m i associated to prototype p i is computed as

m i = (m i ({ω 1 }), . . . , m i ({ω M }), m i (Ω)) T (6a) = (u i 1 s i , . . . , u i M s i , 1 -s i ) T , (6b)
where u i = (u i 1 , . . . , u i M ) is a vector of parameters associated to the prototype p i verifying M j=1 u i j = 1. As illustrated in Figure 1a, Eq. ( 6) can be regarded as computing the activations of units in the second hidden layer L 2 , composed of n modules of M +1 units each. The units of module i are connected to neuron i of the previous layer. The output of module i in the hidden layer corresponds to the belief masses assigned by m i .

Step 3: The n mass functions m i , i = 1, • • • , n, are combined in the final layer based on Dempster's rule as shown in Figure 1b. The vectors of activations

µ i = (µ i 1 , • • • , µ i M +1
), i = 1, . . . , n of the final layer L 3 is defined by the following equations:

µ 1 = m 1 , (7a)
µ i j = µ i-1 j m i ({ω j }) + µ i-1 j m i ({Ω}) + µ i-1 M +1 m i ({ω j }) (7b) for i = 2, • • • , n and j = 1, • • • , M , and
µ i M +1 = µ i-1 M +1 m i ({Ω}) i = 2, • • • , n. (7c)
The

classifier outputs m = (m({ω 1 }), . . . , m({ω M }), m(Ω)) T is finally obtained as m = µ n .
Evidence-Theoretic Rejection Rules Different strategies to make a decision (e.g., assignment to a class or rejection) based on the possible consequences of each action were proposed in [START_REF] Denoeux | Analysis of evidence-theoretic decision rules for pattern classification[END_REF]. For a complete training set X , we consider actions α i , i ∈ {1, • • • , M } assigning the pattern to each class and a rejection action α 0 . Assuming the cost of correct classification to be 0, the cost of misclassification to be 1 and the cost of rejection to be λ 0 , the three conditions for rejection reviewed in [START_REF] Denoeux | Analysis of evidence-theoretic decision rules for pattern classification[END_REF] can be expressed as

Maximum credibility: max j=1,••• ,M m({ω j }) < 1 -λ 0 Maximum plausibility: max j=1,••• ,M m({ω j }) + m(Ω) < 1 -λ 0 Maximum pignistic probability: max j=1,••• ,M m({ω j }) + m(Ω) M < 1 -λ 0 .
Otherwise, the pattern is assigned to class

ω j with j = arg max k=1,••• ,M m({ω k }).
For the maximum plausibility and maximum pignistic probability rules, rejection is possible if and only if 0 ≤ λ 0 ≤ 1 -1/M , whereas a rejection action for the maximum credibility rule only requires 0 ≤ λ 0 ≤ 1.

Convolutional Neural Network

In this section, we provide a brief description of some state-of-the-art techniques for ConvNets including the nonlinear convolutional operation and global average pooling (GAP), which will be implemented in our new model in Section 3. Detailed information about the two structure layers can be found in [START_REF] Lin | Network in network[END_REF].

Nonlinear Convolutional Operation The convolutional layer [START_REF] Lecun | Deep learning[END_REF] is highly efficient for feature extraction and representation. In order to approximate the representations of the latent concepts related to the class membership, a novel convolutional layer has been proposed [START_REF] Lin | Network in network[END_REF], in which nonlinear multilayer perceptron (MLP) operations replace classic convolutional operations to convolve over the input. An MLP layer with nonlinear convolutional operations can be summarized as follows:

f 1 i,j,k = ReLU w 1 k T • x + b 1 k , k = 1, • • • , C (8a)
. . .

f m i,j,k = ReLU (w m k ) T • f m-1 i,j + b m k , k = 1, • • • , C. (8b)
Here, m is the number of layers in an MLP. Matrix x, called receptive field of size i × j × o, is a patch of the input data with the size of (rW -r -p + i) × (rH -r -p + j) × o. An MLP layer with an r stride and a p padding can generate a W × H × C tensor, called feature maps. The size of a feature map is W × H × 1, while the channel number of the feature maps is C. A rectified linear unit (ReLU) is used as an activation function as ReLU (x) = max(0, x). As shown in Eq. ( 8), element-by-element multiplications are first performed between x and the transpositions of the weight matrices

w 1 k (k = 1, • • • , C) in
i,j = (f 1 i,j,1 ,f 1 i,j,2 ,• • •,f 1 i,j,C
). The outputs then flow into the remaining layers in sequence, generating f m i,j of size 1 × 1 × C. After processing all patches by the MLP, the input data is transformed into a W × H × C tensor. As the channel number C of the last MLP in a ConvNet is the same as the input data dimension P in a belief function classifier, a W × H × P tensor is finally generated by a ConvNet.

Global Average Pooling In a traditional ConvNet, the tensor is vectorized and imported into fully connected layers and a softmax layer for a classification task. However, fully connected layers are prone to overfitting, though dropout [START_REF] Hinton | Improving neural networks by preventing co-adaptation of feature detectors[END_REF] and its variation [START_REF] Gomez | Targeted dropout[END_REF] have been proposed. A novel strategy, called global average pooling (GAP), has been proposed to remove traditional fully connected layers [START_REF] Lin | Network in network[END_REF]. A GAP layer transforms the feature tensor W × H × P into a feature vector 1 × 1 × P by taking the average of each feature map as follows:

x k = W i=1 H j=1 f m i,j,k W • H k = 1, • • • , P. (9)
The generated feature vector is used for classification. From the belief function perspective, the feature vector can be used for object representation and classified in one of M classes or rejected by a belief function classifier. Thus, a ConvNet can be regarded as a feature generator.

ConvNet-BF Classifier

In this section, we present a method to combine a belief function classifier and a ConvNet for objection recognition allowing for ambiguous pattern rejection. The architecture of the proposed method, called ConvNet-BF classifier, is illustrated in Figure 2

Connectionist Implementation

In a ConvNet-BF classifier, the Euclidean distance between a feature vector and each prototype is first computed and then used to generate a mass function. To reduce the classification error when P is large, we assign weights to each feature as

d i = P k=1 w i k (x k -p i k ) 2 , (10)
and the weights are normalized by introducing new parameters

ζ i k (ζ i k ∈ R) as w i k = (ζ i k ) 2 P l=1 (ζ i l ) 2 . ( 11
)

Learning

The

E ν (x) = 1 2N I i=1 M q=1
(P re ν,q,i -T ar q,i ) 2 , (12a)

P re ν,q,i = m q,i + νm M +1,i , (12b)
m i = m i M +1 k=1 m i ({ω k }) . (12c)
Here, T ar i = (T ar 1,i , • • • , T ar M,i ) and m i = (m i ({ω 1 }), . . . , m i ({ω M }), m i (Ω)) T are the target output vector and the unnormalized network output vector for pattern x i , respectively. We transform m i to a vector (P re ν,1,i , . . . , P re ν,M,i ) by distributing a fraction ν of m i (Ω) to each class under the constraint 0 ≤ ν ≤ 1.

The numbers P re 1,q,i , P re 0,q,i and P re 1/M,q,i represent, respectively, the credibility, the plausibility, and the pignistic probability of class ω q . The derivatives of E ν (x) w.r.t p i k , w i k , and ζ i k in a belief function classifier can be expressed as

∂E ν (x) ∂p i k = ∂E ν (x) ∂s i ∂s i ∂p i k = ∂E ν (x) ∂s i • 2(η i ) 2 s i • P k=1 w i k (x k -p i k ), ( 13
)
∂E ν (x) ∂w i k = ∂E ν (x) ∂s i ∂s i ∂w i k = ∂E ν (x) ∂s i • η i 2 s i • x k -p i k 2 , (14)
and

∂E ν (x) ∂ζ i k = ∂E ν (x) ∂w i k ∂w i k ∂ζ i k (15a) = 2ζ i k P k=1 ζ i k 2 2 ∂E ν (x) ∂w i k P k=1 (ζ i k ) 2 - P k=1 (ζ i k ) 2 ∂E ν (x) ∂w i k . (15b)
Finally, the derivatives of the error w.r.t. x k , w m i,j,k and b m k in the last MLP are given as

∂E ν (x) ∂x k = ∂E ν (x) ∂s i ∂s i ∂x k = - ∂E ν (x) ∂s i • 2(η i ) 2 s i • P k=1 ω i k (x k -p i k ), ( 16
)
∂E ν (x) ∂w m i,j,k = ∂E ν (x) ∂f m i,j,k • ∂f m i,j,k ∂w m i,j,k = w m i,j,k • ∂E ν (x) ∂f m i,j,k k = 1, • • • , P, (17)
and

∂E ν (x) ∂b m k = ∂E ν (x) ∂f m i,j,k • ∂f m i,j,k ∂b m k = ∂E ν (x) ∂f m i,j,k k = 1, • • • , P (18)
with

∂E ν (x) ∂f m i,j,k = ∂E ν (x) ∂x k • ∂x k ∂f m i,j,k = 1 W • H ∂E ν (x) ∂x k k = 1, • • • , P. (19)
Here, w m i,j,k is the component of the weight matrix w m k , while f m i,j,k is the component of vector f m i,j in Eq. ( 8).

Determination of λ 0 A data-driven method for determining λ 0 to guarantee a ConvNet-BF classifier with a certain rejection rate is shown in Figure 3. We randomly select three-fifths of a training set χ to train a ConvNet-BF classifier, while random one-fifth of the set is used as a validation set. The remaining onefifth of the set is used to draw a λ

(1) 0 -rejection curve. We can determine the value of λ

(1) 0 for a certain rejection rate from the curve. We repeat the process and take the average of λ (i) 0 as the final λ 0 for the desired rejection rate.

Numerical Experiments

In this section, we evaluate ConvNet-BF classifiers on three benchmark datasets: CIFAR-10 [13], CIFAR-100 [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF], and MNIST [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF]. To compare with traditional ConvNets, the architectures and training strategies of the ConvNet parts in ConvNet-BF classifiers are the same as those used in the study of Lin et al, called NIN [START_REF] Lin | Network in network[END_REF]. Feature vectors from the ConvNet parts are imported into a belief function classifier in our method, while they are directly injected into softmax layers in NINs.

In order to make a fair comparison, a probability-based rejection rule is adopted for NINs as max j=1,••• ,M p j < 1 -λ 0 , where p j is the output probability of NINs.

CIFAR-10

The CIFAR-10 dataset [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF] is made up of 60,000 RGB images of size 32 × 32 partitioned in 10 classes. There are 50,000 training images, and we randomly selected 10,000 images as validation data for the ConvNet-BF classifier. We then randomly used 10,000 images of the training set to determine λ 0 .

The test set error rates without rejection of the ConvNet-BF and NIN classifiers are 9.46% and 9.21%, respectively. The difference is small but statistically significant according to McNemar's test (p-value: 0.012). Error rates without rejection mean that we only consider max j=1,••• ,M p j and max j=1,••• ,M m ({ω j }). If the selected class is not the correct one, we regard it as an error. It turns out in our experiment that using a belief function classifier instead of a softmax layer only slightly impacts the classifier performance.

The test set error rates with rejection of the two models are presented in Figure 4a. A rejection decision is not regarded as an incorrect classification. When the rejection rate increases, the test set error decreases, which shows that the belief function classifier rejects a part of incorrect classification. However, the error decreases slightly when the rejection rate is higher than 7.5%. This demonstrates that the belief function classifier rejects more and more correctly classified patterns with the increase of rejection rates. Thus, a satisfactory λ (i) 0 should be determined to guarantee that the ConvNet-BF classifier has a desirable accuracy rate and a low correct-rejection rate. Additionally, compared with the NIN, the ConvNet-BF classifier rejects significantly more incorrectly classified patterns. For example, the p-value of McNemar's test for the difference of error rates between the two classifiers with a 5.0% rejection rate is close to 0. We can conclude that a belief function classifier with an evidence-theoretic rejection rule is more suitable for making a decision allowing for pattern rejection than a softmax layer and the probability-based rejection rule.

Table 1 presents the confusion matrix of the ConvNet-BF classifier with the maximum credibility rule, whose rejection rate is 5.0%. The ConvNet-BF classifier tends to select rejection when there are two or more similar patterns, such as dog and cat, which can lead to incorrect classification. In the view of evidence theory, the ConvNet part provides conflicting evidence when two or more similar patterns exist. The maximally conflicting evidence corresponds to m ({ω i }) = m ({ω j }) = 0.5 [START_REF] Denoeux | Logistic regression, neural networks and Dempster-Shafer theory: A new perspective[END_REF]. Additionally, the additional mass function m (Ω) provides the possibility to verify whether the model is well trained because we have m (Ω) = 1 when the ConvNet part cannot provide any useful evidence.

CIFAR-100

The CIFAR-100 dataset [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF] has the same size and format at the CIFAR-10 dataset, but it contains 100 classes. Thus the number of images in each class is only 100. For CIFAR-100, we also randomly selected 10,000 images of the training set to determine λ 0 . The ConvNet-BF and NIN classifiers achieved, respectively, 40.62% and 39.24% test set error rates without rejection, a small but statistically significant difference (p-value: 0.014). Similarly to CIFAR-10, it turns out that the belief function classifier has a similar error rate as a network with a softmax layer. Figure 4b shows the test set error rates with rejection for the two models. Compared with the rejection performance in CIFAR-10, the ConvNet-BF classifier rejects more incorrect classification results. We can conclude that the evidence-theoretic classifier still performs well when the classification task is difficult and the training set is not adequate. Similarly, Table 2 shows that the ConvNet-BF classifier tends to select the rejection action when two classes are similar, in which case we have m ({ω i }) ≈ m ({ω j }). In contrast, the classifier tends to produce m(Ω) ≈ 1 when the model is not trained well because of an inadequate training set.

MNIST

The MNIST database of handwritten digits consists of a training set of 60,000 examples and a test set of 10,000 examples. The training strategy for the ConvNet-BF classifier was the same as the strategy in CIFAR-10 and CIFAR-100. The test set error rates without rejection of the two models are close (0.88% and 0.82%) and weakly signifiant (p-value: 0.077). Again, using a belief function classifier instead of a softmax layer introduce no negative effect on the network in MNIST.

The test set error rates with rejection of the two models are shown in Figure 4c. The ConvNet-BF classifier rejects a small number of classification results because the feature vectors provided by the ConvNet part include little confusing information.

Conclusion

In this work, we proposed a novel classifier based on ConvNet and DS theory for object recognition allowing for ambiguous pattern rejection, called "ConvNet-BF The mass functions can be used for classification or rejection based on evidencetheoretic rules. Additionally, the novel classifier can be trained in an end-to-end way.

The use of belief function classifiers in ConvNets had no negative effect on the classification performances on the CIFAR-10, CIFAR-100, and MNIST datasets. The combination of belief function classifiers and ConvNet can reduce the errors by rejecting a part of the incorrect classification. This provides a new direction to improve the performance of deep learning for object recognition. The classifier is prone to assign a rejection action when there are conflicting features, which easily yield incorrect classification in the traditional ConvNets. In addition, the proposed method opens a way to explain the relationship between the extracted features in convolutional layers and class membership of each pattern. The mass m(Ω) assigned to the set of classes provides the possibility to verify whether a ConvNet is well trained or not.

Fig. 1 :

 1 Fig. 1: Belief function classifier

  the 1 st layer of the MLP. Each weight matrix w 1 k has the same size as the receptive field. Then the multiplied values are summed, and the bias b 1 k (k = 1, • • • , C) is added to the summed values. The results are transformed by a ReLU function. The output vector is f 1

  . A ConvNet-BF classifier can be divided into three parts: a ConvNet as a feature producer, a belief function classifier as a mass-function generator, and a decision rule. In this classifier, input data are first imported into a ConvNet with nonlinear convolutional layers and a global pooling layer to extract latent features related to the class membership. The features are then imported into a belief function classifier, in which they are converted into mass functions and aggregated by Dempster's rule. Finally, an evidence-theoretic rule is used for pattern classification and rejection based on the aggregated mass functions. As the background of the three parts has been introduced in Section 2, we only provide the details of the combination in this section, including the connectionist implementation and the learning strategy.

Fig. 2 :

 2 Fig. 2: Architecture of a ConvNet-BF classifier

Fig. 3 :

 3 Fig. 3: Illustration of the procedure for determining λ 0

Fig. 4 :

 4 Fig. 4: Rejection-error curves: CIFAR-10 (a), CIFAR-100 (b), and MNIST (c)

  proposed learning strategy to train a ConvNet-BF classifier consists in two parts: (a) an end-to-end training method to train ConvNet and belief function classifier simultaneously; (b) a data-driven method to select λ 0 .

	End-to-End Training Compared with the belief function classifier proposed
	in [6], we have different expressions for the derivatives w.r.t. w i k , ζ i k , and p i k in
	the new belief function classifier. A normalized error E ν (x) is computed as:

Table 1 :

 1 Confusion matrix for Cifar10.

		Airplane Automobile Bird Cat Deer Dog Frog Horse Ship Truck
	Airplane	-	0.03	0.03 0.01 0.02 0.05 0.04 0.01 0.04 0.05
	Automobile 0	-	0.04 0.04 0.08 0.08 0.04 0.06 0.03 0.07
	Bird	0.02	0.04	-	0.05 0.04 0.07 0.03 0.08 0	0.04
	Cat	0.02	0.03	0.13 -	0.06 0.44 0.11 0.04 0.05 0.06
	Deer	0.01	0.04	0.07 0.12 -	0.03 0.12 0.34 0.04 0.08
	Dog	0.02	0.03	0.05 0.49 0.11 -	0.06 0.09 0.01 0.04
	Frog	0.02	0.04	0.08 0.06 0.12 0.06 -	0.06 0.06 0.05
	Horse	0.01	0.02	0.04 0.06 0.31 0.10 0.04 -	0.04 0.04
	Ship	0.04	0.05	0.02 0.04 0.12 0.05 0.04 0.18 -	0.02
	Truck	0.02	0	0.06 0.09 0.03 0.06 0.07 0.06 0.04 -
	Rejection 0.20	0.13	0.14 1.05 0.84 1.07 0.14 1.14 0.18 0.11

Table 2 :

 2 Confusion matrix for the superclass flowers. This new structure consists of a ConvNet with nonlinear convolutional layers and a global pooling layer to extract high-dimensional features and a belief function classifier to convert the features into Dempster-Shafer mass functions.

		Orchids Poppies Roses Sunflowers Tulips
	Orchids	-	0.24	0.23 0.28	0.15
	Poppies	0.14	-	0.43 0.10	0.90
	Roses	0.27	0.12	-	0.16	0.13
	Sunflowers 0.18	0.15	0.12 -	0.22
	Tulips	0.08	1.07	0.76 0.17	-
	Rejection 0.09	0.37	0.63 0.12	0.34

This research was carried out in the framework of the Labex MS2T, which was funded by the French Government, through the program Investments for the future managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02). It was also supported by a scholarship from the China Scholarship Council.


Eric Gouré

Bertrand Gerey

Catalina N Astudillo

Jacques Pécaut

Selim Sirach

Florian Molton

Jérôme Fortage

Marie-Noëlle Collomb
email: marie-noelle.collomb@univ-grenoble-alpes.fr

Self-Assembled Heterometallic Complexes by Incorporation of Calcium or Strontium Ion into a Manganese(II) 12-Metallacrown-3 Framework Supported by a Tripodal Ligand with Pyridine-Carboxylate Motifs: Stability in Their Manganese(III) Oxidized Form

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

Introduction

Heterometallic manganese-calcium and manganese-strontium complexes are of special interest because of their relevance to the structure of the active center of the oxygen-evolving complex (OEC) of Photosystem II (PSII), an asymmetric oxo-bridged Mn 4 CaO 5 -cluster stabilized by six carboxylates ligands and one histidine. [START_REF] Gerey | Manganesecalcium/strontium heterometallic compounds and their relevance for the oxygen-evolving center of photosystem II[END_REF][START_REF] Najafpour | Water-oxidizing complex in Photosystem II: Its structure and relation to manganese-oxide based catalysts[END_REF][START_REF] Luber | S 1 -State Model of the O 2 -Evolving Complex of Photosystem II[END_REF][START_REF] Grundmeier | Structural models of the manganese complex of photosystem II and mechanistic implications[END_REF] Four water molecules directly also coordinated to the cluster (two to the Ca 2+ ion and the other to the outer manganese) may act as substrates for dioxygen formation. [START_REF] Gerey | Manganesecalcium/strontium heterometallic compounds and their relevance for the oxygen-evolving center of photosystem II[END_REF][START_REF] Cox | Current Understanding of the Mechanism of Water Oxidation in Photosystem II and Its Relation to XFEL Data[END_REF] Calcium is essential for the assembly and reassembly processes of the OEC as well as for the oxygen-evolving activity, although its role is not fully elucidated. [START_REF] Lionetti | How calcium affects oxygen formation[END_REF][START_REF] Cox | Electronic structure of the oxygenevolving complex in photosystem II prior to O-O bond formation[END_REF][START_REF] Yocum | The calcium and chloride requirements of the O 2 evolving complex[END_REF][START_REF] Mcevoy | Water-splitting chemistry of photosystem II[END_REF][START_REF] Loll | Towards complete cofactor arrangement in the 3.0A° resolution structure of photosystem II[END_REF][START_REF] Lohmiller | The Basic Properties of the Electronic Structure of the Oxygen-evolving Complex of Photosystem II Are Not Perturbed by Ca 2+ Removal[END_REF][START_REF] Siegbahn | Water oxidation energy diagrams for photosystem II for different protonation states, and the effect of removing calcium[END_REF] Regarding strontium, although this alkaline-earth metal is not present in the native enzyme, its comparable size and Lewis acidity make Sr 2+ the only cation able to reproduce about half of the activity in a PSII from which Ca 2+ has been removed. [START_REF] Boussac | Biosynthetic Ca 2+ /Sr 2+ exchange in the photosystem II oxygen-evolving enzyme of Thermosynechococcus elongatus[END_REF][START_REF] Beckmann | Effects of the Calcium/strontium and chloride/bromide substitution on substrate water exchange rates in photosystem II[END_REF][START_REF] Cox | Effect of Ca 2+ /Sr 2+ Substitution on the Electronic Structure of the Oxygen-Evolving Complex of Photosystem II: A Combined Multifrequency EPR, 55 Mn-ENDOR, and DFT Study of the S 2 State[END_REF][START_REF] Yachandra | Calcium in the oxygen-evolving complex: Structural and mechanistic role determined by X-ray spectroscopy[END_REF] Only few heterometallic manganese-calcium and manganese-strontium compounds have been isolated, and most examples have been published in the last ten years, reflecting a still largely unexplored area of research and also the synthetic difficulty to access such compounds. [START_REF] Gerey | Manganesecalcium/strontium heterometallic compounds and their relevance for the oxygen-evolving center of photosystem II[END_REF][START_REF] Shamsipur | What has biomimicry so far brought on mysterious natural oxygen evolution?[END_REF][START_REF] Paul | Structural models of the biological oxygenevolving complex: achievements, insights, and challenges for biomimicry[END_REF][START_REF] Paul | What Can We Learn from a Biomimetic Model of Nature's Oxygen-Evolving Complex?[END_REF][START_REF] Lee | A CaMn 4 O 2 model of the biological oxygen evolving complex: synthesis via cluster expansion on a low symmetry ligand[END_REF][START_REF] Melnic | Evolution of manganesecalcium cluster structures based on nitrogen and oxygen donor ligands[END_REF][START_REF] Arauzo | Slow magnetic relaxation in a dimeric Mn2Ca2 complex enabled by the large Mn(iii) rhombicity[END_REF][START_REF] Mousazade | A synthetic manganese-calcium cluster similar to the catalyst of Photosystem II: challenges for biomimetic water oxidation[END_REF][START_REF] Chen | Natural and Artificial Mn 4 Ca Cluster for the Water Splitting Reaction[END_REF][START_REF] Koepf | Design of porphyrin-based ligands for the assembly of d-block metal : calcium bimetallic centers[END_REF][START_REF] Alaimo | Structural Diversities in Heterometallic Mn-Ca Cluster Chemistry from the Use of Salicylhydroxamic Acid: {MnIII 4 Ca 2 }, {MnII/III 6 Ca 2 }, {MnIII/IV 8 Ca}, and {MnIII 8 Ca 2 } Complexes with Relevance to Both High-and Low-Valent States of the Oxygen-Evolving Complex[END_REF][START_REF] Alaimo | New insights in Mn-Ca chemistry from the use of oximate-based ligands: {MnII/III 22 Ca 2 } and {MnIV 2 Ca 2 } complexes with relevance to both low-and high-valent states of the oxygenevolving complex[END_REF][START_REF] Chen | Artificial Mn 4 Ca Clusters with Exchangeable Solvent Molecules Mimicking the Oxygen-Evolving Center in Photosynthesis[END_REF][START_REF] Tziotzi | The first amino acid bound manganese-calcium clusters: a {[MnIII 3 Ca] 2 } methylalanine complex, and a [MnIII 6 Ca] trigonal prism[END_REF] These complexes present a large variety of structures, topology and oxidation states, ranging from polymer to high nuclearity Mn 13 Ca 2 clusters and only a few of them are close structural mimics of the OEC. [START_REF] Chen | Natural and Artificial Mn 4 Ca Cluster for the Water Splitting Reaction[END_REF][START_REF] Chen | Artificial Mn 4 Ca Clusters with Exchangeable Solvent Molecules Mimicking the Oxygen-Evolving Center in Photosynthesis[END_REF][START_REF] Mukherjee | Synthetic model of the asymmetric [Mn 3 CaO 4 ] cubane core of the oxygen-evolving complex of photosystem II[END_REF][START_REF] Zhang | A synthetic Mn 4 Cacluster mimicking the oxygen-evolving center of photosynthesis[END_REF][START_REF] Kanady | A Synthetic Model of the Mn 3 Ca Subsite of the Oxygen-Evolving Complex in Photosystem II[END_REF][START_REF] Kanady | Toward Models for the Full Oxygen-Evolving Complex of Photosystem II by Ligand Coordination To Lower the Symmetry of the Mn 3 CaO 4 Cubane: Demonstration That Electronic Effects Facilitate Binding of a Fifth Metal[END_REF] Among the synthetic Mn-Ca complexes, we have succeeded in the isolation of rare examples of carboxylate-bridged manganese(II)-calcium(II) complexes from the deprotonated tripodal tris(2-pyridylmethyl)amine hepta-and hexadentate ligands, tpaa [START_REF] Luber | S 1 -State Model of the O 2 -Evolving Complex of Photosystem II[END_REF]-and tpada 2-, incorporating respectively three and two pyridine-carboxylate units: the tetranuclear [{Mn II (tpaa)} 2 {Ca(OH 2 ) 5 (μ-OH 2 )} 2 ] 2-(Mn 2 (tpaa) 2 Ca 2 (H 2 O) 12 ) and the dinuclear [Mn(tpada)ClCa(OH 2 ) 2.67 (MeOH) 2.33 ] -(Mn(tpada)Ca(H 2 O) 3 ) (Figure 1). These complexes are potentially relevant for mimicking the first stages of the OEC assembly process. [START_REF] Martin-Diaconescu | Ca K-Edge XAS as a Probe of Calcium Centers in Complex Systems[END_REF][START_REF] Gerey | Calcium and heterometallic manganese-calcium complexes supported by tripodal pyridine-carboxylate ligands: structural, EPR and theoretical investigations[END_REF] In these complexes, the coordination of calcium is facilitated by the carboxylate functions incorporated in the periphery of the ligands framework. In Mn(tpada)Ca(H 2 O) 3 , the Mn 2+ and Ca 2+ ions are bridged via two µ 1,1 -carboxylato units, while in Mn 2 (tpaa) 2 Ca 2 (H 2 O) 12 only one µ 1,3 -carboxylate bridge connects each Ca 2+ ion to each Mn(II). The Mn(tpada)Ca(H 2 O) 3 complex was obtained by displacement of the Ca 2+ ion in the neutral dinuclear calcium complex, [Ca(tpada)(OH 2 ) 2 ] 2 (Ca 2 (tpada) 2 (H 2 O) 4 ), with Mn 2+ (Figure 1).

Figure 1:

Structures of the Mn-Ca and Ca complexes previously isolated by our group [START_REF] Gerey | Calcium and heterometallic manganese-calcium complexes supported by tripodal pyridine-carboxylate ligands: structural, EPR and theoretical investigations[END_REF] and of the 8-MC-4 and 12-MC-3 complexes isolated in this work from the carboxylatefunctionalized tpaa 3-and tpada 2-tripodal ligands (Adapted from Ref. [START_REF] Gerey | Calcium and heterometallic manganese-calcium complexes supported by tripodal pyridine-carboxylate ligands: structural, EPR and theoretical investigations[END_REF] , with permission from the Royal Society of Chemistry, Copyright 2015).

On the other hand, multidentate ligands containing one or more carboxylate moieties can also lead to original self-assembled metallamacrocyclic-type structures, also called metallacrowns (MCs), able to trap, in some cases, a cation in their cavity. [START_REF] Mezei | Structural and functional evolution of metallacrowns[END_REF] In such complexes, the metals forming the macrocycle are linked either with µ 1,3 -or µ 1,1 -carboxylato bridges of the ancillary ligand corresponding to -[M-O-C-O] or [M-O] repeat unit. Among the numerous examples of metallamacrocycles, metallacrowns based on such µ-carboxylato bridges are not common. [START_REF] Abufarag | Zinc Complexes of the Ligand Dipicolylglycine[END_REF][START_REF] Ménage | Carboxylatoiron(II) Aggregates: A Novel Fe Complex with Threefold Symmetry[END_REF][START_REF] Mandal | Polynuclear Carboxylato-Bridged Iron(II) Clusters: Synthesis, Structure, and Host-Guest Chemistry[END_REF][START_REF] Zhang | The first lanthanide-templated molecular wheel containing six copper ions[END_REF][START_REF] Wang | Nature of the Reactants and Influence of Water on the Supramolecular Assembly[END_REF] Four examples of the type 12-MC-3, i. e. a 12-membered ring comprising 3 repeating units of -[M-O-C-O] have been isolated from the deprotonated monocarboxylate tripodal ligand bpg -(bis(2-picolyl)glycine) with zinc(II) or iron(II) (Figure 2(A)). The three metal centers are linked by a µ 1,3 -carboxylato of the tripodal ligand. Two of the iron metallacrowns described have encapsulated a Fe 2+ central cation which is linked to the peripheral ions by three oxygen atoms of the ring and by three additional µ 1,3 -acetato [START_REF] Mandal | Polynuclear Carboxylato-Bridged Iron(II) Clusters: Synthesis, Structure, and Host-Guest Chemistry[END_REF] or µ 1,3 -benzoato [START_REF] Ménage | Carboxylatoiron(II) Aggregates: A Novel Fe Complex with Threefold Symmetry[END_REF] bridges (Figure 2(A(c))). The others examples of µ-carboxylato bridges metallacrowns, of higher nuclearity, have been described using copper(II) salts from triscarboxylate ligands derived from glycine. [START_REF] Zhang | The first lanthanide-templated molecular wheel containing six copper ions[END_REF][START_REF] Wang | Nature of the Reactants and Influence of Water on the Supramolecular Assembly[END_REF] Considering metallacrowns with manganese, a series of 12-MC-4 of Mn(III) were obtained with the deprotonated multidentate salicylhydroxamic acid ligand (shi 3-) that does not contain carboxylate units (Figure 2(B)). [START_REF] Mezei | Structural and functional evolution of metallacrowns[END_REF][START_REF] Lah | Isolation and characterization of {Mn II [Mn III (salicylhydroximate)] 4 (acetate) 2 (DMF) 6 }.cntdot.2DMF: an inorganic analog of M 2+ (12-crown-4)[END_REF][START_REF] Gibney | Structural Evaluation and Solution Integrity of Alkali Metal Salt Complexes of the Manganese 12-Metallacrown-4 (12-MC-4) Structural Type[END_REF][START_REF] Dendrinou-Samara | Inter-conversion of 15-MC-5 to 12-MC-4 manganese metallacrowns: structure and bioactivity of metallacrowns hosting carboxylato complexes[END_REF][START_REF] Koumousi | Towards models of the oxygen-evolving complex (OEC) of photosystem II: a Mn 4 Ca cluster of relevance to low oxidation states of the OEC[END_REF][START_REF] Alaimo | Emissive {Mn 4 III Ca} Clusters with Square Pyramidal Topologies: Syntheses and Structural, Spectroscopic, and Physicochemical Characterization[END_REF][START_REF] Azar | Controllable Formation of Heterotrimetallic Coordination Compounds: Systematically Incorporating Lanthanide and Alkali Metal Ions into the Manganese 12-Metallacrown-4 Framework[END_REF][START_REF] Foley | Syntheses and Crystal Structures of a Series of Manganese-Lanthanide-Sodium 12-Metallacrown-4 Dimers[END_REF] In this case, the four Mn(III) ions are linked by the N and O atoms of the oximate group from the shi 3-ligand forming a twelvemembered {M-O-N} 4 MC that can host a large variety of cations in their cavity, including Mn 2+ , Li + , Na + , lanthanides but also Ca 2+ . Similarly to the iron MC mentioned above, the central cation is coordinated either by oxygen atoms of the ring and by additional acetates or benzoates that bridge with the Mn(III) ions of the macrocycle.

In the present study, we report on the isolation and the structural characterization of a new family of manganese(II) MCs, based on the [Mn-O-C-O] repeating motif, from the recently isolated hexadentate tpada 2-ligand. [START_REF] Gerey | Calcium and heterometallic manganese-calcium complexes supported by tripodal pyridine-carboxylate ligands: structural, EPR and theoretical investigations[END_REF] The schematic structure of the complexes of general formula, [{Mn II (tpada)} 3 M II (H 2 O) n ] 2+ (Mn 3 M) with M = Mn 2+ (n = 0), Ca 2+ (n = 1) or Sr 2+ (n = 2), is shown in Figure 1. These self-assembled 12-MC-3 manganese(II) MCs forming a twelve-membered ring with three Mn 2+ centers bridged by μ 1,3 -carboxylato, encapsulate a fourth cation in their central cavity. The first crystallographic structure with a good resolution was obtained with the central unit occupied with a Mn 2+ ion. We then investigated the possibility of replacing this central cation with redox-innocent cations such as calcium and strontium, in order to expand the family of models of the OEC center of PSII. We also explore the molecular architecture that arises upon the self-assembly of the tpada 2-ligand with only Sr 2+ , with the objective of identifying the coordination mode of strontium with this ligand in comparison to calcium, 1 in particular its affinity for aqua ligands. Another type of MC was obtained of the type molecular wheel, namely [Sr(tpada)(OH 2 )] 4 (Sr4) in the form of a 8-MC-4, with four Sr 2+ ions bridged via a µ 1,1 -carboxylato bridge, without any cation in the central cavity. In this article, we will first discuss the synthesis and the X-ray structures of the Mn 3 M and Sr 4 MC complexes. We will then report on solution studies in organic solvent (CH 3 CN) by electrospray mass spectrometry and electrochemistry in combination with UV-visible absorption and EPR spectroscopy of the Mn 3 M complexes. The aim of these solution studies was to establish the stability of the tetranuclear MC structure of these complexes in solution and to investigate their redox properties. Only few studies in solution have been conducted on metallamacrocyclic complexes and to the best of our knowledge, the stability in solution and the redox properties of the family of µ-carboxylato bridged MCs have never been investigated.

Results and discussion

Synthesis and crystal structures of the complexes Mn 3 M (M = Mn 2+ , Ca 2+ or Sr 2+ )

The first metallamacrocyclic 12-MC-3 complex of our series, in which three {Mn(tpada)} units are bridged by μ 1,3 -carboxylate O atoms, was obtained with the central cavity occupied by a Mn 2+ cation. This result prompted us to investigate the insertion of other cations in the central cavity such as divalent Ca 2+ and Sr 2+ ions in order to obtain new heteronuclear MC complexes. To access these complexes, the ligand (H 3 tpada)PF 6 H 2 O was first fully deprotonated in methanol by addition of a base (Scheme 1). The tetranuclear MC with a Mn 2+ in the central cavity (Mn 3 Mn) was then obtained by the spontaneous selfassembly of the tpada 2-ligand with 1.4 equivalents of manganese(II) acetate in methanol at room temperature (Scheme 1(a)). Only one equivalent of manganese(II) acetate per tpada 2- was used for the heteronuclear MCs, Mn 3 Ca and Mn 3 Sr, followed by the addition of an excess (1 to 1.3 equivalents per ligand) of the second metal salt (M = Ca 2+ or Sr 2+ ) in acetonitrile or in acetonitrile/water (Scheme 1(b, c)). The three complexes were isolated in good yields (from 88 to 97 %) as microcrystalline pale yellow powders by partial evaporation of the solvents. The Mn 3 Mn and Mn 3 Ca crystallized in their tetranuclear forms, while two Mn 3 Sr units associate via two bridging μ 1,3 -carboxylates from tpada 2- N tertiary (N tert ) amine are close to coplanar with the Mn 2+ ion, while the axial positions are occupied by the pyridine N atom and the bridging carboxylate O atom (Figure 4(b)). This geometry around each Mn 2+ ion is very similar to that found in the binuclear Mn(tpada)Ca(H 2 O) 3 except that instead of a bridging carboxylate O atom, a chloride ion occupies the axial position (Figure 1). [START_REF] Gerey | Calcium and heterometallic manganese-calcium complexes supported by tripodal pyridine-carboxylate ligands: structural, EPR and theoretical investigations[END_REF] The Mn-N tert bonds are also significantly longer than the Mn-N py and Mn-O bonds (average 2.43(1) Å, versus 2.24(1) Å and 2.24(3) Å (Table S2)) due to the steric constraints imposed by the ligand, but consistent with all Mn(II) centers. [START_REF] Gerey | Calcium and heterometallic manganese-calcium complexes supported by tripodal pyridine-carboxylate ligands: structural, EPR and theoretical investigations[END_REF] The MnMn distances and intermetallic angles within the macrocycle show that the three Mn 2+ ions form an essentially isosceles triangle (Table S9). We can therefore consider that a pseudo axis of symmetry 3 passes through the center of the cavity. The macrocyclic structures of the heteronuclear Mn 3 Ca and {Mn 3 Sr} 2 compounds are similar to that of Mn 3 Mn, and we will therefore only discuss on the differences induced by the substitution of the Mn 2+ ion of the central cavity by the redox-innocent Ca 2+ and Sr 2+ cations.

In Mn 3 Mn, the central Mn 2+ cation is arranged in Δ stereochemical conformation, while in the heteronuclear clusters, Ca 2+ and Sr 2+ are present in the two conformations, Δ and Λ.

The Mn-O/N distances in the {Mn(tpada)} units are rather similar for the different complexes, although with strontium these are slightly longer: for Mn 3 Mn, Mn 3 Ca and {Mn 3 Sr} 2 the Mn-N tert average distances are respectively 2.434(6), 2.459(10) and 2.491(17) Å and those of Mn-N py /O COO -of 2.24(3), 2.23(4) and 2.25(6) Å, respectively (Table S10). On the other hand, the distances between the Mn 2+ ions and the central M 2+ cation (MnM distances) (Table S10) as well as the distances between the M 2+ and the plane formed by the three Mn 2+ ions (1.255(1) (Mn 3 Mn) < 1.562(1) (Mn 3 Ca) < 1.978( 21) Å ({Mn 3 Sr} 2 )) increase significantly with the ionic radius of the M 2+ cation, i. e. from manganese to strontium (83 μm (Mn 2+ ) <100 μm (Ca 2+ ) <118 μm (Sr 2+ )). [START_REF] Shannon | Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides[END_REF] In other words, the larger the size of the cation, the greater its distance from the plane formed by the three manganese ions. This is also reflected by the bond distances of the central M 2+ cation that significantly increase with the size of the cation (average distances of M-O COO -: 2.20 (7) Å (Mn 2+ ), 2.41 (4) Å (Ca 2+ ) and 2.65 (10) Å (Sr 2+ ) (Table S11). Besides, the similarity of the MnMn distances for Mn 3 Ca and {Mn 3 Sr} 2 , although slightly higher than those of Mn 3 Mn (Table S9), indicates that beyond calcium, the structural change of the metallamacrocycle to accommodate the strontium does not simply involve an increase of the MnMn distances, but by a shift of the Sr 2+ cation away from the plane formed by the three Mn 2+ ions.

Finally, regarding the coordination of the central cation, the calcium, capped with a water molecule, is heptacoordinated (Figure 3(c)), adopting an antiprism distorted geometry with a capped face. On the other hand, the strontium is coordinated by two water molecules which act as bridges with the strontium of the second Mn 3 Sr tetranuclear unit (Figure 3(d)). The coordination sphere of the Sr 2+ ion is also supplemented by a O donor from the carboxylate of the neighboring Mn 3 Sr tetranuclear unit resulting in two additional μ 1,3 -carboxylato bridges between the two Sr 2+ cations (Scheme 1(c) and Figure 3(d)). The strontium is thus nonacoordinated as in the [Sr(tpada)(OH 2 )] 4 (Sr 4 ) complex (see below), in a geometry however difficult to determine. Nonacoordinated strontium ions with only oxygen in their coordination sphere are also observed in the Mn-Sr heteronuclear [LMn IV 3 SrO 4 (OAc) 3 (DMF)] 2 cluster isolated by the Agapie group. [START_REF] Tsui | Reduction potentials of heterometallic manganese-oxido cubane complexes modulated by redox-inactive metals[END_REF] The two cubanoid units {Mn IV 3 SrO 4 } 2 that constitute this complex are also associated by two bridging μ 1,1carboxylato and two bridging DMF molecules (DMF = N,N-dimethylformamide).

Synthesis and crystal structures of the Sr 4 complex

We have also explored the molecular architecture that arises upon the self-assembly of the tpada 2- S13). These distances are longer than those observed in the {Ca(tpada)(OH 2 )} units of the neutral dinuclear [Ca(tpada)(OH 2 ) 2 ] 2 which are actually longer than those of the {Mn(tpada)} units in the Mn 3 M complexes, in accordance with the size of the respective cation (Sr 2+ > Ca 2+ > Mn 2+ ) (see above). The aqua ligand of each strontium is in trans position with respect to the two neighbouring aqua ligands and in cis position with respect to the aqua ligand of the strontium facing it; this distribution corresponding to a -4 axis of symmetry. Thus two aqua ligands are located respectively above and below the plane formed by the metallamacrocycle. These aqua ligands are involved in several intra-and inter-molecular hydrogen bonds (Figure 5(d-e) and Table S14).

Electrospray Ionization (ESI) Mass Spectrometry.

To evaluate the structural integrity of the Mn 3 M complexes in solution, electrospray mass spectrometry and electrochemical experiments have been carried out in CH 3 CN.

The positive mode ESI spectra of Mn 3 M obtained in CH 3 CN solution are shown in Figure S1. For all complexes, the two main peaks observed correspond to the di-and monocharged species {Mn 3 (tpada) 3 M} 2+ and [{Mn 3 (tpada) 3 M}(PF 6 )] + . The most intense ones corresponding to the dicationic species are located at m/z 674.1, 666.6 and 690.7 for {Mn 3 (tpada) 3 Mn} 2+ , {Mn 3 (tpada) 3 Ca} 2+ and {Mn 3 (tpada) 3 Sr} 2+ respectively, while the lower intensity peaks belonging to the monocationic species are detected at m/z 1493.1, 1478.1 and 1526.0 for [(Mn 3 (tpada) 3 Mn}(PF 6 )] + , {Mn 3 (tpada) 3 Ca}(PF 6 )] + and [{Mn 3 (tpada) 3 Sr}(PF 6 )] + , respectively. These assignments are confirmed by simulations of the isotopic massifs which reproduce the experimental isotopic distributions (Figure S1, insert). From these results, it is possible to conclude that to the MC tetranuclear structure of the Mn 3 M complexes are maintained in acetonitrile solution. However, the water molecule(s) coordinated to the Ca 2+ and Sr 2+ are not retained in the mass spectrometry conditions. Furthermore, the dimeric structure of the {Mn II 3 Sr} 2 complex is not conserved under mass spectrometry conditions since the only detected species correspond to the tetranuclear Mn II 3 Sr unit.

To determine the fragmentation mode of such MC structure and evaluate its stability under severe ionization, an MS² analysis was carried out on the dication {Mn 3 (tpada) 3 Ca} 2+ at m/z 666.6. The MS² spectrum reveals the formation of two dicationic peaks at m/z 451.0 and 620.1 (Figure S1(D)). The m/z 451.0 species corresponds to the trinuclear {Mn 2 (tpada) 2 Ca} 2+ resulting from the loss of a monomeric unit {Mn(tpada)}. The second fragment, detected at m/z 620.1, originates from the fragmentation of one tpada 2-ligand at the methyl level of the picoline (L'), {Mn II 2 (tpada) 2 L'Ca} 2+ . This fragmentation releases a picolyl radical and generate a rearrangement at the level of the tertiary nitrogen, forming a double bond with the methylene carbon of one of the two picolinates and accompanied by the loss of a proton. These two fragments attest to the high stability of the Mn 3 Ca cluster and the strong coordination of the encapsulated dication since despite the fragmentation of one of the ancillary tpada 2-ligand and the modification of the metallamacrocycle structure by the loss of a {Mn(tpada) unit, the calcium remains coordinated. These studies confirm the stability of the MC structures in CH 3 CN solution even under severe ionization and that the alkaline-earth cations remain coordinated in the metallamacrocycle cavity in solution.

Electrochemical and spectroscopic properties of Mn 3 M in CH 3 CN.

The electrochemical properties of the Mn 3 M complexes have been investigated in CH 3 CN under argon (Figures 6 andS2-S3). The cyclic voltammograms (CV) of all complexes exhibit three successive reversible oxidation waves located between +0.5 and +0.9

V (Table S15) assigned to the successive oxidation of the three Mn(II) ions into Mn(III) of the {Mn II (tpada)} units which forms the macrocycle. No other oxidation event is observed in the region of positive potentials when scanning up to +1.8 V. The presence of three one-electron redox processes, close in potential (E 1/2 = 130 -170 mV) (Table S15), instead of a single three-electron wave, is the result of Coulombic interactions between the three electroactive centers. [START_REF] Romero | A Novel Dimanganese(II) Complex with Two Chloride Bridges. A Two-Electron Oxidation System[END_REF][START_REF] Romain | Multireversible Redox Processes in Pentanuclear Bis(Triple-Helical) Manganese Complexes Featuring an Oxo-Centered triangular {MnII 2 Mn III (μ 3 -O)} 5+ or {Mn II MnIII 2 (μ 3 -O)} 6+ Core Wrapped by Two {MnII 2 (bpp) 3 }[END_REF][START_REF] Gouré | Seven Reversible Redox Processes in a Self-Assembled Cobalt Pentanuclear Bis(triple-stranded helicate): Structural, Spectroscopic, and Magnetic Characterizations in the CoICoII 4 , CoII 5 , and CoII 3 CoIII 2 Redox States[END_REF] Therefore, the oxidation of one Mn(II) into Mn(III) increases the positive field around its neighbors, altering the potential of those neighbors. In addition, the potential values of the reversible waves slightly differ according to the complexes. For example, considering the first reversible oxidation wave, the Mn 3 Mn complex (E 1/2 = +0.62 V) is slightly more difficult to oxidize than complexes with calcium and strontium ions, Mn 3 Ca and {Mn 3 Sr 2 , which are very close in potential (E 1/2 = +0.57 and 0.58 V, respectively). Even if the influence of the nature of M n+ cation inserted in the cavity on the redox potential of the Mn(II) ions forming the metallamacrocycle is very weak, it is consistent with its preservation in the MC cavity. In comparison, a large positive shift of more than 1 V has been observed in the case of the Agapie's series of isostructural cubane clusters [Mn IV 3 MO 4 (L 6 )(O 2 CMe) 3 (DMF)] + for the Mn derivative compared to the Ca or Sr derivatives for which the potential are essentially the same (E 1/2 values of the [MMn IV 3 O 4 ]/[MMn IV 2 Mn III 2 O 4 ] couples of +290 mV for M = Mn 3+ and -940 mV vs Fc/Fc + for M = Ca 2+ or Sr 2+ ). [START_REF] Tsui | Reduction potentials of heterometallic manganese-oxido cubane complexes modulated by redox-inactive metals[END_REF][START_REF] Tsui | Redox-inactive metals modulate the reduction potential in heterometallic manganese-oxido clusters[END_REF] The large difference in potentials observed in this series of complexes, can be explain by the fact that all metal ions are strongly magnetically coupled through oxo bridges. It is worth noting that the oxidation process of the central Mn(II) ion in the Mn 3 Mn complex is not accessible in the solvent electroactivity domain (no signal up to +1.8 V). This Mn 2+ center is very depleted in electrons after the three successive oxidations of the Mn 2+ ions of the macrocycle, and therefore very difficult to oxidize. In the region of the negative potentials, a series of reversible systems very close in potential (E 1/2 of about 100 mV) are also detected between -2.0 and -2.7 V (Figures S2-S3 for CVs recorded on carbon electrode). These ligand-centered reduction processes are reasonably assigned to the successive monoelectronic reduction of the pyridine-carboxylate units of the tpada 2-ligands. Similarly to the reversible oxidation systems, potential shifts are found on these reduction systems, the Mn 3 Mn complex being easier to reduce than the Mn 3 Ca and {Mn 3 Sr} 2 complexes (Table S15). This result highlights that the pyridinecarboxylate motifs, when coordinated to a metal, can be reversibly reduced, similarly to the well-known redox behavior of bipyridine ligands.

The Mn III 3 M species are quantitatively generated by exhaustive electrolyses carried out at a potential of 1.20 V after the exchange of about 2.6-2.9 electrons confirming that each oxidation wave corresponds to the exchange of one-electron. After electrolysis, for the three complexes, all the reversible waves are maintained located either in the positive and negative S3). It is important to note that given the particular geometry of the tpada 2-ligand, the Mn 2+ as well as the Mn 3+ ions in the {Mn(tpada)} units forming the macrocycle in the initial Mn II 3 M and oxidized forms Mn III 3 M are hepta-coordinated, an unusual coordination number especially for a Mn 3+ ion. [START_REF] Gerey | Calcium and heterometallic manganese-calcium complexes supported by tripodal pyridine-carboxylate ligands: structural, EPR and theoretical investigations[END_REF][START_REF] Collomb | Manganese Inorganic & Coordination Chemistry[END_REF] Solutions of the initial and oxidized compounds were studied by UV-visible absorption and EPR spectroscopies (Figures 7 andS4-S5). The yellow pale solutions of the initial Mn II 3 M complexes absorb only in the UV region (shoulders around 350 nm) in accordance with high-spin Mn II species. [START_REF] Collomb | Manganese Inorganic & Coordination Chemistry[END_REF] On the other hand, orange solutions of the Mn III 3 M complexes generated by exhautive electrolyses at 1.2 V exhibit absorption bands in the visible (Figures 7 andS4) in agreement with the generation of Mn(III) species: a narrow band at 450 nm associated with a shoulder at 500 nm and a broad band of lower intensity centered at around 720 nm. [START_REF] Collomb | Redox-Induced μ-Acetato and μ-Oxo Core Interconversions in Dinuclear Manganese Tris(2-methylpyridyl)amine (tpa) Complexes: Isolation and Characterization of [Mn 2 III (μ-O)(μ-O 2 CCH 3 )(tpa) 2 ] 3+[END_REF] The relative intensity of these bands, as well as the value of the  max , slightly vary with the nature of the complexes. This variation, although minimal, confirms the electrochemical results, i. e. the central ions remain also coordinated in the oxidation state Mn III 3 M.

The X-band EPR spectra of the complexes before and after electrolysis at 1.2 V have been recorded at 100 K (Figure S5). In the initial state, the spectra are all comparable and present a main transition centered around g = 2 (340 mT) characteristic of a high spin manganese(II) signal (S = 5/2). Although the spectra are similar, the transition width is variable, suggesting small variations of the D parameter, [START_REF] Duboc | Definition of Magneto-Structural Correlations for the Mn II Ion[END_REF][START_REF] Mantel | Structural characterization and electronic properties determination by high-field and highfrequency EPR of a series of five-coordinated Mn(II) complexes[END_REF] and thus that the central cation has a weak influence on the electron properties of the Mn 2+ ions of the metallamacrocycle.

For the Mn III 3 Ca and Mn III 3 Sr complexes, after exhaustive electrolysis at 1.2 V, no transition is visible on the EPR spectra demonstrating the effective oxidation of the three Mn 2+ of the metallamacrocycle into Mn 3+ . Indeed, the manganese(III), a d 4 ion with integer spin, is EPR silent at 100 K at X-band frequencies. By contrast, the EPR spectrum of the mixed valence

Mn III

3 Mn II species reveals a 6-line EPR signal. This signal, only detected for the Mn III 3 Mn complex, corresponds to the central Mn(II) since the three Mn(II) ions of the macrocycle are oxidized to Mn(III). In addition, the presence of a 6-line signal and not a multiplet for this complex indicates that the central Mn(II) is fully isolated and does no communicate with its Mn(III) neighbors forming the macrocycle. Abs.

 (nm)

Mn 3 Sr Mn 3 Mn Mn 3 Ca

This electrochemical study revealed interesting properties for these tetranuclear

Mn II 3 M complexes featuring a MC structure, namely an excellent stability both in their initial state and in their three-electron oxidized state, Mn III 3 M. We have thus compare this behavior with those of similar manganese(II) complexes already described in the literature. First, it appears that the redox behavior observed is quite unusual for manganese(II) multinuclear species with μ-carboxylato bridges and similar pyridine-based multidentate ancillary ligands.

For example, the cyclic voltammograms of bis-μ-acetato binuclear complexes of the type [Mn II 2 (μ 1,3 -O 2 CCH 3 ) 3 (bpea) 2 ] + and [Mn II 2 (μ 1,3 -O 2 CCH 3 ) 2 (tpa) 2 ] 2+ (bpea = N,N-bis-(2pyridylmethyl)ethylamine) and tpa = tris(2-methylpyridyl)amine) exhibit in acetonitrile fully irreversibly oxidation processes, their oxidation leading to the quantitative formation of the μ- 3+ , respectively. Upon oxidation into the +III oxidation state of manganese, a fast decoordination of an acetato bridge of the complexes with the concomitant formation of an oxo bridge by coordination and deprotonation of a residual water molecule of the solvent occurs. [START_REF] Collomb | Redox-Induced μ-Acetato and μ-Oxo Core Interconversions in Dinuclear Manganese Tris(2-methylpyridyl)amine (tpa) Complexes: Isolation and Characterization of [Mn 2 III (μ-O)(μ-O 2 CCH 3 )(tpa) 2 ] 3+[END_REF][START_REF] Romero | A Dinuclear Manganese(II) Complex with the {Mn 2 (μ-O 2 CCH 3 ) 3 } + Core: Synthesis, Structure, Characterization, Electroinduced Transformation, and Catalase-like Activity[END_REF][START_REF] Collomb | Electro-and Photoinduced Formation and Transformation of Oxido-Bridged Multinuclear Mn Complexes[END_REF] The tendency of Mn(III) ions to form μ-oxo bridges [START_REF] Collomb | Electro-and Photoinduced Formation and Transformation of Oxido-Bridged Multinuclear Mn Complexes[END_REF] has also been demonstrated for other binuclear manganese(II) complexes with μ-carboxylato bridges of the type [Mn II 2 (L) 2 (H 2 O) 2 (L = N-glycyl-N'-methyl/benzyl-N,N'-bis(2-pyridylmethyl)-1,2-ethanediamine) for which the carboxylate bridges are provided by the pentadentate L ligands [START_REF] Baffert | Biologically relevant mono-and di-nuclear manganese II/III/IV complexes of mononegative pentadentate ligands[END_REF] . The latter also exhibit irreversible oxidation processes in acetonitrile. By contrast, the stability of the Mn III 3 M MC structure can be explained by the presence of the M 2+ cation very strongly bound to the macrocycle via six μ-carboxylato bridges from three tpada 2-ligands. In addition, the fact that these carboxylates are provided by the ancillary ligands certainly contributes to the excellent stability of the whole structure even in its oxidized form, by preventing their decoordination and the subsequent formation of μ-oxo bridged species in the +III oxidation state of manganese.

oxo [Mn III 2 (µ-O)(µ 1,3 -O 2 CCH 3 ) 2 (bpea) 2 ] 2+ and [Mn III 2 (μ-O)(μ 1,3 -O 2 CCH 3 )(tpa) 2 ]
These 12-MC-3 Mn 3 M complexes are analogues of the iron complexes [Fe II 3 Fe II (bpg)(O 2 CPh) 3 ] 2+ and [Fe II 3 Fe II (bpg)(O 2 CCH 3 ) 3 ] 2+ synthesized by Que and coworkers with the tetradentate monocarboxylate bpg -ligand (Figure 2A(c)). [START_REF] Ménage | Carboxylatoiron(II) Aggregates: A Novel Fe Complex with Threefold Symmetry[END_REF][START_REF] Mandal | Polynuclear Carboxylato-Bridged Iron(II) Clusters: Synthesis, Structure, and Host-Guest Chemistry[END_REF] Nevertheless, since two carboxylates are already incorporated in the ancillary tpada 2-ligand, the synthesis of Mn 3 M do not require additional carboxylates to coordinate the central metal M. Indeed only one of the two carboxylate units of each tpada 2-ligand is engaged in the macrocycle, the second one being available for the coordination of the cation located in the central cavity. Since, the electrochemical properties of these iron complexes were not investigated, neither those of the other carboxylate bridged metallamacrocycle complexes, it is therefore not possible to compare their stability in their initial and oxidized forms with those of the Mn 3 M complexes.

The inherent structural stability in CH 3 CN solution was also established by mass spectrometry and electrochemistry for the family of structurally related heterometallic pentanuclear Mn III 4 Ca complexes, [Mn III 4 Ca(O 2 CR) 4 (shi) 4 ] 2-(Figure 2B(b-c)). [START_REF] Lah | Isolation and characterization of {Mn II [Mn III (salicylhydroximate)] 4 (acetate) 2 (DMF) 6 }.cntdot.2DMF: an inorganic analog of M 2+ (12-crown-4)[END_REF][START_REF] Koumousi | Towards models of the oxygen-evolving complex (OEC) of photosystem II: a Mn 4 Ca cluster of relevance to low oxidation states of the OEC[END_REF][START_REF] Alaimo | Emissive {Mn 4 III Ca} Clusters with Square Pyramidal Topologies: Syntheses and Structural, Spectroscopic, and Physicochemical Characterization[END_REF] In these complexes featuring a distorted square pyramidal topology with the four Mn(III) atoms forming the square base, the central Ca 2+ in the apical position is maintained through four additional carboxylates (O 2 CR). Due to the pronounced stability of the Mn 4 III Ca MC core induced by the four donor atoms of the salicylhydroximate ligand, these compounds are also very rare examples of metal clusters that retain their solid-state structures in solution (CH 3 CN solvent). The cyclic voltammetry of these complexes exhibits only one reversible oxidation system, attributed by the authors to a mono-electronic oxidation, and poorly defined reduction peaks. This electrochemical behavior is however very different to that of the Mn II 3 M complexes, which underlines the unusual properties of the latter. In addition, the stability of the reduced complex Mn II 4 Ca was not evaluated by bulk electrolysis. Finally, the heterometallic Mn 3 Ca and {Mn 3 Sr] 2 also join a growing, but still small, family of heteronuclear manganese-calcium complexes and even rarer manganese-strontium complexes as models of the native OEC of photosystem II. Table 1 gives an overview of all the synthetic Mn-Ca and Mn-Sr heterometallic compounds structurally characterized to date.

The compounds are classified on the basis of their structures, polymeric compounds or discrete molecules, and on the chemical nature of the bridge(s) between the Mn and Ca/Sr cations, µ-carboxylato, µ-hydroxo, µ-alkoxo, µ-oxo, or a combination of different bridges. If the large majority of these complexes present a structure quite different from that of the OEC, some very close structural models of the native OER and Sr-substituted OEC were obtained by the groups of Agapie and co-workers, [START_REF] Kanady | A Synthetic Model of the Mn 3 Ca Subsite of the Oxygen-Evolving Complex in Photosystem II[END_REF][START_REF] Kanady | Toward Models for the Full Oxygen-Evolving Complex of Photosystem II by Ligand Coordination To Lower the Symmetry of the Mn 3 CaO 4 Cubane: Demonstration That Electronic Effects Facilitate Binding of a Fifth Metal[END_REF][START_REF] Tsui | Reduction potentials of heterometallic manganese-oxido cubane complexes modulated by redox-inactive metals[END_REF] Christou and co-workers [START_REF] Mukherjee | Synthetic model of the asymmetric [Mn 3 CaO 4 ] cubane core of the oxygen-evolving complex of photosystem II[END_REF] and Zhang and coworkers, [START_REF] Chen | Artificial Mn 4 Ca Clusters with Exchangeable Solvent Molecules Mimicking the Oxygen-Evolving Center in Photosynthesis[END_REF][START_REF] Zhang | A synthetic Mn 4 Cacluster mimicking the oxygen-evolving center of photosynthesis[END_REF] with the {Mn 3 CaO 4 }, {Mn 4 CaO 4 }, {Mn 3 Ca 2 O 4 } or {Mn 3 CaAgO 4 } stoichiometry. These complexes containing the {Mn 3 CaO 4 } or {Mn 3 SrO 4 } cubane core and in some case a dangling metal cation (Mn, Ca or Ag) mimicking the fourth external Mn attached through one µ 4 -oxo of the cubane. Although the structure of the Mn 3 Ca and {Mn 3 Sr] 2 compounds are not similar to that of the OEC, the X-ray structures corroborate the high tendency of calcium and strontium to coordinate water molecule as found in the native and Srsubstituted OEC. In addition, such self-assembled Mn 2+ -Ca complexes bridged by carboxylate units could be relevant for mimicking the first stages of the OEC assembly process.

Conclusion

A series of new manganese(II) homo-and heterometallic tetranuclear complexes having a 12-metallacrown-3 type arrangement, including a central cation within their cavity (Mn 2+ , Ca 2+ or Sr 2+ ), have been spontaneously self-assembled at room temperature by reacting Mn 2+ with the hexadentate tripodal tpada 2-ligand two carboxylate functions.

These complexes constitute rare examples of metallacrowns incorporating μ-carboxylate

bridges between the various metals. The heterometallic Mn 3 Ca and {Mn 3 Sr] 2 also join the small family of heteronuclear manganese-calcium complexes and even rarer manganesestrontium complexes as models of the OEC of photosystem II. Solution studies carried out in organic solvent (CH 3 CN), either by mass spectrometry and electrochemistry coupled to UVvisible and EPR spectroscopy confirmed their structural integrity in solution. It appears that these complexes present a very rich redox behavior with up to eight reversible waves detected by cyclic voltammetry, three corresponding to the successive oxidation of the Mn(II) ion constituting the macrocycle into Mn(III) and five to the reduction of the tpada 2-ligand. The Mn III 3 M three-electrons oxidized species have been quantitatively generated by bulk electrolyses at 1.20 V demonstrating the high stability of the tetranuclear structure also in the three-electrons oxidized state. This redox behavior, quite unusual for Mn(II) compounds coordinated with such ligands, can be correlated to the presence of the central M 2+ cation within the cavity strongly linked to the three Mn(II)/Mn(III) ions constituting the ring through six bridging carboxylate ligands of the three tpada 2-ligands, that stabilizes the structure. These 12-MC-3 complexes are analogues of iron complexes synthesized by Que with the tetradentate monocarboxylate ligand Hbpg (Figure 2A(c)), but unlike these complexes, their synthesis does not require the addition of supplementary carboxylates to coordinate the central metal since these are already an integral part of the ancillary ligand. Indeed, only one of the two carboxylate units of each tpada 2-ligand is involved in the macrocycle, the second unit being free to coordinate the central cation. The presence of this second carboxylate within the ligand framework is most probably also at the origin of the excellent stability of these structures in solution, even in their three-electron oxidized form. Anyway, the electrochemical properties of these complexes cannot be compared with those of the few examples of previously published µ-carboxylato bridged MCs, since the electrochemical properties of this family of complexes have never been studied.

Finally, it should be highlighted that due to the particular geometry of the tpada 2- ligand, the Mn 2+ but also the Mn 3+ ions in the Mn II 3 M and Mn III 3 M complexes are heptacoordinated, an unusual coordination number especially for a Mn(III) ion. In addition, the calcium and strontium in the isolated structures incorporate water molecule, as found in the OEC of PSII.

In future works, it will be interesting to explore other experimental conditions in view to test the possibility to access such MC complexes with the tpada 2-ligand without any cation within the central cavity, to determine the affinity of this structure for the coordination of different cations in the context of ionic recognition. We will also explore the synthesis of new complexes with other metal cations by using this tripodal hexadentate tpada 2-ligand that gives rise to original and stable structures, unusual coordination number for metal cations and electrochemical properties with reversibility of the oxidation and reduction processes.

Experimental section

Materials. All solvents and reagents were laboratory reagent grade or better and used as received. Potassium hydroxide (KOH, Sigma-Aldrich, ≥ 85%), tetra-n-butylammonium hydroxide [Bu 4 N]OH (1.0 M in methanol, Alfa Aesar), manganese(II) acetate tetrahydrate (Mn(O 2 CCH 3 ) 2 •4H 2 O, Merck), lithium perchlorate anhydrous (LiClO 4 , ≥ 98%, Sigma-Aldrich), calcium trifluoromethanesulfonate anhydrous (Ca(CF 3 SO 3 ) 2 , Aldrich, 99.9%), strontium nitrate anhydrous (Sr(NO 3 ) 2 , 99+%, Strem) were used as received. Acetonitrile (CH 3 CN, Fisher, HPLC grade), and tetra-n-butyl-ammonium perchlorate ([Bu 4 N]ClO 4 , Fluka) for electrochemical measurements were stored under an argon atmosphere in a dry-glove box.

Synthesis of 6,6'-(pyridin-2-ylmethylazanediyl)bis(methylene)dipicolinic acid (H 2 tpada).

This ligand was synthetized as previously described.{Gerey, 2015 #3350} The fraction isolated corresponds to the triprotonated form, H 3 tpada(PF 6

Synthesis of the complexes

Caution! Perchlorate salts of compounds containing organic ligands are potentially explosive. Although we have encountered no such problems, only small quantities of these compounds should be prepared and handled with care.

Synthesis of [{Mn II (tpada)} 3 Mn II ](X) 2 (Mn 3 Mn(X) 2 ) (X = ClO 4 or PF 6 ).

Figure 2 :

 2 Figure 2: Crystallographic structures of the (A) 12-MC-3 complexes of Zn 2+ (a) and Fe 2+ (bc) with the bpg -ligand providing the O-C-O bridging moiety. A Fe 2+ is encapsulated within the central cavity of the Fe 2+ MC ring (c). (B) 12-MC-4 complexes of Mn 3+ with the shi 3- ligand providing the O-N bridging moiety encapsulating a Mn 2+ ion (a) and a Ca 2+ ion (two views) (b) within the central cavity. Schematic representations of the corresponding ligands and atom connectivity of the metallamacrocycle.

Figure 3 displaysScheme 1 .

 31 Scheme 1. Synthetic routes for the homo-and heteronuclear 12-MC-3 and 8-MC-4 complexes.This arrangement generates a cavity at the center of the structure of six oxygen atoms of the tpada 2-ligands (three of them do not participate in the macrocycle) allowing the molecule to host a fourth metal center. The Mn 2+ cation present in the central cavity is thus coordinated to the six O donors in a distorted octahedral geometry, and linked to each manganese(II) of the {Mn II (tpada)} units via two μ 1,1 -carboxylato bridges (Scheme 1 and Figure3(a)). The

Figure 3 .

 3 Figure 3. Representations of the crystallographic structures obtained by X-ray diffraction of the cationic forms of the complexes Mn 3 Mn (a), Mn 3 Ca (b), Mn 3 Sr (c) and {Mn II 3 Sr} 2 (d). View along the pseudo axis of symmetry 3 (left) and view of the metallamacrocycle cavity (right). Hydrogen atoms have been omitted for clarity.

Figure 4

 4 Figure 4 displays the coordination sphere of one of the three {Mn II (tpada)} units, and of the Mn 2+ present in the cavity. In each {Mn II (tpada)} unit, the Mn 2+ cation is coordinated to all six donor atoms of the tpada 2-ligand and to one carboxylate O atom from the tpada 2-ligand of a neighboring {Mn(tpada)} unit, in an approximately pentagonal bipyramidal geometry (Figures 3(a) and 4(b)). The N-O pairs of the two pyridine-carboxylate tpada 2-ligand and the

Figure 4 .

 4 Figure 4. ORTEP drawing with thermal ellipsoids at 50% probability of the O 6 coordination sphere of the Mn 2+ ion present in the central cavity of Mn 3 Mn (a) and of one of the three {Mn II (tpada)} unit of Mn 3 Mn (b). Hydrogen atoms have been omitted for clarity.

  ligand with Sr 2+ . By reacting the (H 3 tpada)PF 6 H 2 O ligand with two equivalents of Sr(OH) 2 8H 2 O in water, another type of MC was obtained having four -[Sr-O]repeating units, and without any cation in the central cavity (Scheme 1(d) and Figure 5). The crystallographic data and selected bond distances and angles are given in Tables S12-S14. The structure of the neutral [Sr(tpada)(OH 2 )] 4 8CH 3 CN2H 2 O (Sr 4 8CH 3 CN2H 2 O) complex consists of four neutral {Sr(tpada)(OH 2 )} units in which each Sr 2+ cation is bridged to its neighbors by a bidentate carboxylate ligand of a neighbouring ligand, forming an eightmembered metallamacrocycle in the form of a 8-MC-4, which is, to our knowledge, the first strontium-based molecular metallamacrocycle of the type molecular wheel.

Figure 5 .

 5 Figure 5. ORTEP drawings with thermal ellipsoids at 50% probability of the tetranuclear 8-MC-4 [Sr(tpada)(OH 2 )] 4 (Sr 4 8CH 3 CN2H 2 O); view along the axis of symmetry -4 (a) and (b); view of the monomeric unit {Sr(tpada)(OH 2 )} linked to one neighboring strontium (c) and views of the hydrogen bond network perpendicular (d) and parallel (e) of the macrocycle plan. Only relevant hydrogens are shown for clarity.

Figure 6 :

 6 Figure 6: Cyclic voltammograms in CH 3 CN, 0.1 M [Bu 4 N]ClO 4 at a platinum electrode (υ = 100 mV s -1 ) of solutions of (A) Mn 3 M (0.64 mM), (B) Mn 3 Ca (0.57 mM) and (C) {Mn 3 Sr} 2 (0.33 mM): (a) initial solutions and (b) after exhaustive oxidation at 1.20 V. Potentials are referenced to Ag/AgNO 3 (10 mM).

  potential regions with the same intensity and at the same potential, again confirming the excellent stability of the MC structures including the preservation of the cations in the central cavity, even in the three-electrons oxidized state (Figures6 and

Figure 7 :

 7 Figure 7: UV-Visible absorption spectra of solutions of Mn 3 Mn (0.64 mM) (green), Mn 3 Ca (0.57 mM) (blue) and {Mn 3 Sr} 2 (0.33 mM) (red) in CH 3 CN, 0.1 M [Bu 4 N]ClO 4 before and after exhaustive electrolysis at +1.2 V (generation of Mn III 3 Mn II , Mn III 3 Ca and Mn III 3 Sr). Optical path of 1 cm.

Mn 3

 3 Ca tetrahedron similar to the OEC the sp 2 N-atom of the mAla Schiff base ligand mimicks the amide N-atoms present in the OEC Milios 2020 29 a mal = malonate; bipy = 2,2'-bipyridine; H-DCA = dichloroacetic acid; sal = salicylate; prop = propionate; Me-saOH 2 = 2-hydroxyphenylethanone oxime; H 3 -tpaa = 6,6',6''nitrilotris(methylene)tripicolinic acid); H 2 -tpada = 6,6'-(pyridin-2ylmethylazanediyl)bis(methylene)dipicolinic acid ; [MST] 3-= N,N',N''-[2,2',2''-nitrilotris(ethane-2,1diyl)]tris(2,4,6-trimethylbenzenesulfonamido); MeOEtOH = 2-methoxyethanol; H 4 -L S = p-tbutylthiacalixarene; DMF = dimethylformamide; hmp-H = 2-(hydroxymethyl)pyridine; H-CBA = 3-chlorobenzoic acid; shi-H 3 = salicylhydroxamic acid; L 1 -H = 2-naphthoic acid ; L 2 -H = 9-anthracenecarboxylic acid; L 3 -H = 1-pyrenecarboxylic acid; L 4 -H 2 = 2-[[(2-hydroxypropyl)imino]methyl]-6-methoxyphenyl; L 5 -H 3 = 2-(2,3dihydroxypropyliminomethyl)-6-methoxyphenol; py = pyridine; t BuPhCO 2 H = p-tert-buthylbenzoic acid; L 6 -H 3 = 1,3,5-tris(2-di(2'-pyridyl)hydroxymethylphenyl)benzene; THF = tetrahydrofuran; DME = 1,2-dimethoxyethane; L 6' -H 3 = 2-{2,4-bis(2-di(2'-pyridyl)hydroxymethylphenyl)phenyl}benzoic acid; H-BA = benzoic acid; qao-H = quinoline-2-aldoxime; dapdo-H 2 = 2,6-diacetylpyridine dioxime; mAla-H = methylalanine; H 3-L 7 = 2-(β-naphthalideneamino)-2-hydroxyethyl-1-propanol.

  3.0 equiv. of KOH (0.1 M, 19.2 mg, 0.342 mmol) was added to a stirred solution of the ligand (H 3 tpada)PF 6 H 2 O (62 mg, 0.114 mmol) in 8 mL of methanol, To this colorless solution, 2 mL of a methanol solution containing 1.4 equiv of Mn II (O 2 CCH 3 ) 2 4H 2 O (39.3 mg, 0.16 mmol) was added under vigorous stirring. The resulting slightly yellow solution was stirred 30 min and then partially evaporated under reduced pressure until 2 mL leading to the formation of a yellow microcrystalline powder corresponding to [{Mn(tpada)} 3 Mn](PF 6 ) 2 3H 2 O, which was filtered, washed with diisopropyl ether and dried under vacuum. Single crystals suitable for X-ray diffraction were only obtained as perchlorate salts. The yellow powder of [{Mn(tpada)} 3 Mn](PF 6 ) 2 3H 2 O was redissolved in acetonitrile and an excess of LiClO 4 was added. Slow diffusion of diethyl ether or ethyl acetate into the solution leads to the formation of single crystals of [{Mn(tpada)} 3 Mn](ClO 4 ) 2 2CH 3 CN (yield, 60 mg, 97%). Anal. Calcd. for C 60 H 48 N 12 O 12 Mn 4 P 2 F 12 3H 2 O (1692.82): C, 42.57; H, 3.22; N, 9.93. Found: C, 42.44; H, 3.19; N, 9.87. ESI-MS m/z for C 60 H 48 N 12 O 12 Mn 4 P 2F 12 (M): 674.1 [M-2PF 6 ] 2+ , 1493.1 [M-PF 6 ] + . IR (KBr pellet, cm -1 ): 3416 (m), 1623 (µ asCOO-, s), 1594(µ asCOO-, vs), 1466 (sh), 1440 (µ sCOO-, m), 1386 (µ sCOO-, s), 1277 (m), 1183 (w), 1156 (w), 1121 (w), 1102 (w), 1078 (m), 1055 (w), 1019 (m), 980 (w), 955 (w), 845 (PF 6 , vs), 773 (s), 690 (m), 639 (w), 558 (PF 6 , s), 512 (w), 457 (w). Synthesis of [{Mn II (tpada)} 3 Ca(OH 2 )](PF 6 ) 2 (Mn 3 Ca(PF 6 ) 2 ). 3.0 equiv. of n Bu 4 NOH (1.0 M, 342 µL, 0.342 mmol) was added to a stirred solution of (H 3 tpada)PF 6 H 2 O (62 mg, 0.114 mmol) in 8 mL of methanol, 28.0 mg (1.0 equiv., 0.11 mmol) of Mn(O 2 CCH 3 ) 2 4H 2 O was dissolved in 2 mL of MeOH and added to the colorless solution of deprotonated ligand under vigorous stirring. Then, a solution Ca(CF 3 SO 3 ) 2 in 2 mL acetonitrile (38.7 mg, 0.11 mmol) was quickly added to the solution. The resulting slightly yellow solution was stirred 30 min and then partially evaporated under reduced pressure until 2 mL leading to the precipitation of a yellow crystalline

  photosystem II. The 12-MC-3 tetranuclear structures are stable in solution, both in the initial oxidation state, Mn II 3 M, as well as in the three-electrons oxidized state, Mn III 3 M.

ligands and two bridging water molecules leading to the formation of the octanuclear {Mn 3 Sr} 2 (PF

								6 ) 4
	compound (Scheme 1(c)).
		Single crystals of [{Mn II (tpada)} 3 Ca(OH 2 )](PF 6 ) 2 1.75CH 3 CO 2 C 2 H 5 0.25CH 3 CN
	(Mn 3 Ca(PF 6 ) 2 1.75EtOAc0.25CH 3 CN)		and	[{Mn II (tpada)} 3 Sr(µ-
	OH 2 )] 2 (PF 6 ) 4 2CH 3 OH1.5AcOEt0.5H 2 O	({Mn 3 Sr} 2 (PF 6 ) 4 2CH 3 OH1.5AcOEt0.5H 2 O)
	were formed by slow diffusion of ethyl acetate or diethyl ether into acetonitrile solution of the
	respective compound. The crystallographic data of the complexes are summarized in Table S1
	and selected bond lengths and angles in Tables S2-S11. By contrast, single crystals suitable
	for	X-ray	diffraction	analysis	of	the	manganese	derivative,
	[{Mn II (tpada)} 3 Mn II ](ClO 4 )

2 2CH 3 CN (Mn 3 Mn(ClO 4 ) 2 2CH 3 CN), were only obtained by using perchlorate as counter ions. An excess of Li(ClO 4 ) was added to the acetonitrile solution of the Mn 3 Ca(PF 6 ) 2 complex for the crystallization step (see Supporting Information).

Table 1 .

 1 Chemical Formulas of the Structurally Characterized Heterometallic Mn-Ca and Mn-Sr Complexes with Various Ancillary Ligands a Reported to Date Classified on the basis of the Chemical Nature of the Link Between Mn and Ca/Sr Cations and their Main Structural Features Relevant to the Native OEC. Ca 2 O 4 (O 2 C t Bu) 8 ( t BuCO 2 H) 4 ] 1 rst model of the asymmetric Mn 4 CaO 4 OEC cluster: a [Mn IV Ca 2 O 4 ] carboxylate cubane core but with an external Ca attached to one oxo of the cubane -EPR spectra similar to OER S 2 state

						Complex	Structural features relevant to the OEC	Author year ref
						μ-carboxylato bridged Mn(II/III)-Ca(II) polymers
	[Mn II Ca(mal) 2 (H 2 O) 4 ] n	2 water molecules coordinated to Ca and Mn II	Rojo 2000 62
	[Mn II Ca 2 (bipy) 2 (DCA) 6 (H 2 O)(MeOH)] n	2 water molecules coordinated to Mn II	Benniston,
						Bartolomé 2014
						63
	[Mn III	3 CaNa(sal) 6 (H 2 O) 6 ] n	water molecules coordinated to Ca and Mn III (from 1 to	Chen 2010 64
						3)
	{[Mn III	6 Ca 2 O 2 (Me-saO) 6 (prop) 6 (H 2 O) 2 ]} n	[Mn III	3 CaO(OR) 3 (NO) 2 ] 3+ tetrahedron as in the	Milios 2011 65
						Mn 3 CaO 4 distorted-cubane subunit of the OEC
						μ-carboxylato-bridged Mn(II)-Ca(II)/Sr(II) clusters
	[Mn II	2 Ca 2 (tpaa) 2 (OH 2 ) 12 ][Mn II (tpaa)] 2	water molecules coordinated to Ca (from 1 to 7)	Collomb 2015 34-35
	[Mn II Ca(tpada)Cl(OH 2 ) 3 (MeOH) 2 ] -	involved in intra and intermolecular H-bond
						networks
	[Mn II	3 Ca(tpada) 3 (OH 2 )] 2+	water molecules coordinated to Ca and Sr (from 1 to	Collomb This work
	[Mn II	3 Sr(tpada) 3 (OH 2 )] 2 4+	2)
						μ-hydroxo-bridged Mn(III)-Ca(II)/Sr(II) binuclear complexes
	[Mn III Ca(OH)(15-crown-5)(MST)] +	Ca II (μ-OH)Mn III motif proposed to be present in the	Borovik 2011 66-67
						OEC S 1 state
	[Mn III Sr(OH)(15-crown-5)(MST)] +	Borovik 2013 68
						μ-alkoxo-bridged Mn(II/III)-Ca(II) cluster
	[Mn II	4 Ca 2 Cl 4 (μ-OEtOMe) 8 ]	-	Sobota 2007 69
	[Mn 2 II Ca 2 (L S ) 2 (DMF) 2 ] [Mn II Ca 2 (HL S ) 2 (DMF) 2 ]	-	Koutsantonis 2015 70
	[Mn III	2 Ca 2 (hmp) 6 (H 2 O) 4 (MeCN) 2 ] 4+	2 water molecules coordinated to Ca	Arauzo 2017 22
	[Mn III [Mn III	2 Ca 2 (hmp) 6 (NO 3 ) 2 (ClO 4 ) 2 (H 2 O) 2 ] 2 Ca 2 (hmp) 6 (CBA) 2 (H 2 O) 4 ] 2+	CaMn 2 O 3 cuboid structures arrangements -water molecules coordinated to Ca (from 1 to 2)	Shova, Benniston 2017 21
	[Mn III	2 Ca 2 (hmp) 6 (DCA) 2 (NO 3 ) 2 (H 2 O) 2 ]
						μ-carboxylato and μ-alkoxo-bridged Mn(III/IV)-Ca(II) clusters

  ). Anal. Calcd. for C 20 H 19 N 4 O 4 PF 6 H 2 O (542.37): C, 44.29; H, 3.9; N, 10.33. Found: C, 44.23; H, 3.79; N, 10.00.

{Mn II (tpada)} 3 Sr II (OH 2 )] 2 (PF 6 ) 4

  Ca(OH 2 )](PF 6 ) 2 0.25CH 3 CN1.75CH 3 CO 2 C 2 H 5 were obtained by slow diffusion of diethylether or ethyl acetate into an acetonitrile solution of this compound (yield, 60 mg, 88%). Anal. Calcd. for C 60 H 50 N 12 O 13 Mn 3 CaP 2 F 12 (1641,93): C, 43.89; H, 3.07; N, 10.24. Found: C, 43.70; H, 3.13; N, 10.21. ESI-MS m/z for C 60 H 50 N 12 O 13 Mn 3 CaP 2 F 12 (M): 666.6 [M-H 2 O-2PF 6 ] 2+ , 1478.1 [M-H 2 O-PF 6 ] + . IR (KBr pellet, cm -1 ): 3436 (m), 1625 µ asCOO-, s), 1594 (µ asCOO-, vs), 1467 (sh), 1440 (µ sCOO-, m), 1388 (µ sCOO-, s), 1277 (m), 1185 (w), 1157 (w), 1122 (w), 1102 (w), 1078 (m), 1055 (w), 1032 (sh), 1018 (m), 981 (w), 955 (w), 847 (PF 6 , vs), 774 (s), 690 (m), 640 (w), 558 (PF 6 , s), 524 (w), 453 (w). ({Mn 3 Sr} 2 (PF 6 ) 2 ). 3.0 equiv. of n Bu 4 NOH (1.0 M, 342 µL, 0.342 mmol) was added to a stirred solution of (H 3 tpada)PF 6 H 2 O (62 mg, 0.114 mmol) in 8 mL of methanol. 28.0 mg (1.0 eq., 0.11 mmol) of Mn(O 2 CCH 3 ) 2 4H 2 O was dissolved in 2 mL of MeOH and added to the mixture under vigorous stirring. Then, a CH 3 CN/H 2 O (2 mL/0.5 mL) solution of Sr(NO 3 ) 2 (25 mg, 0.12 mmol) was quickly added to the solution. The resulting slightly yellow solution was stirred 30 min and then partially evaporated under reduced pressure until 2 mL leading to the precipitation of a yellow Sr II (OH 2 )] 2 (PF 6 ) 4 2CH 3 OH1.5CH 3 CO 2 C 2 H 5 0.5H 2 O were obtained by slow diffusion of ethyl acetate in acetonitrile (yield, 62 mg, 91%). Anal. Calcd. for C 120 H 100 N 24 O 26 Mn 6 Sr 2 P 4 F 24 4CH 3 CN2H 2 O (3461.05): C, 42.95; H, 3.27; N, 10.96. Found: C, 42.94; H, 3.93; N, 11.49. ESI-MS m/z for C 60 H 50 N 12 O 13 Mn 3 SrP 2 F 12([{Mn II (tpada)} 3 Sr II (OH 2 )](PF 6 ) 2 (M)): 690.7 [M-H 2 O-2PF 6 ] 2+ , 1526.0 [M-H 2 O-PF 6 ] + . IR (KBr pellet, cm -1 ): 3436 (m), 1731 (w), 1626 (µ asCOO-, s), 1591 (µ asCOO-, vs), 1439 (µ sCOO-, m),

		powder.	X-ray	diffraction	suitable	crystals	of
	[{Mn II (tpada)} 3 Synthesis of [crystalline powder.	Single	crystals	suitable	for	X-ray	diffraction	of
	[{Mn II (tpada)} 3 1385 (µ sCOO-, s), 1276 (w), 1188 (w), 1154 (w), 1122 (w), 1077 (w), 1017 (w), 849 (PF 6 , vs),
	773 (vs), 688 (m), 558 (PF 6 , s), 445 (w).

Synthesis of [Sr(tpada)(OH 2 )] 4 (Sr 4 ).

A suspension of Sr(OH) 2 8H 2 O (39.5 mg, 148.6 µmol) in water (0.5 mL) was added dropwise to a suspension of (H 3 tpada)PF 6 H 2 O (38.5 mg, 71

CaO 2 (O 2 CR) 5 L 6' (THF)] rare example of cluster of same metal stoichiometry with oxo bridges, as the OEC Agapie 2017[START_REF] Lee | A CaMn 4 O 2 model of the biological oxygen evolving complex: synthesis via cluster expansion on a low symmetry ligand[END_REF]

ACKNOWLEDGEMENTS

This work has been partially supported by the French National Research Agency through Labex ARCANE and CBH-EUR-GS (ANR-17-EURE-0003) for the project MnCaPSII, as well as ANR-13-BS07-0015-01 (MnCaOEC) including the E.G. postdoctoral grant. B.G. thanks the "Université Grenoble Alpes" for his PhD grant. We thank the IR-RPE CNRS FR3443 RENARD network for EPR facilities. The NanoBio-ICMG platforms (FR 2607) are acknowledged for their support. This work was also supported by COST CM1202 program (PERSPECT H 2 O).

ASSOCIATED CONTENT Supporting Information.

This Supporting Information is available free of charge on the ACS Publications website.

Materials and general experimental details. Additional data for X-ray structure determination, mass spectrometry, electrochemistry, UV-visible absorption and EPR spectroscopy (Tables S1-S15, Figures S1-S5).

Accession Codes

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Notes

The authors declare no competing financial interest